Research Article 3 DOF Spherical Pendulum Oscillations with a Uniform Slewing Pivot Center and a Small Angle Assumption Alexander V. Perig, 1 Alexander N. Stadnik, 2 Alexander I. Deriglazov, 2 and Sergey V. Podlesny 2 1 Manufacturing Processes and Automation Engineering Department, Engineering Automation Faculty, Donbass State Engineering Academy, Shkadinova 72, Donetsk Region, Kramatorsk 84313, Ukraine 2 Department of Technical Mechanics, Engineering Automation Faculty, Donbass State Engineering Academy, Shkadinova 72, Donetsk Region, Kramatorsk 84313, Ukraine Correspondence should be addressed to Alexander V. Perig; [email protected] Received 11 February 2014; Revised 7 May 2014; Accepted 21 May 2014; Published 4 August 2014 Academic Editor: Micka¨ el Lallart Copyright © 2014 Alexander V. Perig et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e present paper addresses the derivation of a 3 DOF mathematical model of a spherical pendulum attached to a crane boom tip for uniform slewing motion of the crane. e governing nonlinear DAE-based system for crane boom uniform slewing has been proposed, numerically solved, and experimentally verified. e proposed nonlinear and linearized models have been derived with an introduction of Cartesian coordinates. e linearized model with small angle assumption has an analytical solution. e relative and absolute payload trajectories have been derived. e amplitudes of load oscillations, which depend on computed initial conditions, have been estimated. e dependence of natural frequencies on the transport inertia forces and gravity forces has been computed. e conservative system, which contains first time derivatives of coordinates without oscillation damping, has been derived. e dynamic analogy between crane boom-driven payload swaying motion and Foucault’s pendulum motion has been grounded and outlined. For a small swaying angle, good agreement between theoretical and averaged experimental results was obtained. 1. Introduction Payload swaying dynamics during crane boom slewing is within the objectives and scope of many academic and industrial research programs in the fields of mechanical, elec- trical, and control engineering, and theoretical mechanics. Mathematical descriptions of relative and absolute payload swaying motion during crane boom rotation require the introduction of design models for a spherical pendulum with a suspension point following a horizontal circular trajectory. Spherical pendula with moving pivot centers are the so-called “eternal” problems that have been posed by ancient builders and civil engineers. Today payload swaying problems attract the great atten- tion of such applied mathematicians and mechanical engi- neers as Abdel-Rahman et al. [1–3], Adamiec-W´ ojcik et al. [4], Al-mousa et al. [5], Allan and Townsend [6], Aston [7], Betsch et al. [8], Blackburn et al. [9, 10], Blajer et al. [11– 17], Cha et al. [18], Chin et al. [19], Ellermann et al. [20], Erneux and Kalm´ ar-Nagy [21, 22], Ghigliazza and Holmes [23], Glossiotis and Antoniadis [24], Grigorov and Mitrev [25], Gusev and Vinogradov [26], Hong and Ngo [27], Hoon et al. [28], Ibrahim [29], Jerman et al. [30–33], Ju et al. [34], Kłosi´ nski [35], Krukowski et al. [36, 37], Lenci et al. [38], Leung and Kuang [39], Loveykin et al. [40], Maleki et al. [41], Maczynski et al. [42–44], Marinovi´ c et al. [45], Masoud et al. [46, 47], Mijailovi´ c[48], Mitrev and Grigorov [49, 50], Morales et al. [51], Nakazono et al. [52], Neitzel et al. [53], Neupert et al. [54], O’Connor and Habibi [55], Omar and Nayfeh [56], Osi´ nski and Wojciech [57], F. Palis and S. Palis [58], Perig et al. [59], Posiadała et al. [60], Ren et al. [61], Safarzadeh et al. [62], Sakawa et al. [63], Sawodny et al. [64], Spathopoulos et al. [65], Schaub [66], Solarz and Tora [67], Uchiyama et al. [68], Urba´ s[69], and others. Hindawi Publishing Corporation Shock and Vibration Volume 2014, Article ID 203709, 32 pages http://dx.doi.org/10.1155/2014/203709
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Research Article3 DOF Spherical Pendulum Oscillations with a UniformSlewing Pivot Center and a Small Angle Assumption
Alexander V Perig1 Alexander N Stadnik2
Alexander I Deriglazov2 and Sergey V Podlesny2
1 Manufacturing Processes and Automation Engineering Department Engineering Automation FacultyDonbass State Engineering Academy Shkadinova 72 Donetsk Region Kramatorsk 84313 Ukraine
2Department of Technical Mechanics Engineering Automation Faculty Donbass State Engineering AcademyShkadinova 72 Donetsk Region Kramatorsk 84313 Ukraine
Correspondence should be addressed to Alexander V Perig olexanderperiggmailcom
Received 11 February 2014 Revised 7 May 2014 Accepted 21 May 2014 Published 4 August 2014
Academic Editor Mickael Lallart
Copyright copy 2014 Alexander V Perig et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The present paper addresses the derivation of a 3 DOF mathematical model of a spherical pendulum attached to a crane boomtip for uniform slewing motion of the crane The governing nonlinear DAE-based system for crane boom uniform slewing hasbeen proposed numerically solved and experimentally verifiedThe proposed nonlinear and linearized models have been derivedwith an introduction of Cartesian coordinates The linearized model with small angle assumption has an analytical solution Therelative and absolute payload trajectories have been derivedThe amplitudes of load oscillations which depend on computed initialconditions have been estimated The dependence of natural frequencies on the transport inertia forces and gravity forces has beencomputed The conservative system which contains first time derivatives of coordinates without oscillation damping has beenderived The dynamic analogy between crane boom-driven payload swaying motion and Foucaultrsquos pendulum motion has beengrounded and outlined For a small swaying angle good agreement between theoretical and averaged experimental results wasobtained
1 Introduction
Payload swaying dynamics during crane boom slewing iswithin the objectives and scope of many academic andindustrial research programs in the fields ofmechanical elec-trical and control engineering and theoretical mechanicsMathematical descriptions of relative and absolute payloadswaying motion during crane boom rotation require theintroduction of design models for a spherical pendulumwitha suspension point following a horizontal circular trajectorySpherical pendula withmoving pivot centers are the so-calledldquoeternalrdquo problems that have been posed by ancient buildersand civil engineers
Today payload swaying problems attract the great atten-tion of such applied mathematicians and mechanical engi-neers as Abdel-Rahman et al [1ndash3] Adamiec-Wojcik et al[4] Al-mousa et al [5] Allan and Townsend [6] Aston [7]
Betsch et al [8] Blackburn et al [9 10] Blajer et al [11ndash17] Cha et al [18] Chin et al [19] Ellermann et al [20]Erneux and Kalmar-Nagy [21 22] Ghigliazza and Holmes[23] Glossiotis and Antoniadis [24] Grigorov and Mitrev[25] Gusev and Vinogradov [26] Hong and Ngo [27]Hoon et al [28] Ibrahim [29] Jerman et al [30ndash33] Ju et al[34] Kłosinski [35] Krukowski et al [36 37] Lenci et al[38] Leung and Kuang [39] Loveykin et al [40] Malekiet al [41] Maczynski et al [42ndash44] Marinovic et al [45]Masoud et al [46 47] Mijailovic [48] Mitrev and Grigorov[49 50] Morales et al [51] Nakazono et al [52] Neitzel et al[53] Neupert et al [54] OrsquoConnor and Habibi [55] Omarand Nayfeh [56] Osinski and Wojciech [57] F Palis and SPalis [58] Perig et al [59] Posiadała et al [60] Ren et al [61]Safarzadeh et al [62] Sakawa et al [63] Sawodny et al [64]Spathopoulos et al [65] Schaub [66] Solarz and Tora [67]Uchiyama et al [68] Urbas [69] and others
Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 203709 32 pageshttpdxdoiorg1011552014203709
2 Shock and Vibration
Applied engineering problems in the field of lifting-and-transport machines mainly deal with rectilinear or rotationalmotion of the spherical pendulum suspension center in deter-mined and stochastic cases Further improvement of rotatingcrane performance and efficiency requires the developmentof mathematical models for an adequate description of pay-load and crane boom tip positions Modern computationalapproaches to the solution of payload positioning problemshave been investigated in the following works
Abdel-Rahman et al [1ndash3] have provided a comprehen-sive review of different types of cranes the essential andwidely used mathematical techniques for models of cranedynamics and classical control methods [1ndash3] However thisresearch [1ndash3] gives inadequate attention to the effects ofCoriolis inertia forces on the payload relative motion duringcrane boom slew motion
Blajer et al [11ndash17] have proposed thirteen index-threedifferential-algebraic equations (DAE) in the rotary cranestate variables and control variables [11ndash17] They have pro-posed governing DAE equations and stable solution tech-niques allowing the rotary crane to execute the prescribedload trajectory and the control commands to implementfeed-forward control In this approach [11ndash17] the governingequations have been derived without consideration of theCoriolis inertia force and non-zero horizontal projections ofthe load gravity force
Hairer et al [70 71] and Oslashksendal [72] have developedcomputational techniques for DAE systems numerical inte-gration
Jerman et al [30ndash33] have applied an enhanced mathe-matical model of slewing the crane motion with load pendu-lation taking into account system stiffness coefficients [30ndash33] All mathematical techniques and governing equations inthese works [30ndash33] have been presented in implicit form
Ju et al have implemented finite element simulation fora flexible crane structure with a spherical pendulum [34] Inthis work [34] the spherical pendulum excitation is inducedby vibration modes of the flexible crane structure
Loveykin et al have derived the law of an optimal controlfor lifting-and-shifting machines under the assumption ofminimization of quadratic performance criteria in the casefor two-phase coordinates control and control rate [40]This work [40] made wide use of variation optimizationtechniques for pendulum oscillations in the vertical planewhich involve the trolleys of crane frames with rectilinearmotion
Maczynski et al have applied a numerically based finiteelement method (FEM) approach for simulation of a ldquocraneboom-payloadrdquo system without an explicit introduction ofinertia forces [42ndash44] Optimization problem of load posi-tioning in this study [42ndash44] has not fully addressed thenatural frequencies estimation for the system ldquocrane boom-payloadrdquo in the case of a fully rigid crane boom model
Mitrev and Grigorov [49 50] have derived governingequations for load relative swaying taking into accountenergy dissipation centrifugal Coriolis inertia and gravityforces [49 50] The Lagrange equations used here allow thesimulation of a spherical pendulum with a movable pivotcenter [49 50] Mitrevrsquos approach [49 50] is based on the
introduction of angular generalized coordinates which resultin nonlinearity of the problems and require a fourth-orderRunge-Kutta fixed-step integration method
However the previously known studies [1ndash100] have giveninadequate attention to the dynamic analysis of a load sway-ing in the horizontal plane of vibrations while accounting forthe effect of the Coriolis force on the trajectory of the relativeload motion of the cable The present research addresses thissituation
It should be noted that spherical pendulum relatedresearch has also been further developed for Foucault pendu-lums in the works of Condurache and Martinusi [77] Gusevand Vinogradov [26] de Icaza-Herrera and Castono [83]Pardy [88] Zanzottera et al [99] Zhuravlev and Petrov [100]and others
At first sight the spherical pendulum with rotatingpivot center and Foucault pendulum are two vastly differ-ent dynamic systems The key difference between the twodynamic systems is that dynamic analysis of the Foucaultpendulum is focused on a load swaying in the field of thecentral gravity force while crane boom slewing problems areposed for the vertical gravity force A commonality of thetwo dynamic systems is in both cases the effect of influenceof normal centrifugal and Coriolis forces on the shape ofthe relative and absolute trajectories The normal centrifugalforces depend on the relative coordinates and Coriolis forcesdepend on the relative velocity of the swaying loadMoreoverthe appearance of Coriolis forces that are dependent onrelative payload velocity retains the dynamic system as aconservative one because Coriolis acceleration remains at alltimes perpendicular to the relative velocity of the load Itmay therefore be concluded that there is a close couplingbetween the spherical pendulum with rotating pivot centerand Foucault pendulum Moreover the close relationshipbetween the two dynamic systems has not been properlyaddressed in all previous known research which emphasizesthe actuality and relevance of the present paper
The present paper is focused on the study of the oscilla-tion processes taking place in the vicinity of a steady equi-librium position of a payload during crane boom uniformslewingThe computational approach is based on the solutionof the initial value problem of particle dynamics for thedetermination of the relative load trajectory in the horizontalplane of the vibrations during crane boom uniform rotationThe approach used here takes into account both the relativerotation of the load vibrations in the vertical plane and theinfluence ofCoriolis acceleration on the formof the trajectoryof the swaying cargo relative motion This paper is alsoaimed at addressing the physically grounded interrelationsbetween the spherical pendulum with rotating pivot centerand Foucault pendulum
2 DAE System for Payload Swaying
It has been shown in Appendices AndashO that the nonlinearDAE system (formulae (C1) (F7)) for the relative coordi-nates119909
1(119905) (Figure 1(a))119910
1(119905) (Figure 1(b)) 119911
1(119905) of a swaying
Shock and Vibration 3
minus002
minus001
05
t
10 15
001
002
x1
(a)
minus002
minus001
05
t
10 15
003
002
001
y1
(b)
minus01
minus02
minus03
minus04
04
03
02
01
0
t
5 10 15x2
(c)
04
03
05
02
01
0
t
105 15
y2
(d)
minus002
minus002
minus001
minus001
x1
x1
y1
y1
002
002
003
001
0 001
(e)
01
02
03
04
05
minus01minus02minus03minus04 01 02 03 040
x2
y2
(0492 + y1) lowast cos(120593) + x1 lowast sin(120593)
(f)
Figure 1 Computational relative (a-b e) and computational absolute (c-d f) coordinates of a swaying payload during uniform crane boomslewing for the following numerical values of system parameters119898 = 01 kg 119897 = 0825m 119877 = 0492m 119892 = 981ms2 and 120596
119890= 0209 rads
4 Shock and Vibration
payload during crane boom slewing (Figures 8ndash10) in thenoninertial reference frameB is as follows
119898(
1198892(1199091(119905))
1198891199052
)
= minus119873 (119905) (
1199091(119905)
119897
) + 119898(
119889 (120593119890(119905))
119889119905
)
2
1199091(119905)
+ 119898(
1198892(120593119890(119905))
1198891199052
) (119877 + 1199101(119905))
+ 2119898(
119889 (120593119890(119905))
119889119905
)(
119889 (1199101(119905))
119889119905
)
119898(
1198892(1199101(119905))
1198891199052
)
= minus119873 (119905) (
1199101(119905)
119897
) + 119898(
119889 (120593119890(119905))
119889119905
)
2
(119877 + 1199101(119905))
minus 119898(
1198892(120593119890(119905))
1198891199052
)1199091(119905)
minus 2119898(
119889 (120593119890(119905))
119889119905
)(
119889 (1199091(119905))
119889119905
)
119898(
1198892(1199111(119905))
1198891199052
) = minus119898119892 + 119873 (119905) (
119897 minus 1199111(119905)
119897
)
1198972= (1199091(119905))2
+ (1199101(119905))2
+ (1199111(119905) minus 119897)
2
120593119890(119905) = 120596
119890119905 120593
119890(0) = 0
119889 (120593119890(0))
119889119905
= 120596119890 119873 (0) = 119898119892
1199091(0) = 0
119889 (1199091(0))
119889119905
= 120596119890119877 119910
1(0) = 0
119889 (1199101(0))
119889119905
= 0 1199111(0) = 0
119889 (1199111(0))
119889119905
= 0
(1)
The absolute coordinates 1199092(119905) (Figure 1(c)) 119910
2(119905)
(Figure 1(d)) 1199112(119905) of payload 119872 in the inertial reference
frame Emay be computed as
1199092(119905) = (119877 + 119910
1(119905)) cos (120593
119890(119905)) + 119909
1(119905) sin (120593
119890(119905))
1199102(119905) = (119877 + 119910
1(119905)) sin (120593
119890(119905)) minus 119909
1(119905) cos (120593
119890(119905))
1199112(119905) = 119911 (119905)
(2)
The absolute coordinates 1199092 1199102in the inertial reference
frameE which dependupon the relative coordinates11990911199101in
the noninertial reference frameB during the swaying of load119872 (Figures 8ndash10) has been defined according to the abovementioned Equations and has been shown in Figures 4 and 5as (- - - 119910
2= 1199102(1199092))
The computational results in Figure 1 derived for DAEproblem (1)-(2) solution coincide with the linearized solu-tion of the payload swaying problem in Appendices AndashI(formula (H1) in the noninertial reference frameB)
It is necessary to note that the posed DAE problem (1)describes payloadmotion in the vicinity of the lower positionof stable dynamical equilibrium assuming rope tension force119873(119905) gt 0 Factually the applied rope is assumed as a unilateralgeometric constraint (Appendix C) in the shape of a torsionfiberThe upper position of themechanical system is unstable(see [73ndash75 78ndash80 91 93]) and corresponds to the case of119873(119905) lt 0 within the parameters of the chosen mechanicalsystem in Figures 8ndash10 Such an upper pendulum positionmight conflict with the hypothesis about the unilateral natureof geometric constraint All of the above mentioned upperpendulum position conditions are outside the objectives andscope of the present paper
3 Experimental Procedure
The experimental procedure has been grounded on the usageof the assembly in Figures 2 and 3
The assembly in Figures 2 and 3 includes the followingmachine components the vertical fixed shaft119874
2119863with height
119867 = 1m the crane boom model 119861119863 with length 119877 = 05mand diameter 6mm the cable119861119872with different fixed lengths1198971= 0825m (in Figure 2 and Figure 5(b)) 119897
2= 0618m (in
Figure 5(a)) 1198973= 0412m (in Figure 4(b)) and 119897
4= 0206m
(in Figure 4(a)) The crane boom model 119861119863 is attached tothe vertical fixed shaft 119874
2119863 by bearing 119863 The cable 119861119872
is attached to the crane boom 119861119863 tip in the point 119861 Thefree or running end 119872 of the cable 119861119872 is the payload 119872
attachment pointThe load119872 is a light emitting diode (LED)with diameter 2mm and the battery with the battery voltage3V The experimental swaying trajectory (mdash 119910
2= 1199102(1199092)) in
Figures 2 4 and 5 is the experimental light emitting diode119872absolute trajectory in the inertial reference frameE
The laser pointer in Figure 2 is also attached to the craneboom 119861119863 tip in the point 119861 The laser pointer is the part ofthe noninertial reference frames B and F and the pointertrajectory is marked as (- - -) in Figure 2 The introductionof laser pointer 119861 allows estimation of dynamic deviation ofpayload119872 The horizontal regular fixed grid with canvas size1200mm times 700mm is placed in the horizontal plane119874
211990921199102
The horizontal grid is formed by the square cells with size20mm times 20mm
The experimental swaying trajectories (mdash 1199102= 1199102(1199092))
in Figures 2 4 and 5 have been written in the obscure roomwith the introduction of an upper digital camera with a longexposure 60 sndash90 s and the camera height above horizontalgrid level is 25m
4 Comparison and Discussion ofDAE-Solution Based Theoretical andExperimental Results
Thecomparison ofDAE-solution based theoretical (- - -1199102=
1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) results is shown
in Figures 4 and 5
Shock and Vibration 5
Light-emitting diode
Laser pointer
Camera
Boom
Slewing axis
Bearing
Lower plate
D
R = 05m
H=1
m
O2
O1
M
B
25
m
Ast
120596e = 0209 rads
l=0825
m
Regular grid 1200mm times 700mmwith square cells 20mm times 20mm
Figure 2 The dimension experimental measurement system for payload119872 swaying motion during crane boom uniform slewing
In order to estimate the relative disagreement of thederived DAE-solution based computational (- - -) andexperimental (mdash) payload 119872 absolute trajectories in theinertial reference frame E we have computed the amplitudediscrepancy 120575 in the polar coordinate system by the followingformula
120575 = ⟨
1
2
((
10038161003816100381610038161003816119903comp minus 119903exper
10038161003816100381610038161003816
119903comp) + (
10038161003816100381610038161003816119903comp minus 119903exper
10038161003816100381610038161003816
119903exper))⟩
(3)
where 119903comp and 119903exper are the magnitudes of the radius-vectors connecting point 119874
2and theoretical (- - -) and
experimental (mdash) curves and computed for the same fixedpolar angle 120593
119890
The amplitude discrepancies 120575 have the following values120575 = 753 for 119897 = 0206m in Figure 4(a) 120575 = 758 for119897 = 0412m in Figure 4(b) 120575 = 740 for 119897 = 0618m inFigure 5(a) and 120575 = 69 for 119897 = 0825m in Figure 5(b)
Numerical computations (- - - 1199102
= 1199102(1199092)) in
Figure 4(a) have been carried out for the following values of
mechanical system parameters 119877 = 0492m 119892 = 981ms2119897 = 0206m 119896 = (119892119897)
05asymp 6901 rads119879 = 14 s120596
119890= 2120587119879 asymp
0449 rads 120593119890= 180
∘ 1205721dyn = 001014 rad 119881
119861= 0221ms
119910dyn = 000209m ]1= 119896 + 120596
119890= 73496 rads ]
2= 119896 minus
120596119890= 64520 rads 119862
1= 001502m and 119862
3= 001711m
(Figure 4(a))Numerical computations (- - - 119910
2= 119910
2(1199092)) in
Figure 4(b) have been carried out for the following values ofmechanical system parameters 119877 = 0492m 119892 = 981ms2119897 = 0412m 119896 = (119892119897)
05asymp 4879 rads119879 = 14 s120596
119890= 2120587119879 asymp
0449 rads 120593119890= 180
∘ 1205721dyn = 001019 rad 119881
119861= 0221ms
119910dyn = 000419m ]1= 119896 + 120596
119890= 53284 rads ]
2= 119896 minus
120596119890= 44308 rads 119862
1= 002072m and 119862
3= 002492m
(Figure 4(b))Numerical computations (- - -119910
2= 1199102(1199092)) in Figure 5(a)
have been carried out for the following values of mechanicalsystem parameters 119877 = 0492m 119892 = 981ms2 119897 =
0618m 119896 = (119892119897)05
asymp 3984 rads 119879 = 20 s 120596119890= 2120587119879 asymp
0314 rads 120593119890
= 180∘ 1205721dyn = 000498 rad 119881
119861=
01546ms 119910dyn = 000308m ]1= 119896 + 120596
119890= 4298 rads
6 Shock and Vibration
(a) (b)
(c) (d)
Figure 3 Photos of dimension experimental measurement system (a)ndash(d) for payload 119872 swaying motion during crane boom uniformslewing
]2
= 119896 minus 120596119890= 3670 rads 119862
1= 001798m and 119862
3=
002106m (Figure 5(a))For further estimation of the relative disagreement of
derived DAE-solution based computational (- - -) andexperimental (mdash) payload 119872 absolute trajectories in theinertial reference frame E we have computed the confidenceintervals in Figures 4 and 5 for the dimensionless parameters119903comp119903exper and confidence probability 095 The Studentrsquos 119905-test results yield the following confidence intervals 09374 le
(119903comp119903exper) le 10315 for 119897 = 0206m in Figure 4(a)09014 le (119903comp119903exper) le 10272 for 119897 = 0412m inFigure 4(b) 09200 le (119903comp119903exper) le 10352 for 119897 = 0618min Figure 5(a) and 09257 le (119903comp119903exper) le 10066 for119897 = 0825m in Figure 5(b)
Both the relative discrepancy 120575 and the confidence inter-vals show the satisfactory agreement between the absolute
payload trajectories (Figures 4 and 5) in the inertial referenceframeE that have been computed with DAE-solution basedtheoreticalmodel (- - -) (1) andmeasured experimentally (mdash)as shown in Figures 4 and 5
It is important to note that the payload motion DAEequations in the nonlinear problem (1) for the noninertialreference frame B may be derived with an introductionof Lagrange equations (Appendices AndashC F) However thediscussion of (1) is more suitable and informative with anintroduction of dynamic Coriolis theorem (Appendices AndashD F) The first terms 1198892119909
11198891199052 1198892119910
11198891199052 in (1) define the
vector a119872B = ar (Appendix B formula (B4)) for the relative
acceleration of load 119872 in the noninertial reference frameB The straight-line terms (D3) formula-based terms in (1)are linearly proportional to the relative payload coordinatesin the noninertial reference frame B These straight-line
Shock and Vibration 7
05
05
04
04
03
03
02
02
00
00
01
01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)
p O2 (p D)
(a)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)Computational y2 = y2(x2)
l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)
p O2 (p D)
(b)
Figure 4 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0206m (a) and fixed cable length 119897 = 0412m (b)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)
2R = 1 (m)
y2 (m)
x2 (m)p O2 (p D)
(a)
05
04
03
02
00
01
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448(rads)
2R = 1 (m)
x2 (m)
y2 (m)
p O2 (p D)
(b)
Figure 5 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0618m (a) and fixed cable length 119897 = 0825m (b)
terms in (1) have been determined by the contribution ofthe normal or centripetal acceleration 120596BE times (120596BE times
r119864119872
) = aen (B6) of transportation for payload119872 and by theappearance of corresponding DrsquoAlembert centrifugal inertiaforce Φe
n= (minus119898)aen (D3) due to crane boom transport
rotation in the noninertial reference frame B Additionaltermsminus2120596
119890(1198891199101119889119905) and 2120596
119890(1198891199091119889119905) in (1) have been defined
by the compound or Coriolis acceleration 2120596BE times V119872B =
acor of payload119872 in the noninertial reference frameB Therectangular Cartesian projections of Coriolis inertia force inthe noninertial reference frameB is defined by formula (B7)
5 Experimental Results for the SphericalPendulum with the Fixed Pivot Center
Experimental load swaying results for the fixed crane boommodel (120596
119890= 0 rads) are shown in Figure 6
Finite motions of a swaying load are shown in Figure 6in the case of 120596
119890= 0 rads In Figure 6 the relative and
absolute load trajectories are identically equal The lingeringremains of elliptical motion in Figure 6 and ellipses semiaxesare defined by the initial conditions
6 Discussion of Analogy between PayloadSwaying and Foucaultrsquos Pendulum Motion
The motion of Foucaultrsquos pendulum is shown in Figure 7where 119909
211991021199112is the heliocentric inertial reference frame
(not shown in Figure 7) 119909111991011199111is the geocentric noninertial
reference frame and 119909119910119911 is a noninertial reference framelocated at the geographic latitude 119910
1119911 119860 st119911 is a local vertical
119861119863 is the radius of the circle of the pinning point 119861119872 isthe cable length and 120596
119890is the angular velocity of the Earth
diurnal rotation
8 Shock and Vibration
(a) (b)
Figure 6 Experimental trajectories of the spherical pendulum with fixed pivot center and fixed cable length 119897 = 0825m (a b)
M
g
yx
z
B
D
x1
Ast
120596e
z1
y1
Figure 7 Foucaultrsquos pendulum schematic view
It is important to note that the structure of linearizedequations (H4)-(H5) for the swaying payload during craneboom uniform rotation in the noninertial reference frameF is analogous to the structure of the governing equationsfor Foucaultrsquos pendulum motion in the known publishedworks by Zhuravlev and Petrov [100] (formula (6) p 36 of[100]) Pardy [88] (formulae (12)ndash(14) pp 850-851 of [88])and Condurache et al [77] (formulae (11)ndash(13) p 743-744of [77]) The above mentioned analogy between linearizedequations (H4)-(H5) and the governing equations in [77 88100] assumes the geometric analogy of relative swaying tra-jectories between Foucaultrsquos pendulum and the boom-drivenpendulum with rotating pivot centers (Figures 1(e) 12 and16) So there is the geometric analogy between computationalrelative trajectories of load 119872 swaying during crane boomuniform slewing (Figures 1(e) 12 and 16) and relative trajec-tories of load119872 swaying in the plane (119860 st119909119910) around p119860 st inFigure 7 shownbyZhuravlev andPetrov [100] (Figure 4 p 39of [100]) and Condurache andMartinusi [77] (Figure 14 at p754 and Figure 16 at p 755 of [77]) The better understandingof Foucaultrsquos pendula dynamics is provided with the studyof computational relative trajectories in Figures 1(e) 12 and16 For Foucaultrsquos pendulum (Figure 7) the major semiaxes
of relative amplitude extremes have angular velocity valuesequal to the angular velocity of the Earth diurnal rotationMoreover for Foucaultrsquos pendulum (Figure 7) the directionof rotation of major semiaxes of relative amplitude extremesis oppositely directed to the direction of the Earth diurnalrotation All the above mentioned means that the plane(119860 st119861119872) of Foucaultrsquos pendula swaying remains fixed in theheliocentric inertial reference frame 119909
211991021199112
The presented analogy between crane boom-driven pay-load swaying and Foucaultrsquos pendulum motion essentiallyincreases knowledge and awareness of the oscillation pro-cesses in such dynamic systems
7 Discussion and MechanicalInterpretation of GoverningEquations for DAE-Based Nonlinearand Linearized Models
A governing nonlinear DAE-based system (1) for crane boomuniform slewing has been proposed and numerically solvedA conservative system (1) and (H4)-(H5) with componentsΦcor119909 andΦcor119910 has been derived Both projectionsΦcor119909 andΦcor119910 coincide in the noninertial reference framesB andF
The occurrence of the first derivatives 119889119909119889119905 119889119910119889119905 ofthe load119872 relative coordinates with respect to time in termsminus2120596119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) determines that there is no
decay of the oscillations in (H4)-(H5) and (1) but onlyredirection of the relative velocity vector V
119872F = Vr of theload119872 in the noninertial reference framesB andF
The introduction of a linearized model (H4)-(H5)allowed the determination of the natural frequencies (]
1=
]2) of free oscillations of payload 119872 ]
1= 119896 + 120596
119890and
]2= 119896 minus 120596
119890
It follows from the forms of the relative payload trajectory119910 = 119910(119909) in the noninertial reference frame F and theabsolute trajectory 119910
2= 1199102(1199092) in the inertial reference frame
E that the resulting motion of the payload 119872 on the cable119872119861 taking into account the Coriolis inertia forceΦcor (D5)will be the sum of two oscillations with natural frequencies ]
1
and ]2 and with periods 119879
1= 2120587]
1and 119879
2= 2120587]
2
It is worth noting that the frequencies ]1and ]
2differ
by 2120596119890 This means that for the small angle assumption
(Appendices H-I) we have ]1
asymp ]2and the trajectory
Shock and Vibration 9
mg
γ
Nℬ
z1z2
e2
e3
ℰ
ℰ
ℰ
ℱ
ℱ
ℬΔ
120574
y1
Vr
At
M
R D
E
Bz
l
120573
O
1205721
1205721dyn
Adyn
ydyn
h
120596e
AstO1
3ẽ
2ẽ
O2
d120593edty
sin(120593e)
cos(120593e)
120596ℬℰ = 120596ℱℰ
3e
2e
1e
Figure 8 Swaying scheme of load119872 on the cable119872119861 which is fixedly attached at the point 119861 on the crane boom 1198611198742 in the vertical plane
(119910119911) for the nonlinear model derivation in Cartesian coordinates
of relative motion 119910 = 119910(119909) in the noninertial referenceframe F on the expiration of the relative oscillations periodtime 2120587119896 looks approximately like an ellipse (Figures 12(b)and 16(b)) with semiaxes which are governed by the initialconditions
The major semiaxis of such an ellipse in Figures 12(b)and 16(b) is defined by the initial velocity of load 119872 in thenoninertial reference frameF and the analytical value of themajor semiaxis is equal to119881
119861119896 = (120596
119890119877)119896 The derived value
of 119881119861119896 is approximately the sum of the absolute values of
the amplitudes in the analytical form (I15) of the linearizedCauchy problem (H4)-(H5)
The minor semiaxis of the above mentioned ellipsein Figures 12(b) and 16(b) is defined by the value 119910dynwhich is approximately the difference of amplitudes in theanalytical form (I15) of the linearizedCauchy problem (H4)-(H5) In this case the major semiaxis is approximately 5ndash20 times larger than the minor semiaxis because the angularvelocity 120596
119890of the transport rotation of crane boom 119861119874
2is
much smaller than the natural frequency 119896 of the sphericaloscillations of payload119872 on the cable119872119861 (Figures 12(b) and16(b))
It is also necessary to note that the major semiaxis of theellipse in the relative motion rotates with an angular velocity120596119890in the opposite direction of the transport rotation of crane
boom 1198611198742in (Figures 12 and 16)
It follows from Figures 1(f) 2 4 and 5 that the trajectoryof absolute motion of load 119872 in the inertial reference frameE is almost a symmetric curve with 119910
2axial symmetry
The initial and the final motions of load 119872 for half-period
essentially differ from its harmonic oscillations neighbor forquarter-period
Due to the negligible quantity of 120572dyn and 119910dyn (G3)ndash(G5) the average deviation of the load119872 from the mechan-ical trajectory of the boom tip 119861 is negligible (Figures1(f) 2 4 and 5) The basic dynamic load on the systemldquocrane boom 119861119874
2-load 119872rdquo is created by the high-frequency
oscillations of load 119872 which are determined by the actionof inertia force Φcor = minus2119898(120596BE times V
119872B) stipulated bythe Coriolis acceleration acor = 2120596BE times V
119872B of load 119872
in the noninertial reference frame F The basic dynamicload in the system defines additional loads and vibrations ofcrane boom 119861119874
2mechanical elements and support bearings
complicates the automatic and manual control systems of theelectromechanical crane boom 119861119874
2 and also makes crane
operation much more difficultIt is also important to note that the stop of crane boom
1198611198742does not lead to instantaneous damping of the load
119872 absolute oscillations in the inertial reference frame E(Figures 2 4 and 5) This phenomenon directly follows fromthe experimental trajectory in Figures 2 4 and 5 Alsothe natural spherical oscillations of load 119872 will occur withthe frequency 119896 and the amplitude (the difference betweenthe final position of load 119872 in the relative motion in thenoninertial reference frame F for half-period of vibrationand static equilibrium 119860 st of load 119872 on the cable) Thefurther oscillations for the stop of the crane boom 119861119874
2
slewing motion are the deviation of the real trajectory fromthe intended final position of load119872 The results of physicalsimulation in Figures 2 4 and 5 show the necessity of add-on
10 Shock and Vibration
N
ℬ
ℬ
e2
e1
ℰ
ℰ
ℱ
ℱ
y2
x2
Ve
Vr
M
E
1ẽ
2ẽ x
x1
O2(D)
AoAst
O
y y1
Adyn
ydyn
1205722
120579
120579
120596e = 120596ℬℰ = 120596ℱℰ
120593e = 120596et
Φcor
B(O1)
Φen
Φeτ
1e
2e
Figure 9 Swaying scheme of load119872 at the cable119872119861 which is fixedly attached in the point 119861 of the crane boom 1198611198742 in the horizontal plane
(11990911199101) for the nonlinear model derivation in Cartesian coordinates
devices development for the efficient suppression of load 119872
final oscillations
8 Discussion and Comparison of Derived andKnown Published Results
In Figure 1 and pp 537-538 of publishedwork by Sakawa et al1981 [63] the small angle between payloadrsquos cable and verticalline has been introduced that confirms proposed nonlinearDAE-based system (1) and linearized model (H4)-(H5)
In Figure 9 of p 278 of published work by Maczynskiand Wojciech 2003 [44] the computational finite elementmethod (FEM)-based results were shown for the abso-lute payload trajectories in the inertial reference frame EMaczynskirsquos Figure 9 in [44] outlines that the angle betweenthe payloadrsquos cable and the vertical line does not exceed 01rad and that agreeswith the proposed linearizedmodel (H4)-(H5) for a small angle assumption Computational trajectoryfor Maczynski-derived small load swaying [44] after craneboom stop qualitatively coincides with the experimentallyobservable load119872motion in Figures 2 4 and 5
In Figure 6 of published work by Ju et al 2006 [34]it was shown that the angle between the payloadrsquos cableand the vertical line does not exceed 16∘ Jursquos formula (15a)in p 382 of [34] assumes that Jursquos angle between payloadrsquoscable and the vertical line has the harmonic law of variationwith an introduction of a small perturbation term Both Jursquosassumptions in [34] confirm the proposed linearized model(H4)-(H5)
Comparison of the derived linearized model (H4)-(H5)with Figure 5 of published work by Mitrev and Grigorov2008 in [49] shows that the Mitrev-derived ranges of payloadswaying angles within Mitrevrsquos nonlinear model does notexceed 48∘ Such small values of swaying angles completelyconfirm the applicability and correctness of the small angleassumption in (H4)-(H5)
9 Final Conclusions
A governing nonlinear DAE-based system for crane boomuniform slewing has been proposed numerically solved andexperimentally verified
Shock and Vibration 11
Fully nonlinear differential equations for 3 DOF sphericalpendulum oscillations with a uniform slewing crane boomaround a fixed vertical axis of rotation have been derivedin relative Cartesian and spherical coordinates The identicallinearized differential equations in relative Cartesian coordi-nates have been derived with an introduction of the Coriolisdynamic theorem and Lagrange equations for a uniformcrane boomrotation and small swaying angle120572
1assumptions
Linearized andnonlinear theoretical problems for relativeswaying of the payload have been formulated in the form ofthe initial value (Cauchy) problem
An analytical solution of the linearized system has beenderived
The influences of inertia forces from the centripetal andcompound accelerations have been estimated
A linearized conservative system which contains firsttime derivatives of coordinates and no damping of oscilla-tions has been derived
The amplitudes of load oscillations which depend oncomputed initial conditions have been estimated within asmall angle assumption The dependence of natural frequen-cies on the transport inertia forces and gravity forces has beencomputed for the linearized systems
The formulae for the association of relative payloadmotion in the noninertial reference frame F and absolutepayload motion in the inertial reference frame E have beenoutlined
The results of the numerical DAE-based investigation andthe performed physical simulation show a satisfactory fit forfrequencies and amplitudes of load oscillations
The dynamic analogy between crane boom-driven pay-load swaying motion and Foucaultrsquos pendulum motion hasbeen grounded and outlined
The results of the present work are the foundation forfurther investigations of payload119872 swaying dynamics duringtelescopic crane boom nonuniform slewing motion withvariable cable length and for the different motions of thependulum pivot center
Appendices
A Velocity Kinematics Analysis inCartesian Coordinates
For the nonlinear problem definition we study the coopera-tive motion of the mechanical system ldquocrane boom 119861119874
2-load
119872rdquo which is shown in Figures 8ndash10In the nomenclature chapter we denote the fixed inertial
frame of reference E as 119909211991021199112 and the moving noninertial
frame of reference B as 119909111991011199111 which is rigidly bounded
with the crane boom1198611198742 Rotation of themoving noninertial
frame of referenceB(119909111991011199111) around the fixed inertial frame
of referenceE(119909211991021199112) defines the transportation motion for
payload 119872 The motion of load 119872 relative to the movingnoninertial frame of referenceB(119909
111991011199111) defines the relative
motion of payload 119872 The point 119860 st with the coordinates1199091= 0 119910
1= 0 and 119911
1= 0 is the steady equilibrium position
for the load119872 when the crane boom 1198611198742is fixed The point
119860dyn with the coordinates 1199091 = 0 1199101= 119910dyn and 1199111 = Δ is the
dynamic equilibrium position for the load119872 for the rotatingcrane boom 119861119874
2 We will assume the point119860dyn as the origin
of the noninertial reference frameF(119909119910119911) The directions ofaxes 119909 119910 119911 of the noninertial reference frameF are parallelto the axes 119909
1 1199101 1199111of the noninertial reference frameB in
Figures 8ndash10 The load 119872 relative motion takes place alongthe sphere surface with the fixed radius equal to the cablelength 119861119872 = 119897
The computational scheme in Figures 8ndash10 for the nonlin-ear model derivation may be described with an introductionof three degrees of freedom For generalized coordinates weassume the rectangular coordinates 119909
1 1199101 and 119911
1of the load
119872 in the moving noninertial frame of reference B and theangle 120593
119890of crane boom 119861119874
2slewing in the horizontal plane
(11990911199101) with the angular velocity 120596
119890around the vertical axis
11987421199112The relative velocity vector Vr = V
119872B of the load 119872 isdefined as Vr = V
119872B = (1198891199091119889119905 119889119910
1119889119905 119889119911
1119889119905)
In order to derive the absolute velocity V119872E of payload
119872 we will apply the vector method for absolute motionassignment We will define r
1198742119872
as the position vector con-necting initial point 119874
2and terminal point119872 in noninertial
reference frameB (119898) consider
r1198742119872
= r1198742119861+ r119861119872
(A1)
where vector components in (1) are defined in nomenclatureIn accordance with Figures 8ndash10 we have the following
unit vectorsrsquo expansions for position vectors r1198742119861
and r119861119872
in (A1)
r1198742119861
= (119877 cos (120593119890)) e1+ (119877 sin (120593
119890)) e2+ (119877 tan (120574)) e
3
(A2)
r119861119872
= 1199091e1+ 1199101e2+ (1199111minus 119897) e3 (A3)
where 119877 120593119890 1199091 and 119910
1notations have been defined in
nomenclature and in Figures 8ndash10For further definition of the position vector r
1198742119872
in iner-tial reference frame E we write unit vectors of noninertialreference frameB in (A3) through the unit vectors of inertialreference frame E
e1= (sin (120593
119890)) e1minus (cos (120593
119890)) e2 (A4)
e2= (cos (120593
119890)) e1minus (sin (120593
119890)) e2 (A5)
e3= e3 (A6)
After substitution of formulae (A2)ndash(A6) in (A1) andsome algebraic manipulations the vector expression (A1) forthe position vector r
1198742119872
in the inertial reference frameEwilltake the following form
r1198742119872
= ((119877 + 1199101) cos (120593
119890) + 1199091sin (120593
119890)) e1
+ ((119877 + 1199101) sin (120593
119890) minus 1199091cos (120593
119890)) e2
+ (119877 tan (120574) + 1199111minus 119897) e3
(A7)
12 Shock and Vibration
The absolute velocity of payload 119872 we define by timedifferentiation of (A7) assuming constancy of unit vectorse1 e2 and e
3of the inertial reference frameE and constancy
119877 and 120574
V119872E = Vabs =
119889 (r1198742119872
)
119889119905
(A8)
After differentiation and algebraic transformations in(A7) and (A8) we have the following vector expression forabsolute payload 119872 velocity in the inertial reference frameE
V119872E
= (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) cos (120593119890)
+ ((
1198891199091
119889119905
) minus (119877 + 1199101) (
119889120593119890
119889119905
)) sin (120593119890)) e1
+ (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) sin (120593119890)
+ ((119877 + 1199101) (
119889120593119890
119889119905
) minus (
1198891199091
119889119905
)) cos (120593119890))
times e2+ (
1198891199111
119889119905
) e3
(A9)
The square of absolute payload119872 velocitymay be writtenas
1198812
abs = (
1198891199091
119889119905
)
2
+ (
1198891199101
119889119905
)
2
+ (
1198891199111
119889119905
)
2
+ (
119889120593119890
119889119905
)
2
times (1199092
1+ (1199101+ 119877)2
) + 2(
119889120593119890
119889119905
)
times (1199091(
1198891199101
119889119905
) minus (1199101+ 119877)(
1198891199091
119889119905
))
(A10)
After algebraic transformations we have the square ofabsolute payload119872 velocity in the form of
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A11)
Algebraic expressions (A10) and (A11) could be derivedon the basis of a velocity addition theorem for payload 119872
compound motion in the inertial reference frame E throughthe unit vectors of the noninertial reference frame B (see[59 76 81 82 84 85 87 90ndash92 94ndash98])
