基于二阶常微分方程的多体系统动力学设计灵敏度分析的伴随变量方法

232Vol.23No.220062Feb.2006ENGINEERINGMECHANICS562004-04-022005-02-27(29903006)*(1978)(E-mail:[e-mail])(1966).1000-4750(2006)02-0056-04*(266071)O313.7AADJOINTVARIABLEMETHODFORDESIGNSENSITIVITYANALYSISOFMULTIBODYSYSTEMDYNAMICSDESCRIBEDBYORDINARYDIFFERENTIALEQUATIONS*DINGJie-yu,PANZhen-kuan(InformationEngineeringCollege,QingdaoUniversity,Qingdao266071,China)Abstract:Generalformulationsfordesignsensitivityanalysisofmultibodysystemdynamicsgovernedbyordinarydifferentialequationswithgeneralobjectivefunctionsinintegralformsareobtainedthroughtheadjointvariablemethod.Itismoreefficientatcomputationthanthedirectdifferentialmethodinthecaseofmanydesignparameters,becausethereisnoneedtocomputeahugeamountofderivativesofgeneralizedcoordinateswithrespecttothedesignparameters.Inordertoimprovecomputationspeedandaccuracy,theintegraltypeofobjectivefunctionanditsderivativeswithrespecttothedesignparametersaretransformedintosimpleordinarydifferentialequations.Anumericalexampleofaplanarmanipulatormodelwithtwolinksispresentedintheend.Keywords:dynamicsandcontrol;multibodysystemdynamics;adjointvariablemethod;designsensitivityanalysis;ordinarydifferentialequationsHaug[1]Haug[2]57Etman[3]BestleSerban[4,5][4][5][6,7]AndersonSohl[8,9]1Tnqqq][21L=qTnbbb][21L=b),,,(),(tbqqQqbqM&&&=(1)1t(1)0bqq0bqΦ==),,(),(11111&ϕ(2)0q0qΦ≠∂∂≠∂∂1111det,det&ϕ2t0),,,(222=tbqq&Ω(3)0dd222222≠∂∂+∂∂+∂∂==ttΩΩΩΩΩqqqq&&&&&∫+=21d),,,,(),,,(222ttttHtGbqqqbqq&&&&Ψ(4)ΨbtHHHHGGGtHGttd)()(dd21222222∫++++++++=bbqbqbqbbqbqbqqqqqb&&&&&&&&&Ψ(5)0dd222222≠∂∂+∂∂+∂∂==tGGGtGGqqqq&&&&&(3)bΩΩΩΩ&&&/][22222bbqbqt++−=qqb(6)(6)(5)2bt∫+++++−++−++−=212222d][]/)([]/)([]/)([dd22222tttHHHHHGGHGGHGGbbqbqbqbbbqqbqqqqqqqb&&&&&&&&&&&&&&&ΩΩΩΩΩΩΨ(7)(7)bbbbbqqqqq,,,,22&&&&2QqMΠ−=&&(1)b0ΠqΠqQqMbbqbqb=++−&&&&(8)µ(8)],[21tt0ΠqΠqQqMµbbqbqb=++−∫tttTd)(21&&&&(9)(2)0qq0ΦqΦbbqbqbbq11=++=[phone]ϕϕϕ&&(10)τη,(10)0τqτqτ0ΦηqΦηbbqbqbbq=++=[phone]ϕϕϕTTTTT&&(11)(9)(11)(7)tHHHHHGGHGGHGGTTttTTTTTTTd])()()[(]/)([)(]/)([]/)([dd211112222112111112222bbbqqbqqbqbbbbbqqbqbqqbqqΠµqΠµqQµqMµτΦηΩΩqτΦηqτqqΩΩb+++++++++++−+++++−++−=∫&&&&&&&&&&&&&&&&&&ϕϕϕΩΩΨ(12)(12)bq&&bq&tHtHtHHGGHtHHHtHHGGHHGGTTTTttTTTTTTTTTTTTTTTTd])2dddd[(]/)([)dd()(]dd/)([]/)([dd2111112222211211111111111111122222222222222bbbqqqqqbbbbbqqqqqbqqbqqqqqbqqqΠµqMµMµMµQµQµΠµτΦηqMµMµQµτΦηqMµτqMµMµQµΩΩqMµb+++++++−+++++−+++++−++−−+−−−−++−++++−=∫&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&ϕϕϕΩΩΩΩΨ(13)58(13)bbbbbqqqqq,,,,1122&&TTTGHHG222222/)(qqqµM&&&&&&−−+=ΩΩ(14a)ΩΩ&&&&&&&&/)(dd)(2222222222TTTTTHGGHtHqqqqqµQMµM+−+−=++(14b)11111µMqq+=TTH&&&τϕ(14c)111111111)(dd111µQMµMτηΦqqqqqTTTTTHtH&&&&&&+−−−=+ϕ(14d)TTTTTHHttqqqqqµQΠMµQMµM−=+++++&&&&&&&&&dd)dd()2((14e)tHHGGttTTTd)(/)(dd21112∫+++++−=bbbbbbΠµτΦηbϕΩΩΨ&&(15)(14a)2µ(14b)2µ&2µ,2µ&(14e)µµµ,,&&&11,µµ&(14c)τ(14d)ηητµ,,(15)bddΨ3∫+=ttsHGtA1d)((16)ψ===)(,)(,dd21tAGtAHtA(17)∫+++++−=ttTTTsHHGGt1d)(/)()(112bbbbbbΠµτΦηBϕΩΩ&&(18)bBτΦηBΠµBbbbbbbdd)(/)()(dd21121ΨΩΩ=+++−=+=tHGGtHtTTTϕ&&(19)(17)(19)ΨbddΨ4121,ll21,mmyx(x1,y1)(x2,y2)l1m1l2m21Fig.1AplanarmanipulatorwithtwolinksT][21θθ=qTmmll][2121=b][diag222111JmmJmm=M2111121lmJ=2222121lmJ=Tgmgm]0000[21−−=Q1m=1kg,2m=2kg,1l=1m,m32=l,TT]2π33π[][12111==θθqtlllld})]sin(sin[)]cos(cos{[22121110221211θθθθθθΨ+++++=∫=Ψ4.9796bddΨ=[1.92991.77570.02250.0257]52θ1θ59[1]HaugEJ,AroraJS.Appliedoptimaldesign:mechanicalandstructuralsystems[M].NewYork:JohnWiley&Sons,Inc.,1979.[2]HaugEJ,ManiNK,KrishnaswamiP.Designsensitivityanalysisandoptimizationofdynamicallydrivensystem[A].HaugEJ.NatoASISeries,Vol.F9,ComputerAidedAnalysisandOptimizationofMechanicalSystems[C].Spring-VerlagBerlin,1984.[3]EtmanLFP.Optimizationofmultibodysystemsusingapproximationconcepts[D].TechniqueUniversityofEindhoven,1997.[4]BestleD,EberhardP.Analyzingandoptimizingmultibodysystems[J].MechanicsofStructuresandMachines,1992,20(1):67~92.[5]SerbanR.Dynamicsandsensitivityanalysisofmultibodysystems[D].TheUniversityofIowa,1998.[6],.[J].(),2002,17(4):1~11.PanZhenkuan,ZhaoWeijia.Designsensitivityanalysisanddynamicoptimizationofmultibodysystemdynamics[J].JournalofQingdaoUniversity(EngineeringandTechnique),2002,17(4):1~11.(inChinese)[7],,,,[J].,2002,34:70~74.PanZhenkuan,ZhaoWeijia,GaoLei,GaoBo.Designsensitivityanalysisofmultibodysystemdynamics[J].JournalofMechanics,2002,34:70~74.(inChinese)[8]AndersonKS,HsuYH.Analyticalfull-recursivesensitivityanalysisformultibodychainsystems[J].MultibodySystemsDynamics,2002,8(1):1~27.[9]SohlGA,BobrowJE.Arecursivemultibodydynamicsandsensitivityalgorithmforbranchedkinematicchains[J].JournalofCryptology,2001,123(3):391~399.[10]LiS,PetzoldL.Softwareandalgorithmsforsensitivityanalysisoflarge-scaledifferential-algebraicsystems[J].JournalofComp.AndApp.Math.,2000,13:131~145.(91)[1],.[J].,2005,20(1):1~6.GuoYanlin,DongQuanli.Researchandapplicationofsteelplateshearwallinhigh-risebuildings[J].SteelStructures,2005,20(1):1~6.(inChinese)[2],.[J].(),2004.196~199.GuoYanlin,MiaoYouwu.Elasticbucklingbehaviorofsteelplateshearwallwithtwoedgeslots[J].BuildingStructures(Sup.),2004.196~199.(inChinese)[3].[D].:,2004.MiaoYouwu.Behaviorstudyofsteel