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Stiff206060808080ANSYSNASTRANADAMSDADSCAE16871775ADAMS-17431788ODEDAE1908-2060-Roberson-WittenburgKane1966R/WR/W1965-N/EN/E2ChaceHaug80-2080Haug2070-123CAE2060z1977InternationalUnionofTheoreticalandAppliedMechanics-IUTAMMagnusz1983NATO-NSF-ARDHaugz1985InternationalAssociationofVehicleSystemADAMSDynamics-IAVSDKortumSchiehlenz1985IUTAMIFTOMMUdineBianchiSchiehlenz1989zz1986z1988z1992z1996z1977WitenburgDynamicsofSystemsofRigidBodiesz1986SchiehlenTechnischeDynamikz1988NikraveshComputer-AidedAnalysisofMechanicalSystemsz1988RobersonSchwertassekDynamicsofMultibodySystemz1989HaugComputer-AidedKinematicsandDynamicsofMechanicalSystemsz1989ShabanaDynamicsofMultibodySystemsz1990HustonMultibodyDynamicsz1992BremerPfeifferElastischeMehrkorpersystemz1992AmiroucheComputationalMethodsforMultibodyDynamicsz1994GarciaBayoKinematicandDynamicSimulationofMultibodySystemsz1994ShabanaComputationalDynamicsz1996StejskalValasekKinematicsandDynamicsofMachinery2CADz..1989z..1996z..1996z..19992060802090Roberson-Wittenburgq),,(),(tqqBqtqA&&&=(2-1)ODEs-OrdinaryDifferentialEquationsABChaceHaugADAMS367Nq3N6N7N⎩⎨⎧=Φ=Φ+0),(tqBqATqλ&&(2-2)ΦqqΦλ-DAEs-DifferentialAlgebraicEquations-Euler-LagrangeEquationsADAMSDADSGarciaBayo1994TTTaqp)(=qa2-12-2qp2-1ODEs--2ODEODE2-2-DAEsDAEqa-DAE2.51--0),,(),(),(=−Φ+tqqQtqqtqMTq&&&λ(2-3)0),(=Φtq(2-4)qq&nRq∈&&mR∈λRt∈nnRM×∈nmqR×∈ΦnRQ∈mR∈Φ(1.4)0),(),(),,(=−Φ=Φtqqtqtqqqυ&&&(2-5)0),,(),(),,,(=−Φ=Φtqqqtqtqqqq&&&&&&&&η(2-6)),(tqtΦ−=υttqtqqqqqΦ−Φ−Φ−=&&&2)(η⎩⎨⎧==00)0()0(qqqq&&(2-7)-z-ODEz(2-3)(2-4)-3(2-3)-(2-6)-1z-z(2-7)(2-4)(25)z-2---q&&λADAMSq&&q&q-ODAEs(2-3)(2-6)q&&λq&&q&q(2-4)(2-5)-(2-6)(2-3)(2-6)-ODAEs-DAEDAE(2-3)(2-4)(2-5)(2-6)-DAEDAEnODAEODAEODAEODAEsODEODEnp-(2-1)ODELUQRSVDLUquvqΦLUq&q&&v&v&&(2-3)(2-1)v&&QRqΦz&-(2.1-1)z&&z&SVDqΦ(2-3)(2-6)(2-4)(2-5)2Gram-SchmidtqΦnmqR×∈ΦnmnRB×−∈)(nnRP×∈Gram-SchmidtPVnRz∈&qVzT&&=z&z&&Iz&Iz&&-Iz&&(2-3)(2-6)--3----LUODAE2.5HaugADAMS--TSDA--RSDA22.12.1ADAMS2.11nbxoyiiiiyox′′′iio′Tiiiyxr],[=iφiTiiiiyxq],,[φ≡2.2nbnbnc×=3TTnbTTqqqq],...,,[21=2iφ2.2nh0)](),...,(),([)(21=ΦΦΦ=ΦTKnhKKKqqqq(2-8)ncnhnhncnhnhncDOF−=1DOF212nhnc−0),(=ΦtqD(2-9)2-82-9ADAMS0),(),(),(=⎥⎦⎤⎢⎣⎡ΦΦ=ΦtqtqtqDK(2-10)2-10ncnc2-10)(tq2(2-10)0),(),(),,(=Φ+Φ=Φtqqtqtqqtq&&&(2-11)),(tqtΦ−=υ0),(),,(=−Φ=Φυqtqtqqq&&&(2-12)qΦ(2-12)q&(2-11)0),(),(2)),((),(),,,(=Φ+Φ+Φ+Φ=Φtqqtqqqtqqtqtqqqttqtqqq&&&&&&&&&&(2-13)ttqtqqqqqΦ−Φ−Φ−=&&&2)(η0),,(),(),,,(=−Φ=Φtqqqtqtqqqq&&&&&&&&η(2-14)qΦ(2-14)q&&(2-12)(2-14)qΦΦmqnqΦnm×jijiqq∂Φ∂=Φ),()(qΦncnc×(2-12)υ(2-14)η3srxoyiiiyox′′′⎪⎩⎪⎨⎧=′=′′TyxTyxssssss],[],[(2-15)Pxoyiiiyox′′′2-3PPPsArsrr′+=+=(2-16)PrPrO′PssrPs′srA⎥⎦⎤⎢⎣⎡−==φφφφφcossinsincos)(AA(2-17)2ABAddAφφφφφφφφ&&&&≡⎥⎦⎤⎢⎣⎡−−−==sincoscossin(2-18)(2-16)xyorPrPsPxyo′rPrPsPoyxxzyz2.32.4(2.2-9)PPPsBrsArr′+=′+=φ&&&&&(2-19)(2.2-12)PPPPPsAsBrsBsBrr′−′+=′+′+=2φφφφ&&&&&&&&&&&&&(2-20)(2-10)(2-12)(2-14(2-16)(2-19)(2-20)412.4zyxo′′′′o′Tzyxr],,[≡zyxo′′′′oxyzADAMS⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡==333231232221131211],,[aaaaaaaaahgfA(2-21)fghzyxo′′′′xo′′yo′′zo′′AA963zyxo′′′′oxyz0=rsrs′ssAs′=(2-22)PPsArr′+=(2-23)PrPoxyzrzyxo′′′′o′oxyzPs′Pzyxo′′′′Azyxo′′′′oxyz(2-23)PPPsrsArrω~+=′+=&&&&(2-24)ω~ω2-38ωzyxo′′′′oxyzTAA&=ω~(2-25)ω(2-22)sAAs~~′=zyxo′′′′ω′AAT&=′ω~(2-26)ωω~~′==AAA&(2-27)(2-24)PPsArr′+=&&&&&&(2-28)AAAωωω~~~+=&&&(2-29)rωrδδπ(2-24)(2-25)(2-26)(2-27)PPPsrsArrπδδδδδ~+=′+=(2-30)TAAδπδ=~(2-31)AATδπδ=′~(2-32)πδπδδ~~′==AAA(2-33)ω′δπoxyzzyxo′′′′oxyzurχzyxo′′′′urχ42⎪⎪⎩⎪⎪⎨⎧≡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡==2sin2cos3210χχueeeee(2-34)14×TTTeeeeeep],,,[],[32100==(2-35)123222120=+++=eeeeppT(2-36)43pA)~(2)12(020eeeeIeAT++−=(2-37)I33×e~e⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−=000~121323eeeeeee(2-38)⎪⎪⎩⎪⎪⎨⎧−=−=−=+=)4/()()4/()()4/()(4/)1(01221303113202332120eaaeeaaeeaaetrAe(2-39)00≠e(2-39)00=e⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧=+=+=+−+=−+=−+=3223323113312112213323222211214444/)21(4/)21(4/)21(eeaaeeaaeeaatrAaetrAaetrAae(2-40)trAAADAMS[]⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−−−−=+−≡0123103223010~,eeeeeeeeeeeeIeeeE(2-41)[]⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−−−−=+−−≡0123103223010~,eeeeeeeeeeeeIeeeG(2-42)TEGA=(2-43)TGEA&&2=(2-44)pE&2=ω(2-45)pG&2=′ω(2-46)ωω′==TTGEp2121&(2-47)(2-44)(2-47)TTGEEGAδδδ+=(2-48)pEδδπ2=(2-49pGδπδ2=′(2-50)πδδπδ′==TTGEp2121(2-51)zo′′xo′′zo′′ψθφ313ψθφψθφ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⋅⋅⋅−⋅⋅+⋅−⋅⋅+⋅⋅⋅⋅−⋅−⋅⋅−⋅=θφθφθθψφθψφψφθψφψθψφθψφψφθψφψccssssccccssscccsssccsscscsccA(2-52)2⎪⎪⎩⎪⎪⎨⎧===−=−==)/arcsin()/arccos()/arcsin()/arccos()1arcsin()arccos(3132132323333θθφθθψθsasasasaaa(2-53),...)1,0(==nnπθ2-53⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧⋅+=⋅−=⋅−=⋅+=)2cos()2sin()2sin()2sin()2sin()2cos()2cos()2cos(4210θφψθφψθφψθφψeeee(2-54)⎪⎩⎪⎨⎧−=−+=+=)/()/(]1)(2arccos[)/()/(120323201203eearctgeearctgeeeearctgeearctgφθψ(2-55)2i⎥⎦⎤⎢⎣⎡=iiiprq(2-56)nbTTnbTTqqqq],...,,[21=(2-57)nb7(2-36)01=−=ΦiTipippnbi,...,2,1=(2-58)0],...,,[21=ΦΦΦ=ΦTpnbppp(2-59)nbnhADAMS0)](),...,(),([)(21=ΦΦΦ=ΦTKnhKKKqqqq(2-60))6(nhnb−0),(=ΦtqD(2-61)0),()()(),(=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡ΦΦΦ≡ΦtqqqtqDKp(2-62)(2.2-55)nb7nb7(2.2--58)(2-51)02=′==ΦiTiTiiTipiGpppπδδδ(2-63)0==Φ′TiTipiGpiπ(2-64)2-632-64Gp=0(2-64)ω′ω&′⎥⎦⎤⎢⎣⎡≡⎥⎦⎤⎢⎣⎡Φ−Φ−=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧′⎥⎥⎦⎤⎢⎢⎣⎡ΦΦ+⎥⎥⎦⎤⎢⎢⎣⎡ΦΦ∑=′′DKDtKtnbiiDKiDrKriiiirυυωππ1&(2-65)KKtυ≡=Φ−0(2-66)(2.2-58)⎥⎦⎤⎢⎣⎡≡⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧′⎥⎥⎦⎤⎢⎢⎣⎡ΦΦ+⎥⎥⎦⎤⎢⎢⎣⎡ΦΦ−⎥⎦⎤⎢⎣⎡ΦΦ−=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧′⎥⎥⎦⎤⎢⎢⎣⎡ΦΦ+⎥⎥⎦⎤⎢⎢⎣⎡ΦΦ∑∑=′′=′′DKnbiiDKiDrKrDttKttnbiiDKi