盘式制动器的有限元仿真分析

Abstract—Inthisstudy,thethermoelasticinstability(TEI)wasanalyzedusingthefiniteelementanalysistechnique.Thegoverningdynamicandheatequationsweredescribed.Threedimensionalthermomechanicalanalysismodelofthediscbrakesystemwerecreated.Anintermediateprocessorbasedonthestaggeredapproachedwasusedtoexchangeresultdata:temperature,frictioncontactpower,nodaldisplacementanddeformation.Discthicknessvariation(DTV)andtemperaturedistributionofthediscwerecalculated,andthetendencyandmeaningofeachresultwerediscussed.IndexTerms—Thermoelasticinstability,Finiteelementanalysistechnique,Thermomechanicalanalysis,Intermediateprocessor,DiscthicknessvariationI.INTRODUCTIONThefrictionheatgeneratedbetweentwoslidingbodiescausesthermoelasticdeformationwhichaltersthecontactpressuredistribution.Thiscoupledthermo-mechanicalprocessisreferredtoasfrictionally-excitedthermoelasticinstabilityorTEI[1].Iftheslidingspeedisaboveonecalledcriticalspeed,theresultingthermo-mechanicalfeedbackisunstable,leadingtothedevelopmentofnon-uniformcontactpressureandlocalhightemperaturewithimportantgradientscalled‘hotspots’[2].Theformationofsuchlocalizedhotspotsisaccompaniedbyhighlocalstressesthatcanleadtomaterialdegradationandeventualfailure[3].Also,thehotspotscanbeasourceofundesirablefrictionalvibrations,knownintheautomotivediscbrakecommunityas‘hotroughness’or‘hotjudder’[4].Inthisstudy,atransientFEanalysismethodwasusedtoanalyzethefullycoupledthermoelasticinstabilityproblemforadiscbrakesystem.Mechanicalandthermalmodelforthediscbrakeweregeneratedseparately,andsolvediterativelyusingthestaggeredapproach[5].Thestaggeredapproachisoneofthemostpopularcomputationtechniquesusedtosolvethehighlycouplednon-linearequations.ThreedimensionalFEmodelofadiscbrakewascreated.Themechanicalmodelofthediscbrakewasassumedtobeinbrakingwithanaccelerationof0.3gfrom160kphto80kph.Thethermalmodelwithaninitialtemperatureof80℃ManuscriptreceivedMarch5,2010.ThisworkwassupportedbyKoreaInstituteforAdvancementofTechnology.S.P.JungisaPhDcandidatestudent.(e-mail:[e-mail])T.W.ParkisaprofessorofAjouUniversity,Suwon,RepublicofKorea.(Phone:[phone];fax:[phone];e-mail:[e-mail]).J.H.LeeandW.H.Kimareinthecourseofmaster’sdegree.(e-mail:[e-mail],[e-mail])W.S.ChungisaseniorresearcherofKoreaAutomotiveTechnologyInstitute.(e-mail:[e-mail]).interactswiththemechanicalmodel,andthefrictionheatbetweenthepadanddiscisgeneratedbythecontactcondition.Duetotheheatgeneration,thematerialofdiscisexpandedandaltersthecontactcondition.BycomparingthesimulationresultstothetestdatainPart.2ofthispaper,thereliabilityoftheFEmodelsandcomputationschemeisverified.II.THEORETICALBACKGROUNDTheequationofmotionofaconstraineddynamicsystemisintroducedbasedonthefiniteelementapproach[6].Thegeneralheatequationisbrieflyreviewedandthebasicstrategytoanalyzethecoupledthermo-mechanicalsystemisdescribedaccordingtothestaggeredapproach.A.DynamicsofaConstrainedFlexibleMultibodySystemTheconstraintsonthesystemareefficientlytakenintoaccountusingtheAugmentedLagrangianmethod.TheaugmentedfunctionalofHamilton’sprincipleis0)2(21dtWpkLttTT(1)wherekisthescalingfactorandpisthepenaltycoefficient.isthevectorofLagrangemultipliersandisthevectorofconstraints.ListheLagrangianofthemechanismdefinedasVTL.TandVarethekineticandpotentialenergiesofthesystem,respectively.Wisthevirtualworkofexternalforces.Usingthevirtualdisplacementprinciple,themotionequationsareobtainedas0),()()(tqkqpkQqLqLdtd(2)whereqisthevectorofgeneralizedcoordinates.Equation(2)canbewritteninthematrixform0),(),,()(tqktqqgkpBqMT(3)whereBisthegradientmatrixoftheconstraintsFiniteElementAnalysisofThemalelasticInstabilityofDiscBrakesS.P.Jung,T.W.Park,J.H.Lee,W.H.Kim,andW.SChungProceedingsoftheWorldCongressonEngineering2010VolIIWCE2010,June30-July2,2010,London,U.K.ISBN:978-988-18210-7-2ISSN:2078-0958(Print);ISSN:2078-0966(Online)WCE2010(qB)andgisthevectorofapparentforce)(qqLqqLQg(4)Thelinearizedformofthemotion(65)canbedescribedas**0000000rqkBkBKqCqMTTT(5)B.TransientComputationofHeatTransferEquationThetemperaturewithininanelementiscomputedbasedonthenodaltemperaturevector)()(),(tTxtxT(7)wherexisthenodalcoordinate,)(xistheinterpolationfunctionvectorand)(tTisthenodaltemperaturevector.Thegoverningequationoftheheattransferproblemis)()()(TQTTKTTC(8)where)(TCand)(TKarethediscretisedsystem’stemperature-dependantheatcapacityandthermalconductivitymatrices,respectively,Tisthenodaltemperaturevector,Tisthetimederivativeofthetemperaturevector,and)(TQistheheatfluxvector.ThesolutionvectorrTofEquation(11)attime,whichislocatedattimeintervalsbetweennandn+1step,canbeexpressedas1)1(nnTTT(9)Thevariationrateofthetemperaturecanbewrittenas)(1nrTTtT(10)C.HeatGenerationDuetoFrictionContactIndiscbrakes,thefrictionheatisgeneratedbetweenthediscandpads.Frictionalheatgenerationperunittimeatthenodeiiscalculatedascicfq(11)whereisthefacto