V119872E = V
119872B + V1198742E + 120596BE times r
1198742119872
(A12)
where payload relative velocity in the noninertial referenceframeBmay be written as
V119872B = Vr = (
1198891199091
119889119905
) e1+ (
1198891199101
119889119905
) e2+ (
1198891199111
119889119905
) e3
(A13)
The velocityV1198742E of the point119874
2= point119864 in the inertial
reference frame E has zero value
V1198742E = 0 (A14)
The last term in (A12) is the vector product 120596BE times
r1198742119872
where crane boom angular velocity vector is 120596BE =
(119889120593119890119889119905)e3(A6) and the position vector r
1198742119872
in the nonin-ertial reference frameB is as follows
r1198742119872
= 1199091e1+ (1199101+ 119877) e
2+ 1199111e3 (A15)
Taking into account (A13)ndash(A15) the vector equation(A12) in the inertial reference frame E written through theunit vectors e
1 e2 e3of the noninertial reference frame B
will have the following form
V119872E = ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101)) e1
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091) e2+ (
1198891199111
119889119905
) e3
(A16)
So on the basis of (A16) we have the following square ofabsolute payload119872 velocity as
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A17)
Independently derived expressions for the scalar prod-uct (V
119872EV119872E) in (A11) and (A17) completely coincidewhich confirms the correctness and accuracy of the pen-dulum absolute velocity V
119872E derivation and shows thatthe scalar product (V
119872EV119872E) is the invariant expressionindependent of choice of reference frame
The second and third terms of (A12) in the noninertialreference frame B determine the vector of the load 119872
transportation velocity as
Ve = V1198742E + 120596BE times r
1198742119872
(A18)
Taking into account (A14) the scalar of the load 119872
transportation velocity is defined as
1003817100381710038171003817Ve
1003817100381710038171003817= 120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817 (A19)
Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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2 Shock and Vibration
Applied engineering problems in the field of lifting-and-transport machines mainly deal with rectilinear or rotationalmotion of the spherical pendulum suspension center in deter-mined and stochastic cases Further improvement of rotatingcrane performance and efficiency requires the developmentof mathematical models for an adequate description of pay-load and crane boom tip positions Modern computationalapproaches to the solution of payload positioning problemshave been investigated in the following works
Abdel-Rahman et al [1ndash3] have provided a comprehen-sive review of different types of cranes the essential andwidely used mathematical techniques for models of cranedynamics and classical control methods [1ndash3] However thisresearch [1ndash3] gives inadequate attention to the effects ofCoriolis inertia forces on the payload relative motion duringcrane boom slew motion
Blajer et al [11ndash17] have proposed thirteen index-threedifferential-algebraic equations (DAE) in the rotary cranestate variables and control variables [11ndash17] They have pro-posed governing DAE equations and stable solution tech-niques allowing the rotary crane to execute the prescribedload trajectory and the control commands to implementfeed-forward control In this approach [11ndash17] the governingequations have been derived without consideration of theCoriolis inertia force and non-zero horizontal projections ofthe load gravity force
Hairer et al [70 71] and Oslashksendal [72] have developedcomputational techniques for DAE systems numerical inte-gration
Jerman et al [30ndash33] have applied an enhanced mathe-matical model of slewing the crane motion with load pendu-lation taking into account system stiffness coefficients [30ndash33] All mathematical techniques and governing equations inthese works [30ndash33] have been presented in implicit form
Ju et al have implemented finite element simulation fora flexible crane structure with a spherical pendulum [34] Inthis work [34] the spherical pendulum excitation is inducedby vibration modes of the flexible crane structure
Loveykin et al have derived the law of an optimal controlfor lifting-and-shifting machines under the assumption ofminimization of quadratic performance criteria in the casefor two-phase coordinates control and control rate [40]This work [40] made wide use of variation optimizationtechniques for pendulum oscillations in the vertical planewhich involve the trolleys of crane frames with rectilinearmotion
Maczynski et al have applied a numerically based finiteelement method (FEM) approach for simulation of a ldquocraneboom-payloadrdquo system without an explicit introduction ofinertia forces [42ndash44] Optimization problem of load posi-tioning in this study [42ndash44] has not fully addressed thenatural frequencies estimation for the system ldquocrane boom-payloadrdquo in the case of a fully rigid crane boom model
Mitrev and Grigorov [49 50] have derived governingequations for load relative swaying taking into accountenergy dissipation centrifugal Coriolis inertia and gravityforces [49 50] The Lagrange equations used here allow thesimulation of a spherical pendulum with a movable pivotcenter [49 50] Mitrevrsquos approach [49 50] is based on the
introduction of angular generalized coordinates which resultin nonlinearity of the problems and require a fourth-orderRunge-Kutta fixed-step integration method
However the previously known studies [1ndash100] have giveninadequate attention to the dynamic analysis of a load sway-ing in the horizontal plane of vibrations while accounting forthe effect of the Coriolis force on the trajectory of the relativeload motion of the cable The present research addresses thissituation
It should be noted that spherical pendulum relatedresearch has also been further developed for Foucault pendu-lums in the works of Condurache and Martinusi [77] Gusevand Vinogradov [26] de Icaza-Herrera and Castono [83]Pardy [88] Zanzottera et al [99] Zhuravlev and Petrov [100]and others
At first sight the spherical pendulum with rotatingpivot center and Foucault pendulum are two vastly differ-ent dynamic systems The key difference between the twodynamic systems is that dynamic analysis of the Foucaultpendulum is focused on a load swaying in the field of thecentral gravity force while crane boom slewing problems areposed for the vertical gravity force A commonality of thetwo dynamic systems is in both cases the effect of influenceof normal centrifugal and Coriolis forces on the shape ofthe relative and absolute trajectories The normal centrifugalforces depend on the relative coordinates and Coriolis forcesdepend on the relative velocity of the swaying loadMoreoverthe appearance of Coriolis forces that are dependent onrelative payload velocity retains the dynamic system as aconservative one because Coriolis acceleration remains at alltimes perpendicular to the relative velocity of the load Itmay therefore be concluded that there is a close couplingbetween the spherical pendulum with rotating pivot centerand Foucault pendulum Moreover the close relationshipbetween the two dynamic systems has not been properlyaddressed in all previous known research which emphasizesthe actuality and relevance of the present paper
The present paper is focused on the study of the oscilla-tion processes taking place in the vicinity of a steady equi-librium position of a payload during crane boom uniformslewingThe computational approach is based on the solutionof the initial value problem of particle dynamics for thedetermination of the relative load trajectory in the horizontalplane of the vibrations during crane boom uniform rotationThe approach used here takes into account both the relativerotation of the load vibrations in the vertical plane and theinfluence ofCoriolis acceleration on the formof the trajectoryof the swaying cargo relative motion This paper is alsoaimed at addressing the physically grounded interrelationsbetween the spherical pendulum with rotating pivot centerand Foucault pendulum
2 DAE System for Payload Swaying
It has been shown in Appendices AndashO that the nonlinearDAE system (formulae (C1) (F7)) for the relative coordi-nates119909
1(119905) (Figure 1(a))119910
1(119905) (Figure 1(b)) 119911
1(119905) of a swaying
Shock and Vibration 3
minus002
minus001
05
t
10 15
001
002
x1
(a)
minus002
minus001
05
t
10 15
003
002
001
y1
(b)
minus01
minus02
minus03
minus04
04
03
02
01
0
t
5 10 15x2
(c)
04
03
05
02
01
0
t
105 15
y2
(d)
minus002
minus002
minus001
minus001
x1
x1
y1
y1
002
002
003
001
0 001
(e)
01
02
03
04
05
minus01minus02minus03minus04 01 02 03 040
x2
y2
(0492 + y1) lowast cos(120593) + x1 lowast sin(120593)
(f)
Figure 1 Computational relative (a-b e) and computational absolute (c-d f) coordinates of a swaying payload during uniform crane boomslewing for the following numerical values of system parameters119898 = 01 kg 119897 = 0825m 119877 = 0492m 119892 = 981ms2 and 120596
119890= 0209 rads
4 Shock and Vibration
payload during crane boom slewing (Figures 8ndash10) in thenoninertial reference frameB is as follows
119898(
1198892(1199091(119905))
1198891199052
)
= minus119873 (119905) (
1199091(119905)
119897
) + 119898(
119889 (120593119890(119905))
119889119905
)
2
1199091(119905)
+ 119898(
1198892(120593119890(119905))
1198891199052
) (119877 + 1199101(119905))
+ 2119898(
119889 (120593119890(119905))
119889119905
)(
119889 (1199101(119905))
119889119905
)
119898(
1198892(1199101(119905))
1198891199052
)
= minus119873 (119905) (
1199101(119905)
119897
) + 119898(
119889 (120593119890(119905))
119889119905
)
2
(119877 + 1199101(119905))
minus 119898(
1198892(120593119890(119905))
1198891199052
)1199091(119905)
minus 2119898(
119889 (120593119890(119905))
119889119905
)(
119889 (1199091(119905))
119889119905
)
119898(
1198892(1199111(119905))
1198891199052
) = minus119898119892 + 119873 (119905) (
119897 minus 1199111(119905)
119897
)
1198972= (1199091(119905))2
+ (1199101(119905))2
+ (1199111(119905) minus 119897)
2
120593119890(119905) = 120596
119890119905 120593
119890(0) = 0
119889 (120593119890(0))
119889119905
= 120596119890 119873 (0) = 119898119892
1199091(0) = 0
119889 (1199091(0))
119889119905
= 120596119890119877 119910
1(0) = 0
119889 (1199101(0))
119889119905
= 0 1199111(0) = 0
119889 (1199111(0))
119889119905
= 0
(1)
The absolute coordinates 1199092(119905) (Figure 1(c)) 119910
2(119905)
(Figure 1(d)) 1199112(119905) of payload 119872 in the inertial reference
frame Emay be computed as
1199092(119905) = (119877 + 119910
1(119905)) cos (120593
119890(119905)) + 119909
1(119905) sin (120593
119890(119905))
1199102(119905) = (119877 + 119910
1(119905)) sin (120593
119890(119905)) minus 119909
1(119905) cos (120593
119890(119905))
1199112(119905) = 119911 (119905)
(2)
The absolute coordinates 1199092 1199102in the inertial reference
frameE which dependupon the relative coordinates11990911199101in
the noninertial reference frameB during the swaying of load119872 (Figures 8ndash10) has been defined according to the abovementioned Equations and has been shown in Figures 4 and 5as (- - - 119910
2= 1199102(1199092))
The computational results in Figure 1 derived for DAEproblem (1)-(2) solution coincide with the linearized solu-tion of the payload swaying problem in Appendices AndashI(formula (H1) in the noninertial reference frameB)
It is necessary to note that the posed DAE problem (1)describes payloadmotion in the vicinity of the lower positionof stable dynamical equilibrium assuming rope tension force119873(119905) gt 0 Factually the applied rope is assumed as a unilateralgeometric constraint (Appendix C) in the shape of a torsionfiberThe upper position of themechanical system is unstable(see [73ndash75 78ndash80 91 93]) and corresponds to the case of119873(119905) lt 0 within the parameters of the chosen mechanicalsystem in Figures 8ndash10 Such an upper pendulum positionmight conflict with the hypothesis about the unilateral natureof geometric constraint All of the above mentioned upperpendulum position conditions are outside the objectives andscope of the present paper
3 Experimental Procedure
The experimental procedure has been grounded on the usageof the assembly in Figures 2 and 3
The assembly in Figures 2 and 3 includes the followingmachine components the vertical fixed shaft119874
2119863with height
119867 = 1m the crane boom model 119861119863 with length 119877 = 05mand diameter 6mm the cable119861119872with different fixed lengths1198971= 0825m (in Figure 2 and Figure 5(b)) 119897
2= 0618m (in
Figure 5(a)) 1198973= 0412m (in Figure 4(b)) and 119897
4= 0206m
(in Figure 4(a)) The crane boom model 119861119863 is attached tothe vertical fixed shaft 119874
2119863 by bearing 119863 The cable 119861119872
is attached to the crane boom 119861119863 tip in the point 119861 Thefree or running end 119872 of the cable 119861119872 is the payload 119872
attachment pointThe load119872 is a light emitting diode (LED)with diameter 2mm and the battery with the battery voltage3V The experimental swaying trajectory (mdash 119910
2= 1199102(1199092)) in
Figures 2 4 and 5 is the experimental light emitting diode119872absolute trajectory in the inertial reference frameE
The laser pointer in Figure 2 is also attached to the craneboom 119861119863 tip in the point 119861 The laser pointer is the part ofthe noninertial reference frames B and F and the pointertrajectory is marked as (- - -) in Figure 2 The introductionof laser pointer 119861 allows estimation of dynamic deviation ofpayload119872 The horizontal regular fixed grid with canvas size1200mm times 700mm is placed in the horizontal plane119874
211990921199102
The horizontal grid is formed by the square cells with size20mm times 20mm
The experimental swaying trajectories (mdash 1199102= 1199102(1199092))
in Figures 2 4 and 5 have been written in the obscure roomwith the introduction of an upper digital camera with a longexposure 60 sndash90 s and the camera height above horizontalgrid level is 25m
4 Comparison and Discussion ofDAE-Solution Based Theoretical andExperimental Results
Thecomparison ofDAE-solution based theoretical (- - -1199102=
1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) results is shown
in Figures 4 and 5
Shock and Vibration 5
Light-emitting diode
Laser pointer
Camera
Boom
Slewing axis
Bearing
Lower plate
D
R = 05m
H=1
m
O2
O1
M
B
25
m
Ast
120596e = 0209 rads
l=0825
m
Regular grid 1200mm times 700mmwith square cells 20mm times 20mm
Figure 2 The dimension experimental measurement system for payload119872 swaying motion during crane boom uniform slewing
In order to estimate the relative disagreement of thederived DAE-solution based computational (- - -) andexperimental (mdash) payload 119872 absolute trajectories in theinertial reference frame E we have computed the amplitudediscrepancy 120575 in the polar coordinate system by the followingformula
120575 = ⟨
1
2
((
10038161003816100381610038161003816119903comp minus 119903exper
10038161003816100381610038161003816
119903comp) + (
10038161003816100381610038161003816119903comp minus 119903exper
10038161003816100381610038161003816
119903exper))⟩
(3)
where 119903comp and 119903exper are the magnitudes of the radius-vectors connecting point 119874
2and theoretical (- - -) and
experimental (mdash) curves and computed for the same fixedpolar angle 120593
119890
The amplitude discrepancies 120575 have the following values120575 = 753 for 119897 = 0206m in Figure 4(a) 120575 = 758 for119897 = 0412m in Figure 4(b) 120575 = 740 for 119897 = 0618m inFigure 5(a) and 120575 = 69 for 119897 = 0825m in Figure 5(b)
Numerical computations (- - - 1199102
= 1199102(1199092)) in
Figure 4(a) have been carried out for the following values of
mechanical system parameters 119877 = 0492m 119892 = 981ms2119897 = 0206m 119896 = (119892119897)
05asymp 6901 rads119879 = 14 s120596
119890= 2120587119879 asymp
0449 rads 120593119890= 180
∘ 1205721dyn = 001014 rad 119881
119861= 0221ms
119910dyn = 000209m ]1= 119896 + 120596
119890= 73496 rads ]
2= 119896 minus
120596119890= 64520 rads 119862
1= 001502m and 119862
3= 001711m
(Figure 4(a))Numerical computations (- - - 119910
2= 119910
2(1199092)) in
Figure 4(b) have been carried out for the following values ofmechanical system parameters 119877 = 0492m 119892 = 981ms2119897 = 0412m 119896 = (119892119897)
05asymp 4879 rads119879 = 14 s120596
119890= 2120587119879 asymp
0449 rads 120593119890= 180
∘ 1205721dyn = 001019 rad 119881
119861= 0221ms
119910dyn = 000419m ]1= 119896 + 120596
119890= 53284 rads ]
2= 119896 minus
120596119890= 44308 rads 119862
1= 002072m and 119862
3= 002492m
(Figure 4(b))Numerical computations (- - -119910
2= 1199102(1199092)) in Figure 5(a)
have been carried out for the following values of mechanicalsystem parameters 119877 = 0492m 119892 = 981ms2 119897 =
0618m 119896 = (119892119897)05
asymp 3984 rads 119879 = 20 s 120596119890= 2120587119879 asymp
0314 rads 120593119890
= 180∘ 1205721dyn = 000498 rad 119881
119861=
01546ms 119910dyn = 000308m ]1= 119896 + 120596
119890= 4298 rads
6 Shock and Vibration
(a) (b)
(c) (d)
Figure 3 Photos of dimension experimental measurement system (a)ndash(d) for payload 119872 swaying motion during crane boom uniformslewing
]2
= 119896 minus 120596119890= 3670 rads 119862
1= 001798m and 119862
3=
002106m (Figure 5(a))For further estimation of the relative disagreement of
derived DAE-solution based computational (- - -) andexperimental (mdash) payload 119872 absolute trajectories in theinertial reference frame E we have computed the confidenceintervals in Figures 4 and 5 for the dimensionless parameters119903comp119903exper and confidence probability 095 The Studentrsquos 119905-test results yield the following confidence intervals 09374 le
(119903comp119903exper) le 10315 for 119897 = 0206m in Figure 4(a)09014 le (119903comp119903exper) le 10272 for 119897 = 0412m inFigure 4(b) 09200 le (119903comp119903exper) le 10352 for 119897 = 0618min Figure 5(a) and 09257 le (119903comp119903exper) le 10066 for119897 = 0825m in Figure 5(b)
Both the relative discrepancy 120575 and the confidence inter-vals show the satisfactory agreement between the absolute
payload trajectories (Figures 4 and 5) in the inertial referenceframeE that have been computed with DAE-solution basedtheoreticalmodel (- - -) (1) andmeasured experimentally (mdash)as shown in Figures 4 and 5
It is important to note that the payload motion DAEequations in the nonlinear problem (1) for the noninertialreference frame B may be derived with an introductionof Lagrange equations (Appendices AndashC F) However thediscussion of (1) is more suitable and informative with anintroduction of dynamic Coriolis theorem (Appendices AndashD F) The first terms 1198892119909
11198891199052 1198892119910
11198891199052 in (1) define the
vector a119872B = ar (Appendix B formula (B4)) for the relative
acceleration of load 119872 in the noninertial reference frameB The straight-line terms (D3) formula-based terms in (1)are linearly proportional to the relative payload coordinatesin the noninertial reference frame B These straight-line
Shock and Vibration 7
05
05
04
04
03
03
02
02
00
00
01
01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)
p O2 (p D)
(a)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)Computational y2 = y2(x2)
l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)
p O2 (p D)
(b)
Figure 4 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0206m (a) and fixed cable length 119897 = 0412m (b)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)
2R = 1 (m)
y2 (m)
x2 (m)p O2 (p D)
(a)
05
04
03
02
00
01
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448(rads)
2R = 1 (m)
x2 (m)
y2 (m)
p O2 (p D)
(b)
Figure 5 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0618m (a) and fixed cable length 119897 = 0825m (b)
terms in (1) have been determined by the contribution ofthe normal or centripetal acceleration 120596BE times (120596BE times
r119864119872
) = aen (B6) of transportation for payload119872 and by theappearance of corresponding DrsquoAlembert centrifugal inertiaforce Φe
n= (minus119898)aen (D3) due to crane boom transport
rotation in the noninertial reference frame B Additionaltermsminus2120596
119890(1198891199101119889119905) and 2120596
119890(1198891199091119889119905) in (1) have been defined
by the compound or Coriolis acceleration 2120596BE times V119872B =
acor of payload119872 in the noninertial reference frameB Therectangular Cartesian projections of Coriolis inertia force inthe noninertial reference frameB is defined by formula (B7)
5 Experimental Results for the SphericalPendulum with the Fixed Pivot Center
Experimental load swaying results for the fixed crane boommodel (120596
119890= 0 rads) are shown in Figure 6
Finite motions of a swaying load are shown in Figure 6in the case of 120596
119890= 0 rads In Figure 6 the relative and
absolute load trajectories are identically equal The lingeringremains of elliptical motion in Figure 6 and ellipses semiaxesare defined by the initial conditions
6 Discussion of Analogy between PayloadSwaying and Foucaultrsquos Pendulum Motion
The motion of Foucaultrsquos pendulum is shown in Figure 7where 119909
211991021199112is the heliocentric inertial reference frame
(not shown in Figure 7) 119909111991011199111is the geocentric noninertial
reference frame and 119909119910119911 is a noninertial reference framelocated at the geographic latitude 119910
1119911 119860 st119911 is a local vertical
119861119863 is the radius of the circle of the pinning point 119861119872 isthe cable length and 120596
119890is the angular velocity of the Earth
diurnal rotation
8 Shock and Vibration
(a) (b)
Figure 6 Experimental trajectories of the spherical pendulum with fixed pivot center and fixed cable length 119897 = 0825m (a b)
M
g
yx
z
B
D
x1
Ast
120596e
z1
y1
Figure 7 Foucaultrsquos pendulum schematic view
It is important to note that the structure of linearizedequations (H4)-(H5) for the swaying payload during craneboom uniform rotation in the noninertial reference frameF is analogous to the structure of the governing equationsfor Foucaultrsquos pendulum motion in the known publishedworks by Zhuravlev and Petrov [100] (formula (6) p 36 of[100]) Pardy [88] (formulae (12)ndash(14) pp 850-851 of [88])and Condurache et al [77] (formulae (11)ndash(13) p 743-744of [77]) The above mentioned analogy between linearizedequations (H4)-(H5) and the governing equations in [77 88100] assumes the geometric analogy of relative swaying tra-jectories between Foucaultrsquos pendulum and the boom-drivenpendulum with rotating pivot centers (Figures 1(e) 12 and16) So there is the geometric analogy between computationalrelative trajectories of load 119872 swaying during crane boomuniform slewing (Figures 1(e) 12 and 16) and relative trajec-tories of load119872 swaying in the plane (119860 st119909119910) around p119860 st inFigure 7 shownbyZhuravlev andPetrov [100] (Figure 4 p 39of [100]) and Condurache andMartinusi [77] (Figure 14 at p754 and Figure 16 at p 755 of [77]) The better understandingof Foucaultrsquos pendula dynamics is provided with the studyof computational relative trajectories in Figures 1(e) 12 and16 For Foucaultrsquos pendulum (Figure 7) the major semiaxes
of relative amplitude extremes have angular velocity valuesequal to the angular velocity of the Earth diurnal rotationMoreover for Foucaultrsquos pendulum (Figure 7) the directionof rotation of major semiaxes of relative amplitude extremesis oppositely directed to the direction of the Earth diurnalrotation All the above mentioned means that the plane(119860 st119861119872) of Foucaultrsquos pendula swaying remains fixed in theheliocentric inertial reference frame 119909
211991021199112
The presented analogy between crane boom-driven pay-load swaying and Foucaultrsquos pendulum motion essentiallyincreases knowledge and awareness of the oscillation pro-cesses in such dynamic systems
7 Discussion and MechanicalInterpretation of GoverningEquations for DAE-Based Nonlinearand Linearized Models
A governing nonlinear DAE-based system (1) for crane boomuniform slewing has been proposed and numerically solvedA conservative system (1) and (H4)-(H5) with componentsΦcor119909 andΦcor119910 has been derived Both projectionsΦcor119909 andΦcor119910 coincide in the noninertial reference framesB andF
The occurrence of the first derivatives 119889119909119889119905 119889119910119889119905 ofthe load119872 relative coordinates with respect to time in termsminus2120596119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) determines that there is no
decay of the oscillations in (H4)-(H5) and (1) but onlyredirection of the relative velocity vector V
119872F = Vr of theload119872 in the noninertial reference framesB andF
The introduction of a linearized model (H4)-(H5)allowed the determination of the natural frequencies (]
1=
]2) of free oscillations of payload 119872 ]
1= 119896 + 120596
119890and
]2= 119896 minus 120596
119890
It follows from the forms of the relative payload trajectory119910 = 119910(119909) in the noninertial reference frame F and theabsolute trajectory 119910
2= 1199102(1199092) in the inertial reference frame
E that the resulting motion of the payload 119872 on the cable119872119861 taking into account the Coriolis inertia forceΦcor (D5)will be the sum of two oscillations with natural frequencies ]
1
and ]2 and with periods 119879
1= 2120587]
1and 119879
2= 2120587]
2
It is worth noting that the frequencies ]1and ]
2differ
by 2120596119890 This means that for the small angle assumption
(Appendices H-I) we have ]1
asymp ]2and the trajectory
Shock and Vibration 9
mg
γ
Nℬ
z1z2
e2
e3
ℰ
ℰ
ℰ
ℱ
ℱ
ℬΔ
120574
y1
Vr
At
M
R D
E
Bz
l
120573
O
1205721
1205721dyn
Adyn
ydyn
h
120596e
AstO1
3ẽ
2ẽ
O2
d120593edty
sin(120593e)
cos(120593e)
120596ℬℰ = 120596ℱℰ
3e
2e
1e
Figure 8 Swaying scheme of load119872 on the cable119872119861 which is fixedly attached at the point 119861 on the crane boom 1198611198742 in the vertical plane
(119910119911) for the nonlinear model derivation in Cartesian coordinates
of relative motion 119910 = 119910(119909) in the noninertial referenceframe F on the expiration of the relative oscillations periodtime 2120587119896 looks approximately like an ellipse (Figures 12(b)and 16(b)) with semiaxes which are governed by the initialconditions
The major semiaxis of such an ellipse in Figures 12(b)and 16(b) is defined by the initial velocity of load 119872 in thenoninertial reference frameF and the analytical value of themajor semiaxis is equal to119881
119861119896 = (120596
119890119877)119896 The derived value
of 119881119861119896 is approximately the sum of the absolute values of
the amplitudes in the analytical form (I15) of the linearizedCauchy problem (H4)-(H5)
The minor semiaxis of the above mentioned ellipsein Figures 12(b) and 16(b) is defined by the value 119910dynwhich is approximately the difference of amplitudes in theanalytical form (I15) of the linearizedCauchy problem (H4)-(H5) In this case the major semiaxis is approximately 5ndash20 times larger than the minor semiaxis because the angularvelocity 120596
119890of the transport rotation of crane boom 119861119874
2is
much smaller than the natural frequency 119896 of the sphericaloscillations of payload119872 on the cable119872119861 (Figures 12(b) and16(b))
It is also necessary to note that the major semiaxis of theellipse in the relative motion rotates with an angular velocity120596119890in the opposite direction of the transport rotation of crane
boom 1198611198742in (Figures 12 and 16)
It follows from Figures 1(f) 2 4 and 5 that the trajectoryof absolute motion of load 119872 in the inertial reference frameE is almost a symmetric curve with 119910
2axial symmetry
The initial and the final motions of load 119872 for half-period
essentially differ from its harmonic oscillations neighbor forquarter-period
Due to the negligible quantity of 120572dyn and 119910dyn (G3)ndash(G5) the average deviation of the load119872 from the mechan-ical trajectory of the boom tip 119861 is negligible (Figures1(f) 2 4 and 5) The basic dynamic load on the systemldquocrane boom 119861119874
2-load 119872rdquo is created by the high-frequency
oscillations of load 119872 which are determined by the actionof inertia force Φcor = minus2119898(120596BE times V
119872B) stipulated bythe Coriolis acceleration acor = 2120596BE times V
119872B of load 119872
in the noninertial reference frame F The basic dynamicload in the system defines additional loads and vibrations ofcrane boom 119861119874
2mechanical elements and support bearings
complicates the automatic and manual control systems of theelectromechanical crane boom 119861119874
2 and also makes crane
operation much more difficultIt is also important to note that the stop of crane boom
1198611198742does not lead to instantaneous damping of the load
119872 absolute oscillations in the inertial reference frame E(Figures 2 4 and 5) This phenomenon directly follows fromthe experimental trajectory in Figures 2 4 and 5 Alsothe natural spherical oscillations of load 119872 will occur withthe frequency 119896 and the amplitude (the difference betweenthe final position of load 119872 in the relative motion in thenoninertial reference frame F for half-period of vibrationand static equilibrium 119860 st of load 119872 on the cable) Thefurther oscillations for the stop of the crane boom 119861119874
2
slewing motion are the deviation of the real trajectory fromthe intended final position of load119872 The results of physicalsimulation in Figures 2 4 and 5 show the necessity of add-on
10 Shock and Vibration
N
ℬ
ℬ
e2
e1
ℰ
ℰ
ℱ
ℱ
y2
x2
Ve
Vr
M
E
1ẽ
2ẽ x
x1
O2(D)
AoAst
O
y y1
Adyn
ydyn
1205722
120579
120579
120596e = 120596ℬℰ = 120596ℱℰ
120593e = 120596et
Φcor
B(O1)
Φen
Φeτ
1e
2e
Figure 9 Swaying scheme of load119872 at the cable119872119861 which is fixedly attached in the point 119861 of the crane boom 1198611198742 in the horizontal plane
(11990911199101) for the nonlinear model derivation in Cartesian coordinates
devices development for the efficient suppression of load 119872
final oscillations
8 Discussion and Comparison of Derived andKnown Published Results
In Figure 1 and pp 537-538 of publishedwork by Sakawa et al1981 [63] the small angle between payloadrsquos cable and verticalline has been introduced that confirms proposed nonlinearDAE-based system (1) and linearized model (H4)-(H5)
In Figure 9 of p 278 of published work by Maczynskiand Wojciech 2003 [44] the computational finite elementmethod (FEM)-based results were shown for the abso-lute payload trajectories in the inertial reference frame EMaczynskirsquos Figure 9 in [44] outlines that the angle betweenthe payloadrsquos cable and the vertical line does not exceed 01rad and that agreeswith the proposed linearizedmodel (H4)-(H5) for a small angle assumption Computational trajectoryfor Maczynski-derived small load swaying [44] after craneboom stop qualitatively coincides with the experimentallyobservable load119872motion in Figures 2 4 and 5
In Figure 6 of published work by Ju et al 2006 [34]it was shown that the angle between the payloadrsquos cableand the vertical line does not exceed 16∘ Jursquos formula (15a)in p 382 of [34] assumes that Jursquos angle between payloadrsquoscable and the vertical line has the harmonic law of variationwith an introduction of a small perturbation term Both Jursquosassumptions in [34] confirm the proposed linearized model(H4)-(H5)
Comparison of the derived linearized model (H4)-(H5)with Figure 5 of published work by Mitrev and Grigorov2008 in [49] shows that the Mitrev-derived ranges of payloadswaying angles within Mitrevrsquos nonlinear model does notexceed 48∘ Such small values of swaying angles completelyconfirm the applicability and correctness of the small angleassumption in (H4)-(H5)
9 Final Conclusions
A governing nonlinear DAE-based system for crane boomuniform slewing has been proposed numerically solved andexperimentally verified
Shock and Vibration 11
Fully nonlinear differential equations for 3 DOF sphericalpendulum oscillations with a uniform slewing crane boomaround a fixed vertical axis of rotation have been derivedin relative Cartesian and spherical coordinates The identicallinearized differential equations in relative Cartesian coordi-nates have been derived with an introduction of the Coriolisdynamic theorem and Lagrange equations for a uniformcrane boomrotation and small swaying angle120572
1assumptions
Linearized andnonlinear theoretical problems for relativeswaying of the payload have been formulated in the form ofthe initial value (Cauchy) problem
An analytical solution of the linearized system has beenderived
The influences of inertia forces from the centripetal andcompound accelerations have been estimated
A linearized conservative system which contains firsttime derivatives of coordinates and no damping of oscilla-tions has been derived
The amplitudes of load oscillations which depend oncomputed initial conditions have been estimated within asmall angle assumption The dependence of natural frequen-cies on the transport inertia forces and gravity forces has beencomputed for the linearized systems
The formulae for the association of relative payloadmotion in the noninertial reference frame F and absolutepayload motion in the inertial reference frame E have beenoutlined
The results of the numerical DAE-based investigation andthe performed physical simulation show a satisfactory fit forfrequencies and amplitudes of load oscillations
The dynamic analogy between crane boom-driven pay-load swaying motion and Foucaultrsquos pendulum motion hasbeen grounded and outlined
The results of the present work are the foundation forfurther investigations of payload119872 swaying dynamics duringtelescopic crane boom nonuniform slewing motion withvariable cable length and for the different motions of thependulum pivot center
Appendices
A Velocity Kinematics Analysis inCartesian Coordinates
For the nonlinear problem definition we study the coopera-tive motion of the mechanical system ldquocrane boom 119861119874
2-load
119872rdquo which is shown in Figures 8ndash10In the nomenclature chapter we denote the fixed inertial
frame of reference E as 119909211991021199112 and the moving noninertial
frame of reference B as 119909111991011199111 which is rigidly bounded
with the crane boom1198611198742 Rotation of themoving noninertial
frame of referenceB(119909111991011199111) around the fixed inertial frame
of referenceE(119909211991021199112) defines the transportation motion for
payload 119872 The motion of load 119872 relative to the movingnoninertial frame of referenceB(119909
111991011199111) defines the relative
motion of payload 119872 The point 119860 st with the coordinates1199091= 0 119910
1= 0 and 119911
1= 0 is the steady equilibrium position
for the load119872 when the crane boom 1198611198742is fixed The point
119860dyn with the coordinates 1199091 = 0 1199101= 119910dyn and 1199111 = Δ is the
dynamic equilibrium position for the load119872 for the rotatingcrane boom 119861119874
2 We will assume the point119860dyn as the origin
of the noninertial reference frameF(119909119910119911) The directions ofaxes 119909 119910 119911 of the noninertial reference frameF are parallelto the axes 119909
1 1199101 1199111of the noninertial reference frameB in
Figures 8ndash10 The load 119872 relative motion takes place alongthe sphere surface with the fixed radius equal to the cablelength 119861119872 = 119897
The computational scheme in Figures 8ndash10 for the nonlin-ear model derivation may be described with an introductionof three degrees of freedom For generalized coordinates weassume the rectangular coordinates 119909
1 1199101 and 119911
1of the load
119872 in the moving noninertial frame of reference B and theangle 120593
119890of crane boom 119861119874
2slewing in the horizontal plane
(11990911199101) with the angular velocity 120596
119890around the vertical axis
11987421199112The relative velocity vector Vr = V
119872B of the load 119872 isdefined as Vr = V
119872B = (1198891199091119889119905 119889119910
1119889119905 119889119911
1119889119905)
In order to derive the absolute velocity V119872E of payload
119872 we will apply the vector method for absolute motionassignment We will define r
1198742119872
as the position vector con-necting initial point 119874
2and terminal point119872 in noninertial
reference frameB (119898) consider
r1198742119872
= r1198742119861+ r119861119872
(A1)
where vector components in (1) are defined in nomenclatureIn accordance with Figures 8ndash10 we have the following
unit vectorsrsquo expansions for position vectors r1198742119861
and r119861119872
in (A1)
r1198742119861
= (119877 cos (120593119890)) e1+ (119877 sin (120593
119890)) e2+ (119877 tan (120574)) e
3
(A2)
r119861119872
= 1199091e1+ 1199101e2+ (1199111minus 119897) e3 (A3)
where 119877 120593119890 1199091 and 119910
1notations have been defined in
nomenclature and in Figures 8ndash10For further definition of the position vector r
1198742119872
in iner-tial reference frame E we write unit vectors of noninertialreference frameB in (A3) through the unit vectors of inertialreference frame E
e1= (sin (120593
119890)) e1minus (cos (120593
119890)) e2 (A4)
e2= (cos (120593
119890)) e1minus (sin (120593
119890)) e2 (A5)
e3= e3 (A6)
After substitution of formulae (A2)ndash(A6) in (A1) andsome algebraic manipulations the vector expression (A1) forthe position vector r
1198742119872
in the inertial reference frameEwilltake the following form
r1198742119872
= ((119877 + 1199101) cos (120593
119890) + 1199091sin (120593
119890)) e1
+ ((119877 + 1199101) sin (120593
119890) minus 1199091cos (120593
119890)) e2
+ (119877 tan (120574) + 1199111minus 119897) e3
(A7)
12 Shock and Vibration
The absolute velocity of payload 119872 we define by timedifferentiation of (A7) assuming constancy of unit vectorse1 e2 and e
3of the inertial reference frameE and constancy
119877 and 120574
V119872E = Vabs =
119889 (r1198742119872
)
119889119905
(A8)
After differentiation and algebraic transformations in(A7) and (A8) we have the following vector expression forabsolute payload 119872 velocity in the inertial reference frameE
V119872E
= (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) cos (120593119890)
+ ((
1198891199091
119889119905
) minus (119877 + 1199101) (
119889120593119890
119889119905
)) sin (120593119890)) e1
+ (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) sin (120593119890)
+ ((119877 + 1199101) (
119889120593119890
119889119905
) minus (
1198891199091
119889119905
)) cos (120593119890))
times e2+ (
1198891199111
119889119905
) e3
(A9)
The square of absolute payload119872 velocitymay be writtenas
1198812
abs = (
1198891199091
119889119905
)
2
+ (
1198891199101
119889119905
)
2
+ (
1198891199111
119889119905
)
2
+ (
119889120593119890
119889119905
)
2
times (1199092
1+ (1199101+ 119877)2
) + 2(
119889120593119890
119889119905
)
times (1199091(
1198891199101
119889119905
) minus (1199101+ 119877)(
1198891199091
119889119905
))
(A10)
After algebraic transformations we have the square ofabsolute payload119872 velocity in the form of
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A11)
Algebraic expressions (A10) and (A11) could be derivedon the basis of a velocity addition theorem for payload 119872
compound motion in the inertial reference frame E throughthe unit vectors of the noninertial reference frame B (see[59 76 81 82 84 85 87 90ndash92 94ndash98])
V119872E = V
119872B + V1198742E + 120596BE times r
1198742119872
(A12)
where payload relative velocity in the noninertial referenceframeBmay be written as
V119872B = Vr = (
1198891199091
119889119905
) e1+ (
1198891199101
119889119905
) e2+ (
1198891199111
119889119905
) e3
(A13)
The velocityV1198742E of the point119874
2= point119864 in the inertial
reference frame E has zero value
V1198742E = 0 (A14)
The last term in (A12) is the vector product 120596BE times
r1198742119872
where crane boom angular velocity vector is 120596BE =
(119889120593119890119889119905)e3(A6) and the position vector r
1198742119872
in the nonin-ertial reference frameB is as follows
r1198742119872
= 1199091e1+ (1199101+ 119877) e
2+ 1199111e3 (A15)
Taking into account (A13)ndash(A15) the vector equation(A12) in the inertial reference frame E written through theunit vectors e
1 e2 e3of the noninertial reference frame B
will have the following form
V119872E = ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101)) e1
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091) e2+ (
1198891199111
119889119905
) e3
(A16)
So on the basis of (A16) we have the following square ofabsolute payload119872 velocity as
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A17)
Independently derived expressions for the scalar prod-uct (V
119872EV119872E) in (A11) and (A17) completely coincidewhich confirms the correctness and accuracy of the pen-dulum absolute velocity V
119872E derivation and shows thatthe scalar product (V
119872EV119872E) is the invariant expressionindependent of choice of reference frame
The second and third terms of (A12) in the noninertialreference frame B determine the vector of the load 119872
transportation velocity as
Ve = V1198742E + 120596BE times r
1198742119872
(A18)
Taking into account (A14) the scalar of the load 119872
transportation velocity is defined as
1003817100381710038171003817Ve
1003817100381710038171003817= 120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817 (A19)
Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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Shock and Vibration
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Shock and Vibration 3
minus002
minus001
05
t
10 15
001
002
x1
(a)
minus002
minus001
05
t
10 15
003
002
001
y1
(b)
minus01
minus02
minus03
minus04
04
03
02
01
0
t
5 10 15x2
(c)
04
03
05
02
01
0
t
105 15
y2
(d)
minus002
minus002
minus001
minus001
x1
x1
y1
y1
002
002
003
001
0 001
(e)
01
02
03
04
05
minus01minus02minus03minus04 01 02 03 040
x2
y2
(0492 + y1) lowast cos(120593) + x1 lowast sin(120593)
(f)
Figure 1 Computational relative (a-b e) and computational absolute (c-d f) coordinates of a swaying payload during uniform crane boomslewing for the following numerical values of system parameters119898 = 01 kg 119897 = 0825m 119877 = 0492m 119892 = 981ms2 and 120596
119890= 0209 rads
4 Shock and Vibration
payload during crane boom slewing (Figures 8ndash10) in thenoninertial reference frameB is as follows
119898(
1198892(1199091(119905))
1198891199052
)
= minus119873 (119905) (
1199091(119905)
119897
) + 119898(
119889 (120593119890(119905))
119889119905
)
2
1199091(119905)
+ 119898(
1198892(120593119890(119905))
1198891199052
) (119877 + 1199101(119905))
+ 2119898(
119889 (120593119890(119905))
119889119905
)(
119889 (1199101(119905))
119889119905
)
119898(
1198892(1199101(119905))
1198891199052
)
= minus119873 (119905) (
1199101(119905)
119897
) + 119898(
119889 (120593119890(119905))
119889119905
)
2
(119877 + 1199101(119905))
minus 119898(
1198892(120593119890(119905))
1198891199052
)1199091(119905)
minus 2119898(
119889 (120593119890(119905))
119889119905
)(
119889 (1199091(119905))
119889119905
)
119898(
1198892(1199111(119905))
1198891199052
) = minus119898119892 + 119873 (119905) (
119897 minus 1199111(119905)
119897
)
1198972= (1199091(119905))2
+ (1199101(119905))2
+ (1199111(119905) minus 119897)
2
120593119890(119905) = 120596
119890119905 120593
119890(0) = 0
119889 (120593119890(0))
119889119905
= 120596119890 119873 (0) = 119898119892
1199091(0) = 0
119889 (1199091(0))
119889119905
= 120596119890119877 119910
1(0) = 0
119889 (1199101(0))
119889119905
= 0 1199111(0) = 0
119889 (1199111(0))
119889119905
= 0
(1)
The absolute coordinates 1199092(119905) (Figure 1(c)) 119910
2(119905)
(Figure 1(d)) 1199112(119905) of payload 119872 in the inertial reference
frame Emay be computed as
1199092(119905) = (119877 + 119910
1(119905)) cos (120593
119890(119905)) + 119909
1(119905) sin (120593
119890(119905))
1199102(119905) = (119877 + 119910
1(119905)) sin (120593
119890(119905)) minus 119909
1(119905) cos (120593
119890(119905))
1199112(119905) = 119911 (119905)
(2)
The absolute coordinates 1199092 1199102in the inertial reference
frameE which dependupon the relative coordinates11990911199101in
the noninertial reference frameB during the swaying of load119872 (Figures 8ndash10) has been defined according to the abovementioned Equations and has been shown in Figures 4 and 5as (- - - 119910
2= 1199102(1199092))
The computational results in Figure 1 derived for DAEproblem (1)-(2) solution coincide with the linearized solu-tion of the payload swaying problem in Appendices AndashI(formula (H1) in the noninertial reference frameB)
It is necessary to note that the posed DAE problem (1)describes payloadmotion in the vicinity of the lower positionof stable dynamical equilibrium assuming rope tension force119873(119905) gt 0 Factually the applied rope is assumed as a unilateralgeometric constraint (Appendix C) in the shape of a torsionfiberThe upper position of themechanical system is unstable(see [73ndash75 78ndash80 91 93]) and corresponds to the case of119873(119905) lt 0 within the parameters of the chosen mechanicalsystem in Figures 8ndash10 Such an upper pendulum positionmight conflict with the hypothesis about the unilateral natureof geometric constraint All of the above mentioned upperpendulum position conditions are outside the objectives andscope of the present paper
