-
realized for a hydraulic mobile crane. In addition to the
structural elements, the mathematical modelling for
Modern mobile cranes, that have been built till today, have oft
a maximal lifting capacityof 3000 tons and incorporate long booms.
Crane structure and drive system must be safe,functionary and as
light as possible. For economic and time reasons it is impossible
to build
Mechanism and Machine Theory 38 (2003) 14891508*Corresponding
author. Tel.: +86 2423900601.hydraulic drive- and control system is
described. The crane rotating simulation for arbitrary working
conditions has been carried out. As a result, a more exact
representation of dynamic behaviour, not only for
the crane structure, but also for the drive system is
achieved.
2003 Elsevier Ltd. All rights reserved.
Keywords: Hydraulic mobile rotary crane; Flexible multibody;
Drive and control system; Dynamic responses
1. IntroductionDynamic responses of hydraulic mobile cranewith
consideration of the drive system
Guangfu Sun a,*, Michael Kleeberger b
a Department of Trac and Mechanical Engineering, Shenyang
Architectural and Civil Engineering University,
Shenyang Hunnan, Shenyang, 110168, Chinab Institute of Material
Handling, Material Flow and Logistics Technical University of
Munich, Germany
Received 13 August 2002; accepted 28 March 2003
Abstract
The dynamic behaviour of mobile cranes is determined not only by
the steel structures and the external
loads but also by the drive- and control systems. In todays
dynamic calculation of mobile cranes, the drivesystems are modelled
through the method of kinematic forcing or by measurements for
outputs of the
drive system. To improve this situation, a new method for
dynamic calculation of mobile cranes has been
developed. In this method, the exible multibody model of the
structure will be coupled with the model ofthe drive system. In
that way the elastic deformation, the rigid body motion of
structures and the dynamic
behaviour of the drive system can be determined in an integrated
model. The calculation method has been
www.elsevier.com/locate/mechmtE-mail address:
[email protected] (G. Sun).
0094-114X/$ - see front matter 2003 Elsevier Ltd. All rights
reserved.doi:10.1016/S0094-114X(03)00099-5
-
1490 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38
(2003) 14891508prototypes for great cranes. Therefore, it is
desirable to determinate the crane dynamic responseswith the
theoretical calculation.Modern mobile cranes include the drive and
the control systems. Control systems send the
feedback signals from the mechanical structure to the drive
systems. In general, they are closed-chain mechanisms with exible
members [1].Rotation, load and boom hoisting are fundamental
motions of the mobile crane. During
transfer of the load as well as at the end of the motion
process, the motor drive forces, the structureinertia forces, the
wind forces and the load inertia forces can result in substantial,
undesired os-cillations in crane. The structure inertia forces and
the load inertia forces can be evaluated withnumerical methods,
such as the nite element method. However, the drive forces are
dicult todescribe. During start-up and braking the output forces of
the drive system signicantly uctuate.To reduce the speed variations
during start-up and braking the controlled motor must producetorque
other than constant [2,3], which in turn aects the performance of
the crane.Several published articles on the dynamic responses of
mobile crane are available in the open
literature. In the mid-seventies Peeken et al. [4] have studied
the dynamic forces of a mobile craneduring rotation of the boom,
using very few degrees of freedom for the dynamic equations andvery
simply spring-mass system for the crane structure. Later Maczynski
et al. [5] studied the loadswing of a mobile crane with a four
mass-model for the crane structure. Posiadala et al. [6]
haveresearched the lifted load motion with consideration for the
change of rotating, booming and loadhoisting. However, only the
kinematics were studied. Later the inuence of the exibility of
thesupport system on the load motion investigated by the same
author [7]. Recently, Kilicaslan et al.[1] have studied the
characteristics of a mobile crane using a exible multibody dynamics
ap-proach. Towarek [16] has concentrated the inuence of exible soil
foundation on the dynamicstability of the boom crane. The drive
forces, however, in all of those studies were presented byusing so
called the method of kinematic forcing [6] with assumed velocities
or accelerations. Inpractice this assumption could not comply with
the motion during start-up and braking.A detailed and accurate
model of a mobile crane can be achieved with the nite element
method. Using non-linear nite element theory Guunthner and
Kleeberger [9] studied the dynamicresponses of lattice mobile
cranes. About 2754 beam elements and 80 truss elements were used
formodelling of the lattice-boom structure. On this basis a ecient
software for mobile crane cal-culationNODYA has been developed.
