Solution Coupled Analysis!

IJMMS 2004:41, 21812197PII. S0161171204305363http://ijmms.hindawi.com Hindawi PublishingCorp.DETERMINATION OF COUPLED SWAY, ROLL, AND YAW MOTIONSOF A FLOATING BODY IN REGULAR WAVESS. N. DAS and S. K. DASReceived 21 May 2003Thispaperinvestigatesthemotionresponseofaoatingbodyintimedomainundertheinuenceofsmall amplituderegularwaves. Thegoverningequationsofmotiondescrib-ing the balance of wave-exciting force with the inertial, damping, and restoring forces aretransformedintofrequencydomainbyapplyingLaplacetransformtechnique. Assumingthe oating body is initially at rest and the waves act perpendicular to the vessel of lateralsymmetry, hydrodynamic coecients were obtained in terms of integrated sectional added-mass, damping, and restoring coecients, derived from Franks close-t curve. A numericalexperimentonavessel of19190tondisplacedmasswascarriedoutforthreedierentwave frequencies, namely, 0.56rad/s, 0.74rad/s, and 1.24rad/s. The damping parameters(i) reveal the system stability criteria, derived from the quartic analysis, corresponding tothe undamped frequencies (i). It is observed that the sway and yaw motions become max-imum for frequency 0.56rad/s, whereas roll motion is maximum for frequency 0.74rad/s.All three motions show harmonic behavior and attain dynamic equilibrium for time t >100seconds. Themathematicalapproachpresentedherewillbeusefultodetermineseawor-thinesscharacteristicsofanyvesselwhenwaveamplitudesaresmallandalsotovalidatecomplex numerical models.2000 Mathematics Subject Classication: 34C15, 70K42, 70K25.1. Introduction. Precise prediction of hydrodynamic behavior and motion responseof a oating body in water waves is essential for proper harbor design. A oating bodyexcited by waves experiences six degrees of motion constituting three translatory andthreerotational motions. Thesemotionscanbedescribedkinematicallyintermsofsurge, sway, and heave, which are translations along the x-, y-, and z-axes, and rotationsabout the same set of axes are roll, pitch, and yaw (Figure 1.1). For a oating body withlateral symmetry in shape and weight distribution, the six coupled equations of motioncan be reduced to two sets of equations, where the rst set consisting of surge, heave,and pitch can be decoupled from the second set consisting of sway, roll, and yaw. Weinvestigate the second set since the roll motion is important with respect to the stabilityof the oating body.Important investigations to understand the hydrodynamic behavior and motion re-sponse of a oating body were rst started by W. Froude with an initial study of rolling.Subsequently, several investigationswerecarriedoutwiththedevelopmentofstriptheory in ship hydrodynamics, in conjunction with the study of ship vibration. In mostofthestudies, shipmotionswereconsideredinacalmwaterenvironmentuntilthelandmarkresearchworkof WeinblumandSt. Denis[10], whichtakesintoaccount2182 S. N. DASANDS. K. DASWaveRoll(4)XSurge(1)Z Heave(3)Yaw(6)Y Sway(2)Pitch(5)O