3 Experimental Procedure
The experimental procedure has been grounded on the usageof the assembly in Figures 2 and 3
The assembly in Figures 2 and 3 includes the followingmachine components the vertical fixed shaft119874
2119863with height
119867 = 1m the crane boom model 119861119863 with length 119877 = 05mand diameter 6mm the cable119861119872with different fixed lengths1198971= 0825m (in Figure 2 and Figure 5(b)) 119897
2= 0618m (in
Figure 5(a)) 1198973= 0412m (in Figure 4(b)) and 119897
4= 0206m
(in Figure 4(a)) The crane boom model 119861119863 is attached tothe vertical fixed shaft 119874
2119863 by bearing 119863 The cable 119861119872
is attached to the crane boom 119861119863 tip in the point 119861 Thefree or running end 119872 of the cable 119861119872 is the payload 119872
attachment pointThe load119872 is a light emitting diode (LED)with diameter 2mm and the battery with the battery voltage3V The experimental swaying trajectory (mdash 119910
2= 1199102(1199092)) in
Figures 2 4 and 5 is the experimental light emitting diode119872absolute trajectory in the inertial reference frameE
The laser pointer in Figure 2 is also attached to the craneboom 119861119863 tip in the point 119861 The laser pointer is the part ofthe noninertial reference frames B and F and the pointertrajectory is marked as (- - -) in Figure 2 The introductionof laser pointer 119861 allows estimation of dynamic deviation ofpayload119872 The horizontal regular fixed grid with canvas size1200mm times 700mm is placed in the horizontal plane119874
211990921199102
The horizontal grid is formed by the square cells with size20mm times 20mm
The experimental swaying trajectories (mdash 1199102= 1199102(1199092))
in Figures 2 4 and 5 have been written in the obscure roomwith the introduction of an upper digital camera with a longexposure 60 sndash90 s and the camera height above horizontalgrid level is 25m
4 Comparison and Discussion ofDAE-Solution Based Theoretical andExperimental Results
Thecomparison ofDAE-solution based theoretical (- - -1199102=
1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) results is shown
in Figures 4 and 5
Shock and Vibration 5
Light-emitting diode
Laser pointer
Camera
Boom
Slewing axis
Bearing
Lower plate
D
R = 05m
H=1
m
O2
O1
M
B
25
m
Ast
120596e = 0209 rads
l=0825
m
Regular grid 1200mm times 700mmwith square cells 20mm times 20mm
Figure 2 The dimension experimental measurement system for payload119872 swaying motion during crane boom uniform slewing
In order to estimate the relative disagreement of thederived DAE-solution based computational (- - -) andexperimental (mdash) payload 119872 absolute trajectories in theinertial reference frame E we have computed the amplitudediscrepancy 120575 in the polar coordinate system by the followingformula
120575 = ⟨
1
2
((
10038161003816100381610038161003816119903comp minus 119903exper
10038161003816100381610038161003816
119903comp) + (
10038161003816100381610038161003816119903comp minus 119903exper
10038161003816100381610038161003816
119903exper))⟩
(3)
where 119903comp and 119903exper are the magnitudes of the radius-vectors connecting point 119874
2and theoretical (- - -) and
experimental (mdash) curves and computed for the same fixedpolar angle 120593
119890
The amplitude discrepancies 120575 have the following values120575 = 753 for 119897 = 0206m in Figure 4(a) 120575 = 758 for119897 = 0412m in Figure 4(b) 120575 = 740 for 119897 = 0618m inFigure 5(a) and 120575 = 69 for 119897 = 0825m in Figure 5(b)
Numerical computations (- - - 1199102
= 1199102(1199092)) in
Figure 4(a) have been carried out for the following values of
mechanical system parameters 119877 = 0492m 119892 = 981ms2119897 = 0206m 119896 = (119892119897)
05asymp 6901 rads119879 = 14 s120596
119890= 2120587119879 asymp
0449 rads 120593119890= 180
∘ 1205721dyn = 001014 rad 119881
119861= 0221ms
119910dyn = 000209m ]1= 119896 + 120596
119890= 73496 rads ]
2= 119896 minus
120596119890= 64520 rads 119862
1= 001502m and 119862
3= 001711m
(Figure 4(a))Numerical computations (- - - 119910
2= 119910
2(1199092)) in
Figure 4(b) have been carried out for the following values ofmechanical system parameters 119877 = 0492m 119892 = 981ms2119897 = 0412m 119896 = (119892119897)
05asymp 4879 rads119879 = 14 s120596
119890= 2120587119879 asymp
0449 rads 120593119890= 180
∘ 1205721dyn = 001019 rad 119881
119861= 0221ms
119910dyn = 000419m ]1= 119896 + 120596
119890= 53284 rads ]
2= 119896 minus
120596119890= 44308 rads 119862
1= 002072m and 119862
3= 002492m
(Figure 4(b))Numerical computations (- - -119910
2= 1199102(1199092)) in Figure 5(a)
have been carried out for the following values of mechanicalsystem parameters 119877 = 0492m 119892 = 981ms2 119897 =
0618m 119896 = (119892119897)05
asymp 3984 rads 119879 = 20 s 120596119890= 2120587119879 asymp
0314 rads 120593119890
= 180∘ 1205721dyn = 000498 rad 119881
119861=
01546ms 119910dyn = 000308m ]1= 119896 + 120596
119890= 4298 rads
6 Shock and Vibration
(a) (b)
(c) (d)
Figure 3 Photos of dimension experimental measurement system (a)ndash(d) for payload 119872 swaying motion during crane boom uniformslewing
]2
= 119896 minus 120596119890= 3670 rads 119862
1= 001798m and 119862
3=
002106m (Figure 5(a))For further estimation of the relative disagreement of
derived DAE-solution based computational (- - -) andexperimental (mdash) payload 119872 absolute trajectories in theinertial reference frame E we have computed the confidenceintervals in Figures 4 and 5 for the dimensionless parameters119903comp119903exper and confidence probability 095 The Studentrsquos 119905-test results yield the following confidence intervals 09374 le
(119903comp119903exper) le 10315 for 119897 = 0206m in Figure 4(a)09014 le (119903comp119903exper) le 10272 for 119897 = 0412m inFigure 4(b) 09200 le (119903comp119903exper) le 10352 for 119897 = 0618min Figure 5(a) and 09257 le (119903comp119903exper) le 10066 for119897 = 0825m in Figure 5(b)
Both the relative discrepancy 120575 and the confidence inter-vals show the satisfactory agreement between the absolute
payload trajectories (Figures 4 and 5) in the inertial referenceframeE that have been computed with DAE-solution basedtheoreticalmodel (- - -) (1) andmeasured experimentally (mdash)as shown in Figures 4 and 5
It is important to note that the payload motion DAEequations in the nonlinear problem (1) for the noninertialreference frame B may be derived with an introductionof Lagrange equations (Appendices AndashC F) However thediscussion of (1) is more suitable and informative with anintroduction of dynamic Coriolis theorem (Appendices AndashD F) The first terms 1198892119909
11198891199052 1198892119910
11198891199052 in (1) define the
vector a119872B = ar (Appendix B formula (B4)) for the relative
acceleration of load 119872 in the noninertial reference frameB The straight-line terms (D3) formula-based terms in (1)are linearly proportional to the relative payload coordinatesin the noninertial reference frame B These straight-line
Shock and Vibration 7
05
05
04
04
03
03
02
02
00
00
01
01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)
p O2 (p D)
(a)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)Computational y2 = y2(x2)
l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)
p O2 (p D)
(b)
Figure 4 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0206m (a) and fixed cable length 119897 = 0412m (b)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)
2R = 1 (m)
y2 (m)
x2 (m)p O2 (p D)
(a)
05
04
03
02
00
01
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448(rads)
2R = 1 (m)
x2 (m)
y2 (m)
p O2 (p D)
(b)
Figure 5 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0618m (a) and fixed cable length 119897 = 0825m (b)
terms in (1) have been determined by the contribution ofthe normal or centripetal acceleration 120596BE times (120596BE times
r119864119872
) = aen (B6) of transportation for payload119872 and by theappearance of corresponding DrsquoAlembert centrifugal inertiaforce Φe
n= (minus119898)aen (D3) due to crane boom transport
rotation in the noninertial reference frame B Additionaltermsminus2120596
119890(1198891199101119889119905) and 2120596
119890(1198891199091119889119905) in (1) have been defined
by the compound or Coriolis acceleration 2120596BE times V119872B =
acor of payload119872 in the noninertial reference frameB Therectangular Cartesian projections of Coriolis inertia force inthe noninertial reference frameB is defined by formula (B7)
5 Experimental Results for the SphericalPendulum with the Fixed Pivot Center
Experimental load swaying results for the fixed crane boommodel (120596
119890= 0 rads) are shown in Figure 6
Finite motions of a swaying load are shown in Figure 6in the case of 120596
119890= 0 rads In Figure 6 the relative and
absolute load trajectories are identically equal The lingeringremains of elliptical motion in Figure 6 and ellipses semiaxesare defined by the initial conditions
6 Discussion of Analogy between PayloadSwaying and Foucaultrsquos Pendulum Motion
The motion of Foucaultrsquos pendulum is shown in Figure 7where 119909
211991021199112is the heliocentric inertial reference frame
(not shown in Figure 7) 119909111991011199111is the geocentric noninertial
reference frame and 119909119910119911 is a noninertial reference framelocated at the geographic latitude 119910
1119911 119860 st119911 is a local vertical
119861119863 is the radius of the circle of the pinning point 119861119872 isthe cable length and 120596
119890is the angular velocity of the Earth
diurnal rotation
8 Shock and Vibration
(a) (b)
Figure 6 Experimental trajectories of the spherical pendulum with fixed pivot center and fixed cable length 119897 = 0825m (a b)
M
g
yx
z
B
D
x1
Ast
120596e
z1
y1
Figure 7 Foucaultrsquos pendulum schematic view
It is important to note that the structure of linearizedequations (H4)-(H5) for the swaying payload during craneboom uniform rotation in the noninertial reference frameF is analogous to the structure of the governing equationsfor Foucaultrsquos pendulum motion in the known publishedworks by Zhuravlev and Petrov [100] (formula (6) p 36 of[100]) Pardy [88] (formulae (12)ndash(14) pp 850-851 of [88])and Condurache et al [77] (formulae (11)ndash(13) p 743-744of [77]) The above mentioned analogy between linearizedequations (H4)-(H5) and the governing equations in [77 88100] assumes the geometric analogy of relative swaying tra-jectories between Foucaultrsquos pendulum and the boom-drivenpendulum with rotating pivot centers (Figures 1(e) 12 and16) So there is the geometric analogy between computationalrelative trajectories of load 119872 swaying during crane boomuniform slewing (Figures 1(e) 12 and 16) and relative trajec-tories of load119872 swaying in the plane (119860 st119909119910) around p119860 st inFigure 7 shownbyZhuravlev andPetrov [100] (Figure 4 p 39of [100]) and Condurache andMartinusi [77] (Figure 14 at p754 and Figure 16 at p 755 of [77]) The better understandingof Foucaultrsquos pendula dynamics is provided with the studyof computational relative trajectories in Figures 1(e) 12 and16 For Foucaultrsquos pendulum (Figure 7) the major semiaxes
of relative amplitude extremes have angular velocity valuesequal to the angular velocity of the Earth diurnal rotationMoreover for Foucaultrsquos pendulum (Figure 7) the directionof rotation of major semiaxes of relative amplitude extremesis oppositely directed to the direction of the Earth diurnalrotation All the above mentioned means that the plane(119860 st119861119872) of Foucaultrsquos pendula swaying remains fixed in theheliocentric inertial reference frame 119909
211991021199112
The presented analogy between crane boom-driven pay-load swaying and Foucaultrsquos pendulum motion essentiallyincreases knowledge and awareness of the oscillation pro-cesses in such dynamic systems
7 Discussion and MechanicalInterpretation of GoverningEquations for DAE-Based Nonlinearand Linearized Models
A governing nonlinear DAE-based system (1) for crane boomuniform slewing has been proposed and numerically solvedA conservative system (1) and (H4)-(H5) with componentsΦcor119909 andΦcor119910 has been derived Both projectionsΦcor119909 andΦcor119910 coincide in the noninertial reference framesB andF
The occurrence of the first derivatives 119889119909119889119905 119889119910119889119905 ofthe load119872 relative coordinates with respect to time in termsminus2120596119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) determines that there is no
decay of the oscillations in (H4)-(H5) and (1) but onlyredirection of the relative velocity vector V
119872F = Vr of theload119872 in the noninertial reference framesB andF
The introduction of a linearized model (H4)-(H5)allowed the determination of the natural frequencies (]
1=
]2) of free oscillations of payload 119872 ]
1= 119896 + 120596
119890and
]2= 119896 minus 120596
119890
It follows from the forms of the relative payload trajectory119910 = 119910(119909) in the noninertial reference frame F and theabsolute trajectory 119910
2= 1199102(1199092) in the inertial reference frame
E that the resulting motion of the payload 119872 on the cable119872119861 taking into account the Coriolis inertia forceΦcor (D5)will be the sum of two oscillations with natural frequencies ]
1
and ]2 and with periods 119879
1= 2120587]
1and 119879
2= 2120587]
2
It is worth noting that the frequencies ]1and ]
2differ
by 2120596119890 This means that for the small angle assumption
(Appendices H-I) we have ]1
asymp ]2and the trajectory
Shock and Vibration 9
mg
γ
Nℬ
z1z2
e2
e3
ℰ
ℰ
ℰ
ℱ
ℱ
ℬΔ
120574
y1
Vr
At
M
R D
E
Bz
l
120573
O
1205721
1205721dyn
Adyn
ydyn
h
120596e
AstO1
3ẽ
2ẽ
O2
d120593edty
sin(120593e)
cos(120593e)
120596ℬℰ = 120596ℱℰ
3e
2e
1e
Figure 8 Swaying scheme of load119872 on the cable119872119861 which is fixedly attached at the point 119861 on the crane boom 1198611198742 in the vertical plane
(119910119911) for the nonlinear model derivation in Cartesian coordinates
of relative motion 119910 = 119910(119909) in the noninertial referenceframe F on the expiration of the relative oscillations periodtime 2120587119896 looks approximately like an ellipse (Figures 12(b)and 16(b)) with semiaxes which are governed by the initialconditions
The major semiaxis of such an ellipse in Figures 12(b)and 16(b) is defined by the initial velocity of load 119872 in thenoninertial reference frameF and the analytical value of themajor semiaxis is equal to119881
119861119896 = (120596
119890119877)119896 The derived value
of 119881119861119896 is approximately the sum of the absolute values of
the amplitudes in the analytical form (I15) of the linearizedCauchy problem (H4)-(H5)
The minor semiaxis of the above mentioned ellipsein Figures 12(b) and 16(b) is defined by the value 119910dynwhich is approximately the difference of amplitudes in theanalytical form (I15) of the linearizedCauchy problem (H4)-(H5) In this case the major semiaxis is approximately 5ndash20 times larger than the minor semiaxis because the angularvelocity 120596
119890of the transport rotation of crane boom 119861119874
2is
much smaller than the natural frequency 119896 of the sphericaloscillations of payload119872 on the cable119872119861 (Figures 12(b) and16(b))
It is also necessary to note that the major semiaxis of theellipse in the relative motion rotates with an angular velocity120596119890in the opposite direction of the transport rotation of crane
boom 1198611198742in (Figures 12 and 16)
It follows from Figures 1(f) 2 4 and 5 that the trajectoryof absolute motion of load 119872 in the inertial reference frameE is almost a symmetric curve with 119910
2axial symmetry
The initial and the final motions of load 119872 for half-period
essentially differ from its harmonic oscillations neighbor forquarter-period
Due to the negligible quantity of 120572dyn and 119910dyn (G3)ndash(G5) the average deviation of the load119872 from the mechan-ical trajectory of the boom tip 119861 is negligible (Figures1(f) 2 4 and 5) The basic dynamic load on the systemldquocrane boom 119861119874
2-load 119872rdquo is created by the high-frequency
oscillations of load 119872 which are determined by the actionof inertia force Φcor = minus2119898(120596BE times V
119872B) stipulated bythe Coriolis acceleration acor = 2120596BE times V
119872B of load 119872
in the noninertial reference frame F The basic dynamicload in the system defines additional loads and vibrations ofcrane boom 119861119874
2mechanical elements and support bearings
complicates the automatic and manual control systems of theelectromechanical crane boom 119861119874
2 and also makes crane
operation much more difficultIt is also important to note that the stop of crane boom
1198611198742does not lead to instantaneous damping of the load
119872 absolute oscillations in the inertial reference frame E(Figures 2 4 and 5) This phenomenon directly follows fromthe experimental trajectory in Figures 2 4 and 5 Alsothe natural spherical oscillations of load 119872 will occur withthe frequency 119896 and the amplitude (the difference betweenthe final position of load 119872 in the relative motion in thenoninertial reference frame F for half-period of vibrationand static equilibrium 119860 st of load 119872 on the cable) Thefurther oscillations for the stop of the crane boom 119861119874
2
slewing motion are the deviation of the real trajectory fromthe intended final position of load119872 The results of physicalsimulation in Figures 2 4 and 5 show the necessity of add-on
10 Shock and Vibration
N
ℬ
ℬ
e2
e1
ℰ
ℰ
ℱ
ℱ
y2
x2
Ve
Vr
M
E
1ẽ
2ẽ x
x1
O2(D)
AoAst
O
y y1
Adyn
ydyn
1205722
120579
120579
120596e = 120596ℬℰ = 120596ℱℰ
120593e = 120596et
Φcor
B(O1)
Φen
Φeτ
1e
2e
Figure 9 Swaying scheme of load119872 at the cable119872119861 which is fixedly attached in the point 119861 of the crane boom 1198611198742 in the horizontal plane
(11990911199101) for the nonlinear model derivation in Cartesian coordinates
devices development for the efficient suppression of load 119872
final oscillations
8 Discussion and Comparison of Derived andKnown Published Results
In Figure 1 and pp 537-538 of publishedwork by Sakawa et al1981 [63] the small angle between payloadrsquos cable and verticalline has been introduced that confirms proposed nonlinearDAE-based system (1) and linearized model (H4)-(H5)
In Figure 9 of p 278 of published work by Maczynskiand Wojciech 2003 [44] the computational finite elementmethod (FEM)-based results were shown for the abso-lute payload trajectories in the inertial reference frame EMaczynskirsquos Figure 9 in [44] outlines that the angle betweenthe payloadrsquos cable and the vertical line does not exceed 01rad and that agreeswith the proposed linearizedmodel (H4)-(H5) for a small angle assumption Computational trajectoryfor Maczynski-derived small load swaying [44] after craneboom stop qualitatively coincides with the experimentallyobservable load119872motion in Figures 2 4 and 5
In Figure 6 of published work by Ju et al 2006 [34]it was shown that the angle between the payloadrsquos cableand the vertical line does not exceed 16∘ Jursquos formula (15a)in p 382 of [34] assumes that Jursquos angle between payloadrsquoscable and the vertical line has the harmonic law of variationwith an introduction of a small perturbation term Both Jursquosassumptions in [34] confirm the proposed linearized model(H4)-(H5)
Comparison of the derived linearized model (H4)-(H5)with Figure 5 of published work by Mitrev and Grigorov2008 in [49] shows that the Mitrev-derived ranges of payloadswaying angles within Mitrevrsquos nonlinear model does notexceed 48∘ Such small values of swaying angles completelyconfirm the applicability and correctness of the small angleassumption in (H4)-(H5)
9 Final Conclusions
A governing nonlinear DAE-based system for crane boomuniform slewing has been proposed numerically solved andexperimentally verified
Shock and Vibration 11
Fully nonlinear differential equations for 3 DOF sphericalpendulum oscillations with a uniform slewing crane boomaround a fixed vertical axis of rotation have been derivedin relative Cartesian and spherical coordinates The identicallinearized differential equations in relative Cartesian coordi-nates have been derived with an introduction of the Coriolisdynamic theorem and Lagrange equations for a uniformcrane boomrotation and small swaying angle120572
1assumptions
Linearized andnonlinear theoretical problems for relativeswaying of the payload have been formulated in the form ofthe initial value (Cauchy) problem
An analytical solution of the linearized system has beenderived
The influences of inertia forces from the centripetal andcompound accelerations have been estimated
A linearized conservative system which contains firsttime derivatives of coordinates and no damping of oscilla-tions has been derived
The amplitudes of load oscillations which depend oncomputed initial conditions have been estimated within asmall angle assumption The dependence of natural frequen-cies on the transport inertia forces and gravity forces has beencomputed for the linearized systems
The formulae for the association of relative payloadmotion in the noninertial reference frame F and absolutepayload motion in the inertial reference frame E have beenoutlined
The results of the numerical DAE-based investigation andthe performed physical simulation show a satisfactory fit forfrequencies and amplitudes of load oscillations
The dynamic analogy between crane boom-driven pay-load swaying motion and Foucaultrsquos pendulum motion hasbeen grounded and outlined
The results of the present work are the foundation forfurther investigations of payload119872 swaying dynamics duringtelescopic crane boom nonuniform slewing motion withvariable cable length and for the different motions of thependulum pivot center
Appendices
A Velocity Kinematics Analysis inCartesian Coordinates
For the nonlinear problem definition we study the coopera-tive motion of the mechanical system ldquocrane boom 119861119874
2-load
119872rdquo which is shown in Figures 8ndash10In the nomenclature chapter we denote the fixed inertial
frame of reference E as 119909211991021199112 and the moving noninertial
frame of reference B as 119909111991011199111 which is rigidly bounded
with the crane boom1198611198742 Rotation of themoving noninertial
frame of referenceB(119909111991011199111) around the fixed inertial frame
of referenceE(119909211991021199112) defines the transportation motion for
payload 119872 The motion of load 119872 relative to the movingnoninertial frame of referenceB(119909
111991011199111) defines the relative
motion of payload 119872 The point 119860 st with the coordinates1199091= 0 119910
1= 0 and 119911
1= 0 is the steady equilibrium position
for the load119872 when the crane boom 1198611198742is fixed The point
119860dyn with the coordinates 1199091 = 0 1199101= 119910dyn and 1199111 = Δ is the
dynamic equilibrium position for the load119872 for the rotatingcrane boom 119861119874
2 We will assume the point119860dyn as the origin
of the noninertial reference frameF(119909119910119911) The directions ofaxes 119909 119910 119911 of the noninertial reference frameF are parallelto the axes 119909
1 1199101 1199111of the noninertial reference frameB in
Figures 8ndash10 The load 119872 relative motion takes place alongthe sphere surface with the fixed radius equal to the cablelength 119861119872 = 119897
The computational scheme in Figures 8ndash10 for the nonlin-ear model derivation may be described with an introductionof three degrees of freedom For generalized coordinates weassume the rectangular coordinates 119909
1 1199101 and 119911
1of the load
119872 in the moving noninertial frame of reference B and theangle 120593
119890of crane boom 119861119874
2slewing in the horizontal plane
(11990911199101) with the angular velocity 120596
119890around the vertical axis
11987421199112The relative velocity vector Vr = V
119872B of the load 119872 isdefined as Vr = V
119872B = (1198891199091119889119905 119889119910
1119889119905 119889119911
1119889119905)
In order to derive the absolute velocity V119872E of payload
119872 we will apply the vector method for absolute motionassignment We will define r
1198742119872
as the position vector con-necting initial point 119874
2and terminal point119872 in noninertial
reference frameB (119898) consider
r1198742119872
= r1198742119861+ r119861119872
(A1)
where vector components in (1) are defined in nomenclatureIn accordance with Figures 8ndash10 we have the following
unit vectorsrsquo expansions for position vectors r1198742119861
and r119861119872
in (A1)
r1198742119861
= (119877 cos (120593119890)) e1+ (119877 sin (120593
119890)) e2+ (119877 tan (120574)) e
3
(A2)
r119861119872
= 1199091e1+ 1199101e2+ (1199111minus 119897) e3 (A3)
where 119877 120593119890 1199091 and 119910
1notations have been defined in
nomenclature and in Figures 8ndash10For further definition of the position vector r
1198742119872
in iner-tial reference frame E we write unit vectors of noninertialreference frameB in (A3) through the unit vectors of inertialreference frame E
e1= (sin (120593
119890)) e1minus (cos (120593
119890)) e2 (A4)
e2= (cos (120593
119890)) e1minus (sin (120593
119890)) e2 (A5)
e3= e3 (A6)
After substitution of formulae (A2)ndash(A6) in (A1) andsome algebraic manipulations the vector expression (A1) forthe position vector r
1198742119872
in the inertial reference frameEwilltake the following form
r1198742119872
= ((119877 + 1199101) cos (120593
119890) + 1199091sin (120593
119890)) e1
+ ((119877 + 1199101) sin (120593
119890) minus 1199091cos (120593
119890)) e2
+ (119877 tan (120574) + 1199111minus 119897) e3
(A7)
12 Shock and Vibration
The absolute velocity of payload 119872 we define by timedifferentiation of (A7) assuming constancy of unit vectorse1 e2 and e
3of the inertial reference frameE and constancy
119877 and 120574
V119872E = Vabs =
119889 (r1198742119872
)
119889119905
(A8)
After differentiation and algebraic transformations in(A7) and (A8) we have the following vector expression forabsolute payload 119872 velocity in the inertial reference frameE
V119872E
= (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) cos (120593119890)
+ ((
1198891199091
119889119905
) minus (119877 + 1199101) (
119889120593119890
119889119905
)) sin (120593119890)) e1
+ (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) sin (120593119890)
+ ((119877 + 1199101) (
119889120593119890
119889119905
) minus (
1198891199091
119889119905
)) cos (120593119890))
times e2+ (
1198891199111
119889119905
) e3
(A9)
The square of absolute payload119872 velocitymay be writtenas
1198812
abs = (
1198891199091
119889119905
)
2
+ (
1198891199101
119889119905
)
2
+ (
1198891199111
119889119905
)
2
+ (
119889120593119890
119889119905
)
2
times (1199092
1+ (1199101+ 119877)2
) + 2(
119889120593119890
119889119905
)
times (1199091(
1198891199101
119889119905
) minus (1199101+ 119877)(
1198891199091
119889119905
))
(A10)
After algebraic transformations we have the square ofabsolute payload119872 velocity in the form of
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A11)
Algebraic expressions (A10) and (A11) could be derivedon the basis of a velocity addition theorem for payload 119872
compound motion in the inertial reference frame E throughthe unit vectors of the noninertial reference frame B (see[59 76 81 82 84 85 87 90ndash92 94ndash98])
V119872E = V
119872B + V1198742E + 120596BE times r
1198742119872
(A12)
where payload relative velocity in the noninertial referenceframeBmay be written as
V119872B = Vr = (
1198891199091
119889119905
) e1+ (
1198891199101
119889119905
) e2+ (
1198891199111
119889119905
) e3
(A13)
The velocityV1198742E of the point119874
2= point119864 in the inertial
reference frame E has zero value
V1198742E = 0 (A14)
The last term in (A12) is the vector product 120596BE times
r1198742119872
where crane boom angular velocity vector is 120596BE =
(119889120593119890119889119905)e3(A6) and the position vector r
1198742119872
in the nonin-ertial reference frameB is as follows
r1198742119872
= 1199091e1+ (1199101+ 119877) e
2+ 1199111e3 (A15)
Taking into account (A13)ndash(A15) the vector equation(A12) in the inertial reference frame E written through theunit vectors e
1 e2 e3of the noninertial reference frame B
will have the following form
V119872E = ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101)) e1
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091) e2+ (
1198891199111
119889119905
) e3
(A16)
So on the basis of (A16) we have the following square ofabsolute payload119872 velocity as
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A17)
Independently derived expressions for the scalar prod-uct (V
119872EV119872E) in (A11) and (A17) completely coincidewhich confirms the correctness and accuracy of the pen-dulum absolute velocity V
119872E derivation and shows thatthe scalar product (V
119872EV119872E) is the invariant expressionindependent of choice of reference frame
The second and third terms of (A12) in the noninertialreference frame B determine the vector of the load 119872
transportation velocity as
Ve = V1198742E + 120596BE times r
1198742119872
(A18)
Taking into account (A14) the scalar of the load 119872
transportation velocity is defined as
1003817100381710038171003817Ve
1003817100381710038171003817= 120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817 (A19)
Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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4 Shock and Vibration
payload during crane boom slewing (Figures 8ndash10) in thenoninertial reference frameB is as follows
119898(
1198892(1199091(119905))
1198891199052
)
= minus119873 (119905) (
1199091(119905)
119897
) + 119898(
119889 (120593119890(119905))
119889119905
)
2
1199091(119905)
+ 119898(
1198892(120593119890(119905))
1198891199052
) (119877 + 1199101(119905))
+ 2119898(
119889 (120593119890(119905))
119889119905
)(
119889 (1199101(119905))
119889119905
)
119898(
1198892(1199101(119905))
1198891199052
)
= minus119873 (119905) (
1199101(119905)
119897
) + 119898(
119889 (120593119890(119905))
119889119905
)
2
(119877 + 1199101(119905))
minus 119898(
1198892(120593119890(119905))
1198891199052
)1199091(119905)
minus 2119898(
119889 (120593119890(119905))
119889119905
)(
119889 (1199091(119905))
119889119905
)
119898(
1198892(1199111(119905))
1198891199052
) = minus119898119892 + 119873 (119905) (
119897 minus 1199111(119905)
119897
)
1198972= (1199091(119905))2
+ (1199101(119905))2
+ (1199111(119905) minus 119897)
2
120593119890(119905) = 120596
119890119905 120593
119890(0) = 0
119889 (120593119890(0))
119889119905
= 120596119890 119873 (0) = 119898119892
1199091(0) = 0
119889 (1199091(0))
119889119905
= 120596119890119877 119910
1(0) = 0
119889 (1199101(0))
119889119905
= 0 1199111(0) = 0
119889 (1199111(0))
119889119905
= 0
(1)
The absolute coordinates 1199092(119905) (Figure 1(c)) 119910
2(119905)
(Figure 1(d)) 1199112(119905) of payload 119872 in the inertial reference
frame Emay be computed as
1199092(119905) = (119877 + 119910
1(119905)) cos (120593
119890(119905)) + 119909
1(119905) sin (120593
119890(119905))
1199102(119905) = (119877 + 119910
1(119905)) sin (120593
119890(119905)) minus 119909
1(119905) cos (120593
119890(119905))
1199112(119905) = 119911 (119905)
(2)
The absolute coordinates 1199092 1199102in the inertial reference
frameE which dependupon the relative coordinates11990911199101in
the noninertial reference frameB during the swaying of load119872 (Figures 8ndash10) has been defined according to the abovementioned Equations and has been shown in Figures 4 and 5as (- - - 119910
2= 1199102(1199092))
The computational results in Figure 1 derived for DAEproblem (1)-(2) solution coincide with the linearized solu-tion of the payload swaying problem in Appendices AndashI(formula (H1) in the noninertial reference frameB)
It is necessary to note that the posed DAE problem (1)describes payloadmotion in the vicinity of the lower positionof stable dynamical equilibrium assuming rope tension force119873(119905) gt 0 Factually the applied rope is assumed as a unilateralgeometric constraint (Appendix C) in the shape of a torsionfiberThe upper position of themechanical system is unstable(see [73ndash75 78ndash80 91 93]) and corresponds to the case of119873(119905) lt 0 within the parameters of the chosen mechanicalsystem in Figures 8ndash10 Such an upper pendulum positionmight conflict with the hypothesis about the unilateral natureof geometric constraint All of the above mentioned upperpendulum position conditions are outside the objectives andscope of the present paper
3 Experimental Procedure
The experimental procedure has been grounded on the usageof the assembly in Figures 2 and 3
The assembly in Figures 2 and 3 includes the followingmachine components the vertical fixed shaft119874
2119863with height
119867 = 1m the crane boom model 119861119863 with length 119877 = 05mand diameter 6mm the cable119861119872with different fixed lengths1198971= 0825m (in Figure 2 and Figure 5(b)) 119897
2= 0618m (in
Figure 5(a)) 1198973= 0412m (in Figure 4(b)) and 119897
4= 0206m
(in Figure 4(a)) The crane boom model 119861119863 is attached tothe vertical fixed shaft 119874
2119863 by bearing 119863 The cable 119861119872
is attached to the crane boom 119861119863 tip in the point 119861 Thefree or running end 119872 of the cable 119861119872 is the payload 119872
attachment pointThe load119872 is a light emitting diode (LED)with diameter 2mm and the battery with the battery voltage3V The experimental swaying trajectory (mdash 119910
2= 1199102(1199092)) in
Figures 2 4 and 5 is the experimental light emitting diode119872absolute trajectory in the inertial reference frameE
The laser pointer in Figure 2 is also attached to the craneboom 119861119863 tip in the point 119861 The laser pointer is the part ofthe noninertial reference frames B and F and the pointertrajectory is marked as (- - -) in Figure 2 The introductionof laser pointer 119861 allows estimation of dynamic deviation ofpayload119872 The horizontal regular fixed grid with canvas size1200mm times 700mm is placed in the horizontal plane119874
211990921199102
The horizontal grid is formed by the square cells with size20mm times 20mm
The experimental swaying trajectories (mdash 1199102= 1199102(1199092))
in Figures 2 4 and 5 have been written in the obscure roomwith the introduction of an upper digital camera with a longexposure 60 sndash90 s and the camera height above horizontalgrid level is 25m
4 Comparison and Discussion ofDAE-Solution Based Theoretical andExperimental Results
Thecomparison ofDAE-solution based theoretical (- - -1199102=
1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) results is shown
in Figures 4 and 5
Shock and Vibration 5
Light-emitting diode
Laser pointer
Camera
Boom
Slewing axis
Bearing
Lower plate
D
R = 05m
H=1
m
O2
O1
M
B
25
m
Ast
120596e = 0209 rads
l=0825
m
Regular grid 1200mm times 700mmwith square cells 20mm times 20mm
Figure 2 The dimension experimental measurement system for payload119872 swaying motion during crane boom uniform slewing
In order to estimate the relative disagreement of thederived DAE-solution based computational (- - -) andexperimental (mdash) payload 119872 absolute trajectories in theinertial reference frame E we have computed the amplitudediscrepancy 120575 in the polar coordinate system by the followingformula
120575 = ⟨
1
2
((
10038161003816100381610038161003816119903comp minus 119903exper
10038161003816100381610038161003816
119903comp) + (
10038161003816100381610038161003816119903comp minus 119903exper
10038161003816100381610038161003816
119903exper))⟩
(3)
where 119903comp and 119903exper are the magnitudes of the radius-vectors connecting point 119874
2and theoretical (- - -) and
experimental (mdash) curves and computed for the same fixedpolar angle 120593
119890
The amplitude discrepancies 120575 have the following values120575 = 753 for 119897 = 0206m in Figure 4(a) 120575 = 758 for119897 = 0412m in Figure 4(b) 120575 = 740 for 119897 = 0618m inFigure 5(a) and 120575 = 69 for 119897 = 0825m in Figure 5(b)
Numerical computations (- - - 1199102
= 1199102(1199092)) in
Figure 4(a) have been carried out for the following values of
mechanical system parameters 119877 = 0492m 119892 = 981ms2119897 = 0206m 119896 = (119892119897)
05asymp 6901 rads119879 = 14 s120596
119890= 2120587119879 asymp
0449 rads 120593119890= 180
∘ 1205721dyn = 001014 rad 119881
119861= 0221ms
119910dyn = 000209m ]1= 119896 + 120596
119890= 73496 rads ]
2= 119896 minus
120596119890= 64520 rads 119862
1= 001502m and 119862
3= 001711m
(Figure 4(a))Numerical computations (- - - 119910
2= 119910
2(1199092)) in
Figure 4(b) have been carried out for the following values ofmechanical system parameters 119877 = 0492m 119892 = 981ms2119897 = 0412m 119896 = (119892119897)
05asymp 4879 rads119879 = 14 s120596
119890= 2120587119879 asymp
0449 rads 120593119890= 180
∘ 1205721dyn = 001019 rad 119881
119861= 0221ms
119910dyn = 000419m ]1= 119896 + 120596
119890= 53284 rads ]
2= 119896 minus
120596119890= 44308 rads 119862
1= 002072m and 119862
3= 002492m
(Figure 4(b))Numerical computations (- - -119910
2= 1199102(1199092)) in Figure 5(a)
have been carried out for the following values of mechanicalsystem parameters 119877 = 0492m 119892 = 981ms2 119897 =
0618m 119896 = (119892119897)05
asymp 3984 rads 119879 = 20 s 120596119890= 2120587119879 asymp
0314 rads 120593119890
= 180∘ 1205721dyn = 000498 rad 119881
119861=
01546ms 119910dyn = 000308m ]1= 119896 + 120596
119890= 4298 rads
6 Shock and Vibration
(a) (b)
(c) (d)
Figure 3 Photos of dimension experimental measurement system (a)ndash(d) for payload 119872 swaying motion during crane boom uniformslewing
]2
= 119896 minus 120596119890= 3670 rads 119862
1= 001798m and 119862
3=
002106m (Figure 5(a))For further estimation of the relative disagreement of
derived DAE-solution based computational (- - -) andexperimental (mdash) payload 119872 absolute trajectories in theinertial reference frame E we have computed the confidenceintervals in Figures 4 and 5 for the dimensionless parameters119903comp119903exper and confidence probability 095 The Studentrsquos 119905-test results yield the following confidence intervals 09374 le
(119903comp119903exper) le 10315 for 119897 = 0206m in Figure 4(a)09014 le (119903comp119903exper) le 10272 for 119897 = 0412m inFigure 4(b) 09200 le (119903comp119903exper) le 10352 for 119897 = 0618min Figure 5(a) and 09257 le (119903comp119903exper) le 10066 for119897 = 0825m in Figure 5(b)
Both the relative discrepancy 120575 and the confidence inter-vals show the satisfactory agreement between the absolute
payload trajectories (Figures 4 and 5) in the inertial referenceframeE that have been computed with DAE-solution basedtheoreticalmodel (- - -) (1) andmeasured experimentally (mdash)as shown in Figures 4 and 5
It is important to note that the payload motion DAEequations in the nonlinear problem (1) for the noninertialreference frame B may be derived with an introductionof Lagrange equations (Appendices AndashC F) However thediscussion of (1) is more suitable and informative with anintroduction of dynamic Coriolis theorem (Appendices AndashD F) The first terms 1198892119909
11198891199052 1198892119910
11198891199052 in (1) define the
vector a119872B = ar (Appendix B formula (B4)) for the relative
acceleration of load 119872 in the noninertial reference frameB The straight-line terms (D3) formula-based terms in (1)are linearly proportional to the relative payload coordinatesin the noninertial reference frame B These straight-line
Shock and Vibration 7
05
05
04
04
03
03
02
02
00
00
01
01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)
p O2 (p D)
(a)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)Computational y2 = y2(x2)
l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)
p O2 (p D)
(b)
Figure 4 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0206m (a) and fixed cable length 119897 = 0412m (b)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)
2R = 1 (m)
y2 (m)
x2 (m)p O2 (p D)
(a)
05
04
03
02
00
01
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448(rads)
2R = 1 (m)
x2 (m)
y2 (m)
p O2 (p D)
(b)
Figure 5 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0618m (a) and fixed cable length 119897 = 0825m (b)
terms in (1) have been determined by the contribution ofthe normal or centripetal acceleration 120596BE times (120596BE times
r119864119872
) = aen (B6) of transportation for payload119872 and by theappearance of corresponding DrsquoAlembert centrifugal inertiaforce Φe
n= (minus119898)aen (D3) due to crane boom transport
rotation in the noninertial reference frame B Additionaltermsminus2120596
119890(1198891199101119889119905) and 2120596
119890(1198891199091119889119905) in (1) have been defined
by the compound or Coriolis acceleration 2120596BE times V119872B =
acor of payload119872 in the noninertial reference frameB Therectangular Cartesian projections of Coriolis inertia force inthe noninertial reference frameB is defined by formula (B7)
5 Experimental Results for the SphericalPendulum with the Fixed Pivot Center
Experimental load swaying results for the fixed crane boommodel (120596
119890= 0 rads) are shown in Figure 6
Finite motions of a swaying load are shown in Figure 6in the case of 120596
119890= 0 rads In Figure 6 the relative and
absolute load trajectories are identically equal The lingeringremains of elliptical motion in Figure 6 and ellipses semiaxesare defined by the initial conditions
6 Discussion of Analogy between PayloadSwaying and Foucaultrsquos Pendulum Motion
The motion of Foucaultrsquos pendulum is shown in Figure 7where 119909
211991021199112is the heliocentric inertial reference frame
(not shown in Figure 7) 119909111991011199111is the geocentric noninertial
reference frame and 119909119910119911 is a noninertial reference framelocated at the geographic latitude 119910
1119911 119860 st119911 is a local vertical
119861119863 is the radius of the circle of the pinning point 119861119872 isthe cable length and 120596
119890is the angular velocity of the Earth
diurnal rotation
8 Shock and Vibration
(a) (b)
Figure 6 Experimental trajectories of the spherical pendulum with fixed pivot center and fixed cable length 119897 = 0825m (a b)
M
g
yx
z
B
D
x1
Ast
120596e
z1
y1
Figure 7 Foucaultrsquos pendulum schematic view
It is important to note that the structure of linearizedequations (H4)-(H5) for the swaying payload during craneboom uniform rotation in the noninertial reference frameF is analogous to the structure of the governing equationsfor Foucaultrsquos pendulum motion in the known publishedworks by Zhuravlev and Petrov [100] (formula (6) p 36 of[100]) Pardy [88] (formulae (12)ndash(14) pp 850-851 of [88])and Condurache et al [77] (formulae (11)ndash(13) p 743-744of [77]) The above mentioned analogy between linearizedequations (H4)-(H5) and the governing equations in [77 88100] assumes the geometric analogy of relative swaying tra-jectories between Foucaultrsquos pendulum and the boom-drivenpendulum with rotating pivot centers (Figures 1(e) 12 and16) So there is the geometric analogy between computationalrelative trajectories of load 119872 swaying during crane boomuniform slewing (Figures 1(e) 12 and 16) and relative trajec-tories of load119872 swaying in the plane (119860 st119909119910) around p119860 st inFigure 7 shownbyZhuravlev andPetrov [100] (Figure 4 p 39of [100]) and Condurache andMartinusi [77] (Figure 14 at p754 and Figure 16 at p 755 of [77]) The better understandingof Foucaultrsquos pendula dynamics is provided with the studyof computational relative trajectories in Figures 1(e) 12 and16 For Foucaultrsquos pendulum (Figure 7) the major semiaxes
of relative amplitude extremes have angular velocity valuesequal to the angular velocity of the Earth diurnal rotationMoreover for Foucaultrsquos pendulum (Figure 7) the directionof rotation of major semiaxes of relative amplitude extremesis oppositely directed to the direction of the Earth diurnalrotation All the above mentioned means that the plane(119860 st119861119872) of Foucaultrsquos pendula swaying remains fixed in theheliocentric inertial reference frame 119909
211991021199112
The presented analogy between crane boom-driven pay-load swaying and Foucaultrsquos pendulum motion essentiallyincreases knowledge and awareness of the oscillation pro-cesses in such dynamic systems
7 Discussion and MechanicalInterpretation of GoverningEquations for DAE-Based Nonlinearand Linearized Models
A governing nonlinear DAE-based system (1) for crane boomuniform slewing has been proposed and numerically solvedA conservative system (1) and (H4)-(H5) with componentsΦcor119909 andΦcor119910 has been derived Both projectionsΦcor119909 andΦcor119910 coincide in the noninertial reference framesB andF
The occurrence of the first derivatives 119889119909119889119905 119889119910119889119905 ofthe load119872 relative coordinates with respect to time in termsminus2120596119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) determines that there is no