However, the inuences of the drive systems must bedetermined by
measuring on hoisting of the load [10], or rotating of the crane
[11]. This is neitherecient nor convenient for computer simulation
of arbitrary crane motions.Studies on the problem of control for
the dynamic response of rotary crane are also available.
Sato et al. [14], derived a control law so that the transfer a
load to a desired position will takeplace that at the end of the
transfer of the swing of the load decays as soon as possible.
Gustafsson[15] described a feedback control system for a rotary
crane to move a cargo without oscillationsand correctly align the
cargo at the nal position. However, only rigid bodies and elastic
jointbetween the boom and the jib in those studies were considered.
The dynamic response of thecrane, for this reason, will be
global.To improve this situation, a new method for dynamic
calculation of mobile cranes will be
presented in this paper. In this method, the exible multibody
model of the steel structure will becoupled with the model of the
drive systems. In that way the elastic deformation, the rigid
body
motion of the structure and the dynamic behaviour of the drive
system can be determined with
-
one integrated model. In this paper this method will be called
complete dynamic calculation fordriven mechanism.On the basis of
exible multibody theory and the Lagrangian equations, the system
equations
for complete dynamic calculation will be established. The drive-
and control system will be de-scribed as dierential equations. The
complete system leads to a non-linear system of
dierentialequations. The calculation method has been realized for a
hydraulic mobile crane. In addition tothe structural elements, the
mathematical modelling of hydraulic drive- and control systems
isdecried. The simulations of crane rotations for arbitrary working
conditions will be carried out.As result, a more exact
representation of dynamic behaviour not only for the crane
structure, butalso for the drive system will be achieved. Based on
the results of these simulations the inuencesof the accelerations,
velocities during start-up and braking of crane motions will be
discussed.
2. The principle of complete dynamic calculation for
mechanism
The principle of complete dynamic calculations, which is based
on integrated model [12] fromexible multibody model of the
mechanism and the mathematical model of the drive system
isshowoutpand d
G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003)
14891508 1491following sections.
2.1. Modelling of the driven mechanism system
Using Lagranges equation, the motion of bodies in the mechanism
is given by
d
dtoTo _qq
T oT
oq
T oC
oq
Tk Qin Qa Qd 1n in Fig. 1. The input is the desired value such
as position or velocity of the mechanism. Theut is the dynamic
response of the complete system, which consists of the mechanism
systemrive system including control. The mathematical models of the
two systems are given in theFig. 1. The principle of complete
dynamic calculation for mechanism.
-
where T is total kinetic energy, q is the vector of the
generalized coordinate,Qin is the vector of theinternal forces, Qa
is the vector of the applied external forces, Qd is the vector of
the drive forcesand k is the vector of Lagrange multiplies.
draulic, the drive forces in general can be expressed through
following algebraic equation
Qd s
moti
reference xxi fxx1i;xx2i;xx3ig and a global inertia reference x
fx1; x2; x3g are selected. Theposit
1492 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38
(2003) 14891508Euler-angle, determinate the location and direction
of the body reference. The motion of body i isgiven by the
following Lagranges equation
1 In this paper, bold characters are used to represent vectors
expressed in the global reference system, and over-barbold cion
vector Ri and the rotation vector Hi, which is usually described
with Euler-parameter orsystems consisting of interconnected rigid
and deformable bodies, can be greatly enhancedthrough dynamic
simulation, provided the deformation eect is incorporated into the
rigid mo-tion.In the exible multibody dynamics, the large rigid
body motion is usually described in the
global inertia reference system. On the other hand, the body
deformation is given in the bodyreference system [17]. For an
arbitrary body i in the system, which is shown in Fig. 2, a
body
T T 1on. The design and performance analysis of cranes, which
can be modeled as multibody3. Flexible multibody formulation
Crane as driven mechanism, their bodies mostly undergo large
translational and rotationalimultaneously.Qd fdz; t 4The equations
of motion for the complete system will be achieved through the
combination of Eq.(1)(4). The fundamental problem of complete
dynamic calculation for mechanism is, withconsideration of the
input and the boundary conditions, to solve all of the time
valuables q, k, z,The constrained conditions of the bodies can be
written in the following vector equation
Cq; t 0 2
2.2. Modelling of the drive system with control
The drive system has a signicant inuence on the dynamic response
of the driven mechanismand should be included in the dynamic model.
