A planar mechanical library in the AMESim simulati

AplanarmechanicallibraryintheAMESimsimulationsoftware.PartI:FormulationofdynamicsequationsWilfridMarquis-Favre*,EricBideaux,SergeScavardaLaboratoired’AutomatiqueIndustrielle,InstitutNationaldesSciencesApplique´esdeLyon,Baˆt.StExupe´ry,25,avenueJeanCapelle,F-69621VilleurbanneCedex,FranceReceived25March2003;receivedinrevisedform17December2004;accepted8February2005Availableonline17March2005AbstractThispaperpresentsthemathematicaldevelopmentsofaplanarmechanicallibraryimple-mentedintheAMESimsimulationtool.Bodyandjointcomponentsarethebasiccomponentsofthislibrary.Duetothelibraryphilosophyrequirements,themathematicalmodelsofthecomponentshaverequiredagenericvectorcalculusbasedformulationoftheconstraintequa-tions.Thisformulationusesasetofdependentgeneralizedcoordinates.Thedynamicsequa-tionsareobtainedfromtheapplicationofJourdain
sprinciplecombinedwiththeLagrangemultipliermethod.Thebodycomponentmathematicalmodelsconsistofdifferentialequationsintermsofthedependentgeneralizedcoordinates.ThejointcomponentmathematicalmodelsarebasedontheBaumgartestabilizationschemesappliedtothegeometrical,kine-maticandaccelerationconstraintequations.TheLagrangemultipliersaretheimplicitsolutionoftheseBaumgartestabilizationschemes.Thefirstmaincontributionofthispaperistheexpressionofgeometricalconstraintsintermsofvectorsandtheirexploitationinthisform.ThesecondimportantcontributionistheadaptationofexistingformulationstotheAMESimphilosophy.
2005ElsevierB.V.Allrightsreserved.1569-190X/$-seefrontmatter
2005ElsevierB.V.Allrightsreserved.doi:10.1016/j.simpat.2005.02.006*Correspondingauthor.Tel.:[phone].E-mailaddress:[e-mail](W.Marquis-Favre).(2006)25–46Keywords:AMESim;Planarmechanics;Dynamicsequations;Constraintequations;Lagrangemulti-pliers;Baumgartestabilization1.IntroductionThispaper,organizedintwoparts,presentsanewlibraryforthesimulationtoolAMESim[2].Thefirstpartisdedicatedtothetheoreticaldevelopmentsofthelibrary.ThesecondpartshowsthecompositionofthelibraryasitwasprimarilyimplementedinAMESimandillustratesitwithanapplicationexampleofaseven-bodymechanism.Thislibraryproposescomponentsbelongingtotheplanarmechanicaldomain.Theobjectivewiththislibrarywasnottocompetewithmulti-bodysystemsoftwaretoolsthatarebetteradaptedtothisdomain.TheobjectivewasmoretoenlargetherangeofindustrialapplicationscapableofbeingtreatedbyAMESim.FromatheoreticalpointofviewthechallengeofimplementingthislibrarywastofitexistingmechanicalformulationstotheinherentrequirementsofAMESimphilosophy.Thesolutionhasbeenfoundbyadaptingthedynamicequa-tionsexpressedfromJourdain
sprincipleandtheLagrangemultipliermethodtogetherwithBaumgarte
sstabilization.Alsoagenericfeatureoftheformulationhasbeenresearchedoverthelibrarycomponents(bodiesandjoints)andonekeycontributionofthispaperisconcernedwiththisgenericfeature.Basicallytheformu-lationconsistsofexpressingthegeometricconstraintsassociatedwithjointsintermsofvectorsandcarryingoutthedevelopmentsofthisform.Theresultisthesetup,forkinematicandaccelerationconstraints,ofauniqueexpressionthatfitseveryjointpresentedinthelibrary.Thegenericfeatureoftheformulationproposedthusenablesthederivationofjointcontraintstobesystematized.Onecanthenimagineanewjointwithitscorre-spondingvectorconstraintandderivestraightforwardlythecorrespondingmathe-maticalmodelbyapplyingtheproposedformulation.Also,inthecontextofpredefinedcomponentmodels,thegivenformulationclearlyshowsthefrontiersofthedifferentmathematicalmodelsintermsofinputsandoutputs.Thereforeitalsohelpstodefineinwhichmodelsoutputequationsmustbeimplemented.Also,theformulationproposedintrinsicallyenablesclosedloopstructurestobedealtwith.AMESim(forAdvancedModelingEnvironmentforperformingSimulations)isorganizedincomponentlibraries.Thecomponents,representedbysymbolicallytechnologicallysuggestedicons,canbeinterconnectedexactlylikethesystemunderstudy.AMESimwasfirstappliedtoelectrohydraulicengineeringsystemswithsimpleone-dimensionalmechanicalsystems(likeinertia,springs,anddampersintransla-tionorinrotation).Itrecentlyopeneditslibrariestoavarietyofothercomponentdomains.Onecannowcarryoutmodeling,analysisandsimulationforsystemscon-sistingofpneumatic,powertrain,hydraulicresistance,thermal,electromagneticandcoolingcomponentsforinstance.Therestrictiontoonlyone-dimensionalmotionforthemechanicalcomponentshasmotivatedthedevelopmentofatwo-dimensionalmechanicallibrary.26W.Marquis-Favreetal./SimulationModellingPracticeandTheory14(2006)25–46Section2presentsanoverviewofsomemultibodycodesandobject-orientedtools,aswellastheenvironmentalrequirementsofAMESim.Theserequirementshavesomeimplicationsonhowthe2Dlibraryisbuilt.Section3detailsthetheoret-icaldevelopmentsthatenabledthemathematicalmodelsofthelibrarycomponentstobesetup.Section4concludesthisfirstpart.2.ConstraintsofAMESimlibraryphilosophyAfterabriefoverviewofmultibodycodeprinciplesandsomeobject-orientedtools,apresentationofAMESimrequirementsisgiven.Concerningmultibodycodesastateoftheartisgivenby[23].Detai