decay of the oscillations in (H4)-(H5) and (1) but onlyredirection of the relative velocity vector V
119872F = Vr of theload119872 in the noninertial reference framesB andF
The introduction of a linearized model (H4)-(H5)allowed the determination of the natural frequencies (]
1=
]2) of free oscillations of payload 119872 ]
1= 119896 + 120596
119890and
]2= 119896 minus 120596
119890
It follows from the forms of the relative payload trajectory119910 = 119910(119909) in the noninertial reference frame F and theabsolute trajectory 119910
2= 1199102(1199092) in the inertial reference frame
E that the resulting motion of the payload 119872 on the cable119872119861 taking into account the Coriolis inertia forceΦcor (D5)will be the sum of two oscillations with natural frequencies ]
1
and ]2 and with periods 119879
1= 2120587]
1and 119879
2= 2120587]
2
It is worth noting that the frequencies ]1and ]
2differ
by 2120596119890 This means that for the small angle assumption
(Appendices H-I) we have ]1
asymp ]2and the trajectory
Shock and Vibration 9
mg
γ
Nℬ
z1z2
e2
e3
ℰ
ℰ
ℰ
ℱ
ℱ
ℬΔ
120574
y1
Vr
At
M
R D
E
Bz
l
120573
O
1205721
1205721dyn
Adyn
ydyn
h
120596e
AstO1
3ẽ
2ẽ
O2
d120593edty
sin(120593e)
cos(120593e)
120596ℬℰ = 120596ℱℰ
3e
2e
1e
Figure 8 Swaying scheme of load119872 on the cable119872119861 which is fixedly attached at the point 119861 on the crane boom 1198611198742 in the vertical plane
(119910119911) for the nonlinear model derivation in Cartesian coordinates
of relative motion 119910 = 119910(119909) in the noninertial referenceframe F on the expiration of the relative oscillations periodtime 2120587119896 looks approximately like an ellipse (Figures 12(b)and 16(b)) with semiaxes which are governed by the initialconditions
The major semiaxis of such an ellipse in Figures 12(b)and 16(b) is defined by the initial velocity of load 119872 in thenoninertial reference frameF and the analytical value of themajor semiaxis is equal to119881
119861119896 = (120596
119890119877)119896 The derived value
of 119881119861119896 is approximately the sum of the absolute values of
the amplitudes in the analytical form (I15) of the linearizedCauchy problem (H4)-(H5)
The minor semiaxis of the above mentioned ellipsein Figures 12(b) and 16(b) is defined by the value 119910dynwhich is approximately the difference of amplitudes in theanalytical form (I15) of the linearizedCauchy problem (H4)-(H5) In this case the major semiaxis is approximately 5ndash20 times larger than the minor semiaxis because the angularvelocity 120596
119890of the transport rotation of crane boom 119861119874
2is
much smaller than the natural frequency 119896 of the sphericaloscillations of payload119872 on the cable119872119861 (Figures 12(b) and16(b))
It is also necessary to note that the major semiaxis of theellipse in the relative motion rotates with an angular velocity120596119890in the opposite direction of the transport rotation of crane
boom 1198611198742in (Figures 12 and 16)
It follows from Figures 1(f) 2 4 and 5 that the trajectoryof absolute motion of load 119872 in the inertial reference frameE is almost a symmetric curve with 119910
2axial symmetry
The initial and the final motions of load 119872 for half-period
essentially differ from its harmonic oscillations neighbor forquarter-period
Due to the negligible quantity of 120572dyn and 119910dyn (G3)ndash(G5) the average deviation of the load119872 from the mechan-ical trajectory of the boom tip 119861 is negligible (Figures1(f) 2 4 and 5) The basic dynamic load on the systemldquocrane boom 119861119874
2-load 119872rdquo is created by the high-frequency
oscillations of load 119872 which are determined by the actionof inertia force Φcor = minus2119898(120596BE times V
119872B) stipulated bythe Coriolis acceleration acor = 2120596BE times V
119872B of load 119872
in the noninertial reference frame F The basic dynamicload in the system defines additional loads and vibrations ofcrane boom 119861119874
2mechanical elements and support bearings
complicates the automatic and manual control systems of theelectromechanical crane boom 119861119874
2 and also makes crane
operation much more difficultIt is also important to note that the stop of crane boom
1198611198742does not lead to instantaneous damping of the load
119872 absolute oscillations in the inertial reference frame E(Figures 2 4 and 5) This phenomenon directly follows fromthe experimental trajectory in Figures 2 4 and 5 Alsothe natural spherical oscillations of load 119872 will occur withthe frequency 119896 and the amplitude (the difference betweenthe final position of load 119872 in the relative motion in thenoninertial reference frame F for half-period of vibrationand static equilibrium 119860 st of load 119872 on the cable) Thefurther oscillations for the stop of the crane boom 119861119874
2
slewing motion are the deviation of the real trajectory fromthe intended final position of load119872 The results of physicalsimulation in Figures 2 4 and 5 show the necessity of add-on
10 Shock and Vibration
N
ℬ
ℬ
e2
e1
ℰ
ℰ
ℱ
ℱ
y2
x2
Ve
Vr
M
E
1ẽ
2ẽ x
x1
O2(D)
AoAst
O
y y1
Adyn
ydyn
1205722
120579
120579
120596e = 120596ℬℰ = 120596ℱℰ
120593e = 120596et
Φcor
B(O1)
Φen
Φeτ
1e
2e
Figure 9 Swaying scheme of load119872 at the cable119872119861 which is fixedly attached in the point 119861 of the crane boom 1198611198742 in the horizontal plane
(11990911199101) for the nonlinear model derivation in Cartesian coordinates
devices development for the efficient suppression of load 119872
final oscillations
8 Discussion and Comparison of Derived andKnown Published Results
In Figure 1 and pp 537-538 of publishedwork by Sakawa et al1981 [63] the small angle between payloadrsquos cable and verticalline has been introduced that confirms proposed nonlinearDAE-based system (1) and linearized model (H4)-(H5)
In Figure 9 of p 278 of published work by Maczynskiand Wojciech 2003 [44] the computational finite elementmethod (FEM)-based results were shown for the abso-lute payload trajectories in the inertial reference frame EMaczynskirsquos Figure 9 in [44] outlines that the angle betweenthe payloadrsquos cable and the vertical line does not exceed 01rad and that agreeswith the proposed linearizedmodel (H4)-(H5) for a small angle assumption Computational trajectoryfor Maczynski-derived small load swaying [44] after craneboom stop qualitatively coincides with the experimentallyobservable load119872motion in Figures 2 4 and 5
In Figure 6 of published work by Ju et al 2006 [34]it was shown that the angle between the payloadrsquos cableand the vertical line does not exceed 16∘ Jursquos formula (15a)in p 382 of [34] assumes that Jursquos angle between payloadrsquoscable and the vertical line has the harmonic law of variationwith an introduction of a small perturbation term Both Jursquosassumptions in [34] confirm the proposed linearized model(H4)-(H5)
Comparison of the derived linearized model (H4)-(H5)with Figure 5 of published work by Mitrev and Grigorov2008 in [49] shows that the Mitrev-derived ranges of payloadswaying angles within Mitrevrsquos nonlinear model does notexceed 48∘ Such small values of swaying angles completelyconfirm the applicability and correctness of the small angleassumption in (H4)-(H5)
9 Final Conclusions
A governing nonlinear DAE-based system for crane boomuniform slewing has been proposed numerically solved andexperimentally verified
Shock and Vibration 11
Fully nonlinear differential equations for 3 DOF sphericalpendulum oscillations with a uniform slewing crane boomaround a fixed vertical axis of rotation have been derivedin relative Cartesian and spherical coordinates The identicallinearized differential equations in relative Cartesian coordi-nates have been derived with an introduction of the Coriolisdynamic theorem and Lagrange equations for a uniformcrane boomrotation and small swaying angle120572
1assumptions
Linearized andnonlinear theoretical problems for relativeswaying of the payload have been formulated in the form ofthe initial value (Cauchy) problem
An analytical solution of the linearized system has beenderived
The influences of inertia forces from the centripetal andcompound accelerations have been estimated
A linearized conservative system which contains firsttime derivatives of coordinates and no damping of oscilla-tions has been derived
The amplitudes of load oscillations which depend oncomputed initial conditions have been estimated within asmall angle assumption The dependence of natural frequen-cies on the transport inertia forces and gravity forces has beencomputed for the linearized systems
The formulae for the association of relative payloadmotion in the noninertial reference frame F and absolutepayload motion in the inertial reference frame E have beenoutlined
The results of the numerical DAE-based investigation andthe performed physical simulation show a satisfactory fit forfrequencies and amplitudes of load oscillations
The dynamic analogy between crane boom-driven pay-load swaying motion and Foucaultrsquos pendulum motion hasbeen grounded and outlined
The results of the present work are the foundation forfurther investigations of payload119872 swaying dynamics duringtelescopic crane boom nonuniform slewing motion withvariable cable length and for the different motions of thependulum pivot center
Appendices
A Velocity Kinematics Analysis inCartesian Coordinates
For the nonlinear problem definition we study the coopera-tive motion of the mechanical system ldquocrane boom 119861119874
2-load
119872rdquo which is shown in Figures 8ndash10In the nomenclature chapter we denote the fixed inertial
frame of reference E as 119909211991021199112 and the moving noninertial
frame of reference B as 119909111991011199111 which is rigidly bounded
with the crane boom1198611198742 Rotation of themoving noninertial
frame of referenceB(119909111991011199111) around the fixed inertial frame
of referenceE(119909211991021199112) defines the transportation motion for
payload 119872 The motion of load 119872 relative to the movingnoninertial frame of referenceB(119909
111991011199111) defines the relative
motion of payload 119872 The point 119860 st with the coordinates1199091= 0 119910
1= 0 and 119911
1= 0 is the steady equilibrium position
for the load119872 when the crane boom 1198611198742is fixed The point
119860dyn with the coordinates 1199091 = 0 1199101= 119910dyn and 1199111 = Δ is the
dynamic equilibrium position for the load119872 for the rotatingcrane boom 119861119874
2 We will assume the point119860dyn as the origin
of the noninertial reference frameF(119909119910119911) The directions ofaxes 119909 119910 119911 of the noninertial reference frameF are parallelto the axes 119909
1 1199101 1199111of the noninertial reference frameB in
Figures 8ndash10 The load 119872 relative motion takes place alongthe sphere surface with the fixed radius equal to the cablelength 119861119872 = 119897
The computational scheme in Figures 8ndash10 for the nonlin-ear model derivation may be described with an introductionof three degrees of freedom For generalized coordinates weassume the rectangular coordinates 119909
1 1199101 and 119911
1of the load
119872 in the moving noninertial frame of reference B and theangle 120593
119890of crane boom 119861119874
2slewing in the horizontal plane
(11990911199101) with the angular velocity 120596
119890around the vertical axis
11987421199112The relative velocity vector Vr = V
119872B of the load 119872 isdefined as Vr = V
119872B = (1198891199091119889119905 119889119910
1119889119905 119889119911
1119889119905)
In order to derive the absolute velocity V119872E of payload
119872 we will apply the vector method for absolute motionassignment We will define r
1198742119872
as the position vector con-necting initial point 119874
2and terminal point119872 in noninertial
reference frameB (119898) consider
r1198742119872
= r1198742119861+ r119861119872
(A1)
where vector components in (1) are defined in nomenclatureIn accordance with Figures 8ndash10 we have the following
unit vectorsrsquo expansions for position vectors r1198742119861
and r119861119872
in (A1)
r1198742119861
= (119877 cos (120593119890)) e1+ (119877 sin (120593
119890)) e2+ (119877 tan (120574)) e
3
(A2)
r119861119872
= 1199091e1+ 1199101e2+ (1199111minus 119897) e3 (A3)
where 119877 120593119890 1199091 and 119910
1notations have been defined in
nomenclature and in Figures 8ndash10For further definition of the position vector r
1198742119872
in iner-tial reference frame E we write unit vectors of noninertialreference frameB in (A3) through the unit vectors of inertialreference frame E
e1= (sin (120593
119890)) e1minus (cos (120593
119890)) e2 (A4)
e2= (cos (120593
119890)) e1minus (sin (120593
119890)) e2 (A5)
e3= e3 (A6)
After substitution of formulae (A2)ndash(A6) in (A1) andsome algebraic manipulations the vector expression (A1) forthe position vector r
1198742119872
in the inertial reference frameEwilltake the following form
r1198742119872
= ((119877 + 1199101) cos (120593
119890) + 1199091sin (120593
119890)) e1
+ ((119877 + 1199101) sin (120593
119890) minus 1199091cos (120593
119890)) e2
+ (119877 tan (120574) + 1199111minus 119897) e3
(A7)
12 Shock and Vibration
The absolute velocity of payload 119872 we define by timedifferentiation of (A7) assuming constancy of unit vectorse1 e2 and e
3of the inertial reference frameE and constancy
119877 and 120574
V119872E = Vabs =
119889 (r1198742119872
)
119889119905
(A8)
After differentiation and algebraic transformations in(A7) and (A8) we have the following vector expression forabsolute payload 119872 velocity in the inertial reference frameE
V119872E
= (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) cos (120593119890)
+ ((
1198891199091
119889119905
) minus (119877 + 1199101) (
119889120593119890
119889119905
)) sin (120593119890)) e1
+ (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) sin (120593119890)
+ ((119877 + 1199101) (
119889120593119890
119889119905
) minus (
1198891199091
119889119905
)) cos (120593119890))
times e2+ (
1198891199111
119889119905
) e3
(A9)
The square of absolute payload119872 velocitymay be writtenas
1198812
abs = (
1198891199091
119889119905
)
2
+ (
1198891199101
119889119905
)
2
+ (
1198891199111
119889119905
)
2
+ (
119889120593119890
119889119905
)
2
times (1199092
1+ (1199101+ 119877)2
) + 2(
119889120593119890
119889119905
)
times (1199091(
1198891199101
119889119905
) minus (1199101+ 119877)(
1198891199091
119889119905
))
(A10)
After algebraic transformations we have the square ofabsolute payload119872 velocity in the form of
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A11)
Algebraic expressions (A10) and (A11) could be derivedon the basis of a velocity addition theorem for payload 119872
compound motion in the inertial reference frame E throughthe unit vectors of the noninertial reference frame B (see[59 76 81 82 84 85 87 90ndash92 94ndash98])
V119872E = V
119872B + V1198742E + 120596BE times r
1198742119872
(A12)
where payload relative velocity in the noninertial referenceframeBmay be written as
V119872B = Vr = (
1198891199091
119889119905
) e1+ (
1198891199101
119889119905
) e2+ (
1198891199111
119889119905
) e3
(A13)
The velocityV1198742E of the point119874
2= point119864 in the inertial
reference frame E has zero value
V1198742E = 0 (A14)
The last term in (A12) is the vector product 120596BE times
r1198742119872
where crane boom angular velocity vector is 120596BE =
(119889120593119890119889119905)e3(A6) and the position vector r
1198742119872
in the nonin-ertial reference frameB is as follows
r1198742119872
= 1199091e1+ (1199101+ 119877) e
2+ 1199111e3 (A15)
Taking into account (A13)ndash(A15) the vector equation(A12) in the inertial reference frame E written through theunit vectors e
1 e2 e3of the noninertial reference frame B
will have the following form
V119872E = ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101)) e1
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091) e2+ (
1198891199111
119889119905
) e3
(A16)
So on the basis of (A16) we have the following square ofabsolute payload119872 velocity as
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A17)
Independently derived expressions for the scalar prod-uct (V
119872EV119872E) in (A11) and (A17) completely coincidewhich confirms the correctness and accuracy of the pen-dulum absolute velocity V
119872E derivation and shows thatthe scalar product (V
119872EV119872E) is the invariant expressionindependent of choice of reference frame
The second and third terms of (A12) in the noninertialreference frame B determine the vector of the load 119872
transportation velocity as
Ve = V1198742E + 120596BE times r
1198742119872
(A18)
Taking into account (A14) the scalar of the load 119872
transportation velocity is defined as
1003817100381710038171003817Ve
1003817100381710038171003817= 120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817 (A19)
Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
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[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
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[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
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[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
International Journal of
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Shock and Vibration
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Shock and Vibration 5
Light-emitting diode
Laser pointer
Camera
Boom
Slewing axis
Bearing
Lower plate
D
R = 05m
H=1
m
O2
O1
M
B
25
m
Ast
120596e = 0209 rads
l=0825
m
Regular grid 1200mm times 700mmwith square cells 20mm times 20mm
Figure 2 The dimension experimental measurement system for payload119872 swaying motion during crane boom uniform slewing
In order to estimate the relative disagreement of thederived DAE-solution based computational (- - -) andexperimental (mdash) payload 119872 absolute trajectories in theinertial reference frame E we have computed the amplitudediscrepancy 120575 in the polar coordinate system by the followingformula
120575 = ⟨
1
2
((
10038161003816100381610038161003816119903comp minus 119903exper
10038161003816100381610038161003816
119903comp) + (
10038161003816100381610038161003816119903comp minus 119903exper
10038161003816100381610038161003816
119903exper))⟩
(3)
where 119903comp and 119903exper are the magnitudes of the radius-vectors connecting point 119874
2and theoretical (- - -) and
experimental (mdash) curves and computed for the same fixedpolar angle 120593
119890
The amplitude discrepancies 120575 have the following values120575 = 753 for 119897 = 0206m in Figure 4(a) 120575 = 758 for119897 = 0412m in Figure 4(b) 120575 = 740 for 119897 = 0618m inFigure 5(a) and 120575 = 69 for 119897 = 0825m in Figure 5(b)
Numerical computations (- - - 1199102
= 1199102(1199092)) in
Figure 4(a) have been carried out for the following values of
mechanical system parameters 119877 = 0492m 119892 = 981ms2119897 = 0206m 119896 = (119892119897)
05asymp 6901 rads119879 = 14 s120596
119890= 2120587119879 asymp
0449 rads 120593119890= 180
∘ 1205721dyn = 001014 rad 119881
119861= 0221ms
119910dyn = 000209m ]1= 119896 + 120596
119890= 73496 rads ]
2= 119896 minus
120596119890= 64520 rads 119862
1= 001502m and 119862
3= 001711m
(Figure 4(a))Numerical computations (- - - 119910
2= 119910
2(1199092)) in
Figure 4(b) have been carried out for the following values ofmechanical system parameters 119877 = 0492m 119892 = 981ms2119897 = 0412m 119896 = (119892119897)
05asymp 4879 rads119879 = 14 s120596
119890= 2120587119879 asymp
0449 rads 120593119890= 180
∘ 1205721dyn = 001019 rad 119881
119861= 0221ms
119910dyn = 000419m ]1= 119896 + 120596
119890= 53284 rads ]
2= 119896 minus
120596119890= 44308 rads 119862
1= 002072m and 119862
3= 002492m
(Figure 4(b))Numerical computations (- - -119910
2= 1199102(1199092)) in Figure 5(a)
have been carried out for the following values of mechanicalsystem parameters 119877 = 0492m 119892 = 981ms2 119897 =
0618m 119896 = (119892119897)05
asymp 3984 rads 119879 = 20 s 120596119890= 2120587119879 asymp
0314 rads 120593119890
= 180∘ 1205721dyn = 000498 rad 119881
119861=
01546ms 119910dyn = 000308m ]1= 119896 + 120596
119890= 4298 rads
6 Shock and Vibration
(a) (b)
(c) (d)
Figure 3 Photos of dimension experimental measurement system (a)ndash(d) for payload 119872 swaying motion during crane boom uniformslewing
]2
= 119896 minus 120596119890= 3670 rads 119862
1= 001798m and 119862
3=
002106m (Figure 5(a))For further estimation of the relative disagreement of
derived DAE-solution based computational (- - -) andexperimental (mdash) payload 119872 absolute trajectories in theinertial reference frame E we have computed the confidenceintervals in Figures 4 and 5 for the dimensionless parameters119903comp119903exper and confidence probability 095 The Studentrsquos 119905-test results yield the following confidence intervals 09374 le
(119903comp119903exper) le 10315 for 119897 = 0206m in Figure 4(a)09014 le (119903comp119903exper) le 10272 for 119897 = 0412m inFigure 4(b) 09200 le (119903comp119903exper) le 10352 for 119897 = 0618min Figure 5(a) and 09257 le (119903comp119903exper) le 10066 for119897 = 0825m in Figure 5(b)
Both the relative discrepancy 120575 and the confidence inter-vals show the satisfactory agreement between the absolute
payload trajectories (Figures 4 and 5) in the inertial referenceframeE that have been computed with DAE-solution basedtheoreticalmodel (- - -) (1) andmeasured experimentally (mdash)as shown in Figures 4 and 5
It is important to note that the payload motion DAEequations in the nonlinear problem (1) for the noninertialreference frame B may be derived with an introductionof Lagrange equations (Appendices AndashC F) However thediscussion of (1) is more suitable and informative with anintroduction of dynamic Coriolis theorem (Appendices AndashD F) The first terms 1198892119909
11198891199052 1198892119910
11198891199052 in (1) define the
vector a119872B = ar (Appendix B formula (B4)) for the relative
acceleration of load 119872 in the noninertial reference frameB The straight-line terms (D3) formula-based terms in (1)are linearly proportional to the relative payload coordinatesin the noninertial reference frame B These straight-line
Shock and Vibration 7
05
05
04
04
03
03
02
02
00
00
01
01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)
p O2 (p D)
(a)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)Computational y2 = y2(x2)
l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)
p O2 (p D)
(b)
Figure 4 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0206m (a) and fixed cable length 119897 = 0412m (b)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)
2R = 1 (m)
y2 (m)
x2 (m)p O2 (p D)
(a)
05
04
03
02
00
01
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448(rads)
2R = 1 (m)
x2 (m)
y2 (m)
p O2 (p D)
(b)
Figure 5 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0618m (a) and fixed cable length 119897 = 0825m (b)
terms in (1) have been determined by the contribution ofthe normal or centripetal acceleration 120596BE times (120596BE times
r119864119872
) = aen (B6) of transportation for payload119872 and by theappearance of corresponding DrsquoAlembert centrifugal inertiaforce Φe
n= (minus119898)aen (D3) due to crane boom transport
rotation in the noninertial reference frame B Additionaltermsminus2120596
119890(1198891199101119889119905) and 2120596
119890(1198891199091119889119905) in (1) have been defined
by the compound or Coriolis acceleration 2120596BE times V119872B =
acor of payload119872 in the noninertial reference frameB Therectangular Cartesian projections of Coriolis inertia force inthe noninertial reference frameB is defined by formula (B7)
5 Experimental Results for the SphericalPendulum with the Fixed Pivot Center
Experimental load swaying results for the fixed crane boommodel (120596
119890= 0 rads) are shown in Figure 6
Finite motions of a swaying load are shown in Figure 6in the case of 120596
119890= 0 rads In Figure 6 the relative and
absolute load trajectories are identically equal The lingeringremains of elliptical motion in Figure 6 and ellipses semiaxesare defined by the initial conditions
6 Discussion of Analogy between PayloadSwaying and Foucaultrsquos Pendulum Motion
The motion of Foucaultrsquos pendulum is shown in Figure 7where 119909
211991021199112is the heliocentric inertial reference frame
(not shown in Figure 7) 119909111991011199111is the geocentric noninertial
reference frame and 119909119910119911 is a noninertial reference framelocated at the geographic latitude 119910
1119911 119860 st119911 is a local vertical
119861119863 is the radius of the circle of the pinning point 119861119872 isthe cable length and 120596
119890is the angular velocity of the Earth
diurnal rotation
8 Shock and Vibration
(a) (b)
Figure 6 Experimental trajectories of the spherical pendulum with fixed pivot center and fixed cable length 119897 = 0825m (a b)
M
g
yx
z
B
D
x1
Ast
120596e
z1
y1
Figure 7 Foucaultrsquos pendulum schematic view
It is important to note that the structure of linearizedequations (H4)-(H5) for the swaying payload during craneboom uniform rotation in the noninertial reference frameF is analogous to the structure of the governing equationsfor Foucaultrsquos pendulum motion in the known publishedworks by Zhuravlev and Petrov [100] (formula (6) p 36 of[100]) Pardy [88] (formulae (12)ndash(14) pp 850-851 of [88])and Condurache et al [77] (formulae (11)ndash(13) p 743-744of [77]) The above mentioned analogy between linearizedequations (H4)-(H5) and the governing equations in [77 88100] assumes the geometric analogy of relative swaying tra-jectories between Foucaultrsquos pendulum and the boom-drivenpendulum with rotating pivot centers (Figures 1(e) 12 and16) So there is the geometric analogy between computationalrelative trajectories of load 119872 swaying during crane boomuniform slewing (Figures 1(e) 12 and 16) and relative trajec-tories of load119872 swaying in the plane (119860 st119909119910) around p119860 st inFigure 7 shownbyZhuravlev andPetrov [100] (Figure 4 p 39of [100]) and Condurache andMartinusi [77] (Figure 14 at p754 and Figure 16 at p 755 of [77]) The better understandingof Foucaultrsquos pendula dynamics is provided with the studyof computational relative trajectories in Figures 1(e) 12 and16 For Foucaultrsquos pendulum (Figure 7) the major semiaxes
of relative amplitude extremes have angular velocity valuesequal to the angular velocity of the Earth diurnal rotationMoreover for Foucaultrsquos pendulum (Figure 7) the directionof rotation of major semiaxes of relative amplitude extremesis oppositely directed to the direction of the Earth diurnalrotation All the above mentioned means that the plane(119860 st119861119872) of Foucaultrsquos pendula swaying remains fixed in theheliocentric inertial reference frame 119909
211991021199112
The presented analogy between crane boom-driven pay-load swaying and Foucaultrsquos pendulum motion essentiallyincreases knowledge and awareness of the oscillation pro-cesses in such dynamic systems
7 Discussion and MechanicalInterpretation of GoverningEquations for DAE-Based Nonlinearand Linearized Models
A governing nonlinear DAE-based system (1) for crane boomuniform slewing has been proposed and numerically solvedA conservative system (1) and (H4)-(H5) with componentsΦcor119909 andΦcor119910 has been derived Both projectionsΦcor119909 andΦcor119910 coincide in the noninertial reference framesB andF
The occurrence of the first derivatives 119889119909119889119905 119889119910119889119905 ofthe load119872 relative coordinates with respect to time in termsminus2120596119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) determines that there is no
decay of the oscillations in (H4)-(H5) and (1) but onlyredirection of the relative velocity vector V
119872F = Vr of theload119872 in the noninertial reference framesB andF
The introduction of a linearized model (H4)-(H5)allowed the determination of the natural frequencies (]
1=
]2) of free oscillations of payload 119872 ]
1= 119896 + 120596
119890and
]2= 119896 minus 120596
119890
It follows from the forms of the relative payload trajectory119910 = 119910(119909) in the noninertial reference frame F and theabsolute trajectory 119910
2= 1199102(1199092) in the inertial reference frame
E that the resulting motion of the payload 119872 on the cable119872119861 taking into account the Coriolis inertia forceΦcor (D5)will be the sum of two oscillations with natural frequencies ]
1
and ]2 and with periods 119879
1= 2120587]
1and 119879
2= 2120587]
2
It is worth noting that the frequencies ]1and ]
2differ
by 2120596119890 This means that for the small angle assumption
(Appendices H-I) we have ]1
asymp ]2and the trajectory
Shock and Vibration 9
mg
γ
Nℬ
z1z2
e2
e3
ℰ
ℰ
ℰ
ℱ
ℱ
ℬΔ
120574
y1
Vr
At
M
R D
E
Bz
l
120573
O
1205721
1205721dyn
Adyn
ydyn
h
120596e
AstO1
3ẽ
2ẽ
O2
d120593edty
sin(120593e)
cos(120593e)
120596ℬℰ = 120596ℱℰ
3e
2e
1e
Figure 8 Swaying scheme of load119872 on the cable119872119861 which is fixedly attached at the point 119861 on the crane boom 1198611198742 in the vertical plane
(119910119911) for the nonlinear model derivation in Cartesian coordinates
of relative motion 119910 = 119910(119909) in the noninertial referenceframe F on the expiration of the relative oscillations periodtime 2120587119896 looks approximately like an ellipse (Figures 12(b)and 16(b)) with semiaxes which are governed by the initialconditions
The major semiaxis of such an ellipse in Figures 12(b)and 16(b) is defined by the initial velocity of load 119872 in thenoninertial reference frameF and the analytical value of themajor semiaxis is equal to119881
119861119896 = (120596
119890119877)119896 The derived value
of 119881119861119896 is approximately the sum of the absolute values of
the amplitudes in the analytical form (I15) of the linearizedCauchy problem (H4)-(H5)
The minor semiaxis of the above mentioned ellipsein Figures 12(b) and 16(b) is defined by the value 119910dynwhich is approximately the difference of amplitudes in theanalytical form (I15) of the linearizedCauchy problem (H4)-(H5) In this case the major semiaxis is approximately 5ndash20 times larger than the minor semiaxis because the angularvelocity 120596
119890of the transport rotation of crane boom 119861119874
2is
much smaller than the natural frequency 119896 of the sphericaloscillations of payload119872 on the cable119872119861 (Figures 12(b) and16(b))
It is also necessary to note that the major semiaxis of theellipse in the relative motion rotates with an angular velocity120596119890in the opposite direction of the transport rotation of crane
boom 1198611198742in (Figures 12 and 16)
It follows from Figures 1(f) 2 4 and 5 that the trajectoryof absolute motion of load 119872 in the inertial reference frameE is almost a symmetric curve with 119910
2axial symmetry
The initial and the final motions of load 119872 for half-period
essentially differ from its harmonic oscillations neighbor forquarter-period
Due to the negligible quantity of 120572dyn and 119910dyn (G3)ndash(G5) the average deviation of the load119872 from the mechan-ical trajectory of the boom tip 119861 is negligible (Figures1(f) 2 4 and 5) The basic dynamic load on the systemldquocrane boom 119861119874
2-load 119872rdquo is created by the high-frequency
oscillations of load 119872 which are determined by the actionof inertia force Φcor = minus2119898(120596BE times V
119872B) stipulated bythe Coriolis acceleration acor = 2120596BE times V
119872B of load 119872
in the noninertial reference frame F The basic dynamicload in the system defines additional loads and vibrations ofcrane boom 119861119874
2mechanical elements and support bearings
complicates the automatic and manual control systems of theelectromechanical crane boom 119861119874
2 and also makes crane
operation much more difficultIt is also important to note that the stop of crane boom
1198611198742does not lead to instantaneous damping of the load
119872 absolute oscillations in the inertial reference frame E(Figures 2 4 and 5) This phenomenon directly follows fromthe experimental trajectory in Figures 2 4 and 5 Alsothe natural spherical oscillations of load 119872 will occur withthe frequency 119896 and the amplitude (the difference betweenthe final position of load 119872 in the relative motion in thenoninertial reference frame F for half-period of vibrationand static equilibrium 119860 st of load 119872 on the cable) Thefurther oscillations for the stop of the crane boom 119861119874
2
slewing motion are the deviation of the real trajectory fromthe intended final position of load119872 The results of physicalsimulation in Figures 2 4 and 5 show the necessity of add-on
10 Shock and Vibration
N
ℬ
ℬ
e2
e1
ℰ
ℰ
ℱ
ℱ
y2
x2
Ve
Vr
M
E
1ẽ
2ẽ x
x1
O2(D)
AoAst
O
y y1
Adyn
ydyn
1205722
120579
120579
120596e = 120596ℬℰ = 120596ℱℰ
120593e = 120596et
Φcor
B(O1)
Φen
Φeτ
1e
2e
Figure 9 Swaying scheme of load119872 at the cable119872119861 which is fixedly attached in the point 119861 of the crane boom 1198611198742 in the horizontal plane
(11990911199101) for the nonlinear model derivation in Cartesian coordinates
devices development for the efficient suppression of load 119872
final oscillations
8 Discussion and Comparison of Derived andKnown Published Results
In Figure 1 and pp 537-538 of publishedwork by Sakawa et al1981 [63] the small angle between payloadrsquos cable and verticalline has been introduced that confirms proposed nonlinearDAE-based system (1) and linearized model (H4)-(H5)
In Figure 9 of p 278 of published work by Maczynskiand Wojciech 2003 [44] the computational finite elementmethod (FEM)-based results were shown for the abso-lute payload trajectories in the inertial reference frame EMaczynskirsquos Figure 9 in [44] outlines that the angle betweenthe payloadrsquos cable and the vertical line does not exceed 01rad and that agreeswith the proposed linearizedmodel (H4)-(H5) for a small angle assumption Computational trajectoryfor Maczynski-derived small load swaying [44] after craneboom stop qualitatively coincides with the experimentallyobservable load119872motion in Figures 2 4 and 5
In Figure 6 of published work by Ju et al 2006 [34]it was shown that the angle between the payloadrsquos cableand the vertical line does not exceed 16∘ Jursquos formula (15a)in p 382 of [34] assumes that Jursquos angle between payloadrsquoscable and the vertical line has the harmonic law of variationwith an introduction of a small perturbation term Both Jursquosassumptions in [34] confirm the proposed linearized model(H4)-(H5)
Comparison of the derived linearized model (H4)-(H5)with Figure 5 of published work by Mitrev and Grigorov2008 in [49] shows that the Mitrev-derived ranges of payloadswaying angles within Mitrevrsquos nonlinear model does notexceed 48∘ Such small values of swaying angles completelyconfirm the applicability and correctness of the small angleassumption in (H4)-(H5)
9 Final Conclusions
A governing nonlinear DAE-based system for crane boomuniform slewing has been proposed numerically solved andexperimentally verified
Shock and Vibration 11
Fully nonlinear differential equations for 3 DOF sphericalpendulum oscillations with a uniform slewing crane boomaround a fixed vertical axis of rotation have been derivedin relative Cartesian and spherical coordinates The identicallinearized differential equations in relative Cartesian coordi-nates have been derived with an introduction of the Coriolisdynamic theorem and Lagrange equations for a uniformcrane boomrotation and small swaying angle120572
1assumptions
Linearized andnonlinear theoretical problems for relativeswaying of the payload have been formulated in the form ofthe initial value (Cauchy) problem
An analytical solution of the linearized system has beenderived
The influences of inertia forces from the centripetal andcompound accelerations have been estimated
A linearized conservative system which contains firsttime derivatives of coordinates and no damping of oscilla-tions has been derived
The amplitudes of load oscillations which depend oncomputed initial conditions have been estimated within asmall angle assumption The dependence of natural frequen-cies on the transport inertia forces and gravity forces has beencomputed for the linearized systems
The formulae for the association of relative payloadmotion in the noninertial reference frame F and absolutepayload motion in the inertial reference frame E have beenoutlined
The results of the numerical DAE-based investigation andthe performed physical simulation show a satisfactory fit forfrequencies and amplitudes of load oscillations
The dynamic analogy between crane boom-driven pay-load swaying motion and Foucaultrsquos pendulum motion hasbeen grounded and outlined
The results of the present work are the foundation forfurther investigations of payload119872 swaying dynamics duringtelescopic crane boom nonuniform slewing motion withvariable cable length and for the different motions of thependulum pivot center
Appendices
A Velocity Kinematics Analysis inCartesian Coordinates
For the nonlinear problem definition we study the coopera-tive motion of the mechanical system ldquocrane boom 119861119874
2-load
119872rdquo which is shown in Figures 8ndash10In the nomenclature chapter we denote the fixed inertial
frame of reference E as 119909211991021199112 and the moving noninertial
frame of reference B as 119909111991011199111 which is rigidly bounded
with the crane boom1198611198742 Rotation of themoving noninertial
frame of referenceB(119909111991011199111) around the fixed inertial frame
of referenceE(119909211991021199112) defines the transportation motion for
payload 119872 The motion of load 119872 relative to the movingnoninertial frame of referenceB(119909
111991011199111) defines the relative
motion of payload 119872 The point 119860 st with the coordinates1199091= 0 119910
1= 0 and 119911
1= 0 is the steady equilibrium position
for the load119872 when the crane boom 1198611198742is fixed The point
119860dyn with the coordinates 1199091 = 0 1199101= 119910dyn and 1199111 = Δ is the
dynamic equilibrium position for the load119872 for the rotatingcrane boom 119861119874
2 We will assume the point119860dyn as the origin
of the noninertial reference frameF(119909119910119911) The directions ofaxes 119909 119910 119911 of the noninertial reference frameF are parallelto the axes 119909
1 1199101 1199111of the noninertial reference frameB in
Figures 8ndash10 The load 119872 relative motion takes place alongthe sphere surface with the fixed radius equal to the cablelength 119861119872 = 119897
The computational scheme in Figures 8ndash10 for the nonlin-ear model derivation may be described with an introductionof three degrees of freedom For generalized coordinates weassume the rectangular coordinates 119909
1 1199101 and 119911
1of the load
119872 in the moving noninertial frame of reference B and theangle 120593
119890of crane boom 119861119874
2slewing in the horizontal plane
(11990911199101) with the angular velocity 120596
119890around the vertical axis
11987421199112The relative velocity vector Vr = V
119872B of the load 119872 isdefined as Vr = V
119872B = (1198891199091119889119905 119889119910
1119889119905 119889119911
1119889119905)
In order to derive the absolute velocity V119872E of payload
119872 we will apply the vector method for absolute motionassignment We will define r
1198742119872
as the position vector con-necting initial point 119874
2and terminal point119872 in noninertial
reference frameB (119898) consider
r1198742119872
= r1198742119861+ r119861119872
(A1)
where vector components in (1) are defined in nomenclatureIn accordance with Figures 8ndash10 we have the following
unit vectorsrsquo expansions for position vectors r1198742119861
and r119861119872
in (A1)
r1198742119861
= (119877 cos (120593119890)) e1+ (119877 sin (120593
119890)) e2+ (119877 tan (120574)) e
3
(A2)
r119861119872
= 1199091e1+ 1199101e2+ (1199111minus 119897) e3 (A3)
where 119877 120593119890 1199091 and 119910
1notations have been defined in
nomenclature and in Figures 8ndash10For further definition of the position vector r
1198742119872
in iner-tial reference frame E we write unit vectors of noninertialreference frameB in (A3) through the unit vectors of inertialreference frame E
e1= (sin (120593
119890)) e1minus (cos (120593
119890)) e2 (A4)
e2= (cos (120593
119890)) e1minus (sin (120593
119890)) e2 (A5)
e3= e3 (A6)
After substitution of formulae (A2)ndash(A6) in (A1) andsome algebraic manipulations the vector expression (A1) forthe position vector r
1198742119872
in the inertial reference frameEwilltake the following form
r1198742119872
= ((119877 + 1199101) cos (120593
119890) + 1199091sin (120593
119890)) e1
+ ((119877 + 1199101) sin (120593
119890) minus 1199091cos (120593
119890)) e2
+ (119877 tan (120574) + 1199111minus 119897) e3
(A7)
12 Shock and Vibration
The absolute velocity of payload 119872 we define by timedifferentiation of (A7) assuming constancy of unit vectorse1 e2 and e
3of the inertial reference frameE and constancy
119877 and 120574
V119872E = Vabs =
119889 (r1198742119872
)
119889119905
(A8)
After differentiation and algebraic transformations in(A7) and (A8) we have the following vector expression forabsolute payload 119872 velocity in the inertial reference frameE
V119872E
= (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) cos (120593119890)
+ ((
1198891199091
119889119905
) minus (119877 + 1199101) (
119889120593119890
119889119905
)) sin (120593119890)) e1
+ (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) sin (120593119890)
+ ((119877 + 1199101) (
119889120593119890
119889119905
) minus (
1198891199091
119889119905
)) cos (120593119890))
times e2+ (
1198891199111
119889119905
) e3
(A9)
The square of absolute payload119872 velocitymay be writtenas
1198812
abs = (
1198891199091
119889119905
)
2
+ (
1198891199101
119889119905
)
2
+ (
1198891199111
119889119905
)
2
+ (
119889120593119890
119889119905
)
2
times (1199092
1+ (1199101+ 119877)2
) + 2(
119889120593119890
119889119905
)
times (1199091(
1198891199101
119889119905
) minus (1199101+ 119877)(
1198891199091
119889119905
))
(A10)
After algebraic transformations we have the square ofabsolute payload119872 velocity in the form of
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A11)
Algebraic expressions (A10) and (A11) could be derivedon the basis of a velocity addition theorem for payload 119872
compound motion in the inertial reference frame E throughthe unit vectors of the noninertial reference frame B (see[59 76 81 82 84 85 87 90ndash92 94ndash98])
V119872E = V
119872B + V1198742E + 120596BE times r
1198742119872
(A12)
where payload relative velocity in the noninertial referenceframeBmay be written as
V119872B = Vr = (
1198891199091
119889119905
) e1+ (
1198891199101
119889119905
) e2+ (
1198891199111
119889119905
) e3
(A13)
The velocityV1198742E of the point119874
2= point119864 in the inertial
reference frame E has zero value
V1198742E = 0 (A14)
The last term in (A12) is the vector product 120596BE times
r1198742119872
where crane boom angular velocity vector is 120596BE =
(119889120593119890119889119905)e3(A6) and the position vector r
1198742119872
in the nonin-ertial reference frameB is as follows
r1198742119872
= 1199091e1+ (1199101+ 119877) e
2+ 1199111e3 (A15)
Taking into account (A13)ndash(A15) the vector equation(A12) in the inertial reference frame E written through theunit vectors e
1 e2 e3of the noninertial reference frame B
will have the following form
V119872E = ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101)) e1
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091) e2+ (
1198891199111
119889119905
) e3
(A16)
So on the basis of (A16) we have the following square ofabsolute payload119872 velocity as
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A17)
Independently derived expressions for the scalar prod-uct (V
119872EV119872E) in (A11) and (A17) completely coincidewhich confirms the correctness and accuracy of the pen-dulum absolute velocity V
119872E derivation and shows thatthe scalar product (V
119872EV119872E) is the invariant expressionindependent of choice of reference frame
The second and third terms of (A12) in the noninertialreference frame B determine the vector of the load 119872
transportation velocity as
Ve = V1198742E + 120596BE times r
1198742119872
(A18)
Taking into account (A14) the scalar of the load 119872
transportation velocity is defined as
1003817100381710038171003817Ve
1003817100381710038171003817= 120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817 (A19)
Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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Shock and Vibration
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Navigation and Observation
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DistributedSensor Networks
International Journal of
6 Shock and Vibration
(a) (b)
(c) (d)
Figure 3 Photos of dimension experimental measurement system (a)ndash(d) for payload 119872 swaying motion during crane boom uniformslewing
]2
= 119896 minus 120596119890= 3670 rads 119862
1= 001798m and 119862
3=
002106m (Figure 5(a))For further estimation of the relative disagreement of