Electronic and hydraulic drive systems with controlare usually used
in mobile cranes. In general the drive system with control can be
described usingthe following rst order explicit dierential
equation
_zz Fz; q; _qq; t 3where z is the state space vector of the
drive system, q is the feedback vector from the
drivenmechanism.According to the corresponding physical law for the
drive system, such as electronic or hy-haracters are used for
vectors that are dened in the body reference system.
-
wherTh
scrib
wherrefer
withand uutransAc
positcleargener
With
G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003)
14891508 1493d
dtoTio _qqi
oTioqi
oCoqi
k Qini Qai Qdi 5
T T T
Fig. 2. A body with rigid and elastic motion.e i 1; 2; . . . ; n
and n is the total number of the mechanism members.e deformable
position vector of the arbitrary point p in the element j of body i
can be de-ed as
rpij Ri upij 6e Ri is a set of Cartesian coordinate that dene
the location of the origin Oi of the bodyence. Clearly,
upij uoij ueij Aiuuoij uueij 7uoij is the position of point p in
the unreformed state; u
eij is the elastic displacement vector; uu
oij
eij are the coordinate of the vectors u
oij and u
eij with respect to the body reference xxi; Ai is the
formation matrix, which is a matrix function of Hi.cording to
the principle of the nite element method, uuoij and uu
eij can be interpolated by nodal
ion coordinate in the undeformed state qqoi and the nodal
displacement qqei , respectively. It is
that rpij can be determined by Ri, Hi and qqei from Eqs. (6) and
(7). Therefore we dene the
alized coordinate qi of the body i as
qi RiT HiT qqei Th iT
8this denition the vector r
pij can be expressed as the function of the generalized
coordinate qi
rpij rpijqi 9
-
culat
withTh
place
respe _
Th
wher
UQ
c
i cavecto
wher
Subs
1494 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38
(2003) 14891508generalized coordinate q qT1 ; qT2 ; . . . ; qTnT,
the new variable u as the system generalized velocityandQini Qei
Qci 20tituting from (11), (16), (17) and (20) into Lagranges
equation (5), introducing the system
Di _qqi; qi Ti DiTi _qqi;TiqiTi 19that are total stiness matrix
and total damping matrix of body i, respectively.The total internal
nodal force vector is given bye
Kiqi TTi KiTiqiTi 18TQei TTi Qe
i TTi KiTiqiTiqi Kiqiqi 16Qci TTi Q
c
i TTi DiTi _qqi;TiqiTi _qqi Di _qqi; qi _qqi 17sing this
transformation matrix the nodal elastic force vectorQi and the
damping force vectorn be transferred to the generalized elastic
force vector Qei and the generalized damping forcer Qci associated
with the generalized coordinate Ri, Hi and qq
ei asTi 0 0 I 15with Ti is the transformation matrix.
eei i
qqei Tiqi 14ct to qqi and qqqqi , Ki and Di are the stiness
matrix and damping matrix, respectively.e nodal displacements qqe
can be written in terms of the generalized coordinate q of body i
asQe
i Kiqqei qqei 12Q
c
i Di _qqqqei ; qqei _qqqqei 13where Q
e
i and Qc
i are the nodal elastic force vector and the nodal damping force
vector withe eMiqi is the total mass matrix of the body i.e elastic
nodal forces and the nodal damping forces can be described by the
nodal dis-ment vector qqei and the nodal velocity vector _qqqq
ei asj1 j1ed using the sum of the all of elements
Ti Xne
Tij 12
_qqTiXne
Mijqi" #
_qqi 1
2_qqTi Miqi _qqi 11where qij is the mass density, Vij is the
element volume and Mijqi is the mass matrix, which ingeneral not
constant, but is dependent on qi. The total kinetic energy of the
body i can be cal-ij2 Vij
ij ij ij ij 2 iij i iThe kinetic energy of the element j is
given by
T 1Z
q _rrp T _rrp dV 1 _qqTM q _qq 10combining with the kinematic
constraints between adjacent bodies, yield
-
_qq u 21
Mq _uu Du; qu Kqq oCoq
Tk Qa Qd Qv 22
Cq; t 0 23where Mq is the system mass matrix, Du; q and Kq are
system damping and stiness matrix.In Eq. (22) Qv is given by
Qv _MMqu 12
ooq
uTMqu T
24
which represents the gyroscopic and Coriolis force
components.
The mathematical model of the hydrostatic closed-loop system,
the controller and the regulatorunit that are often used in mobile
cranes, will be described in the following sections.
