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ADAMSADAMSADAMSADAMSADAMSADAMSDAE331GroundCoordinateSystemADAMS2LocalPartReferenceFrameLPRFADAMS3MarkerSystemADAMSADAMSiBTizyxq],,,,,[ϕθψ=TTnTTqqqq],,,[21L=ADAMSnh12()[(),(),...,()]0KKKKTnhqqqqΦ=ΦΦΦ=4-1nhnc−0),(=ΦtqD(4-2)4-14-20),(),(),(=⎥⎦⎤⎢⎣⎡ΦΦ=ΦtqtqtqDK(4-3)4ADAMS4-3ncnc(4-3)0),(),(),,(=Φ+Φ=Φtqqtqtqqtq&&&(4-4)),(tqtΦ−=υ0),(),,(=−Φ=Φυqtqtqqq&&&(4-5)(4-4)0),(),(2)),((),(),,,(=Φ+Φ+Φ+Φ=Φtqqtqqqtqqtqtqqqttqtqqq&&&&&&&&&&(4-6)ttqtqqqqqΦ−Φ−Φ−=&&&2)(η0),,(),(),,,(=−Φ=Φtqqqtqtqqqq&&&&&&&&η(4-7)qΦΦmqnqΦnm×jijiqq∂Φ∂=Φ),()(qΦncnc×nhncnhncADAMS(,)0nqtΦ=(4-8)ntNewton-Raphson(,)0jqjjnqqtΦ∆+Φ=(4-9)1jjjqqq+∆=−jntADAMSCALAHANMichiganDonaldCalahanHARWELLHARWELLIanDuffCALAHANHARWELLCALAHAN1qtq−=−ΦΦ&(4-10)1()2qqqqtttqqqq−⎡⎤=−ΦΦ+Φ+Φ⎣⎦&&&&&(4-11)ADAMSADAMSB[,,,,,]Tqxyzψθϕ=[],,TRxyz=[],,Tγψθφ=[,]TTTqRγ=coscossincossincossinsincoscossinsinsincoscoscossinsinsincoscoscoscossinsinsinsincoscosgiAψφψθφψφψθφψθψφψθφψφψθφψθθφθφθ−−−⎡⎤⎢⎥=+−+−⎢⎥⎢⎥⎣⎦4-12sinsin0cossincos0sincos10Bθφθθφθθ⎡⎤⎢⎥=−⎢⎥⎢⎥⎣⎦4-13Bωγ=&4-14ADAMSeωeωγ=&4-15ADAMS:1()njiijjjdTTQdtqqqλ=∂∂∂Φ−=+∂∂∂∑&(4-16)TjqjQjqjqADAMSjjTPq∂=∂&(4-17)4ADAMS1njiijCqλ=∂Φ=∂∑4-18(4-16jjjjTPQCq∂−=−∂&(4-19)1122TTTTRMRBJBγγ=+&&&&(4-20)MJ(4-19)RRRRTPQCq∂−=−∂&(4-21)TPQCqγγγγ∂−=−∂&(4-22)()RRddTPMRMVqdtdt⎛⎞∂===⎜⎟∂⎝⎠&&&&0RTq∂=∂(4-21)RRMVQC=−&(4-23)TTPBJBqγγγ⎛⎞∂==⎜⎟⎜⎟∂⎝⎠&&BADAMSPγ&ADAMS15121512ADAMS[][],,,,,,,,,,TxyzTTTeTVVVVRxyzPPPPγψθφψθφωωωωγψθφ⎧⎡⎤=⎣⎦⎪⎪=⎪⎪⎡⎤=⎨⎣⎦⎪⎡⎤⎪=⎣⎦⎪⎪=⎩(4-24)RRTeeMVQCVRTPQCqPBJBγγγγγωωγ⎧=−⎪⎪=⎪∂⎪−=−⎨∂⎪⎪=⎪⎪=⎩&&&&(4-25)ADAMS()