derived DAE-solution based computational (- - -) andexperimental (mdash) payload 119872 absolute trajectories in theinertial reference frame E we have computed the confidenceintervals in Figures 4 and 5 for the dimensionless parameters119903comp119903exper and confidence probability 095 The Studentrsquos 119905-test results yield the following confidence intervals 09374 le
(119903comp119903exper) le 10315 for 119897 = 0206m in Figure 4(a)09014 le (119903comp119903exper) le 10272 for 119897 = 0412m inFigure 4(b) 09200 le (119903comp119903exper) le 10352 for 119897 = 0618min Figure 5(a) and 09257 le (119903comp119903exper) le 10066 for119897 = 0825m in Figure 5(b)
Both the relative discrepancy 120575 and the confidence inter-vals show the satisfactory agreement between the absolute
payload trajectories (Figures 4 and 5) in the inertial referenceframeE that have been computed with DAE-solution basedtheoreticalmodel (- - -) (1) andmeasured experimentally (mdash)as shown in Figures 4 and 5
It is important to note that the payload motion DAEequations in the nonlinear problem (1) for the noninertialreference frame B may be derived with an introductionof Lagrange equations (Appendices AndashC F) However thediscussion of (1) is more suitable and informative with anintroduction of dynamic Coriolis theorem (Appendices AndashD F) The first terms 1198892119909
11198891199052 1198892119910
11198891199052 in (1) define the
vector a119872B = ar (Appendix B formula (B4)) for the relative
acceleration of load 119872 in the noninertial reference frameB The straight-line terms (D3) formula-based terms in (1)are linearly proportional to the relative payload coordinatesin the noninertial reference frame B These straight-line
Shock and Vibration 7
05
05
04
04
03
03
02
02
00
00
01
01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)
p O2 (p D)
(a)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)Computational y2 = y2(x2)
l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)
p O2 (p D)
(b)
Figure 4 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0206m (a) and fixed cable length 119897 = 0412m (b)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)
2R = 1 (m)
y2 (m)
x2 (m)p O2 (p D)
(a)
05
04
03
02
00
01
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448(rads)
2R = 1 (m)
x2 (m)
y2 (m)
p O2 (p D)
(b)
Figure 5 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0618m (a) and fixed cable length 119897 = 0825m (b)
terms in (1) have been determined by the contribution ofthe normal or centripetal acceleration 120596BE times (120596BE times
r119864119872
) = aen (B6) of transportation for payload119872 and by theappearance of corresponding DrsquoAlembert centrifugal inertiaforce Φe
n= (minus119898)aen (D3) due to crane boom transport
rotation in the noninertial reference frame B Additionaltermsminus2120596
119890(1198891199101119889119905) and 2120596
119890(1198891199091119889119905) in (1) have been defined
by the compound or Coriolis acceleration 2120596BE times V119872B =
acor of payload119872 in the noninertial reference frameB Therectangular Cartesian projections of Coriolis inertia force inthe noninertial reference frameB is defined by formula (B7)
5 Experimental Results for the SphericalPendulum with the Fixed Pivot Center
Experimental load swaying results for the fixed crane boommodel (120596
119890= 0 rads) are shown in Figure 6
Finite motions of a swaying load are shown in Figure 6in the case of 120596
119890= 0 rads In Figure 6 the relative and
absolute load trajectories are identically equal The lingeringremains of elliptical motion in Figure 6 and ellipses semiaxesare defined by the initial conditions
6 Discussion of Analogy between PayloadSwaying and Foucaultrsquos Pendulum Motion
The motion of Foucaultrsquos pendulum is shown in Figure 7where 119909
211991021199112is the heliocentric inertial reference frame
(not shown in Figure 7) 119909111991011199111is the geocentric noninertial
reference frame and 119909119910119911 is a noninertial reference framelocated at the geographic latitude 119910
1119911 119860 st119911 is a local vertical
119861119863 is the radius of the circle of the pinning point 119861119872 isthe cable length and 120596
119890is the angular velocity of the Earth
diurnal rotation
8 Shock and Vibration
(a) (b)
Figure 6 Experimental trajectories of the spherical pendulum with fixed pivot center and fixed cable length 119897 = 0825m (a b)
M
g
yx
z
B
D
x1
Ast
120596e
z1
y1
Figure 7 Foucaultrsquos pendulum schematic view
It is important to note that the structure of linearizedequations (H4)-(H5) for the swaying payload during craneboom uniform rotation in the noninertial reference frameF is analogous to the structure of the governing equationsfor Foucaultrsquos pendulum motion in the known publishedworks by Zhuravlev and Petrov [100] (formula (6) p 36 of[100]) Pardy [88] (formulae (12)ndash(14) pp 850-851 of [88])and Condurache et al [77] (formulae (11)ndash(13) p 743-744of [77]) The above mentioned analogy between linearizedequations (H4)-(H5) and the governing equations in [77 88100] assumes the geometric analogy of relative swaying tra-jectories between Foucaultrsquos pendulum and the boom-drivenpendulum with rotating pivot centers (Figures 1(e) 12 and16) So there is the geometric analogy between computationalrelative trajectories of load 119872 swaying during crane boomuniform slewing (Figures 1(e) 12 and 16) and relative trajec-tories of load119872 swaying in the plane (119860 st119909119910) around p119860 st inFigure 7 shownbyZhuravlev andPetrov [100] (Figure 4 p 39of [100]) and Condurache andMartinusi [77] (Figure 14 at p754 and Figure 16 at p 755 of [77]) The better understandingof Foucaultrsquos pendula dynamics is provided with the studyof computational relative trajectories in Figures 1(e) 12 and16 For Foucaultrsquos pendulum (Figure 7) the major semiaxes
of relative amplitude extremes have angular velocity valuesequal to the angular velocity of the Earth diurnal rotationMoreover for Foucaultrsquos pendulum (Figure 7) the directionof rotation of major semiaxes of relative amplitude extremesis oppositely directed to the direction of the Earth diurnalrotation All the above mentioned means that the plane(119860 st119861119872) of Foucaultrsquos pendula swaying remains fixed in theheliocentric inertial reference frame 119909
211991021199112
The presented analogy between crane boom-driven pay-load swaying and Foucaultrsquos pendulum motion essentiallyincreases knowledge and awareness of the oscillation pro-cesses in such dynamic systems
7 Discussion and MechanicalInterpretation of GoverningEquations for DAE-Based Nonlinearand Linearized Models
A governing nonlinear DAE-based system (1) for crane boomuniform slewing has been proposed and numerically solvedA conservative system (1) and (H4)-(H5) with componentsΦcor119909 andΦcor119910 has been derived Both projectionsΦcor119909 andΦcor119910 coincide in the noninertial reference framesB andF
The occurrence of the first derivatives 119889119909119889119905 119889119910119889119905 ofthe load119872 relative coordinates with respect to time in termsminus2120596119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) determines that there is no
decay of the oscillations in (H4)-(H5) and (1) but onlyredirection of the relative velocity vector V
119872F = Vr of theload119872 in the noninertial reference framesB andF
The introduction of a linearized model (H4)-(H5)allowed the determination of the natural frequencies (]
1=
]2) of free oscillations of payload 119872 ]
1= 119896 + 120596
119890and
]2= 119896 minus 120596
119890
It follows from the forms of the relative payload trajectory119910 = 119910(119909) in the noninertial reference frame F and theabsolute trajectory 119910
2= 1199102(1199092) in the inertial reference frame
E that the resulting motion of the payload 119872 on the cable119872119861 taking into account the Coriolis inertia forceΦcor (D5)will be the sum of two oscillations with natural frequencies ]
1
and ]2 and with periods 119879
1= 2120587]
1and 119879
2= 2120587]
2
It is worth noting that the frequencies ]1and ]
2differ
by 2120596119890 This means that for the small angle assumption
(Appendices H-I) we have ]1
asymp ]2and the trajectory
Shock and Vibration 9
mg
γ
Nℬ
z1z2
e2
e3
ℰ
ℰ
ℰ
ℱ
ℱ
ℬΔ
120574
y1
Vr
At
M
R D
E
Bz
l
120573
O
1205721
1205721dyn
Adyn
ydyn
h
120596e
AstO1
3ẽ
2ẽ
O2
d120593edty
sin(120593e)
cos(120593e)
120596ℬℰ = 120596ℱℰ
3e
2e
1e
Figure 8 Swaying scheme of load119872 on the cable119872119861 which is fixedly attached at the point 119861 on the crane boom 1198611198742 in the vertical plane
(119910119911) for the nonlinear model derivation in Cartesian coordinates
of relative motion 119910 = 119910(119909) in the noninertial referenceframe F on the expiration of the relative oscillations periodtime 2120587119896 looks approximately like an ellipse (Figures 12(b)and 16(b)) with semiaxes which are governed by the initialconditions
The major semiaxis of such an ellipse in Figures 12(b)and 16(b) is defined by the initial velocity of load 119872 in thenoninertial reference frameF and the analytical value of themajor semiaxis is equal to119881
119861119896 = (120596
119890119877)119896 The derived value
of 119881119861119896 is approximately the sum of the absolute values of
the amplitudes in the analytical form (I15) of the linearizedCauchy problem (H4)-(H5)
The minor semiaxis of the above mentioned ellipsein Figures 12(b) and 16(b) is defined by the value 119910dynwhich is approximately the difference of amplitudes in theanalytical form (I15) of the linearizedCauchy problem (H4)-(H5) In this case the major semiaxis is approximately 5ndash20 times larger than the minor semiaxis because the angularvelocity 120596
119890of the transport rotation of crane boom 119861119874
2is
much smaller than the natural frequency 119896 of the sphericaloscillations of payload119872 on the cable119872119861 (Figures 12(b) and16(b))
It is also necessary to note that the major semiaxis of theellipse in the relative motion rotates with an angular velocity120596119890in the opposite direction of the transport rotation of crane
boom 1198611198742in (Figures 12 and 16)
It follows from Figures 1(f) 2 4 and 5 that the trajectoryof absolute motion of load 119872 in the inertial reference frameE is almost a symmetric curve with 119910
2axial symmetry
The initial and the final motions of load 119872 for half-period
essentially differ from its harmonic oscillations neighbor forquarter-period
Due to the negligible quantity of 120572dyn and 119910dyn (G3)ndash(G5) the average deviation of the load119872 from the mechan-ical trajectory of the boom tip 119861 is negligible (Figures1(f) 2 4 and 5) The basic dynamic load on the systemldquocrane boom 119861119874
2-load 119872rdquo is created by the high-frequency
oscillations of load 119872 which are determined by the actionof inertia force Φcor = minus2119898(120596BE times V
119872B) stipulated bythe Coriolis acceleration acor = 2120596BE times V
119872B of load 119872
in the noninertial reference frame F The basic dynamicload in the system defines additional loads and vibrations ofcrane boom 119861119874
2mechanical elements and support bearings
complicates the automatic and manual control systems of theelectromechanical crane boom 119861119874
2 and also makes crane
operation much more difficultIt is also important to note that the stop of crane boom
1198611198742does not lead to instantaneous damping of the load
119872 absolute oscillations in the inertial reference frame E(Figures 2 4 and 5) This phenomenon directly follows fromthe experimental trajectory in Figures 2 4 and 5 Alsothe natural spherical oscillations of load 119872 will occur withthe frequency 119896 and the amplitude (the difference betweenthe final position of load 119872 in the relative motion in thenoninertial reference frame F for half-period of vibrationand static equilibrium 119860 st of load 119872 on the cable) Thefurther oscillations for the stop of the crane boom 119861119874
2
slewing motion are the deviation of the real trajectory fromthe intended final position of load119872 The results of physicalsimulation in Figures 2 4 and 5 show the necessity of add-on
10 Shock and Vibration
N
ℬ
ℬ
e2
e1
ℰ
ℰ
ℱ
ℱ
y2
x2
Ve
Vr
M
E
1ẽ
2ẽ x
x1
O2(D)
AoAst
O
y y1
Adyn
ydyn
1205722
120579
120579
120596e = 120596ℬℰ = 120596ℱℰ
120593e = 120596et
Φcor
B(O1)
Φen
Φeτ
1e
2e
Figure 9 Swaying scheme of load119872 at the cable119872119861 which is fixedly attached in the point 119861 of the crane boom 1198611198742 in the horizontal plane
(11990911199101) for the nonlinear model derivation in Cartesian coordinates
devices development for the efficient suppression of load 119872
final oscillations
8 Discussion and Comparison of Derived andKnown Published Results
In Figure 1 and pp 537-538 of publishedwork by Sakawa et al1981 [63] the small angle between payloadrsquos cable and verticalline has been introduced that confirms proposed nonlinearDAE-based system (1) and linearized model (H4)-(H5)
In Figure 9 of p 278 of published work by Maczynskiand Wojciech 2003 [44] the computational finite elementmethod (FEM)-based results were shown for the abso-lute payload trajectories in the inertial reference frame EMaczynskirsquos Figure 9 in [44] outlines that the angle betweenthe payloadrsquos cable and the vertical line does not exceed 01rad and that agreeswith the proposed linearizedmodel (H4)-(H5) for a small angle assumption Computational trajectoryfor Maczynski-derived small load swaying [44] after craneboom stop qualitatively coincides with the experimentallyobservable load119872motion in Figures 2 4 and 5
In Figure 6 of published work by Ju et al 2006 [34]it was shown that the angle between the payloadrsquos cableand the vertical line does not exceed 16∘ Jursquos formula (15a)in p 382 of [34] assumes that Jursquos angle between payloadrsquoscable and the vertical line has the harmonic law of variationwith an introduction of a small perturbation term Both Jursquosassumptions in [34] confirm the proposed linearized model(H4)-(H5)
Comparison of the derived linearized model (H4)-(H5)with Figure 5 of published work by Mitrev and Grigorov2008 in [49] shows that the Mitrev-derived ranges of payloadswaying angles within Mitrevrsquos nonlinear model does notexceed 48∘ Such small values of swaying angles completelyconfirm the applicability and correctness of the small angleassumption in (H4)-(H5)
9 Final Conclusions
A governing nonlinear DAE-based system for crane boomuniform slewing has been proposed numerically solved andexperimentally verified
Shock and Vibration 11
Fully nonlinear differential equations for 3 DOF sphericalpendulum oscillations with a uniform slewing crane boomaround a fixed vertical axis of rotation have been derivedin relative Cartesian and spherical coordinates The identicallinearized differential equations in relative Cartesian coordi-nates have been derived with an introduction of the Coriolisdynamic theorem and Lagrange equations for a uniformcrane boomrotation and small swaying angle120572
1assumptions
Linearized andnonlinear theoretical problems for relativeswaying of the payload have been formulated in the form ofthe initial value (Cauchy) problem
An analytical solution of the linearized system has beenderived
The influences of inertia forces from the centripetal andcompound accelerations have been estimated
A linearized conservative system which contains firsttime derivatives of coordinates and no damping of oscilla-tions has been derived
The amplitudes of load oscillations which depend oncomputed initial conditions have been estimated within asmall angle assumption The dependence of natural frequen-cies on the transport inertia forces and gravity forces has beencomputed for the linearized systems
The formulae for the association of relative payloadmotion in the noninertial reference frame F and absolutepayload motion in the inertial reference frame E have beenoutlined
The results of the numerical DAE-based investigation andthe performed physical simulation show a satisfactory fit forfrequencies and amplitudes of load oscillations
The dynamic analogy between crane boom-driven pay-load swaying motion and Foucaultrsquos pendulum motion hasbeen grounded and outlined
The results of the present work are the foundation forfurther investigations of payload119872 swaying dynamics duringtelescopic crane boom nonuniform slewing motion withvariable cable length and for the different motions of thependulum pivot center
Appendices
A Velocity Kinematics Analysis inCartesian Coordinates
For the nonlinear problem definition we study the coopera-tive motion of the mechanical system ldquocrane boom 119861119874
2-load
119872rdquo which is shown in Figures 8ndash10In the nomenclature chapter we denote the fixed inertial
frame of reference E as 119909211991021199112 and the moving noninertial
frame of reference B as 119909111991011199111 which is rigidly bounded
with the crane boom1198611198742 Rotation of themoving noninertial
frame of referenceB(119909111991011199111) around the fixed inertial frame
of referenceE(119909211991021199112) defines the transportation motion for
payload 119872 The motion of load 119872 relative to the movingnoninertial frame of referenceB(119909
111991011199111) defines the relative
motion of payload 119872 The point 119860 st with the coordinates1199091= 0 119910
1= 0 and 119911
1= 0 is the steady equilibrium position
for the load119872 when the crane boom 1198611198742is fixed The point
119860dyn with the coordinates 1199091 = 0 1199101= 119910dyn and 1199111 = Δ is the
dynamic equilibrium position for the load119872 for the rotatingcrane boom 119861119874
2 We will assume the point119860dyn as the origin
of the noninertial reference frameF(119909119910119911) The directions ofaxes 119909 119910 119911 of the noninertial reference frameF are parallelto the axes 119909
1 1199101 1199111of the noninertial reference frameB in
Figures 8ndash10 The load 119872 relative motion takes place alongthe sphere surface with the fixed radius equal to the cablelength 119861119872 = 119897
The computational scheme in Figures 8ndash10 for the nonlin-ear model derivation may be described with an introductionof three degrees of freedom For generalized coordinates weassume the rectangular coordinates 119909
1 1199101 and 119911
1of the load
119872 in the moving noninertial frame of reference B and theangle 120593
119890of crane boom 119861119874
2slewing in the horizontal plane
(11990911199101) with the angular velocity 120596
119890around the vertical axis
11987421199112The relative velocity vector Vr = V
119872B of the load 119872 isdefined as Vr = V
119872B = (1198891199091119889119905 119889119910
1119889119905 119889119911
1119889119905)
In order to derive the absolute velocity V119872E of payload
119872 we will apply the vector method for absolute motionassignment We will define r
1198742119872
as the position vector con-necting initial point 119874
2and terminal point119872 in noninertial
reference frameB (119898) consider
r1198742119872
= r1198742119861+ r119861119872
(A1)
where vector components in (1) are defined in nomenclatureIn accordance with Figures 8ndash10 we have the following
unit vectorsrsquo expansions for position vectors r1198742119861
and r119861119872
in (A1)
r1198742119861
= (119877 cos (120593119890)) e1+ (119877 sin (120593
119890)) e2+ (119877 tan (120574)) e
3
(A2)
r119861119872
= 1199091e1+ 1199101e2+ (1199111minus 119897) e3 (A3)
where 119877 120593119890 1199091 and 119910
1notations have been defined in
nomenclature and in Figures 8ndash10For further definition of the position vector r
1198742119872
in iner-tial reference frame E we write unit vectors of noninertialreference frameB in (A3) through the unit vectors of inertialreference frame E
e1= (sin (120593
119890)) e1minus (cos (120593
119890)) e2 (A4)
e2= (cos (120593
119890)) e1minus (sin (120593
119890)) e2 (A5)
e3= e3 (A6)
After substitution of formulae (A2)ndash(A6) in (A1) andsome algebraic manipulations the vector expression (A1) forthe position vector r
1198742119872
in the inertial reference frameEwilltake the following form
r1198742119872
= ((119877 + 1199101) cos (120593
119890) + 1199091sin (120593
119890)) e1
+ ((119877 + 1199101) sin (120593
119890) minus 1199091cos (120593
119890)) e2
+ (119877 tan (120574) + 1199111minus 119897) e3
(A7)
12 Shock and Vibration
The absolute velocity of payload 119872 we define by timedifferentiation of (A7) assuming constancy of unit vectorse1 e2 and e
3of the inertial reference frameE and constancy
119877 and 120574
V119872E = Vabs =
119889 (r1198742119872
)
119889119905
(A8)
After differentiation and algebraic transformations in(A7) and (A8) we have the following vector expression forabsolute payload 119872 velocity in the inertial reference frameE
V119872E
= (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) cos (120593119890)
+ ((
1198891199091
119889119905
) minus (119877 + 1199101) (
119889120593119890
119889119905
)) sin (120593119890)) e1
+ (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) sin (120593119890)
+ ((119877 + 1199101) (
119889120593119890
119889119905
) minus (
1198891199091
119889119905
)) cos (120593119890))
times e2+ (
1198891199111
119889119905
) e3
(A9)
The square of absolute payload119872 velocitymay be writtenas
1198812
abs = (
1198891199091
119889119905
)
2
+ (
1198891199101
119889119905
)
2
+ (
1198891199111
119889119905
)
2
+ (
119889120593119890
119889119905
)
2
times (1199092
1+ (1199101+ 119877)2
) + 2(
119889120593119890
119889119905
)
times (1199091(
1198891199101
119889119905
) minus (1199101+ 119877)(
1198891199091
119889119905
))
(A10)
After algebraic transformations we have the square ofabsolute payload119872 velocity in the form of
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A11)
Algebraic expressions (A10) and (A11) could be derivedon the basis of a velocity addition theorem for payload 119872
compound motion in the inertial reference frame E throughthe unit vectors of the noninertial reference frame B (see[59 76 81 82 84 85 87 90ndash92 94ndash98])
V119872E = V
119872B + V1198742E + 120596BE times r
1198742119872
(A12)
where payload relative velocity in the noninertial referenceframeBmay be written as
V119872B = Vr = (
1198891199091
119889119905
) e1+ (
1198891199101
119889119905
) e2+ (
1198891199111
119889119905
) e3
(A13)
The velocityV1198742E of the point119874
2= point119864 in the inertial
reference frame E has zero value
V1198742E = 0 (A14)
The last term in (A12) is the vector product 120596BE times
r1198742119872
where crane boom angular velocity vector is 120596BE =
(119889120593119890119889119905)e3(A6) and the position vector r
1198742119872
in the nonin-ertial reference frameB is as follows
r1198742119872
= 1199091e1+ (1199101+ 119877) e
2+ 1199111e3 (A15)
Taking into account (A13)ndash(A15) the vector equation(A12) in the inertial reference frame E written through theunit vectors e
1 e2 e3of the noninertial reference frame B
will have the following form
V119872E = ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101)) e1
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091) e2+ (
1198891199111
119889119905
) e3
(A16)
So on the basis of (A16) we have the following square ofabsolute payload119872 velocity as
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A17)
Independently derived expressions for the scalar prod-uct (V
119872EV119872E) in (A11) and (A17) completely coincidewhich confirms the correctness and accuracy of the pen-dulum absolute velocity V
119872E derivation and shows thatthe scalar product (V
119872EV119872E) is the invariant expressionindependent of choice of reference frame
The second and third terms of (A12) in the noninertialreference frame B determine the vector of the load 119872
transportation velocity as
Ve = V1198742E + 120596BE times r
1198742119872
(A18)
Taking into account (A14) the scalar of the load 119872
transportation velocity is defined as
1003817100381710038171003817Ve
1003817100381710038171003817= 120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817 (A19)
Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
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[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
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[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
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[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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International Journal of
Shock and Vibration 7
05
05
04
04
03
03
02
02
00
00
01
01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)
p O2 (p D)
(a)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
2R = 1 (m)
y2 (m)
x2 (m)
Experimental y2 = y2(x2)Computational y2 = y2(x2)
l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)
p O2 (p D)
(b)
Figure 4 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0206m (a) and fixed cable length 119897 = 0412m (b)
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
05
04
03
02
00
01
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)
2R = 1 (m)
y2 (m)
x2 (m)p O2 (p D)
(a)
05
04
03
02
00
01
Experimental y2 = y2(x2)
Computational y2 = y2(x2)
0504030200 01minus05 minus04 minus03 minus02 minus01
l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448(rads)
2R = 1 (m)
x2 (m)
y2 (m)
p O2 (p D)
(b)
Figure 5 The comparison of the small swaying angle 1205721theoretical (- - - 119910
2= 1199102(1199092)) and experimental (mdash 119910
2= 1199102(1199092)) load absolute
trajectories in the inertial reference frameE for fixed cable length 119897 = 0618m (a) and fixed cable length 119897 = 0825m (b)
terms in (1) have been determined by the contribution ofthe normal or centripetal acceleration 120596BE times (120596BE times
r119864119872
) = aen (B6) of transportation for payload119872 and by theappearance of corresponding DrsquoAlembert centrifugal inertiaforce Φe
n= (minus119898)aen (D3) due to crane boom transport
rotation in the noninertial reference frame B Additionaltermsminus2120596
119890(1198891199101119889119905) and 2120596
119890(1198891199091119889119905) in (1) have been defined
by the compound or Coriolis acceleration 2120596BE times V119872B =
acor of payload119872 in the noninertial reference frameB Therectangular Cartesian projections of Coriolis inertia force inthe noninertial reference frameB is defined by formula (B7)
5 Experimental Results for the SphericalPendulum with the Fixed Pivot Center
Experimental load swaying results for the fixed crane boommodel (120596
119890= 0 rads) are shown in Figure 6
Finite motions of a swaying load are shown in Figure 6in the case of 120596
119890= 0 rads In Figure 6 the relative and
absolute load trajectories are identically equal The lingeringremains of elliptical motion in Figure 6 and ellipses semiaxesare defined by the initial conditions
6 Discussion of Analogy between PayloadSwaying and Foucaultrsquos Pendulum Motion
The motion of Foucaultrsquos pendulum is shown in Figure 7where 119909
211991021199112is the heliocentric inertial reference frame
(not shown in Figure 7) 119909111991011199111is the geocentric noninertial
reference frame and 119909119910119911 is a noninertial reference framelocated at the geographic latitude 119910
1119911 119860 st119911 is a local vertical
119861119863 is the radius of the circle of the pinning point 119861119872 isthe cable length and 120596
119890is the angular velocity of the Earth
diurnal rotation
8 Shock and Vibration
(a) (b)
Figure 6 Experimental trajectories of the spherical pendulum with fixed pivot center and fixed cable length 119897 = 0825m (a b)
M
g
yx
z
B
D
x1
Ast
120596e
z1
y1
Figure 7 Foucaultrsquos pendulum schematic view
It is important to note that the structure of linearizedequations (H4)-(H5) for the swaying payload during craneboom uniform rotation in the noninertial reference frameF is analogous to the structure of the governing equationsfor Foucaultrsquos pendulum motion in the known publishedworks by Zhuravlev and Petrov [100] (formula (6) p 36 of[100]) Pardy [88] (formulae (12)ndash(14) pp 850-851 of [88])and Condurache et al [77] (formulae (11)ndash(13) p 743-744of [77]) The above mentioned analogy between linearizedequations (H4)-(H5) and the governing equations in [77 88100] assumes the geometric analogy of relative swaying tra-jectories between Foucaultrsquos pendulum and the boom-drivenpendulum with rotating pivot centers (Figures 1(e) 12 and16) So there is the geometric analogy between computationalrelative trajectories of load 119872 swaying during crane boomuniform slewing (Figures 1(e) 12 and 16) and relative trajec-tories of load119872 swaying in the plane (119860 st119909119910) around p119860 st inFigure 7 shownbyZhuravlev andPetrov [100] (Figure 4 p 39of [100]) and Condurache andMartinusi [77] (Figure 14 at p754 and Figure 16 at p 755 of [77]) The better understandingof Foucaultrsquos pendula dynamics is provided with the studyof computational relative trajectories in Figures 1(e) 12 and16 For Foucaultrsquos pendulum (Figure 7) the major semiaxes
of relative amplitude extremes have angular velocity valuesequal to the angular velocity of the Earth diurnal rotationMoreover for Foucaultrsquos pendulum (Figure 7) the directionof rotation of major semiaxes of relative amplitude extremesis oppositely directed to the direction of the Earth diurnalrotation All the above mentioned means that the plane(119860 st119861119872) of Foucaultrsquos pendula swaying remains fixed in theheliocentric inertial reference frame 119909
211991021199112
The presented analogy between crane boom-driven pay-load swaying and Foucaultrsquos pendulum motion essentiallyincreases knowledge and awareness of the oscillation pro-cesses in such dynamic systems
7 Discussion and MechanicalInterpretation of GoverningEquations for DAE-Based Nonlinearand Linearized Models
A governing nonlinear DAE-based system (1) for crane boomuniform slewing has been proposed and numerically solvedA conservative system (1) and (H4)-(H5) with componentsΦcor119909 andΦcor119910 has been derived Both projectionsΦcor119909 andΦcor119910 coincide in the noninertial reference framesB andF
The occurrence of the first derivatives 119889119909119889119905 119889119910119889119905 ofthe load119872 relative coordinates with respect to time in termsminus2120596119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) determines that there is no
decay of the oscillations in (H4)-(H5) and (1) but onlyredirection of the relative velocity vector V
119872F = Vr of theload119872 in the noninertial reference framesB andF
The introduction of a linearized model (H4)-(H5)allowed the determination of the natural frequencies (]
1=
]2) of free oscillations of payload 119872 ]
1= 119896 + 120596
119890and
]2= 119896 minus 120596
119890
It follows from the forms of the relative payload trajectory119910 = 119910(119909) in the noninertial reference frame F and theabsolute trajectory 119910
2= 1199102(1199092) in the inertial reference frame
E that the resulting motion of the payload 119872 on the cable119872119861 taking into account the Coriolis inertia forceΦcor (D5)will be the sum of two oscillations with natural frequencies ]
1
and ]2 and with periods 119879
1= 2120587]
1and 119879
2= 2120587]
2
It is worth noting that the frequencies ]1and ]
2differ
by 2120596119890 This means that for the small angle assumption
(Appendices H-I) we have ]1
asymp ]2and the trajectory
Shock and Vibration 9
mg
γ
Nℬ
z1z2
e2
e3
ℰ
ℰ
ℰ
ℱ
ℱ
ℬΔ
120574
y1
Vr
At
M
R D
E
Bz
l
120573
O
1205721
1205721dyn
Adyn
ydyn
h
120596e
AstO1
3ẽ
2ẽ
O2
d120593edty
sin(120593e)
cos(120593e)
120596ℬℰ = 120596ℱℰ
3e
2e
1e
Figure 8 Swaying scheme of load119872 on the cable119872119861 which is fixedly attached at the point 119861 on the crane boom 1198611198742 in the vertical plane
(119910119911) for the nonlinear model derivation in Cartesian coordinates
of relative motion 119910 = 119910(119909) in the noninertial referenceframe F on the expiration of the relative oscillations periodtime 2120587119896 looks approximately like an ellipse (Figures 12(b)and 16(b)) with semiaxes which are governed by the initialconditions
The major semiaxis of such an ellipse in Figures 12(b)and 16(b) is defined by the initial velocity of load 119872 in thenoninertial reference frameF and the analytical value of themajor semiaxis is equal to119881
119861119896 = (120596
119890119877)119896 The derived value
of 119881119861119896 is approximately the sum of the absolute values of
the amplitudes in the analytical form (I15) of the linearizedCauchy problem (H4)-(H5)
The minor semiaxis of the above mentioned ellipsein Figures 12(b) and 16(b) is defined by the value 119910dynwhich is approximately the difference of amplitudes in theanalytical form (I15) of the linearizedCauchy problem (H4)-(H5) In this case the major semiaxis is approximately 5ndash20 times larger than the minor semiaxis because the angularvelocity 120596
119890of the transport rotation of crane boom 119861119874
2is
much smaller than the natural frequency 119896 of the sphericaloscillations of payload119872 on the cable119872119861 (Figures 12(b) and16(b))
It is also necessary to note that the major semiaxis of theellipse in the relative motion rotates with an angular velocity120596119890in the opposite direction of the transport rotation of crane
boom 1198611198742in (Figures 12 and 16)
It follows from Figures 1(f) 2 4 and 5 that the trajectoryof absolute motion of load 119872 in the inertial reference frameE is almost a symmetric curve with 119910
2axial symmetry
The initial and the final motions of load 119872 for half-period
essentially differ from its harmonic oscillations neighbor forquarter-period
Due to the negligible quantity of 120572dyn and 119910dyn (G3)ndash(G5) the average deviation of the load119872 from the mechan-ical trajectory of the boom tip 119861 is negligible (Figures1(f) 2 4 and 5) The basic dynamic load on the systemldquocrane boom 119861119874
2-load 119872rdquo is created by the high-frequency
oscillations of load 119872 which are determined by the actionof inertia force Φcor = minus2119898(120596BE times V
119872B) stipulated bythe Coriolis acceleration acor = 2120596BE times V
119872B of load 119872
in the noninertial reference frame F The basic dynamicload in the system defines additional loads and vibrations ofcrane boom 119861119874
2mechanical elements and support bearings
complicates the automatic and manual control systems of theelectromechanical crane boom 119861119874
2 and also makes crane
operation much more difficultIt is also important to note that the stop of crane boom
1198611198742does not lead to instantaneous damping of the load
119872 absolute oscillations in the inertial reference frame E(Figures 2 4 and 5) This phenomenon directly follows fromthe experimental trajectory in Figures 2 4 and 5 Alsothe natural spherical oscillations of load 119872 will occur withthe frequency 119896 and the amplitude (the difference betweenthe final position of load 119872 in the relative motion in thenoninertial reference frame F for half-period of vibrationand static equilibrium 119860 st of load 119872 on the cable) Thefurther oscillations for the stop of the crane boom 119861119874
2
slewing motion are the deviation of the real trajectory fromthe intended final position of load119872 The results of physicalsimulation in Figures 2 4 and 5 show the necessity of add-on
10 Shock and Vibration
N
ℬ
ℬ
e2
e1
ℰ
ℰ
ℱ
ℱ
y2
x2
Ve
Vr
M
E
1ẽ
2ẽ x
x1
O2(D)
AoAst
O
y y1
Adyn
ydyn
1205722
120579
120579
120596e = 120596ℬℰ = 120596ℱℰ
120593e = 120596et
Φcor
B(O1)
Φen
Φeτ
1e
2e
Figure 9 Swaying scheme of load119872 at the cable119872119861 which is fixedly attached in the point 119861 of the crane boom 1198611198742 in the horizontal plane
(11990911199101) for the nonlinear model derivation in Cartesian coordinates
devices development for the efficient suppression of load 119872
final oscillations
8 Discussion and Comparison of Derived andKnown Published Results
In Figure 1 and pp 537-538 of publishedwork by Sakawa et al1981 [63] the small angle between payloadrsquos cable and verticalline has been introduced that confirms proposed nonlinearDAE-based system (1) and linearized model (H4)-(H5)
In Figure 9 of p 278 of published work by Maczynskiand Wojciech 2003 [44] the computational finite elementmethod (FEM)-based results were shown for the abso-lute payload trajectories in the inertial reference frame EMaczynskirsquos Figure 9 in [44] outlines that the angle betweenthe payloadrsquos cable and the vertical line does not exceed 01rad and that agreeswith the proposed linearizedmodel (H4)-(H5) for a small angle assumption Computational trajectoryfor Maczynski-derived small load swaying [44] after craneboom stop qualitatively coincides with the experimentallyobservable load119872motion in Figures 2 4 and 5
In Figure 6 of published work by Ju et al 2006 [34]it was shown that the angle between the payloadrsquos cableand the vertical line does not exceed 16∘ Jursquos formula (15a)in p 382 of [34] assumes that Jursquos angle between payloadrsquoscable and the vertical line has the harmonic law of variationwith an introduction of a small perturbation term Both Jursquosassumptions in [34] confirm the proposed linearized model(H4)-(H5)
Comparison of the derived linearized model (H4)-(H5)with Figure 5 of published work by Mitrev and Grigorov2008 in [49] shows that the Mitrev-derived ranges of payloadswaying angles within Mitrevrsquos nonlinear model does notexceed 48∘ Such small values of swaying angles completelyconfirm the applicability and correctness of the small angleassumption in (H4)-(H5)
9 Final Conclusions
A governing nonlinear DAE-based system for crane boomuniform slewing has been proposed numerically solved andexperimentally verified
Shock and Vibration 11
Fully nonlinear differential equations for 3 DOF sphericalpendulum oscillations with a uniform slewing crane boomaround a fixed vertical axis of rotation have been derivedin relative Cartesian and spherical coordinates The identicallinearized differential equations in relative Cartesian coordi-nates have been derived with an introduction of the Coriolisdynamic theorem and Lagrange equations for a uniformcrane boomrotation and small swaying angle120572
1assumptions
Linearized andnonlinear theoretical problems for relativeswaying of the payload have been formulated in the form ofthe initial value (Cauchy) problem
An analytical solution of the linearized system has beenderived
The influences of inertia forces from the centripetal andcompound accelerations have been estimated
A linearized conservative system which contains firsttime derivatives of coordinates and no damping of oscilla-tions has been derived
The amplitudes of load oscillations which depend oncomputed initial conditions have been estimated within asmall angle assumption The dependence of natural frequen-cies on the transport inertia forces and gravity forces has beencomputed for the linearized systems
The formulae for the association of relative payloadmotion in the noninertial reference frame F and absolutepayload motion in the inertial reference frame E have beenoutlined
The results of the numerical DAE-based investigation andthe performed physical simulation show a satisfactory fit forfrequencies and amplitudes of load oscillations
The dynamic analogy between crane boom-driven pay-load swaying motion and Foucaultrsquos pendulum motion hasbeen grounded and outlined
The results of the present work are the foundation forfurther investigations of payload119872 swaying dynamics duringtelescopic crane boom nonuniform slewing motion withvariable cable length and for the different motions of thependulum pivot center
Appendices
A Velocity Kinematics Analysis inCartesian Coordinates
For the nonlinear problem definition we study the coopera-tive motion of the mechanical system ldquocrane boom 119861119874
2-load
119872rdquo which is shown in Figures 8ndash10In the nomenclature chapter we denote the fixed inertial
frame of reference E as 119909211991021199112 and the moving noninertial
frame of reference B as 119909111991011199111 which is rigidly bounded
with the crane boom1198611198742 Rotation of themoving noninertial
frame of referenceB(119909111991011199111) around the fixed inertial frame
of referenceE(119909211991021199112) defines the transportation motion for
payload 119872 The motion of load 119872 relative to the movingnoninertial frame of referenceB(119909
111991011199111) defines the relative
motion of payload 119872 The point 119860 st with the coordinates1199091= 0 119910
1= 0 and 119911
1= 0 is the steady equilibrium position
for the load119872 when the crane boom 1198611198742is fixed The point
119860dyn with the coordinates 1199091 = 0 1199101= 119910dyn and 1199111 = Δ is the
dynamic equilibrium position for the load119872 for the rotatingcrane boom 119861119874
2 We will assume the point119860dyn as the origin
of the noninertial reference frameF(119909119910119911) The directions ofaxes 119909 119910 119911 of the noninertial reference frameF are parallelto the axes 119909
1 1199101 1199111of the noninertial reference frameB in
Figures 8ndash10 The load 119872 relative motion takes place alongthe sphere surface with the fixed radius equal to the cablelength 119861119872 = 119897
The computational scheme in Figures 8ndash10 for the nonlin-ear model derivation may be described with an introductionof three degrees of freedom For generalized coordinates weassume the rectangular coordinates 119909
1 1199101 and 119911
1of the load
119872 in the moving noninertial frame of reference B and theangle 120593
119890of crane boom 119861119874
2slewing in the horizontal plane
(11990911199101) with the angular velocity 120596
119890around the vertical axis
11987421199112The relative velocity vector Vr = V
119872B of the load 119872 isdefined as Vr = V
119872B = (1198891199091119889119905 119889119910
1119889119905 119889119911
1119889119905)
In order to derive the absolute velocity V119872E of payload
119872 we will apply the vector method for absolute motionassignment We will define r
1198742119872
as the position vector con-necting initial point 119874
2and terminal point119872 in noninertial
reference frameB (119898) consider
r1198742119872
= r1198742119861+ r119861119872
(A1)
where vector components in (1) are defined in nomenclatureIn accordance with Figures 8ndash10 we have the following
unit vectorsrsquo expansions for position vectors r1198742119861
and r119861119872
in (A1)
r1198742119861
= (119877 cos (120593119890)) e1+ (119877 sin (120593
119890)) e2+ (119877 tan (120574)) e
3
(A2)
r119861119872
= 1199091e1+ 1199101e2+ (1199111minus 119897) e3 (A3)
where 119877 120593119890 1199091 and 119910
1notations have been defined in
nomenclature and in Figures 8ndash10For further definition of the position vector r
1198742119872
in iner-tial reference frame E we write unit vectors of noninertialreference frameB in (A3) through the unit vectors of inertialreference frame E
e1= (sin (120593
119890)) e1minus (cos (120593
119890)) e2 (A4)
e2= (cos (120593
119890)) e1minus (sin (120593
119890)) e2 (A5)
e3= e3 (A6)
After substitution of formulae (A2)ndash(A6) in (A1) andsome algebraic manipulations the vector expression (A1) forthe position vector r
1198742119872
in the inertial reference frameEwilltake the following form
r1198742119872
= ((119877 + 1199101) cos (120593
119890) + 1199091sin (120593
119890)) e1
+ ((119877 + 1199101) sin (120593
119890) minus 1199091cos (120593
119890)) e2
+ (119877 tan (120574) + 1199111minus 119897) e3
(A7)
12 Shock and Vibration
The absolute velocity of payload 119872 we define by timedifferentiation of (A7) assuming constancy of unit vectorse1 e2 and e
3of the inertial reference frameE and constancy
119877 and 120574
V119872E = Vabs =
119889 (r1198742119872
)
119889119905
(A8)
After differentiation and algebraic transformations in(A7) and (A8) we have the following vector expression forabsolute payload 119872 velocity in the inertial reference frameE
V119872E
= (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) cos (120593119890)
+ ((
1198891199091
119889119905
) minus (119877 + 1199101) (
119889120593119890
119889119905
)) sin (120593119890)) e1
+ (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) sin (120593119890)
+ ((119877 + 1199101) (
119889120593119890
119889119905
) minus (
1198891199091
119889119905
)) cos (120593119890))
times e2+ (
1198891199111
119889119905
) e3
(A9)
The square of absolute payload119872 velocitymay be writtenas
1198812
abs = (
1198891199091
119889119905
)
2
+ (
1198891199101
119889119905
)
2
+ (
1198891199111
119889119905
)
2
+ (
119889120593119890
119889119905
)
2
times (1199092
1+ (1199101+ 119877)2
) + 2(
119889120593119890
119889119905
)
times (1199091(
1198891199101
119889119905
) minus (1199101+ 119877)(
1198891199091
119889119905
))
(A10)
After algebraic transformations we have the square ofabsolute payload119872 velocity in the form of
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A11)
Algebraic expressions (A10) and (A11) could be derivedon the basis of a velocity addition theorem for payload 119872