OGzcFigure1.1. Signconventionfortranslatoryandangulardisplacementsofaoating body with the direction of wave propagation.sea environment. Cummins [1] formulated linearized equations of motion of a oatingbody in transient seaway. Tasai [6] introduced a strip theory to calculate sway-roll-yawmotions for a ship in oblique waves with zero forward speed. Vugts [9] reported exper-imental observations on sway and roll amplitudes for various cylinder shapes in beamwaves. Salvesen et al. [4] presented a new strip theory for predicting heave, pitch, sway,roll, andyawmotionsaswell aswave-inducedvertical andhorizontal shearforces,bending moments, and torsional moments for a ship in arbitrary heading waves withconstant forward speed. The motion of a oating horizontal cylinder in a uniform in-viscid uid at irregular wave frequencies was studied by Ursell [8], considering it as aclassical potential ow problem. Mulk and Falzarano [3] studied nonlinear ship rollingmotioninsixdegreesoffreedombyusingnumericalpath-followingtechniquesandnumerical integration. Faltinsenetal. [2] studiednonlinearwaveloadsonaverticalcylinder. Theeectofnonlineardampingandrestoringinshiprollingandshipsta-bilityindynamicenvironmentscanbefoundintherecentworksofTaylan[7] andSurendran and Reddy [5].The present study attempts to develop a mathematical approach to determine sway,roll,andyawmotionsandconsidersthegeneralformulationgivenbySalvesenetal.[4] to describe harmonic response of a oating body. This approach is useful for threemain purposes: (i) to get insight into the eect of various parameters and its relativeimportancewhilewaveforcesactinconcurrence, (ii) tovalidatecomplexnumericalmodels by providing useful benchmarking, and (iii) to provide accurate mathematicaltools to supplement detailed model testing.2. Problemformulation. Let (x, y, z) be a right-handed coordinate systemxed withrespect to the mean position of the body and z-axis considered vertically upward. TheDETERMINATIONOFCOUPLEDSWAY, ROLL, ANDYAWMOTIONS . . . 2183origin O lies in the undisturbed free surface. Let the translatory displacements in x, y,and z directions with respect to the origin be 1, 2, and 3 indicating surge, sway, andheave, respectively. In rotational motion, the angular displacements about the same setof axes are 4, 5, and 6 indicating roll, pitch, and yaw, respectively. The denition ofsix motions of any oating body with sign convention is shown in Figure 1.1.In order to construct the governing equations of motion, the following assumptionsare made.(i) The oating body is slender and rigid with symmetric distribution of mass.(ii) Motion amplitude is small so that equations can be linearized.(iii) Except in roll motion, the eect of viscosity is neglected.(iv) Incident waves are unidirectional and of single periodicity.(v) Due to lateral symmetry, longitudinal and transverse motions are decoupled.Undertheaboveassumptions, sixlinearlycoupleddierentialequationsofmotionscan be written as (see [4])6_k=1__Mjk+Ajk_ k+Bjk k+Cjkk_=Fj, j =1, 2, . . . , 6, (2.1)where Mjkarethecomponentsofthegeneralizedmassmatrixoftheship, AjkandBjkarethefrequency-dependentadded-massanddampingcoecients, respectively,Cjk are the hydrostatic restoring coecients, and Fjare the external exciting forces ormoments. The generalized form of mass matrix coecient is given byMjk=___________M 0 0 0 Mzc00 M 0 Mzc0 00 0 M 0 0 00 Mzc0 I40 I46Mzc0 0 0 I500 0 0 I460 I6___________, (2.2)where M is the mass of the oating body, Ij is the moment of inertia in the jth mode ofmotion, and Ijkis the product of inertia for the kth mode of motion coupled with thejth mode. The coordinate of the center of gravity G is (O