0,0(,,)TTqTPHFqTPquqqtFfuqtλ∂⎧−+Φ+=∂⎪⎪∂=⎪∂⎪⎨=⎪⎪Φ=⎪=⎪⎩&&&(4-26)PHADAMS4ADAMS4-1MI2L,OOXYOpXpYpOXYxy()22212TMxMyIθ=++&&&TMxxTMyyTIθθ⎧∂=⎪∂⎪∂⎪=⎨∂⎪⎪∂=⎪∂⎩&&&&&&00THFMg⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦cos0sin0xLyLθθ−=⎧⎨−=⎩ADAMS10sin01cosqLLθθ⎡⎤Φ=⎢⎥−⎣⎦12λλλ⎡⎤=⎢⎥⎣⎦1212000sincos0xTTqyMVTPHFMVMgqPLLθλλλλθλθ⎡⎤+=⎢⎥∂−+Φ+=⇒+−=⎢⎥∂⎢⎥+−=⎣⎦&&&&TPPIqθθω∂=⇒=∂&000xyVxuqVyθωθ−=⎡⎤⎢⎥=⇒−=⎢⎥⎢⎥−=⎣⎦&&&&121200sincos00000cos0sin0xyxyMVMVMgPLLPIVxVyxLyLθθθθλλλθλθωωθθθ⎡⎤+=⎢⎥+−=⎢⎥⎢⎥+−=⎢⎥−=⎢⎥⎢⎥−=⎢⎥−=⎢⎥⎢⎥−=⎢⎥⎢⎥−=⎢⎥−=⎣⎦&&&&&&12TxyxyVVPθθθωλλ⎡⎤⎣⎦4ADAMSADAMS1Minimize()001()2TCqqWqq=−−Subjectto()0qΦ=qWq0()()()T001q-qWq-q+Φq2TLλ=(4-38)L0,0LLqλ∂∂==∂∂()0Φ()0Φq0TWqqqλ⎧⎡⎤∂−+=⎪⎢⎥∂⎨⎣⎦⎪=⎩(4-39)Newton-Raphson[][]()()00TTWqWqqqqqqλλ⎡⎤⎡⎤∂Φ⎧⎫⎢⎥⎡⎤∂Φ⎢⎥∆−+∂⎪⎪⎧⎫⎢⎥⎣⎦⎢⎥=∂⎨⎬⎨⎬⎣⎦⎢⎥∆⎩⎭⎡⎤∂Φ⎪⎪⎢⎥Φ⎢⎥⎩⎭⎢⎥∂⎣⎦⎣⎦(4-40)2Minimize()001()2TCqqWqq=−−&&&&Subjectto0qqt⎡⎤∂Φ∂Φ+=∂∂⎢⎥⎣⎦&0q&ADAMSW0q&()()T001q-qWq-q+q2TLqtλ⎛⎞⎡⎤∂Φ∂Φ=+⎜⎟∂∂⎢⎥⎣⎦⎝⎠&&&&&(4-41)L0Φ()00TWqqqqqtλ⎧⎡⎤∂−+=⎪⎢⎥∂⎪⎣⎦⎨⎡⎤∂Φ∂Φ⎪+=⎢⎥⎪∂∂⎣⎦⎩&&&(4-42)q[][]00TWWqqqtqλ⎡⎤⎡⎤∂Φ⎢⎥⎧⎫⎢⎥∂⎧⎫⎪⎪⎢⎥⎣⎦=∂Φ⎨⎬⎨⎬⎡⎤⎢⎥⎩⎭⎡⎤∂Φ⎪⎪⎢⎥⎢⎥∂⎣⎦⎩⎭⎢⎥⎢⎥∂⎣⎦⎣⎦&(4-43)3(4-26)ADAMSDAEDAEODE4.5DAE=uq&Index3I3()0,0(,,)TTqTPHFqTPquqqtFfuqtλ∂⎧−+Φ+=∂⎪⎪∂=⎪∂⎪⎨=⎪⎪Φ=⎪=⎪⎩&&&(4-44)4ADAMS()λuqJacobianNewton-Rapsonh0JacobianIndexreductionmethodsADAMSSI2(Stabilized-IndexTwo)SI1(Stabilized-IndexOne)SI2()()00;(0),0,,0(,,)TTqTqTPHFqTPquqqtqutFfuqtλµµ∂⎧−+Φ+=∂⎪⎪∂=⎪∂⎪⎪−+Φ==⎨⎪Φ=⎪⎪Φ=⎪=⎪⎩&&&&(4-45)ΦΦ&h0JacobianSI1()()00;,0,,0(,,)TTqTqTPHFqTPquqqtqutFfuqtηξ∂⎧−+Φ+=∂⎪⎪∂=⎪∂⎪⎪−+Φ=⎨⎪Φ=⎪⎪Φ=⎪=⎪⎩&&&&&&(4-46)η&ζ&,ηλζµ==&&1η&ζ&SI1quηζSI1ADAMS()00,0(,,)TTqTHFqTPquqtFfuqtλ∂⎧−