compound motion in the inertial reference frame E throughthe unit vectors of the noninertial reference frame B (see[59 76 81 82 84 85 87 90ndash92 94ndash98])
V119872E = V
119872B + V1198742E + 120596BE times r
1198742119872
(A12)
where payload relative velocity in the noninertial referenceframeBmay be written as
V119872B = Vr = (
1198891199091
119889119905
) e1+ (
1198891199101
119889119905
) e2+ (
1198891199111
119889119905
) e3
(A13)
The velocityV1198742E of the point119874
2= point119864 in the inertial
reference frame E has zero value
V1198742E = 0 (A14)
The last term in (A12) is the vector product 120596BE times
r1198742119872
where crane boom angular velocity vector is 120596BE =
(119889120593119890119889119905)e3(A6) and the position vector r
1198742119872
in the nonin-ertial reference frameB is as follows
r1198742119872
= 1199091e1+ (1199101+ 119877) e
2+ 1199111e3 (A15)
Taking into account (A13)ndash(A15) the vector equation(A12) in the inertial reference frame E written through theunit vectors e
1 e2 e3of the noninertial reference frame B
will have the following form
V119872E = ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101)) e1
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091) e2+ (
1198891199111
119889119905
) e3
(A16)
So on the basis of (A16) we have the following square ofabsolute payload119872 velocity as
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A17)
Independently derived expressions for the scalar prod-uct (V
119872EV119872E) in (A11) and (A17) completely coincidewhich confirms the correctness and accuracy of the pen-dulum absolute velocity V
119872E derivation and shows thatthe scalar product (V
119872EV119872E) is the invariant expressionindependent of choice of reference frame
The second and third terms of (A12) in the noninertialreference frame B determine the vector of the load 119872
transportation velocity as
Ve = V1198742E + 120596BE times r
1198742119872
(A18)
Taking into account (A14) the scalar of the load 119872
transportation velocity is defined as
1003817100381710038171003817Ve
1003817100381710038171003817= 120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817 (A19)
Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
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DistributedSensor Networks
International Journal of
8 Shock and Vibration
(a) (b)
Figure 6 Experimental trajectories of the spherical pendulum with fixed pivot center and fixed cable length 119897 = 0825m (a b)
M
g
yx
z
B
D
x1
Ast
120596e
z1
y1
Figure 7 Foucaultrsquos pendulum schematic view
It is important to note that the structure of linearizedequations (H4)-(H5) for the swaying payload during craneboom uniform rotation in the noninertial reference frameF is analogous to the structure of the governing equationsfor Foucaultrsquos pendulum motion in the known publishedworks by Zhuravlev and Petrov [100] (formula (6) p 36 of[100]) Pardy [88] (formulae (12)ndash(14) pp 850-851 of [88])and Condurache et al [77] (formulae (11)ndash(13) p 743-744of [77]) The above mentioned analogy between linearizedequations (H4)-(H5) and the governing equations in [77 88100] assumes the geometric analogy of relative swaying tra-jectories between Foucaultrsquos pendulum and the boom-drivenpendulum with rotating pivot centers (Figures 1(e) 12 and16) So there is the geometric analogy between computationalrelative trajectories of load 119872 swaying during crane boomuniform slewing (Figures 1(e) 12 and 16) and relative trajec-tories of load119872 swaying in the plane (119860 st119909119910) around p119860 st inFigure 7 shownbyZhuravlev andPetrov [100] (Figure 4 p 39of [100]) and Condurache andMartinusi [77] (Figure 14 at p754 and Figure 16 at p 755 of [77]) The better understandingof Foucaultrsquos pendula dynamics is provided with the studyof computational relative trajectories in Figures 1(e) 12 and16 For Foucaultrsquos pendulum (Figure 7) the major semiaxes
of relative amplitude extremes have angular velocity valuesequal to the angular velocity of the Earth diurnal rotationMoreover for Foucaultrsquos pendulum (Figure 7) the directionof rotation of major semiaxes of relative amplitude extremesis oppositely directed to the direction of the Earth diurnalrotation All the above mentioned means that the plane(119860 st119861119872) of Foucaultrsquos pendula swaying remains fixed in theheliocentric inertial reference frame 119909
211991021199112
The presented analogy between crane boom-driven pay-load swaying and Foucaultrsquos pendulum motion essentiallyincreases knowledge and awareness of the oscillation pro-cesses in such dynamic systems
7 Discussion and MechanicalInterpretation of GoverningEquations for DAE-Based Nonlinearand Linearized Models
A governing nonlinear DAE-based system (1) for crane boomuniform slewing has been proposed and numerically solvedA conservative system (1) and (H4)-(H5) with componentsΦcor119909 andΦcor119910 has been derived Both projectionsΦcor119909 andΦcor119910 coincide in the noninertial reference framesB andF
The occurrence of the first derivatives 119889119909119889119905 119889119910119889119905 ofthe load119872 relative coordinates with respect to time in termsminus2120596119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) determines that there is no
decay of the oscillations in (H4)-(H5) and (1) but onlyredirection of the relative velocity vector V
119872F = Vr of theload119872 in the noninertial reference framesB andF
The introduction of a linearized model (H4)-(H5)allowed the determination of the natural frequencies (]
1=
]2) of free oscillations of payload 119872 ]
1= 119896 + 120596
119890and
]2= 119896 minus 120596
119890
It follows from the forms of the relative payload trajectory119910 = 119910(119909) in the noninertial reference frame F and theabsolute trajectory 119910
2= 1199102(1199092) in the inertial reference frame
E that the resulting motion of the payload 119872 on the cable119872119861 taking into account the Coriolis inertia forceΦcor (D5)will be the sum of two oscillations with natural frequencies ]
1
and ]2 and with periods 119879
1= 2120587]
1and 119879
2= 2120587]
2
It is worth noting that the frequencies ]1and ]
2differ
by 2120596119890 This means that for the small angle assumption
(Appendices H-I) we have ]1
asymp ]2and the trajectory
Shock and Vibration 9
mg
γ
Nℬ
z1z2
e2
e3
ℰ
ℰ
ℰ
ℱ
ℱ
ℬΔ
120574
y1
Vr
At
M
R D
E
Bz
l
120573
O
1205721
1205721dyn
Adyn
ydyn
h
120596e
AstO1
3ẽ
2ẽ
O2
d120593edty
sin(120593e)
cos(120593e)
120596ℬℰ = 120596ℱℰ
3e
2e
1e
Figure 8 Swaying scheme of load119872 on the cable119872119861 which is fixedly attached at the point 119861 on the crane boom 1198611198742 in the vertical plane
(119910119911) for the nonlinear model derivation in Cartesian coordinates
of relative motion 119910 = 119910(119909) in the noninertial referenceframe F on the expiration of the relative oscillations periodtime 2120587119896 looks approximately like an ellipse (Figures 12(b)and 16(b)) with semiaxes which are governed by the initialconditions
The major semiaxis of such an ellipse in Figures 12(b)and 16(b) is defined by the initial velocity of load 119872 in thenoninertial reference frameF and the analytical value of themajor semiaxis is equal to119881
119861119896 = (120596
119890119877)119896 The derived value
of 119881119861119896 is approximately the sum of the absolute values of
the amplitudes in the analytical form (I15) of the linearizedCauchy problem (H4)-(H5)
The minor semiaxis of the above mentioned ellipsein Figures 12(b) and 16(b) is defined by the value 119910dynwhich is approximately the difference of amplitudes in theanalytical form (I15) of the linearizedCauchy problem (H4)-(H5) In this case the major semiaxis is approximately 5ndash20 times larger than the minor semiaxis because the angularvelocity 120596
119890of the transport rotation of crane boom 119861119874
2is
much smaller than the natural frequency 119896 of the sphericaloscillations of payload119872 on the cable119872119861 (Figures 12(b) and16(b))
It is also necessary to note that the major semiaxis of theellipse in the relative motion rotates with an angular velocity120596119890in the opposite direction of the transport rotation of crane
boom 1198611198742in (Figures 12 and 16)
It follows from Figures 1(f) 2 4 and 5 that the trajectoryof absolute motion of load 119872 in the inertial reference frameE is almost a symmetric curve with 119910
2axial symmetry
The initial and the final motions of load 119872 for half-period
essentially differ from its harmonic oscillations neighbor forquarter-period
Due to the negligible quantity of 120572dyn and 119910dyn (G3)ndash(G5) the average deviation of the load119872 from the mechan-ical trajectory of the boom tip 119861 is negligible (Figures1(f) 2 4 and 5) The basic dynamic load on the systemldquocrane boom 119861119874
2-load 119872rdquo is created by the high-frequency
oscillations of load 119872 which are determined by the actionof inertia force Φcor = minus2119898(120596BE times V
119872B) stipulated bythe Coriolis acceleration acor = 2120596BE times V
119872B of load 119872
in the noninertial reference frame F The basic dynamicload in the system defines additional loads and vibrations ofcrane boom 119861119874
2mechanical elements and support bearings
complicates the automatic and manual control systems of theelectromechanical crane boom 119861119874
2 and also makes crane
operation much more difficultIt is also important to note that the stop of crane boom
1198611198742does not lead to instantaneous damping of the load
119872 absolute oscillations in the inertial reference frame E(Figures 2 4 and 5) This phenomenon directly follows fromthe experimental trajectory in Figures 2 4 and 5 Alsothe natural spherical oscillations of load 119872 will occur withthe frequency 119896 and the amplitude (the difference betweenthe final position of load 119872 in the relative motion in thenoninertial reference frame F for half-period of vibrationand static equilibrium 119860 st of load 119872 on the cable) Thefurther oscillations for the stop of the crane boom 119861119874
2
slewing motion are the deviation of the real trajectory fromthe intended final position of load119872 The results of physicalsimulation in Figures 2 4 and 5 show the necessity of add-on
10 Shock and Vibration
N
ℬ
ℬ
e2
e1
ℰ
ℰ
ℱ
ℱ
y2
x2
Ve
Vr
M
E
1ẽ
2ẽ x
x1
O2(D)
AoAst
O
y y1
Adyn
ydyn
1205722
120579
120579
120596e = 120596ℬℰ = 120596ℱℰ
120593e = 120596et
Φcor
B(O1)
Φen
Φeτ
1e
2e
Figure 9 Swaying scheme of load119872 at the cable119872119861 which is fixedly attached in the point 119861 of the crane boom 1198611198742 in the horizontal plane
(11990911199101) for the nonlinear model derivation in Cartesian coordinates
devices development for the efficient suppression of load 119872
final oscillations
8 Discussion and Comparison of Derived andKnown Published Results
In Figure 1 and pp 537-538 of publishedwork by Sakawa et al1981 [63] the small angle between payloadrsquos cable and verticalline has been introduced that confirms proposed nonlinearDAE-based system (1) and linearized model (H4)-(H5)
In Figure 9 of p 278 of published work by Maczynskiand Wojciech 2003 [44] the computational finite elementmethod (FEM)-based results were shown for the abso-lute payload trajectories in the inertial reference frame EMaczynskirsquos Figure 9 in [44] outlines that the angle betweenthe payloadrsquos cable and the vertical line does not exceed 01rad and that agreeswith the proposed linearizedmodel (H4)-(H5) for a small angle assumption Computational trajectoryfor Maczynski-derived small load swaying [44] after craneboom stop qualitatively coincides with the experimentallyobservable load119872motion in Figures 2 4 and 5
In Figure 6 of published work by Ju et al 2006 [34]it was shown that the angle between the payloadrsquos cableand the vertical line does not exceed 16∘ Jursquos formula (15a)in p 382 of [34] assumes that Jursquos angle between payloadrsquoscable and the vertical line has the harmonic law of variationwith an introduction of a small perturbation term Both Jursquosassumptions in [34] confirm the proposed linearized model(H4)-(H5)
Comparison of the derived linearized model (H4)-(H5)with Figure 5 of published work by Mitrev and Grigorov2008 in [49] shows that the Mitrev-derived ranges of payloadswaying angles within Mitrevrsquos nonlinear model does notexceed 48∘ Such small values of swaying angles completelyconfirm the applicability and correctness of the small angleassumption in (H4)-(H5)
9 Final Conclusions
A governing nonlinear DAE-based system for crane boomuniform slewing has been proposed numerically solved andexperimentally verified
Shock and Vibration 11
Fully nonlinear differential equations for 3 DOF sphericalpendulum oscillations with a uniform slewing crane boomaround a fixed vertical axis of rotation have been derivedin relative Cartesian and spherical coordinates The identicallinearized differential equations in relative Cartesian coordi-nates have been derived with an introduction of the Coriolisdynamic theorem and Lagrange equations for a uniformcrane boomrotation and small swaying angle120572
1assumptions
Linearized andnonlinear theoretical problems for relativeswaying of the payload have been formulated in the form ofthe initial value (Cauchy) problem
An analytical solution of the linearized system has beenderived
The influences of inertia forces from the centripetal andcompound accelerations have been estimated
A linearized conservative system which contains firsttime derivatives of coordinates and no damping of oscilla-tions has been derived
The amplitudes of load oscillations which depend oncomputed initial conditions have been estimated within asmall angle assumption The dependence of natural frequen-cies on the transport inertia forces and gravity forces has beencomputed for the linearized systems
The formulae for the association of relative payloadmotion in the noninertial reference frame F and absolutepayload motion in the inertial reference frame E have beenoutlined
The results of the numerical DAE-based investigation andthe performed physical simulation show a satisfactory fit forfrequencies and amplitudes of load oscillations
The dynamic analogy between crane boom-driven pay-load swaying motion and Foucaultrsquos pendulum motion hasbeen grounded and outlined
The results of the present work are the foundation forfurther investigations of payload119872 swaying dynamics duringtelescopic crane boom nonuniform slewing motion withvariable cable length and for the different motions of thependulum pivot center
Appendices
A Velocity Kinematics Analysis inCartesian Coordinates
For the nonlinear problem definition we study the coopera-tive motion of the mechanical system ldquocrane boom 119861119874
2-load
119872rdquo which is shown in Figures 8ndash10In the nomenclature chapter we denote the fixed inertial
frame of reference E as 119909211991021199112 and the moving noninertial
frame of reference B as 119909111991011199111 which is rigidly bounded
with the crane boom1198611198742 Rotation of themoving noninertial
frame of referenceB(119909111991011199111) around the fixed inertial frame
of referenceE(119909211991021199112) defines the transportation motion for
payload 119872 The motion of load 119872 relative to the movingnoninertial frame of referenceB(119909
111991011199111) defines the relative
motion of payload 119872 The point 119860 st with the coordinates1199091= 0 119910
1= 0 and 119911
1= 0 is the steady equilibrium position
for the load119872 when the crane boom 1198611198742is fixed The point
119860dyn with the coordinates 1199091 = 0 1199101= 119910dyn and 1199111 = Δ is the
dynamic equilibrium position for the load119872 for the rotatingcrane boom 119861119874
2 We will assume the point119860dyn as the origin
of the noninertial reference frameF(119909119910119911) The directions ofaxes 119909 119910 119911 of the noninertial reference frameF are parallelto the axes 119909
1 1199101 1199111of the noninertial reference frameB in
Figures 8ndash10 The load 119872 relative motion takes place alongthe sphere surface with the fixed radius equal to the cablelength 119861119872 = 119897
The computational scheme in Figures 8ndash10 for the nonlin-ear model derivation may be described with an introductionof three degrees of freedom For generalized coordinates weassume the rectangular coordinates 119909
1 1199101 and 119911
1of the load
119872 in the moving noninertial frame of reference B and theangle 120593
119890of crane boom 119861119874
2slewing in the horizontal plane
(11990911199101) with the angular velocity 120596
119890around the vertical axis
11987421199112The relative velocity vector Vr = V
119872B of the load 119872 isdefined as Vr = V
119872B = (1198891199091119889119905 119889119910
1119889119905 119889119911
1119889119905)
In order to derive the absolute velocity V119872E of payload
119872 we will apply the vector method for absolute motionassignment We will define r
1198742119872
as the position vector con-necting initial point 119874
2and terminal point119872 in noninertial
reference frameB (119898) consider
r1198742119872
= r1198742119861+ r119861119872
(A1)
where vector components in (1) are defined in nomenclatureIn accordance with Figures 8ndash10 we have the following
unit vectorsrsquo expansions for position vectors r1198742119861
and r119861119872
in (A1)
r1198742119861
= (119877 cos (120593119890)) e1+ (119877 sin (120593
119890)) e2+ (119877 tan (120574)) e
3
(A2)
r119861119872
= 1199091e1+ 1199101e2+ (1199111minus 119897) e3 (A3)
where 119877 120593119890 1199091 and 119910
1notations have been defined in
nomenclature and in Figures 8ndash10For further definition of the position vector r
1198742119872
in iner-tial reference frame E we write unit vectors of noninertialreference frameB in (A3) through the unit vectors of inertialreference frame E
e1= (sin (120593
119890)) e1minus (cos (120593
119890)) e2 (A4)
e2= (cos (120593
119890)) e1minus (sin (120593
119890)) e2 (A5)
e3= e3 (A6)
After substitution of formulae (A2)ndash(A6) in (A1) andsome algebraic manipulations the vector expression (A1) forthe position vector r
1198742119872
in the inertial reference frameEwilltake the following form
r1198742119872
= ((119877 + 1199101) cos (120593
119890) + 1199091sin (120593
119890)) e1
+ ((119877 + 1199101) sin (120593
119890) minus 1199091cos (120593
119890)) e2
+ (119877 tan (120574) + 1199111minus 119897) e3
(A7)
12 Shock and Vibration
The absolute velocity of payload 119872 we define by timedifferentiation of (A7) assuming constancy of unit vectorse1 e2 and e
3of the inertial reference frameE and constancy
119877 and 120574
V119872E = Vabs =
119889 (r1198742119872
)
119889119905
(A8)
After differentiation and algebraic transformations in(A7) and (A8) we have the following vector expression forabsolute payload 119872 velocity in the inertial reference frameE
V119872E
= (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) cos (120593119890)
+ ((
1198891199091
119889119905
) minus (119877 + 1199101) (
119889120593119890
119889119905
)) sin (120593119890)) e1
+ (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) sin (120593119890)
+ ((119877 + 1199101) (
119889120593119890
119889119905
) minus (
1198891199091
119889119905
)) cos (120593119890))
times e2+ (
1198891199111
119889119905
) e3
(A9)
The square of absolute payload119872 velocitymay be writtenas
1198812
abs = (
1198891199091
119889119905
)
2
+ (
1198891199101
119889119905
)
2
+ (
1198891199111
119889119905
)
2
+ (
119889120593119890
119889119905
)
2
times (1199092
1+ (1199101+ 119877)2
) + 2(
119889120593119890
119889119905
)
times (1199091(
1198891199101
119889119905
) minus (1199101+ 119877)(
1198891199091
119889119905
))
(A10)
After algebraic transformations we have the square ofabsolute payload119872 velocity in the form of
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A11)
Algebraic expressions (A10) and (A11) could be derivedon the basis of a velocity addition theorem for payload 119872
compound motion in the inertial reference frame E throughthe unit vectors of the noninertial reference frame B (see[59 76 81 82 84 85 87 90ndash92 94ndash98])
V119872E = V
119872B + V1198742E + 120596BE times r
1198742119872
(A12)
where payload relative velocity in the noninertial referenceframeBmay be written as
V119872B = Vr = (
1198891199091
119889119905
) e1+ (
1198891199101
119889119905
) e2+ (
1198891199111
119889119905
) e3
(A13)
The velocityV1198742E of the point119874
2= point119864 in the inertial
reference frame E has zero value
V1198742E = 0 (A14)
The last term in (A12) is the vector product 120596BE times
r1198742119872
where crane boom angular velocity vector is 120596BE =
(119889120593119890119889119905)e3(A6) and the position vector r
1198742119872
in the nonin-ertial reference frameB is as follows
r1198742119872
= 1199091e1+ (1199101+ 119877) e
2+ 1199111e3 (A15)
Taking into account (A13)ndash(A15) the vector equation(A12) in the inertial reference frame E written through theunit vectors e
1 e2 e3of the noninertial reference frame B
will have the following form
V119872E = ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101)) e1
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091) e2+ (
1198891199111
119889119905
) e3
(A16)
So on the basis of (A16) we have the following square ofabsolute payload119872 velocity as
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A17)
Independently derived expressions for the scalar prod-uct (V
119872EV119872E) in (A11) and (A17) completely coincidewhich confirms the correctness and accuracy of the pen-dulum absolute velocity V
119872E derivation and shows thatthe scalar product (V
119872EV119872E) is the invariant expressionindependent of choice of reference frame
The second and third terms of (A12) in the noninertialreference frame B determine the vector of the load 119872
transportation velocity as
Ve = V1198742E + 120596BE times r
1198742119872
(A18)
Taking into account (A14) the scalar of the load 119872
transportation velocity is defined as
1003817100381710038171003817Ve
1003817100381710038171003817= 120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817 (A19)
Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
International Journal of
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Shock and Vibration
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International Journal of
Shock and Vibration 9
mg
γ
Nℬ
z1z2
e2
e3
ℰ
ℰ
ℰ
ℱ
ℱ
ℬΔ
120574
y1
Vr
At
M
R D
E
Bz
l
120573
O
1205721
1205721dyn
Adyn
ydyn
h
120596e
AstO1
3ẽ
2ẽ
O2
d120593edty
sin(120593e)
cos(120593e)
120596ℬℰ = 120596ℱℰ
3e
2e
1e
Figure 8 Swaying scheme of load119872 on the cable119872119861 which is fixedly attached at the point 119861 on the crane boom 1198611198742 in the vertical plane
(119910119911) for the nonlinear model derivation in Cartesian coordinates
of relative motion 119910 = 119910(119909) in the noninertial referenceframe F on the expiration of the relative oscillations periodtime 2120587119896 looks approximately like an ellipse (Figures 12(b)and 16(b)) with semiaxes which are governed by the initialconditions
The major semiaxis of such an ellipse in Figures 12(b)and 16(b) is defined by the initial velocity of load 119872 in thenoninertial reference frameF and the analytical value of themajor semiaxis is equal to119881
119861119896 = (120596
119890119877)119896 The derived value
of 119881119861119896 is approximately the sum of the absolute values of
the amplitudes in the analytical form (I15) of the linearizedCauchy problem (H4)-(H5)
The minor semiaxis of the above mentioned ellipsein Figures 12(b) and 16(b) is defined by the value 119910dynwhich is approximately the difference of amplitudes in theanalytical form (I15) of the linearizedCauchy problem (H4)-(H5) In this case the major semiaxis is approximately 5ndash20 times larger than the minor semiaxis because the angularvelocity 120596
119890of the transport rotation of crane boom 119861119874
2is
much smaller than the natural frequency 119896 of the sphericaloscillations of payload119872 on the cable119872119861 (Figures 12(b) and16(b))
It is also necessary to note that the major semiaxis of theellipse in the relative motion rotates with an angular velocity120596119890in the opposite direction of the transport rotation of crane
boom 1198611198742in (Figures 12 and 16)
It follows from Figures 1(f) 2 4 and 5 that the trajectoryof absolute motion of load 119872 in the inertial reference frameE is almost a symmetric curve with 119910
2axial symmetry
The initial and the final motions of load 119872 for half-period
essentially differ from its harmonic oscillations neighbor forquarter-period
Due to the negligible quantity of 120572dyn and 119910dyn (G3)ndash(G5) the average deviation of the load119872 from the mechan-ical trajectory of the boom tip 119861 is negligible (Figures1(f) 2 4 and 5) The basic dynamic load on the systemldquocrane boom 119861119874
2-load 119872rdquo is created by the high-frequency
oscillations of load 119872 which are determined by the actionof inertia force Φcor = minus2119898(120596BE times V
119872B) stipulated bythe Coriolis acceleration acor = 2120596BE times V
119872B of load 119872
in the noninertial reference frame F The basic dynamicload in the system defines additional loads and vibrations ofcrane boom 119861119874
2mechanical elements and support bearings
complicates the automatic and manual control systems of theelectromechanical crane boom 119861119874
2 and also makes crane
operation much more difficultIt is also important to note that the stop of crane boom
1198611198742does not lead to instantaneous damping of the load
119872 absolute oscillations in the inertial reference frame E(Figures 2 4 and 5) This phenomenon directly follows fromthe experimental trajectory in Figures 2 4 and 5 Alsothe natural spherical oscillations of load 119872 will occur withthe frequency 119896 and the amplitude (the difference betweenthe final position of load 119872 in the relative motion in thenoninertial reference frame F for half-period of vibrationand static equilibrium 119860 st of load 119872 on the cable) Thefurther oscillations for the stop of the crane boom 119861119874
2
slewing motion are the deviation of the real trajectory fromthe intended final position of load119872 The results of physicalsimulation in Figures 2 4 and 5 show the necessity of add-on
10 Shock and Vibration
N
ℬ
ℬ
e2
e1
ℰ
ℰ
ℱ
ℱ
y2
x2
Ve
Vr
M
E
1ẽ
2ẽ x
x1
O2(D)
AoAst
O
y y1
Adyn
ydyn
1205722
120579
120579
120596e = 120596ℬℰ = 120596ℱℰ
120593e = 120596et
Φcor
B(O1)
Φen
Φeτ
1e
2e
Figure 9 Swaying scheme of load119872 at the cable119872119861 which is fixedly attached in the point 119861 of the crane boom 1198611198742 in the horizontal plane
(11990911199101) for the nonlinear model derivation in Cartesian coordinates
devices development for the efficient suppression of load 119872
final oscillations
8 Discussion and Comparison of Derived andKnown Published Results
In Figure 1 and pp 537-538 of publishedwork by Sakawa et al1981 [63] the small angle between payloadrsquos cable and verticalline has been introduced that confirms proposed nonlinearDAE-based system (1) and linearized model (H4)-(H5)
In Figure 9 of p 278 of published work by Maczynskiand Wojciech 2003 [44] the computational finite elementmethod (FEM)-based results were shown for the abso-lute payload trajectories in the inertial reference frame EMaczynskirsquos Figure 9 in [44] outlines that the angle betweenthe payloadrsquos cable and the vertical line does not exceed 01rad and that agreeswith the proposed linearizedmodel (H4)-(H5) for a small angle assumption Computational trajectoryfor Maczynski-derived small load swaying [44] after craneboom stop qualitatively coincides with the experimentallyobservable load119872motion in Figures 2 4 and 5
In Figure 6 of published work by Ju et al 2006 [34]it was shown that the angle between the payloadrsquos cableand the vertical line does not exceed 16∘ Jursquos formula (15a)in p 382 of [34] assumes that Jursquos angle between payloadrsquoscable and the vertical line has the harmonic law of variationwith an introduction of a small perturbation term Both Jursquosassumptions in [34] confirm the proposed linearized model(H4)-(H5)
Comparison of the derived linearized model (H4)-(H5)with Figure 5 of published work by Mitrev and Grigorov2008 in [49] shows that the Mitrev-derived ranges of payloadswaying angles within Mitrevrsquos nonlinear model does notexceed 48∘ Such small values of swaying angles completelyconfirm the applicability and correctness of the small angleassumption in (H4)-(H5)
9 Final Conclusions
A governing nonlinear DAE-based system for crane boomuniform slewing has been proposed numerically solved andexperimentally verified
Shock and Vibration 11
Fully nonlinear differential equations for 3 DOF sphericalpendulum oscillations with a uniform slewing crane boomaround a fixed vertical axis of rotation have been derivedin relative Cartesian and spherical coordinates The identicallinearized differential equations in relative Cartesian coordi-nates have been derived with an introduction of the Coriolisdynamic theorem and Lagrange equations for a uniformcrane boomrotation and small swaying angle120572
1assumptions
Linearized andnonlinear theoretical problems for relativeswaying of the payload have been formulated in the form ofthe initial value (Cauchy) problem
An analytical solution of the linearized system has beenderived
The influences of inertia forces from the centripetal andcompound accelerations have been estimated
A linearized conservative system which contains firsttime derivatives of coordinates and no damping of oscilla-tions has been derived
The amplitudes of load oscillations which depend oncomputed initial conditions have been estimated within asmall angle assumption The dependence of natural frequen-cies on the transport inertia forces and gravity forces has beencomputed for the linearized systems
The formulae for the association of relative payloadmotion in the noninertial reference frame F and absolutepayload motion in the inertial reference frame E have beenoutlined
The results of the numerical DAE-based investigation andthe performed physical simulation show a satisfactory fit forfrequencies and amplitudes of load oscillations
The dynamic analogy between crane boom-driven pay-load swaying motion and Foucaultrsquos pendulum motion hasbeen grounded and outlined
The results of the present work are the foundation forfurther investigations of payload119872 swaying dynamics duringtelescopic crane boom nonuniform slewing motion withvariable cable length and for the different motions of thependulum pivot center
Appendices
A Velocity Kinematics Analysis inCartesian Coordinates
For the nonlinear problem definition we study the coopera-tive motion of the mechanical system ldquocrane boom 119861119874
2-load
119872rdquo which is shown in Figures 8ndash10In the nomenclature chapter we denote the fixed inertial
frame of reference E as 119909211991021199112 and the moving noninertial
frame of reference B as 119909111991011199111 which is rigidly bounded
with the crane boom1198611198742 Rotation of themoving noninertial
frame of referenceB(119909111991011199111) around the fixed inertial frame
of referenceE(119909211991021199112) defines the transportation motion for
payload 119872 The motion of load 119872 relative to the movingnoninertial frame of referenceB(119909
111991011199111) defines the relative
motion of payload 119872 The point 119860 st with the coordinates1199091= 0 119910
1= 0 and 119911
1= 0 is the steady equilibrium position
for the load119872 when the crane boom 1198611198742is fixed The point
119860dyn with the coordinates 1199091 = 0 1199101= 119910dyn and 1199111 = Δ is the
dynamic equilibrium position for the load119872 for the rotatingcrane boom 119861119874
2 We will assume the point119860dyn as the origin
of the noninertial reference frameF(119909119910119911) The directions ofaxes 119909 119910 119911 of the noninertial reference frameF are parallelto the axes 119909
1 1199101 1199111of the noninertial reference frameB in
Figures 8ndash10 The load 119872 relative motion takes place alongthe sphere surface with the fixed radius equal to the cablelength 119861119872 = 119897
The computational scheme in Figures 8ndash10 for the nonlin-ear model derivation may be described with an introductionof three degrees of freedom For generalized coordinates weassume the rectangular coordinates 119909
1 1199101 and 119911
1of the load
119872 in the moving noninertial frame of reference B and theangle 120593
119890of crane boom 119861119874
2slewing in the horizontal plane
(11990911199101) with the angular velocity 120596
119890around the vertical axis
11987421199112The relative velocity vector Vr = V
119872B of the load 119872 isdefined as Vr = V
119872B = (1198891199091119889119905 119889119910
1119889119905 119889119911
1119889119905)
In order to derive the absolute velocity V119872E of payload
119872 we will apply the vector method for absolute motionassignment We will define r
1198742119872
as the position vector con-necting initial point 119874
2and terminal point119872 in noninertial
reference frameB (119898) consider
r1198742119872
= r1198742119861+ r119861119872
(A1)
where vector components in (1) are defined in nomenclatureIn accordance with Figures 8ndash10 we have the following
unit vectorsrsquo expansions for position vectors r1198742119861
and r119861119872
in (A1)
r1198742119861
= (119877 cos (120593119890)) e1+ (119877 sin (120593
119890)) e2+ (119877 tan (120574)) e
3
(A2)
r119861119872
= 1199091e1+ 1199101e2+ (1199111minus 119897) e3 (A3)
where 119877 120593119890 1199091 and 119910
1notations have been defined in
nomenclature and in Figures 8ndash10For further definition of the position vector r
1198742119872
in iner-tial reference frame E we write unit vectors of noninertialreference frameB in (A3) through the unit vectors of inertialreference frame E
e1= (sin (120593
119890)) e1minus (cos (120593
119890)) e2 (A4)
e2= (cos (120593
119890)) e1minus (sin (120593
119890)) e2 (A5)
e3= e3 (A6)
After substitution of formulae (A2)ndash(A6) in (A1) andsome algebraic manipulations the vector expression (A1) forthe position vector r
1198742119872
in the inertial reference frameEwilltake the following form
r1198742119872
= ((119877 + 1199101) cos (120593
119890) + 1199091sin (120593
119890)) e1
+ ((119877 + 1199101) sin (120593
119890) minus 1199091cos (120593
119890)) e2
+ (119877 tan (120574) + 1199111minus 119897) e3
(A7)
12 Shock and Vibration
The absolute velocity of payload 119872 we define by timedifferentiation of (A7) assuming constancy of unit vectorse1 e2 and e
3of the inertial reference frameE and constancy
119877 and 120574
V119872E = Vabs =
119889 (r1198742119872
)
119889119905
(A8)
After differentiation and algebraic transformations in(A7) and (A8) we have the following vector expression forabsolute payload 119872 velocity in the inertial reference frameE
V119872E
= (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) cos (120593119890)
+ ((
1198891199091
119889119905
) minus (119877 + 1199101) (
119889120593119890
119889119905
)) sin (120593119890)) e1
+ (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) sin (120593119890)
+ ((119877 + 1199101) (
119889120593119890
119889119905
) minus (
1198891199091
119889119905
)) cos (120593119890))
times e2+ (
1198891199111
119889119905
) e3
(A9)
The square of absolute payload119872 velocitymay be writtenas
1198812
abs = (
1198891199091
119889119905
)
2
+ (
1198891199101
119889119905
)
2
+ (
1198891199111
119889119905
)
2
+ (
119889120593119890
119889119905
)
2
times (1199092
1+ (1199101+ 119877)2
) + 2(
119889120593119890
119889119905
)
times (1199091(
1198891199101
119889119905
) minus (1199101+ 119877)(
1198891199091
119889119905
))
(A10)
After algebraic transformations we have the square ofabsolute payload119872 velocity in the form of
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A11)
Algebraic expressions (A10) and (A11) could be derivedon the basis of a velocity addition theorem for payload 119872
compound motion in the inertial reference frame E throughthe unit vectors of the noninertial reference frame B (see[59 76 81 82 84 85 87 90ndash92 94ndash98])
V119872E = V
119872B + V1198742E + 120596BE times r
1198742119872
(A12)
where payload relative velocity in the noninertial referenceframeBmay be written as
V119872B = Vr = (
1198891199091
119889119905
) e1+ (
1198891199101
119889119905
) e2+ (
1198891199111
119889119905
) e3
(A13)
The velocityV1198742E of the point119874
2= point119864 in the inertial
reference frame E has zero value
V1198742E = 0 (A14)
The last term in (A12) is the vector product 120596BE times
r1198742119872
where crane boom angular velocity vector is 120596BE =
(119889120593119890119889119905)e3(A6) and the position vector r
1198742119872
in the nonin-ertial reference frameB is as follows
r1198742119872
= 1199091e1+ (1199101+ 119877) e
2+ 1199111e3 (A15)
Taking into account (A13)ndash(A15) the vector equation(A12) in the inertial reference frame E written through theunit vectors e
1 e2 e3of the noninertial reference frame B
will have the following form
V119872E = ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101)) e1
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091) e2+ (
1198891199111
119889119905
) e3
(A16)
So on the basis of (A16) we have the following square ofabsolute payload119872 velocity as
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A17)
Independently derived expressions for the scalar prod-uct (V
119872EV119872E) in (A11) and (A17) completely coincidewhich confirms the correctness and accuracy of the pen-dulum absolute velocity V
119872E derivation and shows thatthe scalar product (V
119872EV119872E) is the invariant expressionindependent of choice of reference frame
The second and third terms of (A12) in the noninertialreference frame B determine the vector of the load 119872
transportation velocity as
Ve = V1198742E + 120596BE times r
1198742119872
(A18)
Taking into account (A14) the scalar of the load 119872
transportation velocity is defined as
1003817100381710038171003817Ve
1003817100381710038171003817= 120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817 (A19)
Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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Shock and Vibration
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International Journal of
10 Shock and Vibration
N
ℬ
ℬ
e2
e1
ℰ
ℰ
ℱ
ℱ
y2
x2
Ve
Vr
M
E
1ẽ
2ẽ x
x1
O2(D)
AoAst
O
y y1
Adyn
ydyn
1205722
120579
120579
120596e = 120596ℬℰ = 120596ℱℰ
120593e = 120596et
Φcor
B(O1)
Φen
Φeτ
1e
2e
Figure 9 Swaying scheme of load119872 at the cable119872119861 which is fixedly attached in the point 119861 of the crane boom 1198611198742 in the horizontal plane
(11990911199101) for the nonlinear model derivation in Cartesian coordinates
devices development for the efficient suppression of load 119872
final oscillations
8 Discussion and Comparison of Derived andKnown Published Results
In Figure 1 and pp 537-538 of publishedwork by Sakawa et al1981 [63] the small angle between payloadrsquos cable and verticalline has been introduced that confirms proposed nonlinearDAE-based system (1) and linearized model (H4)-(H5)
In Figure 9 of p 278 of published work by Maczynskiand Wojciech 2003 [44] the computational finite elementmethod (FEM)-based results were shown for the abso-lute payload trajectories in the inertial reference frame EMaczynskirsquos Figure 9 in [44] outlines that the angle betweenthe payloadrsquos cable and the vertical line does not exceed 01rad and that agreeswith the proposed linearizedmodel (H4)-(H5) for a small angle assumption Computational trajectoryfor Maczynski-derived small load swaying [44] after craneboom stop qualitatively coincides with the experimentallyobservable load119872motion in Figures 2 4 and 5
In Figure 6 of published work by Ju et al 2006 [34]it was shown that the angle between the payloadrsquos cableand the vertical line does not exceed 16∘ Jursquos formula (15a)in p 382 of [34] assumes that Jursquos angle between payloadrsquoscable and the vertical line has the harmonic law of variationwith an introduction of a small perturbation term Both Jursquosassumptions in [34] confirm the proposed linearized model(H4)-(H5)
Comparison of the derived linearized model (H4)-(H5)with Figure 5 of published work by Mitrev and Grigorov2008 in [49] shows that the Mitrev-derived ranges of payloadswaying angles within Mitrevrsquos nonlinear model does notexceed 48∘ Such small values of swaying angles completelyconfirm the applicability and correctness of the small angleassumption in (H4)-(H5)
9 Final Conclusions
A governing nonlinear DAE-based system for crane boomuniform slewing has been proposed numerically solved andexperimentally verified
Shock and Vibration 11
Fully nonlinear differential equations for 3 DOF sphericalpendulum oscillations with a uniform slewing crane boomaround a fixed vertical axis of rotation have been derivedin relative Cartesian and spherical coordinates The identicallinearized differential equations in relative Cartesian coordi-nates have been derived with an introduction of the Coriolisdynamic theorem and Lagrange equations for a uniformcrane boomrotation and small swaying angle120572
1assumptions
Linearized andnonlinear theoretical problems for relativeswaying of the payload have been formulated in the form ofthe initial value (Cauchy) problem
An analytical solution of the linearized system has beenderived
The influences of inertia forces from the centripetal andcompound accelerations have been estimated
A linearized conservative system which contains firsttime derivatives of coordinates and no damping of oscilla-tions has been derived
The amplitudes of load oscillations which depend oncomputed initial conditions have been estimated within asmall angle assumption The dependence of natural frequen-cies on the transport inertia forces and gravity forces has beencomputed for the linearized systems
The formulae for the association of relative payloadmotion in the noninertial reference frame F and absolutepayload motion in the inertial reference frame E have beenoutlined
The results of the numerical DAE-based investigation andthe performed physical simulation show a satisfactory fit forfrequencies and amplitudes of load oscillations
The dynamic analogy between crane boom-driven pay-load swaying motion and Foucaultrsquos pendulum motion hasbeen grounded and outlined
The results of the present work are the foundation forfurther investigations of payload119872 swaying dynamics duringtelescopic crane boom nonuniform slewing motion withvariable cable length and for the different motions of thependulum pivot center
Appendices
A Velocity Kinematics Analysis inCartesian Coordinates
For the nonlinear problem definition we study the coopera-tive motion of the mechanical system ldquocrane boom 119861119874
2-load
119872rdquo which is shown in Figures 8ndash10In the nomenclature chapter we denote the fixed inertial
frame of reference E as 119909211991021199112 and the moving noninertial
frame of reference B as 119909111991011199111 which is rigidly bounded
with the crane boom1198611198742 Rotation of themoving noninertial
frame of referenceB(119909111991011199111) around the fixed inertial frame
of referenceE(119909211991021199112) defines the transportation motion for
payload 119872 The motion of load 119872 relative to the movingnoninertial frame of referenceB(119909
111991011199111) defines the relative
motion of payload 119872 The point 119860 st with the coordinates1199091= 0 119910
1= 0 and 119911
1= 0 is the steady equilibrium position
for the load119872 when the crane boom 1198611198742is fixed The point
119860dyn with the coordinates 1199091 = 0 1199101= 119910dyn and 1199111 = Δ is the
dynamic equilibrium position for the load119872 for the rotatingcrane boom 119861119874
2 We will assume the point119860dyn as the origin
of the noninertial reference frameF(119909119910119911) The directions ofaxes 119909 119910 119911 of the noninertial reference frameF are parallelto the axes 119909
1 1199101 1199111of the noninertial reference frameB in
Figures 8ndash10 The load 119872 relative motion takes place alongthe sphere surface with the fixed radius equal to the cablelength 119861119872 = 119897
The computational scheme in Figures 8ndash10 for the nonlin-ear model derivation may be described with an introductionof three degrees of freedom For generalized coordinates weassume the rectangular coordinates 119909
1 1199101 and 119911
1of the load
119872 in the moving noninertial frame of reference B and theangle 120593
119890of crane boom 119861119874
2slewing in the horizontal plane
(11990911199101) with the angular velocity 120596
119890around the vertical axis
11987421199112The relative velocity vector Vr = V
119872B of the load 119872 isdefined as Vr = V
119872B = (1198891199091119889119905 119889119910
1119889119905 119889119911
1119889119905)
In order to derive the absolute velocity V119872E of payload
119872 we will apply the vector method for absolute motionassignment We will define r
1198742119872
as the position vector con-necting initial point 119874
2and terminal point119872 in noninertial
reference frameB (119898) consider
r1198742119872
= r1198742119861+ r119861119872
(A1)