G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003)
14891508 14954. Mathematical models for hydraulic drive system with
control
The hydrostatic closed-loop system of the type shown in Fig. 3,
which is usually used in mobilecranes, will be considered. A
variable pump that is driven by a diesel engine, capable of
pumpingin either direction, is directly ported to a bidirectional
motor. Using this system, smooth transitionfrom forward to reverse
through the zero rotating speed and full hydrostatic braking action
ineither direction will be available. To prevent excessive loop
pressure, cross-port relief valves arecommonly used as shown in
Fig. 3. The small auxiliary supercharge pump that is usually
mounteddirectly on the main pump is to provide enough ow to take
care of pump and motor internalleakage. There is sometimes a need
for control of motor in mobile crane, such as speed feedbacksystem
with controller and regulator unit, which are also shown in Fig. 3.
The controller uses theresponse signals from the mechanism and
compares them with their desired values in its task ofdetermine an
appropriate action fR to the regulator unit that can create the
regulating value xs toadjust the pump. The signal ow is given in
Fig. 4.Fig. 3. Hydraulic closed circuit system in mobile crane.
-
4.1. M
since
to th
1496 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38
(2003) 14891508in out
with
Qout Qh Qr 27where Qr is the ow through the relief valve, which
can be described as following
Qr 0 jphj6 jpsetjKrph pset jphj > jpsetj
28wherTh
withdisplTh
within out
e Kirchhos low, should be identical
Q Q 26h
dt
ChQch Ch Qh Qm 25
where Ch is the hydraulic capacity of the high-pressure side, Qh
and Qm are the input ow to themotor and the output ow from the
motor, respectively.Because of the neglecting of all leakage, the
input ow Q and the output ow Q , accordingdp 1 1pressure line [18].
The leakage will be neglected.According to basic dierential
equations of the hydraulic circuit [19], we havean auxiliary pump
and relief valves are typically used to assure constant pressure in
the low-For simplication it is assumed that the ow pressure p0 of
the low-pressure side is constantwhile the high pressure ph of the
high-pressure side is variable. This is general true in
practice,odelling the hydrostatic closed-loop systemFig. 4. Signal
ow.e Kr is the constant of the relief valve and pset is the
prescribed limiting pressure.e pumping ow Qin from the variable
pump is given by
Qin V1maxn1x1max xs 29
V1max is the maximum pumping ow of the variable pump, x1max the
maximum adjustableacement, n1 is the pump rotation speed, which is
assumed to be constant.e ow from the constant motor of the system
can be expressed as
Qm n2V2 30n2 is the rotating speed of the motor, V2 is the motor
volume.
-
The output torque of the motor is given by
M2 ph p0V22p
31From Eqs. (25)(30) the dierential equation of the pressure ph
is given by
_pph 1Ch xsn1v1maxx1max
Qr n2V2
32
Because the motor speed n2 is a component of the generalized
velocity u, substituting Eq. (28) intoEq. (32), we can write Eq.
(32) in the general form as
_pph fpxs; ph; u 33The generalized driving forces, which can be
obtained from the output torque of the motor in Eq.(31), can be
expressed in a function form of the pressure ph as
Qd fdph 34
show
G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003)
14891508 1497Using Newtons law the motion of the control valve
piston is given by the following dierentialequation
mvxxv fR dv _xxv cvxv 35where mv is mass of the valve piston, dv
is the damping constant of the valve, cv is the springconstant, fR
is regulating force from the controller and xv is the motion of the
valve piston.s a typical regulator unit system.4.2. Modelling the
regulator unit system
The regulator unit system usually consists of a cylinder and a
4/3-way control valve. Fig. 5Fig. 5. Regulator unit system.
