Multibody System Dynamics-Roots and Perspectives

MultibodySystemDynamics1:149–188,1997.149c1997KluwerAcademicPublishers.PrintedintheNetherlands.MultibodySystemDynamics:RootsandPerspectivesW.SCHIEHLENInstituteBofMechanics,UniversityofStuttgart,D-70550Stuttgart,Germany(Received:21January1997;acceptedinrevisedform:15April1997)Abstract.Thepaperreviewstheroots,thestate-of-the-artandperspectivesofmultibodysystemdynamics.Somehistoricalremarksshowthatmultibodysystemdynamicsisbasedonclassicalmechanicsanditsengineeringapplicationsrangingfrommechanisms,gyroscopes,satellitesandrobotstobiomechanics.Thestate-of-the-artinrigidmultibodysystemsispresentedwithreferencetotextbooksandproceedings.Multibodysystemdynamicsischaracterizedbyalgorithmsorformalisms,respectively,readyforcomputerimplementation.Asaresultsimulationandanimationaremostimportant.Thestate-of-the-artinflexiblemultibodysystemsisconsideredinacompanionreviewbyShabana.Futureresearchfieldsinmultibodydynamicsareidentifiedasstandardizationofdata,couplingwithCADsystems,parameteridentification,real-timeanimation,contactandimpactproblems,extensiontocontrolandmechatronicsystems,optimalsystemdesign,strengthanalysisandinterac-tionwithfluids.Further,thereisastronginterestonmultibodysystemsinanalyticalandnumericalmathematicsresultinginreductionmethodsforrigoroustreatmentofsimplemodelsandspecialinte-grationcodesforODEandDAErepresentationssupportingthenumericalefficiency.Newsoftwareengineeringtoolswithmodularapproachespromiseimprovedefficiencystillrequiredforthemoredemandingneedsinbiomechanics,roboticsandvehicledynamics.Keywords:dynamicsofrigidbodies,multibodysystems,computationalmethods,datamodels,parameteridentification,optimaldesign,strengthanalysis,DAEintegrationcodes.1.HistoricalRemarksThedynamicsofmultibodysystemsisbasedonclassicalmechanics.ThemostsimpleelementofamultibodysystemisafreeparticlewhichcanbetreatedbyNewton’sequationspublishedin1686inhis“PhilosophiaeNaturalisPrincipiaMathematica”[111].Theprincipalelement,therigidbody,wasintroducedin1775byEulerinhiscontributionentitled“Novamethodusmotumcorporumrigidarumdeterminandi”[43].Forthemodelingofconstraintsandjoints,Euleralreadyusedthefreebodyprincipleresultinginreactionforces.TheequationsobtainedareknowninmultibodydynamicsasNewton–Eulerequations.Asystemofconstrainedrigidbodieswasconsideredin1743byd’Alembertinhis“Trait´edeDynamique”[32]wherehedistinguishedbetweenappliedandreac-tionforces.D’Alembertcalledthereactionforces“lostforces”havingtheprincipleofvirtualworkinmind.Amathematicalconsistentformulationofd’Alembert’sprincipleisduetoLagrange[89]combiningd’Alembert’sfundamentalideawith150W.SCHIEHLENtheprincipleofvirtualwork.Asaresultaminimalsetofordinarydifferentialequations(ODE)ofsecondorderisfound.Asystematicanalysisofconstrainedmechanicalsystemswasestablishedin1788byLagrange[89],too.ThevariationalprincipleappliedtothetotalkineticandpotentialenergyofthesystemconsideringitskinematicalconstraintsandthecorrespondinggeneralizedcoordinatesresultintheLagrangianequationsofthefirstandthesecondkind.Lagrange’sequationsofthefirstkindrepresentasetofdifferential-algebraicalequations(DAE)whilethesecondkindleadstoaminimalsetofordinarydifferentialequations(ODE).Anextensionofd’Alembert’sprinciplevalidforholonomicsystemsonlywaspresentedin1913byJourdain[76].Fornonholonomicsystemsthevariationswithrespecttothetranslationalandrotationalvelocitiesresultingingeneralizedvelocitiesarerequired.Then,aminimalsetofordinarydifferentialequations(ODE)offirstorderisobtained.Theapproachofgeneralizedvelocities,identifiedaspartialvelocities,wasalsointroducedbyKaneandLevinson[81].TheresultingKane’sequationsrepresentacompactdescriptionofmultibodysystems.MoredetailsonthehistoryofclassicalmechanicsincludingrigidbodydynamicscanbefoundinP¨asler[120]andSzab´o[181].Thefirstapplicationsofthedynamicsofrigidbodiesarerelatedtogyrodynam-ics,mechanismtheoryandbiomechanics.Euler’sequationsforthekinematicsanddynamicsofasinglegyrodatebackto1758.Formorethanacentury,theresearchonthesolutionofEuler’sequationsattractedmathematiciansandmechanicians.Atthebeginningofthiscenturytheengineeringapplicationsofthesinglegyro-scopegotmoreimportant.Then,gyroscopicsystemsreceivedalsosomeattention.Grammelmentionedin1920inthefirsteditionofhisbook“DerKreisel–SeineTheorieundseineAnwendungen”[51]atwo-gyrosystembuthedidnotdiscussitsdynamics.Thirtyyearslaterinthesecondeditionofthesamebookasmallsectionwasalreadydevotedtogyroscopicsystems.Magnuspresentedin1971inhisbook“Kreisel”[99]alargesectionongyrosystemsincludingarigorousstabilitytheory.Forexample,acardanicsuspendedgyrohastobemodeledaccu-ratelyasathree-bodysystem(Figure1).In1977Magnus[100]organizedthefirstIUTAMSymposiumonDynamicsofMultibodySystemswithquiteanumberofcontributionstogyroscopicproblems.Mechanismtheorydealsalsowiththemotionofconstrainedmechanicalsys-tems.However,theapplicationofthepowerfulgraphicalmethods,developed,e.g.,in1913byWittenbauer[194],wasrestrictedtoplanarmechanisms.Laterin1955,matrixmethodswereintroducedbyDenavitandHartenberg[33]forspatialkine-maticswhichformedthebasisforthedynamicalana