where vector components in (1) are defined in nomenclatureIn accordance with Figures 8ndash10 we have the following
unit vectorsrsquo expansions for position vectors r1198742119861
and r119861119872
in (A1)
r1198742119861
= (119877 cos (120593119890)) e1+ (119877 sin (120593
119890)) e2+ (119877 tan (120574)) e
3
(A2)
r119861119872
= 1199091e1+ 1199101e2+ (1199111minus 119897) e3 (A3)
where 119877 120593119890 1199091 and 119910
1notations have been defined in
nomenclature and in Figures 8ndash10For further definition of the position vector r
1198742119872
in iner-tial reference frame E we write unit vectors of noninertialreference frameB in (A3) through the unit vectors of inertialreference frame E
e1= (sin (120593
119890)) e1minus (cos (120593
119890)) e2 (A4)
e2= (cos (120593
119890)) e1minus (sin (120593
119890)) e2 (A5)
e3= e3 (A6)
After substitution of formulae (A2)ndash(A6) in (A1) andsome algebraic manipulations the vector expression (A1) forthe position vector r
1198742119872
in the inertial reference frameEwilltake the following form
r1198742119872
= ((119877 + 1199101) cos (120593
119890) + 1199091sin (120593
119890)) e1
+ ((119877 + 1199101) sin (120593
119890) minus 1199091cos (120593
119890)) e2
+ (119877 tan (120574) + 1199111minus 119897) e3
(A7)
12 Shock and Vibration
The absolute velocity of payload 119872 we define by timedifferentiation of (A7) assuming constancy of unit vectorse1 e2 and e
3of the inertial reference frameE and constancy
119877 and 120574
V119872E = Vabs =
119889 (r1198742119872
)
119889119905
(A8)
After differentiation and algebraic transformations in(A7) and (A8) we have the following vector expression forabsolute payload 119872 velocity in the inertial reference frameE
V119872E
= (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) cos (120593119890)
+ ((
1198891199091
119889119905
) minus (119877 + 1199101) (
119889120593119890
119889119905
)) sin (120593119890)) e1
+ (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) sin (120593119890)
+ ((119877 + 1199101) (
119889120593119890
119889119905
) minus (
1198891199091
119889119905
)) cos (120593119890))
times e2+ (
1198891199111
119889119905
) e3
(A9)
The square of absolute payload119872 velocitymay be writtenas
1198812
abs = (
1198891199091
119889119905
)
2
+ (
1198891199101
119889119905
)
2
+ (
1198891199111
119889119905
)
2
+ (
119889120593119890
119889119905
)
2
times (1199092
1+ (1199101+ 119877)2
) + 2(
119889120593119890
119889119905
)
times (1199091(
1198891199101
119889119905
) minus (1199101+ 119877)(
1198891199091
119889119905
))
(A10)
After algebraic transformations we have the square ofabsolute payload119872 velocity in the form of
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A11)
Algebraic expressions (A10) and (A11) could be derivedon the basis of a velocity addition theorem for payload 119872
compound motion in the inertial reference frame E throughthe unit vectors of the noninertial reference frame B (see[59 76 81 82 84 85 87 90ndash92 94ndash98])
V119872E = V
119872B + V1198742E + 120596BE times r
1198742119872
(A12)
where payload relative velocity in the noninertial referenceframeBmay be written as
V119872B = Vr = (
1198891199091
119889119905
) e1+ (
1198891199101
119889119905
) e2+ (
1198891199111
119889119905
) e3
(A13)
The velocityV1198742E of the point119874
2= point119864 in the inertial
reference frame E has zero value
V1198742E = 0 (A14)
The last term in (A12) is the vector product 120596BE times
r1198742119872
where crane boom angular velocity vector is 120596BE =
(119889120593119890119889119905)e3(A6) and the position vector r
1198742119872
in the nonin-ertial reference frameB is as follows
r1198742119872
= 1199091e1+ (1199101+ 119877) e
2+ 1199111e3 (A15)
Taking into account (A13)ndash(A15) the vector equation(A12) in the inertial reference frame E written through theunit vectors e
1 e2 e3of the noninertial reference frame B
will have the following form
V119872E = ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101)) e1
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091) e2+ (
1198891199111
119889119905
) e3
(A16)
So on the basis of (A16) we have the following square ofabsolute payload119872 velocity as
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A17)
Independently derived expressions for the scalar prod-uct (V
119872EV119872E) in (A11) and (A17) completely coincidewhich confirms the correctness and accuracy of the pen-dulum absolute velocity V
119872E derivation and shows thatthe scalar product (V
119872EV119872E) is the invariant expressionindependent of choice of reference frame
The second and third terms of (A12) in the noninertialreference frame B determine the vector of the load 119872
transportation velocity as
Ve = V1198742E + 120596BE times r
1198742119872
(A18)
Taking into account (A14) the scalar of the load 119872
transportation velocity is defined as
1003817100381710038171003817Ve
1003817100381710038171003817= 120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817 (A19)
Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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Shock and Vibration
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International Journal of
Shock and Vibration 11
Fully nonlinear differential equations for 3 DOF sphericalpendulum oscillations with a uniform slewing crane boomaround a fixed vertical axis of rotation have been derivedin relative Cartesian and spherical coordinates The identicallinearized differential equations in relative Cartesian coordi-nates have been derived with an introduction of the Coriolisdynamic theorem and Lagrange equations for a uniformcrane boomrotation and small swaying angle120572
1assumptions
Linearized andnonlinear theoretical problems for relativeswaying of the payload have been formulated in the form ofthe initial value (Cauchy) problem
An analytical solution of the linearized system has beenderived
The influences of inertia forces from the centripetal andcompound accelerations have been estimated
A linearized conservative system which contains firsttime derivatives of coordinates and no damping of oscilla-tions has been derived
The amplitudes of load oscillations which depend oncomputed initial conditions have been estimated within asmall angle assumption The dependence of natural frequen-cies on the transport inertia forces and gravity forces has beencomputed for the linearized systems
The formulae for the association of relative payloadmotion in the noninertial reference frame F and absolutepayload motion in the inertial reference frame E have beenoutlined
The results of the numerical DAE-based investigation andthe performed physical simulation show a satisfactory fit forfrequencies and amplitudes of load oscillations
The dynamic analogy between crane boom-driven pay-load swaying motion and Foucaultrsquos pendulum motion hasbeen grounded and outlined
The results of the present work are the foundation forfurther investigations of payload119872 swaying dynamics duringtelescopic crane boom nonuniform slewing motion withvariable cable length and for the different motions of thependulum pivot center
Appendices
A Velocity Kinematics Analysis inCartesian Coordinates
For the nonlinear problem definition we study the coopera-tive motion of the mechanical system ldquocrane boom 119861119874
2-load
119872rdquo which is shown in Figures 8ndash10In the nomenclature chapter we denote the fixed inertial
frame of reference E as 119909211991021199112 and the moving noninertial
frame of reference B as 119909111991011199111 which is rigidly bounded
with the crane boom1198611198742 Rotation of themoving noninertial
frame of referenceB(119909111991011199111) around the fixed inertial frame
of referenceE(119909211991021199112) defines the transportation motion for
payload 119872 The motion of load 119872 relative to the movingnoninertial frame of referenceB(119909
111991011199111) defines the relative
motion of payload 119872 The point 119860 st with the coordinates1199091= 0 119910
1= 0 and 119911
1= 0 is the steady equilibrium position
for the load119872 when the crane boom 1198611198742is fixed The point
119860dyn with the coordinates 1199091 = 0 1199101= 119910dyn and 1199111 = Δ is the
dynamic equilibrium position for the load119872 for the rotatingcrane boom 119861119874
2 We will assume the point119860dyn as the origin
of the noninertial reference frameF(119909119910119911) The directions ofaxes 119909 119910 119911 of the noninertial reference frameF are parallelto the axes 119909
1 1199101 1199111of the noninertial reference frameB in
Figures 8ndash10 The load 119872 relative motion takes place alongthe sphere surface with the fixed radius equal to the cablelength 119861119872 = 119897
The computational scheme in Figures 8ndash10 for the nonlin-ear model derivation may be described with an introductionof three degrees of freedom For generalized coordinates weassume the rectangular coordinates 119909
1 1199101 and 119911
1of the load
119872 in the moving noninertial frame of reference B and theangle 120593
119890of crane boom 119861119874
2slewing in the horizontal plane
(11990911199101) with the angular velocity 120596
119890around the vertical axis
11987421199112The relative velocity vector Vr = V
119872B of the load 119872 isdefined as Vr = V
119872B = (1198891199091119889119905 119889119910
1119889119905 119889119911
1119889119905)
In order to derive the absolute velocity V119872E of payload
119872 we will apply the vector method for absolute motionassignment We will define r
1198742119872
as the position vector con-necting initial point 119874
2and terminal point119872 in noninertial
reference frameB (119898) consider
r1198742119872
= r1198742119861+ r119861119872
(A1)
where vector components in (1) are defined in nomenclatureIn accordance with Figures 8ndash10 we have the following
unit vectorsrsquo expansions for position vectors r1198742119861
and r119861119872
in (A1)
r1198742119861
= (119877 cos (120593119890)) e1+ (119877 sin (120593
119890)) e2+ (119877 tan (120574)) e
3
(A2)
r119861119872
= 1199091e1+ 1199101e2+ (1199111minus 119897) e3 (A3)
where 119877 120593119890 1199091 and 119910
1notations have been defined in
nomenclature and in Figures 8ndash10For further definition of the position vector r
1198742119872
in iner-tial reference frame E we write unit vectors of noninertialreference frameB in (A3) through the unit vectors of inertialreference frame E
e1= (sin (120593
119890)) e1minus (cos (120593
119890)) e2 (A4)
e2= (cos (120593
119890)) e1minus (sin (120593
119890)) e2 (A5)
e3= e3 (A6)
After substitution of formulae (A2)ndash(A6) in (A1) andsome algebraic manipulations the vector expression (A1) forthe position vector r
1198742119872
in the inertial reference frameEwilltake the following form
r1198742119872
= ((119877 + 1199101) cos (120593
119890) + 1199091sin (120593
119890)) e1
+ ((119877 + 1199101) sin (120593
119890) minus 1199091cos (120593
119890)) e2
+ (119877 tan (120574) + 1199111minus 119897) e3
(A7)
12 Shock and Vibration
The absolute velocity of payload 119872 we define by timedifferentiation of (A7) assuming constancy of unit vectorse1 e2 and e
3of the inertial reference frameE and constancy
119877 and 120574
V119872E = Vabs =
119889 (r1198742119872
)
119889119905
(A8)
After differentiation and algebraic transformations in(A7) and (A8) we have the following vector expression forabsolute payload 119872 velocity in the inertial reference frameE
V119872E
= (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) cos (120593119890)
+ ((
1198891199091
119889119905
) minus (119877 + 1199101) (
119889120593119890
119889119905
)) sin (120593119890)) e1
+ (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) sin (120593119890)
+ ((119877 + 1199101) (
119889120593119890
119889119905
) minus (
1198891199091
119889119905
)) cos (120593119890))
times e2+ (
1198891199111
119889119905
) e3
(A9)
The square of absolute payload119872 velocitymay be writtenas
1198812
abs = (
1198891199091
119889119905
)
2
+ (
1198891199101
119889119905
)
2
+ (
1198891199111
119889119905
)
2
+ (
119889120593119890
119889119905
)
2
times (1199092
1+ (1199101+ 119877)2
) + 2(
119889120593119890
119889119905
)
times (1199091(
1198891199101
119889119905
) minus (1199101+ 119877)(
1198891199091
119889119905
))
(A10)
After algebraic transformations we have the square ofabsolute payload119872 velocity in the form of
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A11)
Algebraic expressions (A10) and (A11) could be derivedon the basis of a velocity addition theorem for payload 119872
compound motion in the inertial reference frame E throughthe unit vectors of the noninertial reference frame B (see[59 76 81 82 84 85 87 90ndash92 94ndash98])
V119872E = V
119872B + V1198742E + 120596BE times r
1198742119872
(A12)
where payload relative velocity in the noninertial referenceframeBmay be written as
V119872B = Vr = (
1198891199091
119889119905
) e1+ (
1198891199101
119889119905
) e2+ (
1198891199111
119889119905
) e3
(A13)
The velocityV1198742E of the point119874
2= point119864 in the inertial
reference frame E has zero value
V1198742E = 0 (A14)
The last term in (A12) is the vector product 120596BE times
r1198742119872
where crane boom angular velocity vector is 120596BE =
(119889120593119890119889119905)e3(A6) and the position vector r
1198742119872
in the nonin-ertial reference frameB is as follows
r1198742119872
= 1199091e1+ (1199101+ 119877) e
2+ 1199111e3 (A15)
Taking into account (A13)ndash(A15) the vector equation(A12) in the inertial reference frame E written through theunit vectors e
1 e2 e3of the noninertial reference frame B
will have the following form
V119872E = ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101)) e1
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091) e2+ (
1198891199111
119889119905
) e3
(A16)
So on the basis of (A16) we have the following square ofabsolute payload119872 velocity as
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A17)
Independently derived expressions for the scalar prod-uct (V
119872EV119872E) in (A11) and (A17) completely coincidewhich confirms the correctness and accuracy of the pen-dulum absolute velocity V
119872E derivation and shows thatthe scalar product (V
119872EV119872E) is the invariant expressionindependent of choice of reference frame
The second and third terms of (A12) in the noninertialreference frame B determine the vector of the load 119872
transportation velocity as
Ve = V1198742E + 120596BE times r
1198742119872
(A18)
Taking into account (A14) the scalar of the load 119872
transportation velocity is defined as
1003817100381710038171003817Ve
1003817100381710038171003817= 120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817 (A19)
Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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Shock and Vibration
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International Journal of
12 Shock and Vibration
The absolute velocity of payload 119872 we define by timedifferentiation of (A7) assuming constancy of unit vectorse1 e2 and e
3of the inertial reference frameE and constancy
119877 and 120574
V119872E = Vabs =
119889 (r1198742119872
)
119889119905
(A8)
After differentiation and algebraic transformations in(A7) and (A8) we have the following vector expression forabsolute payload 119872 velocity in the inertial reference frameE
V119872E
= (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) cos (120593119890)
+ ((
1198891199091
119889119905
) minus (119877 + 1199101) (
119889120593119890
119889119905
)) sin (120593119890)) e1
+ (((
1198891199101
119889119905
) + 1199091(
119889120593119890
119889119905
)) sin (120593119890)
+ ((119877 + 1199101) (
119889120593119890
119889119905
) minus (
1198891199091
119889119905
)) cos (120593119890))
times e2+ (
1198891199111
119889119905
) e3
(A9)
The square of absolute payload119872 velocitymay be writtenas
1198812
abs = (
1198891199091
119889119905
)
2
+ (
1198891199101
119889119905
)
2
+ (
1198891199111
119889119905
)
2
+ (
119889120593119890
119889119905
)
2
times (1199092
1+ (1199101+ 119877)2
) + 2(
119889120593119890
119889119905
)
times (1199091(
1198891199101
119889119905
) minus (1199101+ 119877)(
1198891199091
119889119905
))
(A10)
After algebraic transformations we have the square ofabsolute payload119872 velocity in the form of
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A11)
Algebraic expressions (A10) and (A11) could be derivedon the basis of a velocity addition theorem for payload 119872
compound motion in the inertial reference frame E throughthe unit vectors of the noninertial reference frame B (see[59 76 81 82 84 85 87 90ndash92 94ndash98])
V119872E = V
119872B + V1198742E + 120596BE times r
1198742119872
(A12)
where payload relative velocity in the noninertial referenceframeBmay be written as
V119872B = Vr = (
1198891199091
119889119905
) e1+ (
1198891199101
119889119905
) e2+ (
1198891199111
119889119905
) e3
(A13)
The velocityV1198742E of the point119874
2= point119864 in the inertial
reference frame E has zero value
V1198742E = 0 (A14)
The last term in (A12) is the vector product 120596BE times
r1198742119872
where crane boom angular velocity vector is 120596BE =
(119889120593119890119889119905)e3(A6) and the position vector r
1198742119872
in the nonin-ertial reference frameB is as follows
r1198742119872
= 1199091e1+ (1199101+ 119877) e
2+ 1199111e3 (A15)
Taking into account (A13)ndash(A15) the vector equation(A12) in the inertial reference frame E written through theunit vectors e
1 e2 e3of the noninertial reference frame B
will have the following form
V119872E = ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101)) e1
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091) e2+ (
1198891199111
119889119905
) e3
(A16)
So on the basis of (A16) we have the following square ofabsolute payload119872 velocity as
(V119872EV119872E) = 119881
2
abs
= ((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+ ((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
(A17)
Independently derived expressions for the scalar prod-uct (V
119872EV119872E) in (A11) and (A17) completely coincidewhich confirms the correctness and accuracy of the pen-dulum absolute velocity V
119872E derivation and shows thatthe scalar product (V
119872EV119872E) is the invariant expressionindependent of choice of reference frame
The second and third terms of (A12) in the noninertialreference frame B determine the vector of the load 119872
transportation velocity as
Ve = V1198742E + 120596BE times r
1198742119872
(A18)
Taking into account (A14) the scalar of the load 119872
transportation velocity is defined as
1003817100381710038171003817Ve
1003817100381710038171003817= 120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817 (A19)
Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
International Journal of
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Shock and Vibration
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Shock and Vibration 13
N
ℬ
ℬ
e3
ℰ
ℱ
z1
z2
Ve
R
3ẽ
2ẽ
y2
Ast
M
zH
D
B
Adyn
1205721
120596ℬℰ = 120596ℱℰ
ℰ
ℰ
120596e
O2
120593e
x1
x2
l
ℱx
mgVr
Φcor
119834cor
119834120591e
e2 ℱ
ℬ
e1y
y1
1ẽO1
O
Δ
1205722
Φeτ
Φen
119834en
1e
2e
3e
= d120593edt
Figure 10 Computational spatial scheme of spherical pendulum 119872 swaying on the cable 119872119861 during crane boom 1198611198742slewing motion for
the nonlinear model derivation in Cartesian coordinates
The transportation velocity vector Ve is perpendicular tor1198742119872
and r119863119872
(Figures 8ndash10) that is
(Ve r1198742119872
) = (V1198742E + 120596BE times r
1198742119872
r1198742119872
) = 0
(Ve r119863119872) = (V1198742E + 120596BE times r
1198742119872
r119863119872
) = 0
(A20)In nomenclature and in Figure 9 we denote the current
angle 120579 = ang(e2 (r1198742119872
)11990911199101
) where
sin (120579) =1199091
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
cos (120579) =(119877 + 119910
1)
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817
(A21)
In the noninertial reference frame B the vector of theload119872 transportation velocity Ve is defined as
Ve = (minus1003817100381710038171003817Ve
1003817100381710038171003817cos (120579)) e
1+ (
1003817100381710038171003817Ve
1003817100381710038171003817sin (120579)) e
2 (A22)
Taking into account (A19) (A22) takes the followingform
Ve = (minus120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817cos (120579)) e
1
+ (120596BE
10038171003817100381710038171003817r1198742119872
10038171003817100381710038171003817sin (120579)) e
2
(A23)
Assuming (A21) equation (A23) will take the followingform in the noninertial reference frameB
Ve = (minus(
119889120593119890
119889119905
) (119877 + 1199101)) e1+ ((
119889120593119890
119889119905
) 1199091) e2 (A24)
Taking into account (A13) and (A24) the formula (A12)yields again (A16) and (A17)
So the square of absolute payload 119872 velocity (A10)(A11) and (A17) has been derived with three independentmethods which confirms the accuracy and correctness ofexpressions (A10) (A11) and (A17)
B Acceleration Kinematics Analysis
Further dynamic analysis with the introduction of Newtonrsquossecond law requires us to study the accelerations of payload119872 shown in Figure 10
14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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Shock and Vibration
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14 Shock and Vibration
The standard vector equation for the acceleration addi-tion for payload 119872 in the inertial reference frame E has theform (see [59 76 81 82 84 85 87 90ndash92 94ndash98])
a119872E = a
119872B + a119864E + 120572BE times r
119864119872
+ 120596BE times (120596BE times r119864119872
)
+ 2120596BE times V119872B
(B1)
In our case we assume point 119864 as the pole for payloadtransportation motion located at the vertical axis 119874
21199112 So
the second term in (B1) for the inertial reference frame Eand in the noninertial reference frameB takes the form
a119864E = a
119864B = 0 (B2)
So taking into account the nomenclature and (B2) (B1)in the inertial reference frameE takes the following form
aabs = ar + ae120591+ ae
n+ acor (B3)
Equations (B1) and (B2) contain the following accelera-tions
The vector of payload 119872 relative acceleration is definedin the noninertial reference frameB as
ar = a119872B
= (
11988921199091
1198891199052) e1+ (
11988921199101
1198891199052) e2+ (
11988921199111
1198891199052) e3
(B4)
The vector of tangential acceleration ae120591 for transporta-tion of payload119872 has the same direction as the vector of theload 119872 transportation velocity Ve that is ae120591 uarruarr Ve and isdefined in the noninertial reference frameB as
ae120591= 120572BE times r
119864119872
= (minus(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1+ ((
1198892120593119890
1198891199052)1199091) e2
(B5)
The vector of the normal or centripetal acceleration aenof transportation for payload 119872 is directed towards the axis11987421199112and at the same time ae120591 and aen are the coplanar
vectors located in the horizontal plane (11990921199102) where
aen= 120596BE times (120596BE times r
119864119872)
= (minus(
119889120593119890
119889119905
)
2
1199091) e1minus ((
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(B6)
The vector of the Coriolis (compound) acceleration ofpayload119872 is directed in accordance with the vector productlaw
acor = 2120596BE times V119872B
= (minus2(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1+ (2(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(B7)
C The Geometric ConstraintsImposed on the Payload
The geometric constraint imposed on the payload 119872 isshown in Figures 8ndash10 in the formof the cable119861119872The length119897 = 119897119861119872
= r119861119872
of the cable 119861119872 determines the geometricalconstraint in this problem
1198972= 1199092
1+ 1199102
1+ (1199111minus 119897)2
(C1)
On the basis of Figure 10 we may derive that
1198972= (119897 sin (120572
1) cos (120572
2))2
+ (119897 sin (1205721) sin (120572
2))2
+ (119897 cos (1205721))2
(C2)
where angles 1205721and 120572
2in Figures 8ndash10 are the spherical
coordinates of spherical pendulum119872The comparison of formulae (C1) and (C2) allows us
to determine the Cartesian coordinates 1199091 1199101 and 119911
1in the
noninertial reference frameB as
1199091= 119897 sin (120572
1) cos (120572
2)
1199101= 119897 sin (120572
1) sin (120572
2)
1199111= 119897 minus 119897 cos (120572
1)
(C3)
The geometric constraint of (C1)ndash(C3) can be derived onthe basis of (C3) as
1198972sin2 (120572
1) = 1199092
1+ 1199102
1
1199111= 119897 minus 119897 cos (120572
1)
(C4)
Equations (C4) yield the following partial derivatives of1205721with respect to Cartesian coordinates 119909
1 1199101 and 119911
1in the
noninertial reference frameB as
1205971205721
1205971199091
=
1199091
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199101
=
1199101
1198972 sin (120572
1) cos (120572
1)
1205971205721
1205971199111
=
1
119897 sin (1205721)
(C5)
Absolute coordinates 1199092 1199102 and 119911
2in inertial reference
frame E which depend upon the relative coordinates 1199091 1199101
and 1199111in noninertial reference frame B during the swaying
of load 119872 are defined according to the following equations(Figures 8ndash10)
1199092= (119877 + 119910
1) cos (120593
119890) + 1199091sin (120593
119890)
1199102= (119877 + 119910
1) sin (120593
119890) minus 1199091cos (120593
119890)
1199112= 1199111= 119897 minus 119897 cos (120572
1)
(C6)
Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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Shock and Vibration 15
D Forces Imposed on the Payload
Among the forces imposed on the payload 119872 (Figure 3(a))we have an active force of gravitymg the cable reaction forceN the tangential inertial forceΦe120591 the normal or centrifugalinertial forceΦe
n and the Coriolis inertial forceΦcor Takinginto account formulae (B1)ndash(B7) we will express below allimposed forces in the noninertial reference frameB
mg= (minus119898119892) e3 (D1)
N = minus119873(
1199091
119897
) e1minus 119873(
1199101
119897
) e2minus 119873(
(1199111minus 119897)
119897
) e3 (D2)
Φen= (minus119898) ae
n= (119898(
119889120593119890
119889119905
)
2
1199091) e1
+ (119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)) e2
(D3)
Φe120591= (minus119898) ae
120591= (119898(
1198892120593119890
1198891199052) (119877 + 119910
1)) e1
minus (119898(
1198892120593119890
1198891199052)1199091) e2
(D4)
Φcor = (minus119898) acor = (2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)) e1
minus (2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)) e2
(D5)
where minus(1199091119897) minus(119910
1119897) and minus((119911
1minus 119897)119897) are the direction
cosines of the cable reaction force N in the noninertialreference frame B The force N is directed from point 119872 topointB that is the forceN uarrdarr r
119861119872is oppositely directed to
the r119861119872
(A3)
E Forces Imposed on the CraneBoom-Payload System
The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figures 8ndash10 is governed by the vector
equation for the rate of change of moment of momentumH11987423
for the system ldquocrane boom 1198611198742-load 119872rdquo with respect
to point 1198742in the inertial reference frameE
E119889H11987423119889119905
= sumM1198742 (E1)
The vector equation (E1) contains the following compo-nents
H11987423
= 1198681198742
33120596BE (E2)
1198681198742
33= (1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
) (E3)
sumM1198742 = M1198742119863119879
minusM1198742119865119879
= (1198721198742
119863119879minus1198721198742
119865119879) e3 (E4)
where (119868119874233)1198611198742
is the element of mass moment of inertia forthe crane boom119861119874
2in inertial fixed on earth reference frame
E with respect to axis e3119898(1199092
1+ (119877 + 119910
1)2) is the element of
mass moment of inertia for the payload119872 in inertial fixed onearth reference frameE with respect to axis e
3
The external moment of gravitational forceM1198742(mg) = 0
in (E4) becausemguarrdarr e3
For the system ldquocrane boom-payloadrdquo the cable reactionforce N is the internal force So in (E1) and (E4) we haveM1198742(N) = 0
Substitution of (E2) (E3) and (E4) into (E1) yields thefollowing scalar equation for the rate of change of momentof momentumH1198742
3for the system ldquocrane boom 119861119874
2-load119872rdquo
with respect to point 1198742in the inertial reference frameE
119889
119889119905
(((1198681198742
33)1198611198742
+ 119898(1199092
1+ (119877 + 119910
1)2
)) (
119889120593119890
119889119905
))
= 1198721198742
119863119879minus1198721198742
119865119879
(E5)
where driving1198721198742119863119879
and frictional1198721198742119865119879
torques (see nomen-clature section) are the technically defined functions forspecific electric drive systems (see [1ndash3 42ndash44 59 63 67 7681 82 84 85 87 90ndash92 94ndash98])
F Derivation of the Fully NonlinearEquations in Relative Cartesian Coordinatesof the Noninertial Reference frame B
Thevector differential equation for relativemotion of payload119872 in the noninertial reference frameB is as follows
119898a119872B = mg + N +Φe
n+Φe120591+Φcor (F1)
119898a119872B = mg + N + (minus119898) (120596BE times (120596BE times r
119864119872))
+ (minus119898) (120572BE times r119864119872
)+(minus119898) (2120596BE times V119872B)
(F2)The vector differential equation (F1)-(F2) yields three
scalar ordinary differential equations (ODEs) for payload119872
swaying motionWe will project (F1) and (F2) to the axes 119909
1 1199101 and 119911
1in
the noninertial reference frameB
119898(
11988921199091
1198891199052) = minus119873(
1199091
119897
) + 119898(
119889120593119890
119889119905
)
2
1199091
+ 119898(
1198892120593119890
1198891199052) (119877 + 119910
1) + 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
)
119898(
11988921199101
1198891199052) = minus119873(
1199101
119897
) + 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
minus 119898(
1198892120593119890
1198891199052)1199091minus 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F3)
16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
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[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
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[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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16 Shock and Vibration
ℰ
ℱℬ
O2(E D)
y2
120593e(0) = 0 rad
2ẽ
1ẽ
x2y1y
x1
e2
e1
x
ydyn
Adyn
VMℬ(0)
120596ℬℰ(0)
120596ℬℰ(0) times r
r
O2M(0)
B Ast M
t0 = 0 s
O2M(0) = R
2e
1e
Figure 11 The computational scheme for initial conditions assignment
The derived system (F3) and (E5) is the nonlinear ODEsystem The nonlinearity of (F3) is determined by the pres-ence of the unknown function 119873 = 119873(119905) variable boomslewing angle 120593
119890 variable boom slewing angular velocity
119889120593119890119889119905 and variable angular acceleration 119889
21205931198901198891199052
In order to verify the correctness of the derived system(F3) we will utilize second-kind Lagrange equations Takinginto account equations (A10) (A11) and (A17) for thesquare of absolute payload119872 velocity (V
119872E V119872E) and byadding the kinetic energy (see [59 76 81 82 84 85 87 90ndash92 94ndash98]) for a slewing crane boom according to (E2)-(E3) we will have the following expression for ldquocrane boom-payloadrdquo kinetic energy
119879 =
119898
2
(((
1198891199091
119889119905
) minus (
119889120593119890
119889119905
) (119877 + 1199101))
2
+((
1198891199101
119889119905
) + (
119889120593119890
119889119905
) 1199091)
2
+ (
1198891199111
119889119905
)
2
)
+
1198681198742
33
2
(
119889120593119890
119889119905
)
2
(F4)
Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N and the equations forgeometric constraints (C1)ndash(C5) we have the following for-mulae for the generalized forces in the noninertial referenceframeB
1198761199091
= minus119873(
1199091
119897
)
1198761199101
= minus119873(
1199101
119897
)
1198761199111
= +119873(
119897 minus 1199111
119897
) minus 119898119892
(F5)
We will derive the same nonlinear differential equations(F3) for relative system motion with an introduction of
the following Lagrange equations (see [59 76 81 82 84 8587 90ndash92 94ndash98]) in the noninertial reference frameB
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199091
= 1198761199091
119889
119889119905
(
120597119879
120597 1199101
) minus
120597119879
1205971199101
= 1198761199101
119889
119889119905
(
120597119879
1205971
) minus
120597119879
1205971199111
= 1198761199111
(F6)
Taking into account (F4)-(F5) (F6) in the noninertialreference frameB will finally take the following form
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091minus 119898(
1198892120593119890
1198891199052) (119877 + 119910
1)
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119898(
1198892120593119890
1198891199052)1199091
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(F7)
The derived ODE system (F7) coincides with the system(F3) which confirms the accuracy and correctness of the fullynonlinear equations (F3) in relative Cartesian coordinates forpayload119872 swayingmotion in the noninertial reference frameB
G Uniform Crane Boom RotationAssumption The Introduction ofthe Noninertial Reference Frame F
We now address the case of uniform crane boom slewingWeassume that the crane boom 119861119874
2rotates with the constant
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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International Journal of
Shock and Vibration 17
angular velocity 120596119890around vertical axis119863119874
2 that is that the
noninertial reference frame B uniformly rotates around theunit vector e
3of the inertial reference frame E Such case
takes place when the right-hand side of (E5) is zero that isfor the steady state of crane boom rotation with
1198892120593119890
1198891199052
= 0
119889120593119890
119889119905
= 120596119890= const (G1)
So taking into account (G1) the third terms containing11988921205931198901198891199052 vanish in the 1st and 2nd equations of system (F7)
The second terms of (F7) are linearly coordinate-dependenton 1199091 1199101 and the forth terms of (F7) are linearly velocity-
dependent on 1198891199091119889119905 and 119889119910
1119889119905 So in the noninertial
reference frameB we have
119898(
11988921199091
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
1199091
minus 2119898(
119889120593119890
119889119905
)(
1198891199101
119889119905
) = minus119873(
1199091
119897
)
119898(
11988921199101
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 1199101)
+ 2119898(
119889120593119890
119889119905
)(
1198891199091
119889119905
) = minus119873(
1199101
119897
)
119898(
11988921199111
1198891199052) = minus119898119892 + 119873(
119897 minus 1199111
119897
)
(G2)
The nonlinear system (G2) has been presented in thenoninertial reference frameB for the case of uniform craneboom slewing
For further physical analysis of the swaying problem andfor ease of building the analytical solution we introduce thenoninertial reference frameF The origin of the noninertialreference frame F we connect with the so-called point119860dyn of dynamic equilibrium for load 119872 where the 119910-distance between the noninertial references frames B andF we define through the numerical solution of the followingrelative equilibrium transcendental equation
119898119892119897 sdot sin (1205721dyn) = 119898120596
2
119890(119877 + 119897 sdot sin (120572
1dyn)) sdot 119897 sdot cos (1205721dyn) (G3)
This equation (G3) has been derived as the momentumsum for the gravitational force mg and the normal inertialforceΦe
n about point 119861 that is119872119861(mg) = 119872
119861(Φe
n)
Taking into account the numerically derived angle 1205721dyn
from (G3) we find the value of the horizontal distance 119910dynbetween points 119860 st and 119860dyn (Figures 8ndash10) according to thefollowing equation
119910dyn = 119860 st119860dyn = 119897 sin (1205721dyn) (G4)
Relative coordinates 119909 119910 and 119911 in noninertial referenceframe F have been connected with the relative coordinates
1199091 1199101 and 119911
1in noninertial reference frameB according to
the following equations (Figures 8ndash10)
1199091= 119909
1199101= 119910dyn + 119910
1199111= Δ + 119911
(G5)
With (G3)ndash(G5) the nonlinear system (G2) in thenoninertial reference frame F takes the following form(Figures 8ndash10)
119898(
1198892119909
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
119909
minus 2119898(
119889120593119890
119889119905
)(
119889119910
119889119905
) = minus119873(
119909
119897
)
119898(
1198892119910
1198891199052) minus 119898(
119889120593119890
119889119905
)
2
(119877 + 119910dyn + 119910)
+ 2119898(
119889120593119890
119889119905
)(
119889119909
119889119905
) = minus119873(
119910dyn + 119910
119897
)
119898(
1198892119911
1198891199052) = minus119898119892 + 119873(
119897 minus Δ minus 119911
119897
)
(G6)
H The Small Swaying Angle 1205721
Assumption
We now address the case of the small swaying angle 1205721 In
the case of small swaying angle 1205721the system (C3) defines 119909
1
and 1199101as the small variables and 119911
1= 0 Having 119911
1= 0 we
conclude that Δ = 0 and all time derivatives are zero that isthe vertical load velocity 119889119911
1119889119905 = 0 and the vertical payload
acceleration 119889211991111198891199052= 0
Thus the 3rd equation of system (F7) yields that the cablereaction force N approximately coincides the gravitationalforce mg that is N asymp mg As a result the system (G2)containing three ODEs transforms into a linearized systemwith two independent equations for the relative Cartesiancoordinates 119909
1and 119910
1in the noninertial reference frame B
We then cancel the mass 119898 of load 119872 from the system of(G6) Consider
11988921199091
1198891199052
minus (
119889120593119890
119889119905
)
2
1199091+ 119892(
1199091
119897
) minus 2 (
119889120593119890
119889119905
)(
1198891199101
119889119905
) = 0
11988921199101
1198891199052
minus (
119889120593119890
119889119905
)
2
(119877 + 1199101) + 119892 (
1199101
119897
) + 2 (
119889120593119890
119889119905
)(
1198891199091
119889119905
) = 0
(H1)
So we have derived the system (H1) of differentialequations for relative motion of load 119872 on cable 119861119872 with amovable suspension center 119861 which is attached to the craneboom 119861119874
2in the noninertial reference frame B We then
transfer the origin of coordinate system 119874119909119910119911 (Figures 8ndash10)to the point 119860dyn of dynamic equilibrium for load119872 that is
18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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Shock and Vibration
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18 Shock and Vibration
we make the transition from the noninertial reference frameB to the noninertial reference frameF and simplify (G5) to
1199091= 119909
1199101= 119910dyn + 119910
1199111= 119911
(H2)
The second equation of system (H1) defines the amountof dynamic deflection that is the 119910-distance between thenoninertial reference framesB andF as
119910dyn = 119860 st119860dyn =1205962
119890119877119897
119892 minus 1205962
119890119897
= minus
1205962
119890119877
(1205962
119890minus (119892119897))
(H3)
where 120596119890= 119889120593119890119889119905
Then with the introduction of (H2) and (H3) into (H1)we have the normal system of two linear homogeneousdifferential equations of second order for relative motion ofthe load119872 in the noninertial reference frameF
1198892119909
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119909 minus 2(
119889120593119890
119889119905
)(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus ((
119889120593119890
119889119905
)
2
minus (
119892
119897
))119910 + 2(
119889120593119890
119889119905
)(
119889119909
119889119905
) = 0
(H4)
The same system in the noninertial reference frameF canbe derived from (G6) for the case of the small swaying angle1205721 Taking into account that 120596
119890= 119889120593119890119889119905 = const we will
write (H4) as
1198892119909
1198891199052minus (1205962
119890minus (
119892
119897
)) 119909 minus 2120596119890(
119889119910
119889119905
) = 0
1198892119910
1198891199052minus (1205962
119890minus (
119892
119897
)) 119910 + 2120596119890(
119889119909
119889119905
) = 0
(H5)
Taking into account (C6) the absolute coordinates11990921199102
and 1199112in inertial reference frameE which depend upon the
relative coordinates 119909 119910 and 119911 in the noninertial referenceframe F during the swaying of load 119872 will be definedaccording to the following equations (Figures 8ndash10)
1199092= (119877 + 119910dyn + 119910) cos (120593
119890) + 119909 sin (120593
119890)
1199102= (119877 + 119910dyn + 119910) sin (120593
119890) minus 119909 cos (120593
119890)
1199112= 119911
(H6)
So the linearized ODE system (H4)-(H5) has beenderived for the case of uniform crane boom slewing (G1) andthe small swaying angle 120572
1(H2)
It is important to note that the left-hand sides of payloadmotion equations in linearized problem (H4)-(H5) for thenoninertial reference frameFmay be derived with an intro-duction of Lagrange equations (F6) However the discussionof (H4)-(H5) is more suitable and informative with anintroduction of dynamic Coriolis theorem (F1)-(F2) Thefirst terms 11988921199091198891199052 11988921199101198891199052 in (H4)-(H5) define the vector
a119872F = ar (B4) for the relative acceleration of load 119872 inthe noninertial reference frame F The straight-line terms(120596119890
2minus (119892119897))119909 and (120596
119890
2minus (119892119897))119910 in (H4)-(H5) are linear
proportional to the relative payload coordinates in the nonin-ertial reference frameF These straight-line terms have beendetermined by the contribution of the normal or centripetalacceleration120596BEtimes(120596BEtimesr119864119872) = aen of transportation forpayload 119872 and by appearance of corresponding DrsquoAlembertcentrifugal inertia force Φe
n= (minus119898)aen due to crane boom
transport rotation in the noninertial reference frame BThe third terms minus2120596
119890(119889119910119889119905) and 2120596
119890(119889119909119889119905) in (H4)-(H5)
have been defined by the compound or Coriolis acceleration2120596BE times V
119872B = acor of payload 119872 in the noninertialreference frame F The rectangular Cartesian projections ofCoriolis inertia force in the noninertial reference frameF aredefined by formula (D5)
I Analytical Solution of the Linearized System
After generation of the determinant of natural frequenciesmatrix for the system (H4)-(H5) we will write the followingcharacteristic biquadratic equation of fourth order for system(H4)-(H5) in the form
1205824+ 2 ((
119892
119897
) + 1205962
119890) 1205822+ ((
119892
119897
) minus 1205962
119890)
2
= 0 (I1)
Using (I1) we adjust the roots 1205821and 120582
2of the secular
equation
1205821= plusmn]1119894 120582
2= plusmn]2119894
]1= 119896 + 120596
119890 ]
2= 119896 minus 120596
119890
119896 = radic
119892
119897
120596119890
= 119896
(I2)
Taking into account (I2) the law of relative load 119872
motion in the noninertial reference frame F takes thefollowing form
119909 (119905) = 1198621sin (]1119905) + 119862
2cos (]
1119905)
+ 1198623sin (]2119905) + 119862
4cos (]
2119905)
119910 (119905) = 1198621cos (]
1119905) minus 119862
2sin (]1119905)
minus 1198623cos (]
2119905) + 119862
4sin (]2119905)
(I3)
Time derivatives of (I3) yield the following projections ofrelative payload119872 velocity in the noninertial reference frameF
119889 (119909 (119905))
119889119905
= +1198621]1cos (]
1119905) minus 119862
2]1sin (]1119905)
+ 1198623]2cos (]
2119905) minus 119862
4]2sin (]2119905)
119889 (119910 (119905))
119889119905
= minus1198621]1sin (]1119905) minus 119862
2]1cos (]
1119905)
+ 1198623]2sin (]2119905) + 119862
4]2cos (]
2119905)
(I4)
Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
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[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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Shock and Vibration 19
In order to determine the arbitrary constants of integra-tion 119862
1 119862
4in (I3)-(I4) we define the initial conditions
of the problem according to Figures 8ndash10 and Figure 11In the initial time 119905 = 0 load 119872 has the vertical 119861119860 st in-
line position (Figures 8ndash10) that is the load119872 initial positioncoincides with the static equilibrium position119860 st for the load119872 on the cable119872119861This means that the initial coordinates ofpayload119872 are as follows
point 119860 st (119877 0 0) in the reference frame E (I5)
point 119860 st (0 0 0) in the reference frame B (I6)
point 119860 st (0 minus119910dyn 0) in the reference frameF (I7)
At time 119905 = 0 the load119872 has zero absolute velocity in theinertial reference frame E that is V
119872E(0) = 0In order to determine the load 119872 relative velocity in the
noninertial reference frame F which is equal to the load119872 relative velocity in the noninertial reference frameB weaddress the velocity addition theorem (A12) and formula(A14)