-
Th
with
both
withvolumTh
sprinIf
1498 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38
(2003) 14891508_xx6 1m fR dvx6 cvx2_xx2 x6_xx3 Eolva Q1 Q2 Asx5
_xx4 Eolvb Q4 Q3 Asx5
_xx5 1ms x3 x4As dsx5 csx1 f0
40yields the following state equations for the regulator
unit
_xx1 x5s s 0
g, ds is the damping constant of the cylinder and xs is the
servo-moton.we dene the state variables
x1 xs; x2 xv; x3 pa; x4 pb; x5 _xxs; x6 _xxvmsxxs pa pbAs ds
_xxs csxs f0 39where m is the piston mass, c is the spring
constant, f is the initial elastic force of the cylinderAs is the
area of the cylinder piston, vm is the initial system volume of the
cylinder with thees of all drillings, pipelines of the unit, as the
piston is at the position xs 0.e motion of the cylinder piston can
be written asva vm Asxsvb vm Asxs
38ol a b
side of a and b, which are given by_ppb Eolvb Q4 Q3 As
_xxs37
where E is eective bulk modulus of the oil, v and v ,
respectively, are the oil volumes in the_ppa olva Q1 Q2 As _xxsThe
pressure rates in the both cylinder chambers can be calculated by
the continuity equations
Ee ows through the control valve are given by the turbulent
equations [20] as
Q1 Bv2xv x0 jxv x0j
jps paj
psignps pa
Q2 Bv2xv x0 jxv x0j
jpaj
psignpa
Q3 Bv2xv x0 jxv x0j
jpbj
psignpb
Q4 Bv2xv x0 jxv x0j
jps pbj
psignps pb
36
x0 is the valve lapping, ps is system supply pressure and Bv is
the ow coecient of the valve.v
-
1 2 6 s
vector x, which is used in Eq. (33), can be directly from x
obtained
4.3. M
d
desirbe re
T
ow oq ou ow oq ou
The t
G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003)
14891508 1499Having determined, in the preceding sections, the
equations of the mechanism, the hydraulic
close5. Total system equationsfR frq; u; xi; k;Qa;Qd 48With
consideration of (45)(47) and (22) the regulating force can be
written in the followinggeneral formerm _uu in (47) can be
calculated using Eq. (22). Clearly, it is a function of q, u, k, Qa
and Qd.we have
fR Kp xd
1Tp
xi Tv _xxd
46
with
_xxd oxd _ww oxd _qq oxd _uu oxd _ww oxd u oxd _uu 47i
_xxi xd 45p
where Kp is the proportional parameter, Tp and Tv are time
constants. It is clear that the error xd isa function of the
observed value q, u and the desired value w.Assuming a new variable
x and letfR Kp xd 1 xd dt Tv _xxd 44which can be described with
following equation Z
ed value, the output of controller is the controlled signal fR,
with which the regulator unit willalized (see Fig. 4). The commonly
used controller in mobile cranes is the PID controller,The input
value of the controller is x , which is the dierence of the
response value and theodelling the controllerwhere Ts is the
Boolean translation vector. Substituting (42) into (33), the
pressure rate can bewritten in the general form by
_pph fpph; x; u 43xs Tsx 42Using the vector form, the state
equation of the regulator unit can be written as
_xx fxx; fR 41with x x ; x ; . . . ; x T is the state vector of
the regulator actuator. The component x of the stated system and
the controller system one can write the total system equations
as
-
where
correvenie
6. Nu
Mtatiothe ccraneshowTh
nismmechTh
(2) Tt
(3) A
1500 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38
(2003) 14891508he boom is exible, the suspension ties and ropes are
elastic, but the superstructure, coun-erweight and crane base are
regarded as rigid bodies.(1) The load is regarded as a point
mass.merical simulation of a mobile hydraulic rotary crane
obile cranes are widely used in industry. The fundamental motion
of a rotary crane is ro-n by which the load pendulum action due to
this motion is a strong dynamic phenomenon ofrane, which must be
taken into account. In this section the numerical simulation of a
rotarythat is driven by a closed hydraulic system with speed
control will be carried out. Fig. 6s this crane with coordinate
systems.e whole superstructure of the crane can be rotated about
z-axis driven by a rotary mecha-. The global technical data of the
crane are given in Table 1. The technical data of the rotaryanism
and hydraulic close system are shown in Table 2.e following
assumptions will be used:sponding dynamic responses of the total
crane system can be obtained. This is very con-nt for computer
simulation of crane motions.fmq; u; k;Qa;Qd Du; qu Kqq @CoqT
k Qa Qd Qv
The system equations (49) are a mixed system of dierential and
algebraic equations that have tobe solved simultaneously. By
solving Eq. (49) not only the responses of the mechanism q, u and
k,but also the state values of the drive system x, fR, xi, ph and
the output force Q
d can be simul-taneously obtained. By arbitrary input of desired
values and parameters of the controller themy dydt
fy; t 49
with
y qT uT xT xi ph fR QdT kT T
fy; t
u
fmq; u; k;Qa;Qdfxx; fRxdw; q; ufpph; x; u
fR frq; u; xi; k;Qa;QdQd fdph
Cq; t
266666666664
377777777775
my diagI;Mq; I; 1; 1; 0; 0; 0ll frictional and damping forces
will be neglected.