, 0, zc), where O

and zcarethe x-coordinate and the z-coordinate of the center of gravity, respectively (Figure 1.1).The added-mass and damping coecient matrices are expressed asAjk=___________A110 A130 A1500 A220 A240 A26A310 A330 A3500 A420 A440 A46A510 A530 A5500 A620 A640 A66___________, Bjk=___________B110 B130 B1500 B220 B240 B26B310 B330 B3500 B420 B440 B46B510 B530 B5500 B620 B640 B66___________.(2.3)2184 S. N. DASANDS. K. DAS3. Sway, roll, andyawmotions. Followingtheassumptions, sway, roll, andyawmotions are described as__Mij_+_Aij___ i_+_Bij__ i_+_Cij__i_=_Fj_, (3.1)where_Mij_=____M Mzc0MzcI4I460 I46I6____,_Aij_=____A22A24A26A42A44A46A62A64A66____,_Bij_=____B22B24B26B42B44B46B62B64B66____,_Cij_=____0 0 00 C4400 0 0____,_ i_=____ 2 4 6____,_ i_=____ 2 4 6____,_i_=____246____,_Fj_=____F2F4F6____,(3.2) iand ibeing the velocity and acceleration in the ith mode of motion, respectively.Substituting (3.2) in (3.1), the governing equations are obtained as_A22+M_ 2+B22 2+_A24Mzc_ 4+B24 4+A26 6+B26 6+=F2, (3.3)_A42Mzc_ 2+B42 2+_A44+I4_ 4+B44 4+C444+_A46I46_ 6+B46 6=F4, (3.4)A62 2+B62 2+_A64I46_ 4+B64 4+_A66+I6_ 6+B66 6=F6. (3.5)Consideringtheforwardspeedequal tozero, theexpressionsforadded-massanddampingcoecientsaregivenintheappendix. Themomentofinertia(Ij) andtheproduct of inertia (Ijk) can be determined for any particular oating body. The restor-ing coecient C44which appears in the roll equation (3.4) can be expressed asC44=gG M, (3.6)where isthedisplacedvolumeofthebodyincalmwater,G Misthemetacentricheight, is the mass density of water, and g is the gravitational acceleration. The waveexciting forces and moments areFi= Fisin(t +), i =2, 4, 6, (3.7)whereF2,F4, andF6 are the amplitudes of the sway exciting force, roll exciting moment,andyawexcitingmoment, respectively, andisthephaseangle. representstheencountering frequency. Since there is no forward speed of the body, the amplitudes ofsway exciting force, roll exciting moments, and yaw exciting moments can be obtainedDETERMINATIONOFCOUPLEDSWAY, ROLL, ANDYAWMOTIONS . . . 2185as (see [4])F2=_ _f2+h2_d,F4=_ _f4+h4_d,F6=_ _f2+h2_d,(3.8)where is the amplitude of the incident wave, fi and hi represent the sectional Froude-Krilo force and sectional diraction force, respectively, and is a variable of integra-tion in x direction. The integration has been taken over the length of the body.4. Method of solution. The coupled sway, roll, and yaw motions can be rewritten inthefollowingformafternormalizingbytherespectivecoecientoftheaccelerationterm: 2+a2 2+b1 4+b2 4+c1 6+c2 6=K2sint,a4 2+a5 2+ 4+b5 4+b64+c4 6+c5 6=K4sint,a7 2+a8 2+b7 4+b8 4+ 6+c8 6=K6sint.(4.1)Foragivenfrequency,theadded-massanddampingcoecientsareconsideredcon-stantforsmallamplitudemotion.Intheabsenceofaphaseangle,weset = 0.TheLaplace transform of (4.1) giveslif2(s)+mif4(s)+nif6(s) =rj(i =1, . . . ,3, j =1, . . . ,4), (4.2)wherefi(s) =_0esti(t)dt. (4.3)The expressions for li, mi, ni, and rjare given in the appendix. Equation (4.2) containsthree unknowns, namely, f2(s), f4(s), and f6(s), which can be solved in the frequencydomain by using Cramers rule:f2(s) = D1D , f4(s) = D2D , f6(s) = D3D , (4.4)whereD1=r1m1n1r2m2n2r3m3n3, D2=l1r1n1l2r2n2l3r3n3,D3=l1m1r1l2m2n2l3m3n3, D =l1m1n1l2m2n2l3m3n3.(4.5)To evaluate (4.4), the denominator term Dshould be nonsingular, that is, D 0. Nowexpanding determinant D, we obtainD =l1_m2n3n2m3_+l2_m3n1n3m1_+l3_m1n2n1m2_. (4.6)2186 S. N. DASANDS. K. DASSubstituting the expressions for li, mi, and niin (4.6), one can writeD =g4s2_s4+r3s3+r2s2+r1s +r0_=g4s2Q, (4.7)where Q is the quartic with respect to the frequency-dependent variable s. ExpressingQ as the product of two quadratic factors, we further writeQ=_s2+211s +21__s2+222s +22_. (4.8)Similarly, determinant D1in (4.5) can be expressed as (see the appendix)D1=r1_m2n3n2m3_+r2_m3n1n3m1_+r3_m1n2n1m2_= s

6i=0D1isi_s2+2_, (4.9)where D1i = f{ai, bi, ci, , Ki, 2i(0), 2i(0)}. Substituting (4.7), (4.8), and (4.9) in (4.4),we obtainf2(s) =s