0 = V119872B (0) + 120596BE (0) times r
1198742119872
(0) (I8)
The vector equation (I8) determines the initial relativevelocityV
119872B(0) of the load119872 in the inertial reference frameE as
V119872B (0) = V
119872F (0) = minus120596BE (0) times r1198742119872
(0)
= minus120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e2
(I9)
The direction of the vector for the initial relative velocityV119872B(0) is directed opposite to the unit vector e
2in the
inertial reference frameE that is V119872B(0) uarrdarr e
2 The initial
direction of the vector V119872B(0) is invariant in all reference
systems E B and F So in the noninertial reference frameB we have V
119872119861(0) uarruarr e
1
V119872B (0) = V
119872F (0) = +120596BE (0) times
10038171003817100381710038171003817r1198742119872
(0)
10038171003817100381710038171003817e1 (I10)
Taking into account that 120596BE(0) = 120596119890
andr1198742119872
(0) = 119877 in (I10) we have the following scalarequation in the noninertial reference framesB andF
V119872B (0) = V
119872F (0) =
119889119909
119889119905
(0) = 120596119890119877 (I11)
Thus with an introduction of (I5)ndash(I11) the initialconditions for ODE system (H4)-(H5) in the noninertialreference frameF are as follows
119909 (0) = 0
119889119909
119889119905
(0) = 120596119890119877
119910 (0) = minus119910dyn119889119910
119889119905
(0) = 0
(I12)
The substitution of (I12) into (I3) and (I4) yields thefollowing system for derivation the analytical expressions forthe arbitrary constants of integration 119862
1 119862
4
0 = 1198622+ 1198624
119881119861= 1198621]1+ 1198622]2
minus119910dyn = 1198621minus 1198623
0 = minus1198622]1+ 1198624]2
(I13)
The algebraic expressions for the arbitrary constantsof integration 119862
1 119862
4can be derived as the following
solution of the system (I13)
1198621=
119881119861minus 119910dyn]2]1+ ]2
1198622= 0
1198623=
119881119861+ 119910dyn]1]1+ ]2
1198624= 0
or 1198621=
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
1198622= 0
1198623=
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
1198624= 0
(I14)
So the relative motion equations for pendulum119872 sway-ing take the final form as stated below
119909 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) sin (]1119905)
+ (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) sin (]2119905)
119910 (119905) = (
119881119861minus 119910dyn (119896 minus 120596
119890)
2119896
) cos (]1119905)
minus (
119881119861+ 119910dyn (119896 + 120596
119890)
2119896
) cos (]2119905)
(I15)
The obtained analytical equations (I15) derived for thelinearized model (H4)-(H5) determine the shape of the rel-ative payload 119872 swaying trajectory as is shown in Figure 12Numerical computations in Figure 12 have been carried outfor the following values of mechanical system parameters119877 = 0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)
05asymp
3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120593
119890= 180
∘1205721dyn = 000221 rad 119881
119861= 0103ms 119910dyn = 000182m
]1= 119896 + 120596
119890= 36578 rads ]
2= 119896 minus 120596
119890= 32388 rads
1198621
= 001408m and 1198623
= 001591m (Figure 12) Theabove mentioned numerical values of analytical relations(I15) subject to the initial conditions (I5)ndash(I12) define atheoretically computed law of relative motion for load 119872After the elimination of variable 119905 from the system of (I15)we will have the relative trajectory for load119872 (Figure 12)Thesystem (H6) and the computational relative trajectories forload 119872 swaying in Figure 12 show that there is the contra-rotation of the noninertial reference framesB andF to craneboom 119861119874
2slewing
20 Shock and Vibration
minus0002minus001minus002 001 002
0002
0007
x
y
t1 = 125 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
Relative trajectory of load M swayingfor linearized model
(a)
0 001 002
x
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
minus0005
minus001minus002
0005
0010
minus0010
t2 = 3s
Relative trajectory of load M swayingfor linearized model
(b)
minus001
minus001
minus002
minus002
001
0010
002
002
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
Relative trajectory of load M swayingfor linearized model
(c)
minus001minus002 0010 002
x
minus001
minus002
001
002
y
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(d)
Figure 12The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with linearized model introduction
J Velocity Kinematics Analysis inRelative Spherical Coordinates ofthe Noninertial Reference Frame BThe Introduction of the NoninertialReference Frames C D and G
The fully nonlinear equations (F7) in relative Cartesiancoordinates of the noninertial reference frame B have theunknown function 119873 = 119873(119905) that is undeterminedLagrangian coefficient The reaction forceN of the cable 119861119872in Figures 8ndash10 and Figure 13 is the nonconservative force
So for derivation of solvable nonlinear equations weshould remove the cable tension forceN from the payload119872motion equations with the introduction of Newtonrsquos secondlaw and the natural noninertial comoving reference framesC D and G which correspond to the spherical angularcoordinates (120572
1 1205722 and 120593
119890) and moving together with
payload119872 (Figure 13) The advantage of the introduction ofthe natural noninertial comoving reference framesCD andG is grounded on the orthogonality of the reaction forceN tothe unit vectors e
1205911205721
and e1205911205722
that is on the fact thatN perp e1205911205721
and N perp e1205911205722
The vector equation (A12) for velocity addition will take
the form
V119872E = V
119872C + V119872D + 120596BE times r
1198742119872
(J1)
where V119872C = V
1205721
and V119872D = V
1205722
are the relative payloadvelocities in the natural noninertial comoving referenceframesC andD
In Figure 13 and by using (J1) we have that the naturalcomponents of payload119872 velocity in the natural noninertial
Shock and Vibration 21
O2(D E )
ℰ
ℬ
ℰ
y1
e120591120593
ℬ
120593e
y2
x2
2ẽ
119970
1ẽ
119967
119967
119966
119966
119970
120579
120579
x1
1205722
1205722
1205722
1205721
119966
en1205721en1205722
M
1205722V
1205721V
V120593
eb1205721en120593
eb1205722 eb120593
e1205911205722
e1205911205721cos 1205721
cos 1205721
sin 1205721
B(O1)
1e
2e
Figure 13 Swaying scheme for velocity kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached inthe point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
comoving reference framesCD andG have the followingvalues
1198811205721
= 119897
1198891205721
119889119905
1198811205722
= 119897 sin (1205721)
1198891205722
119889119905
119881120593= 1198742119872
119889120593119890
119889119905
(J2)
K Acceleration KinematicsAnalysis in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
The vector equation (B1) for acceleration addition will takethe following form (Figure 14)
a119872E = a
119872C + a119872D + 2120596CE times V
119872C
+ 2120596DE times V119872D
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K1)
where acor(V119872C) = acor(V1205721
) = 2120596CEtimesV119872C = 2120596CEtimesV1205721
and acor(V119872D) = acor(V1205722
) = 2120596DE timesV119872D = 2120596DE timesV
1205722
in Figure 14
So formula (K1) will take the following form
a119872E = a
119872C + a119872D + acor (V120572
1
) + acor (V1205722
)
+ 120596DE times (120596DE times r1198742119872
) + 120572DE times r1198742119872
(K2)
The directions of vectors have been shown in Figures 1314 and 15 The scalar values of vectors 120596CE and 120596DE are asfollows
120596CE =
119889120593119890
119889119905
+
1198891205722
119889119905
120596DE =
119889120593119890
119889119905
(K3)
Taking into account (K3) the magnitudes of Coriolisacceleration vectors in the natural noninertial comovingreference framesCD andG in Figure 14 have the followingvalues
acor (1198811205721
) = 2119897 (
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721) (K4)
acor (1198811205722
) = 2119897 (
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721) (K5)
22 Shock and Vibration
ℰ
ℰ
y2 x1
O2(D E )
y1
x2
1ẽ
120579
120579
120579
1205721
ℬ
ℬ
2ẽ
120593e1205721
1205722
1205722
1205722
M
1198341205911205722
1198341205911205721
119834n1205722
119834n1205721
cos 1205721
sin 1205721
119834120591120593
119834n120593B(O1)
119834cor(V1205721)
119834cor(V1205722)
2e
1e
Figure 14 Swaying scheme for acceleration kinematics analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attachedin the point 119861 to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
y1
ℬ
y2
ℰ
ℰx2
120593e
1205791ẽ
ℬ
1205722
1205722
1205722
2ẽ
N
x1
cos 1205721
120579
120579
120579
120579
MΦ1205911205721
Φ1205911205722
Φ120591120593
Φn120593
Φn1205722
Φn1205721sin 1205721
B(O1)
Φcor(V1205722)
Φcor(V1205721)
O2(D E)
2e
1e
Figure 15 Swaying scheme for dynamic analysis in spherical coordinates of load119872 at the cable119872119861 which is fixedly attached in the point 119861to the crane boom 119861119874
2 in the horizontal plane (119909
11199101) for nonlinear model derivation in relative spherical coordinates
Shock and Vibration 23
The tangential accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861205911205721
= (
11988921205721
1198891199052) 119897
1198861205911205722
= (
11988921205722
1198891199052) 119897 sin (120572
1)
119886120591120593
= (
1198892120593119890
1198891199052) (1198742119872)
(K6)
The normal accelerations in the natural noninertialcomoving reference framesCDG have the values as statedbelow
1198861198991205721
= (
1198891205721
119889119905
)
2
119897
1198861198991205722
= (
1198891205722
119889119905
)
2
119897 sin (1205721)
119886119899120593
= (
119889120593119890
119889119905
)
2
(1198742119872)
(K7)
L Forces Imposed on the Payload in RelativeSpherical Coordinates in the NoninertialReference Frames C D and G
Among the forces imposed on the payload 119872 in Figure 15we have an active force of gravitymg the cable reaction forceN the tangential inertial forces Φ
1205911205721
Φ1205911205722
Φ120591120593 the normal
or centrifugal inertial forcesΦ1198991205721
Φ1198991205722
Φ119899120593 and the Coriolis
inertial forcesΦcor(1198811205721
) andΦcor(1198811205722
)Taking into account formulae (K3)ndash(K7) we will express
below all imposed forces in the noninertial reference framesCDG
Φ1205911205721
= 1198981198861205911205721
= 119898(
11988921205721
1198891199052) 119897
Φ1205911205722
= 1198981198861205911205722
= 119898(
11988921205722
1198891199052) 119897 sin (120572
1)
Φ120591120593
= 119898119886120591120593
= 119898(
1198892120593119890
1198891199052)100381710038171003817100381711987421198721003817100381710038171003817
Φ1198991205721
= 1198981198861198991205721
= 119898(
1198891205721
119889119905
)
2
119897
Φ1198991205722
= 1198981198861198991205722
= 119898(
1198891205722
119889119905
)
2
119897 sin (1205721)
Φ119899120593
= 119898119886119899120593
= 119898(
119889120593119890
119889119905
)
2
100381710038171003817100381711987421198721003817100381710038171003817
Φcor (1198811205721
) = 119898119886cor (1198811205721
)
= 2119898119871(
1198891205721
119889119905
)(
119889120593119890
119889119905
+
1198891205722
119889119905
) cos (1205721)
Φcor (1198811205722
) = 119898119886cor (1198811205722
)
= 2119898119871(
1198891205722
119889119905
)(
119889120593119890
119889119905
) sin (1205721)
(L1)
M Derivation of the Fully NonlinearEquations in Relative SphericalCoordinates in the Noninertial ReferenceFrames C D and G
Taking into account derived expressions (L1) the introduc-tion of DrsquoAlembertrsquos principle in the noninertial referenceframes C D G yields the following scalar equations in theprojections to the axes e
1205911205721
and e1205911205722
e1205911205721
0 = minusΦ1205911205721
+ Φ1198991205722
cos (1205721) + Φcor (119881120572
2
) cos (1205721)
+ Φ119899120593sin (120572
2+ 120579) cos (120572
1)
+ Φ120591120593cos (120572
2+ 120579) cos (120572
1) minusmg sin (120572
1) = 0
e1205911205722
0 = minusΦ1205911205722
minus Φcor (1198811205721
)
+ Φ119899120593cos (120572
2+ 120579) minus Φ
120591120593sin (120572
2+ 120579)
(M1)
where
sin (120579) =119897 sin (120572
1) cos (120572
2)
100381710038171003817100381711987421198721003817100381710038171003817
cos (120579) =(119877 + 119897 sin (120572
1) sin (120572
2))
100381710038171003817100381711987421198721003817100381710038171003817
(M2)
N Uniform Crane Boom Rotation Assumptionwith Nonlinear Equations in RelativeSpherical Coordinates in the NoninertialReference Frames B
Assuming formulae (G1) we will have the following non-linear ODE system in the noninertial reference frame B
24 Shock and Vibration
0010 002minus001minus002
0008
0005
0002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t1 = 125 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(a)
0 002001
0010
0015
0005
minus001minus002minus0005
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t2 = 3s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t3 = 10 s
0
y
x
001
001
002
003
002minus001
minus001
minus002
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(c)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 0209 rads
t4 = 25 s
0 001 002minus001minus002
001
002
003
minus001
minus002
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(d)
Figure 16The relative trajectories for load119872 swaying on the cable during uniform rotationalmotion of the crane boom1198611198742for themoments
of times 1199051= 125 s (a) 119905
2= 3 s (b) 119905
3= 10 s (c) and 119905
4= 25 s (d) derived with nonlinear model introduction
describing payload119872 relative swaying in the case of uniformcrane boom slewing
(
11988921205721
1198891199052) minus
1
2
(
1198891205722
119889119905
)
2
sin (21205721) minus
1
2
1205962
119890sin (2120572
1)
minus (
119877
119897
)1205962
119890cos (120572
1) sin (120572
2) = minus (
119892
119897
) sin (1205721)
(
11988921205722
1198891199052) sin (120572
1) + 2 (
1198891205721
119889119905
)(120596119890+ (
1198891205722
119889119905
)) cos (1205721)
minus (
119877
119897
)1205962
119890cos (120572
2) = 0
(N1)
The initial conditions for the nonlinear system (N1)may be formulated on the basis of the initial conditions in
the noninertial reference frame F and with introduction ofconstraint equations (C3)
119897 sin (1205721(0)) cos (120572
2(0)) = 0
119897 sin (1205721(0)) sin (120572
2(0)) = minus119910dyn
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) cos (120572
2(0))
minus 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) sin (120572
2(0)) = 120596
119890119877
119897 (
1198891205721(0)
119889119905
) cos (1205721(0)) sin (120572
2(0))
+ 119897 (
1198891205722(0)
119889119905
) sin (1205721(0)) cos (120572
2(0)) = 0
(N2)
Shock and Vibration 25
minus002
minus002
minus004
minus004
minus006
minus006
minus008
minus008
y
x
008
008
006
006
004
004
002
0 002
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
(a)
minus002minus004minus006minus008
minus002
minus004
minus006
008
006
004
002
0080060040020
y(x) l = 0825m k asymp 3448 rads 120596e asymp 06 rads
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(b)
minus005
minus005
minus010
minus010
minus015
minus015
015
015
010
010
005
0 005
y
x
Relative trajectory of load M swayingfor linearized model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
(c)
minus005minus010minus015
minus01
005 010 015
02
01
0
Relative trajectory of load M swayingfor nonlinear model
t4 = 25 s
y(x) l = 0825m k asymp 3448 rads 120596e asymp 12 rads
0825 lowast sin(1205721) lowast cos(1205722)
(d)
minus01
minus01
minus02
minus02
minus03
minus03
03
02
01
0201 030
y
x
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(e)
03
02
02
01
010
Relative trajectory of load M swayingfor nonlinear model
y(x) l = 0825m k asymp 3448 rads 120596e asymp 18 rads
minus01
minus01
minus02
t4 = 25 s
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 17 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 2871 (a-b) 120596
1198901205961198900asymp 5742 (c-d) and 120596
1198901205961198900asymp 8612 (e-f)
26 Shock and Vibration
minus02
minus02
minus04
minus04
minus06
x
y
02
0 02
04
04
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
(a)
minus02minus01
minus02
minus03
minus04 0
05
04
03
02
01
06
0402
y(x) l = 0825m k asymp 3448 rads 120596e asymp 24 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
0825 lowast sin(1205721) lowast cos(1205722)
(b)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus02minus02
minus04
minus04
minus06
minus06
minus08
minus08
y
x
06
06
04
04
02
020
08
08
(c)
04
04
02
020
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 30 rads
Relative trajectory of load M swayingfor nonlinear model
minus04
minus02
minus02minus04minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
t4 = 25 s
(d)
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor linearized model
minus05
minus1
minus12
minus10
minus08
minus06
minus04
minus02
y
x
05
1
06
04
020
(e)
04
04
02
0 02
06
06
08
y(x) l = 0825m k asymp 3448 rads 120596e asymp 36 rads
t4 = 25 s
Relative trajectory of load M swayingfor nonlinear model
minus02
minus02
minus04
minus04
minus06
minus06
0825 lowast sin(1205721) lowast cos(1205722)
(f)
Figure 18 The comparison of relative trajectories of load119872 swaying for linearized (a) (c) (e) and nonlinear (b) (d) and (f) models in thecases of 120596
1198901205961198900asymp 11483 (a-b) 120596
1198901205961198900asymp 14354 (c-d) and 120596
1198901205961198900asymp 17225 (e-f)
Shock and Vibration 27
The solution of the system (N2) defines the initialconditions for the nonlinear ODE system (N1) for payloadspherical coordinatesrsquo 120572
1and 120572
2 The solution set member of
the system (N2) has the following form
1205721(0) = arcsin(
119910dyn
119897
)
1205722(0) = minus
120587
2
1198891205721(0)
119889119905
= 0
1198891205722(0)
119889119905
=
120596119890119877
119910dyn
(N3)
The numerical solution of the Cauchy problem (N1)ndash(N3) for the nonlinear problem in spherical coordinatesdetermines the shape of the relative payload 119872 swayingtrajectory as is shown in Figures 16 17(b) 17(d) 17(f) 18(b)18(d) and 18(f) Numerical plots in Figure 16 have beencarried out for the following values of mechanical systemparameters 119877 = 0492m 119892 = 981ms2 119897 = 0825m119896 = (119892119897)
05asymp 3448 rads 119879 = 30 s 120596
119890= 2120587119879 asymp 0209 rads
120593119890
= 180∘ 1205721(0) asymp 000221 rad 120572
2(0) asymp minus157079 rad
119881119861= 0103ms 119910dyn = 000182m 119889(120572
1(0))119889119905 = 0 rads
and 119889(1205722(0))119889119905 = 56565608 rads (Figure 16)
O Comparative Estimation andDiscussion of the Amplitude ResponsesComputed from the Linear andNonlinear Models
For comparison of the linear (H4)-(H5) and nonlinear (N1)models we introduce the relative dimensionless amplitudediscrepancy 120575
1between averaged radii 119903nonlinear (Figures
17(b) 17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized(Figures 17(a) 17(c) 17(e) 18(a) 18(c) and 18(e)) of compu-tational relative trajectories of payload 119872 in the noninertialreference frameF (Figures 17-18)
1205751= ⟨
1
2
((
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903nonlinear)
+(
1003816100381610038161003816119903nonlinear minus 119903linearized
1003816100381610038161003816
119903linearized))⟩
(O1)
The results of numerical amplitude comparison com-puted from the linear and nonlinear models in accordancewith the formula (O1) and Figures 12 16 17 and 18 are shownin Figure 19
The computational relative trajectories of load 119872 in thenoninertial reference frame F shown in Figures 12 16 17and 18 outline that both averaged radii 119903nonlinear (Figures 17(b)17(d) 17(f) 18(b) 18(d) and 18(f)) and 119903linearized (Figures 17(a)17(c) 17(e) 18(a) 18(c) and 18(e)) and amplitudes of relativetrajectories increase with increasing dimensionless craneboom slewing velocity 120596
1198901205961198900 where 120596
1198900asymp 0209 rads is
10
10 12 14 162 4 6 8
20
30
40
50
Dimensionless crane boom slewing velocity 120596e120596e0
Am
plitu
de d
iscre
panc
y1205751
()
l = 0825m k asymp 3448 rads
Figure 19 The computational points (IIII) and regressiondiagram (mdash) for the relative dimensionless amplitude discrepancy1205751between averaged radii of computational relative trajectories
of payload 119872 in the noninertial reference frame F derived forlinearized and nonlinear models in the case of cable length 119897 =
0825m
the experimentally chosen technological crane boom slewingvelocity
It is shown in Figures 12 16 17 and 18 that there is apersistent trend 119903linearized gt 119903nonlinear
The computational diagram in Figure 19 shows that therelative dimensionless amplitude discrepancy 120575
1between
the linear and nonlinear models increases with increasingdimensionless crane boom slewing velocity 120596
1198901205961198900 It is
shown in Figure 19 that 1205751lt 11 for the 120596
119890lt 08 rads that
is for 120596119890lt 3828 120596
1198900 It is shown in Figure 19 that 120575
1lt 11
for the 1205721lt 0145 rad
So we draw a conclusion that the small angle assumptionfor the linearized model (H4)-(H5) adequately describesnonlinear load swaying problem (N1) during uniform craneboom slewing for the above mentioned limiting values of 120596
119890
and 1205721
Nomenclature
DOF Degree of freedomODE Ordinary differential equationDAE Differential algebraic equationpoint 119860 st Point of static equilibrium of payload
119872 at the cable 119861119872point 119860dyn Point of dynamic equilibrium of
payload119872 at the cable 119861119872point119872 Point of position of payload119872 in the
current moment of time
28 Shock and Vibration
point 119864 Initial point of crane boom 119861119864
point 119861 Terminal point of crane boom 119861119864
point119863 Center of circle for trajectory oftransport and absolute motion for thepoint 119861
119898 Mass of payload119872 (kg)119892 Scalar value of gravitational
acceleration (ms2)120587 Ratio of perimeter of circle to
diameter1198621 1198622 1198623and 119862
4 Constants of integration (m)
1205821 1205822 Roots of secular equation
119905 Swaying time for payload119872 (s)E Inertial reference frame (119874
2119909211991021199112)
fixed on earthe1 e2 e3 Orthogonal unit vectors defining the
inertial reference frame E1199092 1199102 1199112 The absolute coordinates of payload
119872 with respect to Earth (m)B Noninertial reference frame
(1198741119909111991011199111) which is connected with
rotating crane boom 1198742119861 and the
origin of coordinates point 1198741
coincides with point 119860 st of staticequilibrium for load119872 at the cable119861119872
1199091 1199101 1199111 The relative coordinates of payload119872
with respect to crane boom 1198742119861 (m)
e1 e2 e3 The orthogonal unit vectors defining
the noninertial reference frameBF Noninertial reference frame (119874119909119910119911)
with coordinate axes parallel to(1198741119909111991011199111) and the origin of
coordinates point 119874 coincides withpoint 119860dyn of dynamic equilibriumfor load119872 at the cable 119861119872
e1 e2 e3 Orthogonal unit vectors defining the
noninertial reference frameFC Natural noninertial comoving
reference frame which correspondsto the swaying angle 120572
1and moving
together with payload119872e1205911205721
e1198991205721
e1198871205721
Orthogonal unit vectors defining thenoninertial reference frameC
D Natural noninertial comovingreference frame which correspondsto the swaying angle 120572
2and moving
together with payload119872e1205911205722
e1198991205722
e1198871205722
Orthogonal unit vectors defining thenoninertial reference frameD
G Natural noninertial comovingreference frame which corresponds tothe angle of crane boom rotation 120593
119890
and moving together with payload119872
e120591120593 e119899120593 e119887120593 Orthogonal unit vectors defining the
noninertial reference frameGr119861119872
Position vector connecting initialpoint 119861 and terminal point119872 innoninertial reference frameB (m)
r1198742119861 Position vector connecting initial
point 1198742and terminal point 119861 in
noninertial reference frameB (m)119877 = 119861119863 Boom projection to the horizontal
plane (e1 e2) (m)
119897 = 119897119861119872
= r119861119872
Length of the cable that is themagnitude of the position vectorr119861119872
(m)119896 = (119892119897)
05 Natural frequency of undampedoscillations of payload119872 at the cable119861119872 (1s)
120591 = 2120587119896 Period of undamped oscillations ofpayload119872 at the cable 119861119872 (s)
120593119890= ang(e1 e2) Angle of rotation of noninertial
reference frameB around axis 11987421199112
of inertial reference frame E that isthe angle of transport rotation forcrane boom 119861119874
2around the vertical
axis 11987421199112 that is crane boom slewing
angle (rad)120574 Angle between the position vector
r1198742119861
and horizontal plane (e1 e2) or
(e1 e2) that is inclination angle of
crane boom 1198742119861 with horizontal
plane (e1 e2) or (e
1 e2) (rad)
1205721= ang(e3 r119861119872
) Swaying angle between the verticalaxis 119874
11199111and the position vector
r119861119872
that is the first sphericalcoordinate of spherical pendulum119872
(rad)1205722 Swaying angle between the planes
(11990911199111) and (119861119872119874
1) that is the
second spherical coordinate ofspherical pendulum119872 (rad)
120573 = ang(119860119905119861119860dyn) Angle of ascent of payload119872 over
dynamical equilibrium level 119860dyn(rad)
120579 Angle between the vertical planeconnected with crane boom 119874
2119861 and
the vertical plane connected withposition vector r
1198742119872
of payload119872in inertial reference frame E (rad)
ℎ = 119897(1 minus cos(1205721)) =
1199111
Vertical coordinate of payload innoninertial reference frameB (m)
1205721dyn Angular distance between the axis
119860 st1199111 and the line 119861119860dyn (rad)119910dyn = 119897 sin(120572
1dyn) Length of perpendicular from point119860dyn to the axis 119860 st1199111 (m)
Δ =
119897(1 minus cos(1205721dyn))
Vertical distance between the point119860 st and the point 119860dyn that is thevertical distance between thehorizontal planes (e
1e2) and (e
1e2)
of the noninertial reference framesBandF correspondingly (m)
E120596
B= 120596BE Angular velocity vector of reference
frameB with respect to frame E(rads)
Shock and Vibration 29
E120596
F= 120596FE Angular velocity vector of reference
frameF with respect to frame E(rads)
120596BE = 120596FE =
120596119890= 119889120593119890119889119905
Scalar of transport angular velocity ofreference framesB andF withrespect to frame E (rads)
120596BE = 120596FE =
120596119890= (119889120593
119890119889119905)e3
Transport angular velocity vector ofreference framesB andF withrespect to frame E (rads)
E120576
B= 120576BE =
120572BEAngular acceleration vector ofreference frameB with respect toframe E (rads2)
E120576
F= 120576FE =
120572FEAngular acceleration vector ofreference frameF with respect toframe E (rads2)
120572BE = 120572FE =
120576119890= 11988921205931198901198891199052
Scalar of transport angularacceleration of reference framesBandF with respect to frame E(rads2)
120576BE = 120576FE = 120576e =
(11988921205931198901198891199052)e3
Transport angular acceleration vectorof reference framesB andF withrespect to frame E (rads2)
EV119872 = V119872E =
VabsVelocity of point119872 in inertial fixedon earth reference frameE that isabsolute velocity of payload119872 (ms)
BV119872 = V119872B =
FV119872 = V119872F =
Vr
Velocity of point119872 in noninertialreference framesB andF that isrelative velocity of payload119872 (ms)
Ve = V1198742E +
120596BE times r1198742119872
Transport velocity of point119872 ininertial reference frame E (ms)
1198811199091
1198811199101
1198811199111
1199091- 1199101- 1199111-projections of payload119872
velocity in noninertial referenceframeB (ms)
119881119909 119881119910 119881119911 119909- 119910- 119911-projections of payload119872
velocity in noninertial referenceframeF (ms)
Ea119872 = a119872E = aabs Acceleration of point119872 in inertial
fixed on earth reference frameE thatis absolute acceleration of payload119872(ms2)
Ba119872 = a119872B =
Fa119872 = a119872F = ar
Acceleration of point119872 innoninertial reference framesB andF that is relative acceleration ofpayload119872 (ms2)
Ea119864 = a119864E =
Ea119863 = a119863E = 0
Acceleration of points 119864 and119863 ininertial fixed on earth referenceframe E that is the absoluteacceleration of points 119864 and119863located at rotation axis 119864119863 (ms2)
ae120591 = 120572BE times r119864119872
Tangential acceleration oftransportation for payload119872 (ms2)
119879 Kinetic energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
Π Potential energy of the system ldquocraneboom 119861119874
2-load119872rdquo (J = N-m)
1198761199091
1198761199101
1198761199111
Generalized forces (N)]1 ]2 First and second natural frequencies
of payload relative oscillations (1s)1205911 1205912 First and second periods of payload
relative oscillations (s)mg Gravitational force (N)N Reaction force of the cable 119861119872 (N)Φe120591= (minus119898)ae120591 Tangential inertial force (tangential
force of moving space) for payload119872(N)
Φen= (minus119898)ae120591 Normal or centrifugal inertial force
(normal force of moving space) forpayload119872 (N)
Φcor = (minus119898)acor Coriolis inertial force (compoundcentrifugal force) for payload119872 (N)
1198681198742
33 Element of mass moment of inertia
for the system ldquocrane boom 1198611198742-load
119872rdquo in inertial fixed on earthreference frame E with respect toaxis e3(kg-m2)
H11987423
= 1198681198742
33120596BE Moment of momentum for the
system ldquocrane boom 1198611198742-load119872rdquo
that is vector component of theangular momentum for ldquocrane boom1198611198742-load119872rdquo with respect to point
1198742in inertial fixed on earth reference
frame E (kg-m2s)sumM1198742 = 119872
1198742 e3 Vector of the resultant external
moment codirectional to axis e3with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
1198721198742
119863119879 Driving torque for the system ldquocrane
boom 1198611198742-load119872rdquo with respect to
point 1198742 where point 119874
2is fixed in
the inertial fixed on earth referenceframe E (N-m)
M1198742119863119879
= 1198721198742
119863119879e3 Vector of the driving torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
1198721198742
119865119879 Frictional torque for the system
ldquocrane boom 1198611198742-load119872rdquo with
respect to point 1198742 where point 119874
2is
fixed in the inertial fixed on earthreference frame E (N-m)
M1198742119865119879
= 1198721198742
119865119879e3 Vector of the frictional torque for the
system ldquocrane boom 1198611198742-load119872rdquo
codirectional to axis e3with respect
to point1198742 where point119874
2is fixed in
the inertial fixed on earth referenceframeE with respect to axis e
3(N-m)
30 Shock and Vibration
120575 Relative dimensionless amplitudediscrepancy between computationaland experimental absolutetrajectories of payload119872 in theinertial reference frame E
1205751 Relative dimensionless amplitude
discrepancy between averaged radiiof computational relative trajectoriesof payload119872 in the noninertialreference frameF derived forlinearized and nonlinear models
Disclosure
The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors
Conflict of Interests
The authors Alexander V Perig Alexander N StadnikAlexander I Deriglazov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper
References
[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002
[2] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003
[3] E M Abdel-Rahman and A H Nayfeh ldquo Two-dimensionalcontrol for ship-mounted cranes a feasibility studyrdquo Journal ofVibration and Control vol 9 no 12 pp 1327ndash1342 2003
[4] I Adamiec-Wojcik P Fałat A Maczynski and S WojciechldquoLoad Stabilisation in an a-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009
[5] A A Al-mousa A H Nayfeh and P Kachroo ldquoControl ofrotary cranes using fuzzy logicrdquo Shock and Vibration vol 10no 2 pp 81ndash95 2003
[6] A P Allan and M A Townsend ldquoMotions of a constrainedspherical pendulum in an arbitrarily moving reference framethe automobile seatbelt inertial sensorrdquo Shock and Vibrationvol 2 no 3 pp 227ndash236 1995
[7] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquo Computer Methods in Applied Mechanics and Engi-neering vol 170 no 3-4 pp 343ndash353 1999
[8] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009
[9] D Blackburn J Lawrence J Danielson W Singhose T Kamoiand A Taura ldquoRadial-motion assisted command shapers for
nonlinear tower crane rotational slewingrdquo Control EngineeringPractice vol 18 no 5 pp 523ndash531 2010
[10] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo JVC Journal of Vibration andControl vol 16 no 4 pp 477ndash501 2010
[11] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004
[12] W Blajer and K Kołodziejczyk ldquoDynamics and control ofrotary cranes executing a load prescribed motionrdquo Journal ofTheoretical and Applied Mechanics vol 44 no 4 pp 929ndash9482006
[13] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007
[14] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008
[15] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing a loadprescribed motionrdquoThe Archive of Mechanical Engineering vol56 no 3 pp 209ndash222 2009
[16] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010
[17] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011
[18] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-Body Dynamics vol 224 no 4 pp 365ndash373 2010
[19] C Chin A H Nayfeh and E Abdel-Rahman ldquoNonlineardynamics of a boom cranerdquo Journal of Vibration and Controlvol 7 no 2 pp 199ndash220 2001
[20] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002
[21] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007
[22] T Erneux ldquoStrongly nonlinear oscillators subject to delayrdquoJournal of Vibration and Control vol 16 no 7-8 pp 1141ndash11492010
[23] R M Ghigliazza and P Holmes ldquoOn the dynamics of cranes orspherical pendula with moving supportsrdquo International Journalof Non-Linear Mechanics vol 37 no 7 pp 1211ndash1221 2002
[24] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K Journalof Multi-body Dynamics vol 217 no 2 pp 99ndash109 2003
[25] B Grigorov and R Mitrev ldquoA Newton-Euler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009
[26] A V Gusev andM P Vinogradov ldquoAngular velocity of rotationof the swing plane of a spherical pendulum with an anisotropicsuspensionrdquoMeasurement Techniques vol 36 no 10 pp 1078ndash1082 1993
Shock and Vibration 31
[27] K-S Hong and Q H Ngo ldquoDynamics of the container craneon amobile harborrdquoOcean Engineering vol 53 pp 16ndash24 2012
[28] J J Hoon K Sung-Ha and J Eun Tae ldquoPendulation reductionon ship-mounted container crane via T-S fuzzy modelrdquo Journalof Central South University vol 19 pp 163ndash167 2012
[29] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006
[30] B Jerman P Podrzaj and J Kramar ldquoAn investigation ofslewing-crane dynamics during slewing motionmdashdevelopmentand verification of amathematical modelrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 729ndash750 2004
[31] B Jerman ldquoAn enhanced mathematical model for investigatingthe dynamic loading of a slewing cranerdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 220 no 4 pp 421ndash433
[32] B Jerman ldquoThe comparison of the accuracy of two mathemat-ical models concerning dynamics of the slewing cranesrdquo FMETransactions vol 34 pp 185ndash192 2006
[33] B Jerman and J Kramar ldquoA study of the horizontal inertialforces acting on the suspended load of slewing cranesrdquo Interna-tional Journal ofMechanical Sciences vol 50 no 3 pp 490ndash5002008
[34] F Ju Y S Choo and F S Cui ldquoDynamic response of tower craneinduced by the pendulummotion of the payloadrdquo InternationalJournal of Solids and Structures vol 43 no 2 pp 376ndash389 2006
[35] J Kłosinski ldquoFuzzy logicmdashbased control of a mobile craneslewingmotionrdquoMechanics andMechanical Engineering vol 15no 4 pp 73ndash80 2011
[36] J Krukowski A Maczynski and M Szczotka ldquoThe influenceof a shock absorber on dynamics of an offshore pedestal cranerdquoJournal of Theoretical and Applied Mechanics vol 50 no 4 pp953ndash966 2012
[37] J Krukowski and A Maczynski ldquoApplication of the rigid finiteelement method for modeling an offshore pedestal cranerdquoArchive of Mechanical Engineering vol 60 no 3 pp 451ndash4712013
[38] S Lenci E Pavlovskaia G Rega andMWiercigroch ldquoRotatingsolutions and stability of parametric pendulum by perturbationmethodrdquo Journal of Sound and Vibration vol 310 no 1-2 pp243ndash259 2008
[39] A Y T Leung and J L Kuang ldquoOn the chaotic dynamics of aspherical pendulum with a harmonically vibrating suspensionrdquoNonlinear Dynamics vol 43 no 3 pp 213ndash238 2006
[40] V S Loveykin Romasevich and O Yu ldquoOptimum manage-ment of dynamic systems taking into account higher derivatesof management functionrdquo Scientific Herald National Forest-Technical University of Ukraine vol 21 no 15 pp 304ndash314 2011(Ukrainian)
[41] E Maleki and W Singhose ldquoDynamics and control of a small-scale boom cranerdquo Journal of Computational and NonlinearDynamics vol 6 no 3 Article ID 031015 8 pages 2011
[42] A Maczynski and M Szczotka ldquoComparison of models fordynamic analysis of a mobile telescopic cranerdquo Journal ofTheoretical and Applied Mechanics vol 40 no 4 pp 1051ndash10742002
[43] A Maczynski ldquoLoad positioning and minimization of loadoscillations in rotary cranesrdquo Journal of Theoretical and AppliedMechanics vol 41 no 4 pp 873ndash885 2003
[44] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003
[45] I Marinovic D Sprecic and B Jerman ldquoA slewing cranepayload dynamicsrdquo Tehnicki VjesnikmdashTechnical Gazette vol 19no 4 pp 907ndash916 2012
[46] Z N Masoud M F Daqaq and N A Nayfeh ldquoPendulationreduction on small ship-mounted telescopic cranesrdquo Journal ofVibration and Control vol 10 no 8 pp 1167ndash1179 2004
[47] Z N Masoud A H Nayfeh and D T Mook ldquoCargo pendu-lation reduction of ship-mounted cranesrdquoNonlinear Dynamicsvol 35 no 3 pp 299ndash311 2004
[48] R Mijailovic ldquoModelling the dynamic behaviour of the truck-cranerdquo Transport vol 26 no 4 pp 410ndash417 2011
[49] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008
[50] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity SCIENCE A vol 8 no 9 pp 1395ndash1400 2007
[51] D O Morales S Westerberg P X La Hera U Mettin L Frei-dovich and A S Shiriaev ldquoIncreasing the level of automationin the forestry logging process with crane trajectory planningand controlrdquo Journal of Field Robotics vol 31 no 3 pp 343ndash3632014
[52] K Nakazono K Ohnishi and H Kinjo ldquoVibration control ofload for rotary crane system using neural network with GA-based trainingrdquoArtificial Life and Robotics vol 13 no 1 pp 98ndash101 2008
[53] R L Neitzel N S Seixas and K K Ren ldquoA review of cranesafety in the construction industryrdquo Applied Occupational andEnvironmental Hygiene vol 16 no 12 pp 1106ndash1117 2001
[54] J Neupert E Arnold K Schneider and O Sawodny ldquoTrackingand anti-sway control for boom cranesrdquo Control EngineeringPractice vol 18 no 1 pp 31ndash44 2010
[55] W OrsquoConnor and H Habibi ldquoGantry crane control of a double-pendulum distributed-mass load using mechanical wave con-ceptsrdquoMechanical Sciences vol 4 pp 251ndash261 2013
[56] H M Omar and A H Nayfeh ldquoAnti-swing control of gantryand tower cranes using fuzzy and time-delayed feedback withfriction compensationrdquo Shock and Vibration vol 12 no 2 pp73ndash89 2005
[57] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998
[58] F Palis and S Palis ldquoHigh performance tracking control ofautomated slewing cranesrdquo in Robotics and Automation inConstruction C Balaguer and M Abderrahim Eds InTech2008
[59] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014
[60] B Posiadała B Skalmierski and L Tomski ldquoMotion of thelifted load brought by a kinematic forcing of the crane telescopicboomrdquoMechanism and Machine Theory vol 25 no 5 pp 547ndash556 1990
[61] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008
[62] D Safarzadeh S Sulaiman F A Aziz D B Ahmad andG H Majzoobi ldquoAn approach to determine effect of cranehook on payload swayrdquo International Journal of Mechanics andApplications vol 1 no 1 pp 30ndash38 2011
32 Shock and Vibration
[63] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981
[64] O Sawodny J Neupert and E Arnold ldquoActual trends in craneautomationmdashdirections for the futurerdquo FME Transactions vol37 pp 167ndash174 2009 httpwwwmasbgacrs mediaistrazi-vanjefmevol37402 osawodnypdf
[65] M P Spathopoulos D Fragopoulos A Isidori F Lamnabhi-Lagarrigue and W Respondek ldquoControl design of a crane foroffshore lifting operationsrdquo in Nonlinear Control in the Year2000 A Isidori F Lamnabhi-Lagarrigue and W RespondekEds vol 2 of Lecture Notes in Control and Information Sciencespp 469ndash486 2001
[66] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008
[67] W Solarz and G Tora ldquoSimulation of control drives in atower cranerdquo Transport Problems vol 6 no 4 pp 69ndash78 2011httptransportproblemspolslplplArchiwum2011zeszyt42011t6z4 08pdf
[68] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 223 no 4 pp 575ndash580 2009
[69] A Urbas ldquoAnalysis of flexibility of the support and its influenceon dynamics of the grab cranerdquo Latin American Journal of Solidsand Structures vol 10 no 1 pp 109ndash121 2013
[70] E Hairer S P Noslashrsett and GWanner Solving Ordinary Differ-ential Equations I Nonstiff Problems vol 8 of Springer Series inComputational Mathematics Springer Berlin Germany 1993
[71] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems SpringerBerlin Germany 2002
[72] B Oslashksendal Stochastic Differential Equations An Introduc-tion with Applications Springer Amsterdam The Netherlands2005
[73] M Bar-Yam Dynamics of Complex Systems Westview PressBoulder Colo USA 1997
[74] C S Bertuglia and F VaioNonlinearity Chaos and ComplexityOxford University Press New York NY USA 2005
[75] Y Chen and A Y T Leung Bifurcation and Chaos in Engineer-ing Springer London UK 1998
[76] W L Cleghorn Mechanics of Machines Oxford UniversityPress New York NY USA 2005
[77] D Condurache and V Martinusi ldquoFoucault Pendulum-likeproblems a tensorial approachrdquo International Journal of Non-Linear Mechanics vol 43 no 8 pp 743ndash760 2008
[78] J M Cushing R F Costantino B Dennis R A Desharnaisand S M Henson Chaos in Ecology Experimental NonlinearDynamics Academic Press Elsevier Science San Diego CalifUSA 2003
[79] R L Devaney An Introduction To Chaotic Dynamical SystemsWestview Press Boulder Colo USA 2003
[80] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Second edition 2003
[81] A Ghosal Robotics Fundamental Concepts and AnalysisOxford University Press New Delhi India 2006
[82] D T Greenwood Advanced Dynamics Cambridge UniversityPress New York NY USA 2006
[83] M de Icaza-Herrera and V M Castano ldquoGeneralizedLagrangian of the parametric foucault pendulum withdissipative forcesrdquo Acta Mechanica vol 218 no 1-2 pp 45ndash642011
[84] O D Johns Analytical Mechanics for Relativity and QuantumMechanics Oxford University Press New York NY USA 2005
[85] J V Jose and E J Saletan Classical Dynamics A ContemporaryApproach CambridgeUniversity Press CambridgeMass USA1998
[86] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dynamicsvol 224 no 4 pp 341ndash352 2010
[87] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001
[88] M Pardy ldquoBoundmotion of bodies and particles in the rotatingsystemsrdquo International Journal ofTheoretical Physics vol 46 no4 pp 848ndash859 2007
[89] A Preumont Vibration Control of Active Structures vol 179of Solid Mechanics and its Applications Kluwer AcademicDordrecht The Netherlands 2011
[90] W F Smith Waves and Oscillations A Prelude to QuantumMechanics Oxford University Press New York NY USA 2010
[91] F Scheck Mechanics From Newtons Laws to DeterministicChaos Graduate Texts in Physics Springer Heidelberg Ger-many 2010
[92] A Smaili and F MradAppliedMechatronics Oxford UniversityPress New York NY USA 2008
[93] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress Boulder Colo US 2000
[94] B H Tongue Principles of Vibration Oxford University PressNew York NY USA 2002
[95] A Tozeren Human Body Dynamics Classical Mechanics andHuman Movement Springer New York NY USA 2000
[96] F E Udwadia and R E Kalaba Analytical Dynamics A NewApproach Cambridge University Press New York NY USA1996
[97] J Uicker G Pennock and J Shigley Theory of Machines andMechanisms Oxford University Press New York NY USA2003
[98] M A Wahab Dynamics and Vibration An Introduction JohnWiley amp Sons The Atrium UK 2008
[99] A Zanzottera G Mingotti R Castelli and M Dellnitz ldquoLow-energy Earth-to-halo transfers in the Earth-Moon scenariowith Sun-perturbationrdquo in Nonlinear and Complex DynamicsApplications in Physical Biological and Financial Systems JA T Machado D Baleanu and A C J Luo Eds pp 39ndash51Springer New York NY USA 2011
[100] V F Zhuravlev and A G Petrov ldquoThe lagrange top and thefoucault pendulum in observed variablesrdquoDoklady Physics vol59 no 1 pp 35ndash39 2014
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