-
G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003)
14891508 1501(4) T(5) T
Twcoordto thloaddescrselec
Theerencpensistaticrotathe oil is compressible, but the oil
leakage of the hydraulic system is not considered.here are not any
load hoisting or boom hoisting during the rotary motions of the
crane.
o reference systems: the global inertial coordinate system x, y
and z and the body referenceinate system xx, yy and zz are used.
The vertical axises of the both systems z and zz are identicale
rotary axis as shown in Fig. 6(b). The global coordinate R and h
dene the location of theand the orientation of the rotary body
reference system. The elastic nodal coordinate qqe
ibe the rotary body deformation with respect to the body
reference xx, yy and zz. Therefore theted generalised coordinate
are given by
q RT h qqeT T 50
boom system is divided into eight Timoshenko boom elements [13].
The element local ref-e system xxj, yyj and zzj is used to dene the
element cross-section j, see Fig. 6(b). The sus-on ties are
described as one truss element. The ropes are represented as a
spring element. Thenodal displacements of the boom system which are
initial conditions for the simulation ofion are calculated under
acting load and the boom gravity force.
Fig. 6. (a) Rotary mobile crane and (b) schematic model with
coordinate systems.
-
Table 2
Technical data of the rotary mechanism and hydraulic close
system
Name Symbol Value
Rotating speed of the pump (rpm) n1 1680.0Forder volume of the
variable pump (cm3) V1max 71.0Volume of the hydraulic motor (cm3)
V2 90.0Translate factor between hydraulic motor and rotary
mechanism i 3500.0Capacity of the hydraulic system (cm3/bar) Ch
0.08125Factor of the safe valve (cm3/bar) Kr 185.0Maximal
controlled displacement of the variable pump (mm) xmax 43.62The
limiting pressure of the system (bar) jpsetj 300.0Flow coecient of
the valve (l/(min bar1=2 mm)) Bv 3.641System pressure for the
control valve (bar) ps 100.0Constant low pressure of the closed
circuit (bar) p0 20.0Piston area of the regulator cylinder (cm2) Ak
1.3Volume in chamber A of the regulator cylinder (cm3) va
70.0Volume in chamber B of the regulator cylinder (cm3) vb
70.0Spring constant in the regulator cylinder r (N/mm) ck
19.0Damping constant in the regulator cylinder (N s/mm) dk
0.042272Initial spring force in the regulator cylinder (N) f0
66.424Eective uid balk module (bar) Eol 16000Lapping of the control
valve (mm) x0 1.0Piston mass of the regulator cylinder (kg) mk
1.0Mass of the control valve piston (kg) mv 0.1Spring constant in
control valve (N/mm) cv 300.0Damping constant in control valve (N
s/mm) dv 0.029709
Table 1
Technical data of the crane structure
Name Value
Load radius (m) 12.0
Length of the boom (m) 96.0
Mass of the boom (kg/m) 550.0
Mass of the superstructure (t) 53.0
Mass of the load (t) 55.0
Mass of the load hook (t) 5.526
Mass of the counterweight (t) 150.0
Eective boom inertia moment about yyj-axis of the element
reference (m4) 0.0838Eective boom inertia moment about zzj-axis of
the element reference (m4) 0.0665Reduced boom shear area about
yyj-axis of the element reference (m2) 0.0034Reduced boom shear
area about zzj-axis of the element reference (m2) 0.0032Eective
area of the boom (m2) 0.0443
Area of the suspension ties (m2) 0.0084
Area of the ropes (m2) 0.0004
Length of the ropes (m) 71.51
Youngs modulus (N/m2) 1.08 1011
1502 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38
(2003) 14891508
-
The desired rotary process of the superstructure with speed of
0.5 rpm and braking at t 10:0 scan be described using following
step function:
dspeed 0:5 06 t < 10:00:0 10:06 t < 1
51
The following rotary processes of the superstructure under
dierent control parameters Kp, Tpand Tv are carried out:
rotary 1 : Kp 5; Tp 0:10; Tv 0:4rotary 2 : Kp 10; Tp 0:12; Tv
0:4rotary 3 : Kp 25; Tp 0:128; Tv 0:4
Fig. 7 shows the corresponding rotary speed responses. The
dierent controlled speed curvesare achieved. All curves are
focussed to the intended rotary process (51), but with dierent
ac-celerations and decelerations. The curve with rotary 1 is
focussed at 4.0 and 14.0 s, the curve with
G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003)
14891508 1503rotary 2 and 3 are at 2.0 and 12.0 s, respectively.
