On the Theory of Nonlinear Dynamics and its Applic

Thisarticlewasdownloadedby:[UniversityofTokyo/TOKYODAIGAKU]On:9July2008AccessDetails:[subscriptionnumber778576937]Publisher:Taylor&FrancisInformaLtdRegisteredinEnglandandWalesRegisteredNumber:1072954Registeredoffice:MortimerHouse,37-41MortimerStreet,LondonW1T3JH,UKVehicleSystemDynamicsInternationalJournalofVehicleMechanicsandMobilityPublicationdetails,includinginstructionsforauthorsandsubscriptioninformation:~content=t713659010OntheTheoryofNonlinearDynamicsanditsApplicationsinVehicleSystemsDynamicsHansTrueOnlinePublicationDate:01June1999TocitethisArticle:True,Hans(1999)'OntheTheoryofNonlinearDynamicsanditsApplicationsinVehicleSystemsDynamics',VehicleSystemDynamics,31:5,393—421Tolinktothisarticle:DOI:10.1076/vesd.31.5.393.8361URL:::[UniversityofTokyo/TOKYODAIGAKU]At:16:399July2008.VehicleSystemDynamics,311999,pp.393–4210042-3114r99r3105-393$15.00qSwets&ZeitlingerOntheTheoryofNonlinearDynamicsanditsApplicationsinVehicleSystemsDynamicsHANSTRUE),1SUMMARYWepresentabriefoutlineofnonlineardynamicsanditsapplicationstovehiclesystemsdynamicsproblems.Theconceptofaphasespaceisintroducedinordertoillustratethedynamicsofnonlinearsystemsinawaythatiseasytoperceive.Variousequilibriumstatesaredefined,andtheimportantcaseofmultipleequilibriumstatesandtheirdependenceonaparameterisdiscussed.Itisarguedthattheanalysisofnonlineardynamicproblemsalwaysshouldstartwithananalysisoftheequilibriumstatesofthefullnonlinearproblemwherebygreatcaremustbetakeninthechoiceofthenumericalsolvers.Whentheequilibriumstatesareknowncertainlinearizationsaroundonechosenstatemaybeappliedcarefullyinordertofacilitateorspeedupthenumericalsolutionofthedynamicalproblem.Itisargued,however,thatcertainproblemscannotbelinearized.Theapplicationsofnonlineardynamicsinvehiclesimulationsisdiscussed,anditisarguedthatitisnecessarytoknowtheequilibriumstatesofthefullnonlinearsystembeforethesimulationcalculationsareperformed.1.INTRODUCTIONPracticallyallkinematicandconstitutiverelationsinvehiclesystemsdynamicsarenonlinear.Themodellingofsuchsystemsthereforeleadstononlineardynamicalsystems,whichmustbeinvestigatedwithrespecttotheirpropertiesandevolution.Themathematicaltheoryofnonlineardynamicalsystemsdifferinseveralandfundamentalwaysfromthemathematicaltheoryoflinearsystems.Everyengineerlearnedlinearanalysisintheuniversity,andithasformorethanhundredyearsyieldedimportantcontributionstothedevelopmentoftheengineeringsciences.Nowithappensmoreandmoreoftenthattheapplicationoflinearmathematicalmethodsleadstoerroneousresults-eventhoughthenonlinearmechanicsorphysicsismodelledcorrectly.Thequestionsthennaturallyarise:Whenandwhyisitnecessarytoapplynonlineardynamicstotheinvestigationsofvehiclesystems?Wereallthesuc-cessesoftheapplicationsoflinearmathematicsinthepastaccidental?Mathematicscananswertheseandmanymorequestionsaboutthelimitationsoflineardynamics.Inthisarticlesomeofthesequestionswillbeaddressedinthecontextofapplicationstovehiclesystems,andthefundamentalissueofthe)Correspondingauthor1TheTechnicalUniversityofDenmark,DepartmentofMathematicalModelling,Bldg.321,DK-2800Lyngby,Denmark.DownloadedBy:[UniversityofTokyo/TOKYODAIGAKU]At:16:399July2008HANSTRUE394necessityofknowingatleastthebasictheoryofnonlineardynamicswillbestressed.Nonlinearmechanicsdealswiththeformulationormodellingofprocesseswhilenonlineardynamicsisamathematicaldiscipline.Thetopicsthereforearedifferent.Weshalldealwithnonlinearanddeterministicdynamicsinthisarticle.Stochasticvariablesareexcluded.Firstsomedifferencesbetweenlinearandnonlineardynamicswillbedemon-stratedthroughsimpleexamplesandcomparisonsbetweentheresponses.Theseexamplesincludethenon-validityoftheprincipleofsuperposition,theamplitudedependentfrequencyofnonlinearoscillationsandthelackofuniquenessofsolutionstosomenonlinearproblems.Allareviolationsofthefundamentalprinciplesinthelinearworldinwhichweallgrewup.Thebasicnotionsofa‘phasespace’andtheconceptof‘trajectories’willbeintroducedbecausetheyleadtoavisualisationofthesolutionstodynamical.problems,theiruniquenessorlackofsameandstability.Thegeometricaldescriptionofthesolutionstodynamicalsystemshavebecomeatopicofmathematicalresearchonitsownright,andwetellhowtheequilibriumsolutionsinthephasespacecon