6i=0D1isig4s2_s2+2__s2+211s +21__s2+222s +22_. (4.10)The partial fraction (4.10) givesf2(s) =1g4_1s+2s +3s2+2 +4s +5s2+211s +21+6s +7s2+222s +22_, (4.11)where i=f{D1i, , i, i}. Similarly, the expressions for f4(s) and f6(s) aref4(s) =1g4_

1s +

2s2+2 +

3s +

4s2+211s +21+

5s +

6s2+222s +22_,f6(s) =1g4_

1s+

2s +

3s2+2+

4s +

5s2+211s +21+

6s +

7s2+222s +22_,(4.12)where i,

i, and

iareunknowncoecientsrequiredtobedetermined. Equatingthe like powers of s, a set of linear algebraic equations involving the above unknowncoecients are obtained which are then solved by using the Gauss elimination method.5. Numericalexperiment. Togetaninsightintotheeectofvariousparameterson sway, roll, and yaw motions, a ship of length = 150m, beam = 20.06m, draught =9.88m, and mass =19190tons was assumed, for which the beam-draft ratio becomesnearly equal to two. The location of center of gravity Gis considered at the point O

,which is 1maway fromthe origin O. The monochromatic sinusoidal waves act perpen-dicular to the longitudinal axis of the ship with three dierent frequencies of 0.56rad/s,0.74rad/s, and 1.24rad/s, corresponding to the wave height of 1.0m. The coecientsrelated to sectional added mass, sectional damping, and sectional wave exciting forcewere used from the experimental results of Vugts [9] and Franks close-t curve (Table5.1), for a cylinder with rectangular cross-section with identical beam-draft ratio.A computer program, SIPCOEF, was developed to generate the relevant coecientscorresponding to the set of linear equations for each mode of motion. In order to obtainDETERMINATIONOFCOUPLEDSWAY, ROLL, ANDYAWMOTIONS . . . 2187Table 5.1. Computed sectional coecients of the oating body.WaveFrequency (Rad/s) 0.56 0.74 1.24Period (s) 11.2 8.5 5.1SectionalcoecientsSway added mass 1.6 0.65 0.05Roll added mass 0.07 0.055 0.035Sway-roll added mass 0.25 0.13 0.02Sway damping 0.6 1.0 0.7Roll damping 0.01 0.02 0.012Sway-roll damping 0.07 0.16 0.1Sway exciting force 2.25 1.5 0.34Roll exciting moment 1.9 1.2 0.28Principal dimensions of theoating bodySec. coe. for added mass,damping, and wave excitingforceInitial conditionsWave parametersSIPCOEFCalculation of the coe. aijof the system ofequations j aijj = diin transformed domainCalculation of damping factor (i) and undampedsystem frequency (i)GAUSEUSolves the system of equations to get j,

j,

j .CALMOTCalculation of sway, roll, and yaw motionsTime historySway Roll YawFigure 5.1. Schematic diagram of the mathematical model development.unknowncoecientsinthetransformeddomain,theGausseliminationmethodwasused. Acomputerprogram, GAUSEU,wasusedtosolvethesystemofequations.Finally, the time evolution of sway, roll, and yawmotions for a particular wave frequencywas obtained fromthe programCALMOT. The schematic diagramof the mathematicalmodel development is shown in Figure 5.1.6. Systemstability and quartic analysis. In order to obtain the inverse of the Laplacetransform, determinant D which appears as the denominator is set equal to zero. Thisleadstotheconditionthateither Qor g4s2isequaltozero.For Q = 0,therootsofthe characteristic equation are obtained in the frequency domain. The variables i andi(i = 1, 2)whichappearinthebiquadraticsareknownasdampingfactorsandun-damped natural frequencies of the damped system, respectively. As the factor g4s2is2188 S. N. DASANDS. K. DASTable 6.1. Computed damping coecients.Wave frequency(rad/s)Damping coecients120.56 1.13 0.0530.74 1.29 0.041.24 2.3 0.077setequaltozeroin D,thecorresponding 3alsobecomeszero.Thismanifestspureoscillation, without damping or building up. As the numerical values for i are of primeimportance, their role on system stability is summarized as follows:(i) i1 implies that the motion will be a nonoscillatory divergence;(ii)1