The dynamic displacements of the boom tip indirection yy of the
body reference system xx, yy and zz for dierent controlled speed
curves are given inFig. 8. For comparison with the method of
kinematic forcing that was used by some literature[6,8], two types
of assumed trapezoidal velocity proles with the same acceleration
and deceler-ation times 2.0 and 4.0 s are considered, see Fig. 7.
It should be noted that these assumed inputvelocity proles could
not occur in the simulation of the real rotary control process.
However, forcomparison with the current study, the corresponding
dynamic responses of the boom tip withthese velocity proles have
been calculated in this paper using the method of kinematic
forcing[6]. The results are also shown in Fig. 8.The inuences of
dierent rotary speeds by the same accelerations and decelerations
under the
same control parameters Kp 5, Tp 0:10 and Tv 0:4 are also
studied, which are given in Fig.10. It is observed that the rotary
speed with the same acceleration and deceleration has alsoFig. 7.
Dierent rotary speeds of the crane superstructure.
-
1504 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38
(2003) 14891508signicant inuence on the dynamic response of crane.
With higher rotary speeds the super-structure of crane can have
more kinetic energy, which will be transformed to dynamic response
inbraking process. The relationship between the rotary speed and
the maximum displacement of theboom tip is approximately linear,
which is given in Fig. 10.Based on these results the following
observations can be made:
1. At the same rotary speed, the acceleration time and braking
time, which can be controlled by Kpand Tp, have strong inuence on
the dynamic response. Using the assumed trapezoidal velocity
Fig. 8. Dynamic response of the boom tip.
Fig. 9. Dierent accelerations and decelerations.
-
G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003)
14891508 1505proles with the same acceleration and deceleration
times, the results are lower as the real ve-locity proles. The
reason is that during start-up of the rotary process and braking
the shockacceleration and deceleration are very higher than the
assumed trapezoidal velocity proles andinduce strong dynamic
responses (see Fig. 9).It is clear that the assumed trapezoidal
velocity proles with the same acceleration and de-celeration times
for sudden start-up and braking, in comparison to the current
study, could notmore accurately describe the acceleration and
deceleration histories during start-up andbraking of the rotary
mechanism.
2. With the same rotating acceleration and deceleration, the
rotary speed has also signicant in-
Fig. 10. Maximum boom tip displacements with dierent rotary
speeds.uence on the dynamic response of crane.3. The maximum
responses occur during the braking processes.
In addition to the dynamic responses of the crane structure, the
state parameters and theoutputs of the hydraulic drive system can
be determined easily. Fig. 11 shows the motor outputmoments during
various rotary processes.As expected, during start-up and braking
the motor outputs the maximum shock moments. The
safety relief valves ensure that these shock moments remain
lower than the allowable values. Thepressure histories in the
chambers of the piston are given in Fig. 12. It is observed that
the pressurecurves of chamber a and b are symmetric to the value of
50 bar.
7. Conclusion
A simulation method for determining the dynamic response of
rotary cranes with considerationof the hydraulic drive system is
developed. With the steel structure, hydraulic drive and
controlsystem, a comprehensive crane model is established. Based on
the theory of exible multibody
-
1506 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38
(2003) 14891508system, the nite element method, the hydraulic and
control theory, the system equations forcomplete dynamic
calculations are established.With this method, simulations of a
hydraulic mobile crane in rotating processes are carried out.
The inputs for the simulations are the desired rotary speeds and
control parameters. In com-parison with the assumed velocity proles
in the method of kinematic forcing, the currentmethod is very
convenient for crane simulations. The results show that the dynamic
responsesduring start-up and braking of the crane rotation with the
assumed trapezoidal velocity prolesfor sudden acceleration and
deceleration could not more accurately described. The results
support
Fig. 11. Motor output moments.
Fig. 12. Pressures pa and pb in chamber a and b of the
piston.
-
[4] H. Peeken, K. Menninger, Dynamische Kraafte beim Drehen
eines Mobilkranes, Foordern und Heben 24 (12) (1974)11431147.
169178.
[9] W.A. Guunthner, M. Kleeberger, Zum Stand der Berechnung von
Gittermast-Fahrzeugkranen, dhf 43 (3) (1997)
G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003)
14891508 15075661.
[10] M. Kleeberger, Nichtlineare dynamische Berechnung von
Gittermast-Fahrzeugkranen, Muunchen, TechnischeUniversitaat,
Dissertation, 1996.