+Φ+=∂⎪⎪∂=⎪∂⎪⎨=⎪⎪Φ=⎪=⎪⎩&4-47Newton-Rapson,qquu∗∗==0MuCuKquq++=⎧⎨=⎩&&4-48MCK4.41ADAMSNewton-RaphsonLUCALAHANHARWELLBDF(BackwardsDifferentiationFormulation)(PECEPredict-Evaluate-Correct-Evaluate)Index3SI2SI14ADAMSNewton-Raphson(High-Frequencies)(Coordinate-PartitionedEquation)DAEODEABAMAdams-Bashforth-Adams-MoultonRKF45ADAMS●CALAHANHARWELL●Newton-Raphson●DAEBDFSI2GSTIFFWSTIFFCONSTANT_BDFSI1GSTIFFWSTIFFCONSTANT_BDFI3GSTIFFWSTIFFDstiffCONSTANT_BDF●ODE()ABAMRKF451DAEADAMSDAEBDF1Gear--221212!nnnnyyyyhhtt+∂∂=+++∂∂L4-491nnhtt+=−Gear1k+()10111knniniiyhyyβα++−+==−+∑&4-501ny+()yt1ntt+=0βiαGear111101[]knniniiyyyhαβ++−+=−=−∑&4-51ADAMS2●G(,,)0Gyyt=&y●Newton-Raphsony∆yG1(,,)nJyGyyt+∆=&J●Newton-Raphson1:kkkyyyy+=+∆●y∆3●●hn●ttt=+∆2ADAMSABAM(AdamsBashforthandAdams-Moulton)RKF45-ABAMRKF45ABAM1{}q{}iq{}dq{}{}Tidqqq=2Adams-Bashforth1nt+{}PiqP3Adams-Moulton{}CiqC454ADAMS1DAEI3JacobianSI2JacobianJacobianI3I3SI2I3SI1JacobianSI24-1Index3SI2SI121GstiffGstiff6DAEI3SI2SI1ADAMSI3JacobianSI2SI1Jacobian2WstiffWstiff6DAEI3SI2SI1NDFNewtonDividedDifferenceGstiff3Dstiff3DstiffDstiff6DAEADAMSI3NDFNewtonDividedDifferenceGstiffDASSLPetzold4Constant_BDFConstant_BDF6DAEI3SI2SI1NDFNewtonDividedDifferenceSI2GstiffGstiffGstiffWstiffHmaxHmax5ABAMABAM(12)ABAMDAEODEL.F.ShampineM.K.Gordon6RKF45RKF45ABAML.FShampineH.A.WattsDDERKF200204ADAMSADAMSABAMRK45GstiffWstiffDstiffConstant_BDFABAMRK452.5.2k110N/mmk21000000N/mmm1=1kgm2=1kg11.25Hz7117.6Hzc10.1N/mm/sc2100000N/mm/sADAMSABAMRKF45GstiffWstiffDstiffConstant_BDFABAMRKF45GstiffWstiffDstiffConstant_BDF