[11] J. Maier, Untersuchung zur nichtlinearen Berechnung
dynamischer Belas-tungsvorga nge an Turmdrehktanen,
Muunchen, Technische Universitaat, Dissertation, 1999.[12] G.
Sun, Berechnung von Gittermast-Fahrzeugkranen unter
Beruucksichtigung der Antriebs- und Regelungssysteme,
Muunchen, Technische Universitaat, Dissertation, 2001.[13] M.
Guunthner, Statische Berechnung vom Gittermast-Auslegerkranen mit
Hilfe niter Turmelemente unter
Beruucksigtigung der Elastizitaat des Kranwagens und von
Messungen, Muunchen, Technische Universitaat,Dissertation, 1985.[5]
K. Maczynski, S. Wojciech, Bin diskretes Modell fuur
Teleskopdrehkrane zur Anylyse der Bewegung der Last beimDrehen des
Kranes, Hebezeuge und Foordermittel 21 (1981) 333337.
[6] B. Posiadala, B. Skalmierski, L. Tomski, Motion of the
lifted load brought by a kinematic forcing of the crane
telescopic boom, Mech. Mach. Theory 25 (1990) 547555.
[7] B. Posiadala, Inuence of crane support system on motion of
the lifted load, Mech. Mach. Theory 32 (1) (1997) 9
20.
[8] W.S.M. Lau, K.H. Low, Motion analysis of a suspended mass
attached to a crane, Comput. Struct. 52 (1) (1994)the proposition
that a more accurate and convenient calculating instead of the
kinematic forc-ing for the dynamic response of crane system is
expected. The simulation results also show that,in addition to the
accelerations and decelerations of crane movements, the rotary
speed has alsosignicant inuence on crane dynamic responses.With
this method, not only the dynamic responses of the structure, but
also the state param-
eters of the drive system, such as motor output moment history,
dynamic response of valves, oilpressure, oil ow and control
stability, which are signicant for design of crane drive system,
canbe studied simultaneously.
Acknowledgements
The rst author appreciates the nancial support provided by The
Institute of MaterialHandling, Material Flow and Logistics,
Technical University of Munich, Germany, for thisproject. The
authors are very grateful to Prof. Dr. -Ing, Willibald A. Guunthner
for supporting thiswork.
References
[1] S. Kilicaslan, T. Balkan, S.K. Ider, Tipping load of mobile
cranes with exible booms, J. Sound Vib. 223 (4) (1999)
645657.
[2] A. Myklebust, Dynamic response of an electric motor-linkage
system during start-up, J. Mech. Des. 104 (1982)
137142.
[3] A. Smaili, M. Kopparapu, M. Sannah, Elastodynamic response
of a d.c. motor driven exible mechanism system
with compliant drive train components during start-up, Mech.
Mach. Theory 31 (5) (1996) 659672.[14] K. Sato, Y. Sakawa,
Modelling and control of a exible rotary crane, Int. J. Control 48
(5) (1988) 20852105.
-
[15] T. Gustafsson, Modelling and control of a rotary crane. In:
Proceedings of 3rd European Control Conference,
Rome Italy, September 1995, pp. 30853810.
[16] Z. Towarek, The dynamic stability of a crane standing on
soil during the rotation of the boom, Int. J. Mech. Sci. 40
(6) (1998) 557574.
[17] A.A. Shabana, Dynamics of Multibody Systems, Cambridge
University Press, 1998.
[18] N.D. Manring, G.R. Luecke, Modelling and designing a
hydrostatic transmission with a xed-displacement motor,
J. Dyn. Syst. Meas. Control 120 (45) (1998) 4549.
[19] P. Dreanseld, Hydraulic Control Systems. Design and
Analysis of their Dynamics, Springer-Verlag, 1981.
[20] H.E. Merritt, Hydraulic Control Systems, John Wiley, New
York, 1967.
1508 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38
(2003) 14891508
Dynamic responses of hydraulic mobile crane with consideration
of the drive systemIntroductionThe principle of complete dynamic
calculation for mechanismModelling of the driven mechanism
systemModelling of the drive system with control
Flexible multibody formulationMathematical models for hydraulic
drive system with controlModelling the hydrostatic closed-loop
systemModelling the regulator unit systemModelling the
controller
Total system equationsNumerical simulation of a mobile hydraulic
rotary craneConclusionAcknowledgementsReferences