Modeling-tire_road-contact-using-piecewise-Ritz-pr

Modelingtire/roadcontactusingpiecewiseRitzprocedureYanhaiXua,*,LipingJiab,JianwuZhangbaSchoolofAutomobileEngineering,XiHuaUniversity,Chengdu,Sichuan610039,PRChinabSchoolofMechanicalEngineering,ShanghaiJiaoTongUniversity,Shanghai200030,PRChinaAbstractBasedontheReddy’shigherorderrefinedsheardeformationshell,apiecewiseRitztypenumericalschemeispresentedfortheanalysisofthetire/roadinteractions.BoththegeometryandthedisplacementfieldsoftheinnersurfaceofreferenceofthetirearedescribedbyBezierpolynomials.c0andc1continuitiesareimposedoncommonboundariesbetweensurfacepatches.Thecontactconstraints,includingforceconstraintsanddisplacementconstraints,aredirectlyenforcedintothegoverningequationstoreducethenumberofunknownvariables.Fromthenumericalexamples,itisshownthatthefootprintshapeandthestressdistributioninthebeltspredictedbythepresentmethodareveryclosetotheresultsbythefiniteelementmethodwhilelessunknownvariablesareinuse.
2004ISTVS.PublishedbyElsevierLtd.Allrightsreserved.1.IntroductionTiresaretheonlycomponentsofavehicleincontactwiththeroad.Theinter-actionbetweenthetireandroadgeneratestheforcesfordrivingandbraking.Theshapeandareaofthetirefootprintdeterminethecharacteristicsoftiresandhan-dlingstabilityofvehicles.Theyalsoprovideparameterstoamelioratetherelationsbetweenthetirestructureandhandlingstabilityofvehicles.Increasingly,highper-formancetiresareneededinmodernautomotiveengineeringandthereisanurgentneedtomodeltheinteractionoftire/roadunderdifferentloadcases.*Correspondingauthor.0022-4898/$20.00
2004ISTVS.PublishedbyElsevierLtd.Allrightsreserved.doi:10.1016/j.jterra.2004.04.001(2005)99–113Currently,themostwidelyusedapproachisanumericalsolution,suchasFiniteElementMethodandBoundaryElementMethod.Withtherapiddevelopmentofcomputertechnologyandcomputationaltechnique,designinganddevelopingnewtiremodelsthatcanpredictthestaticanddynamiccharacteristicsoftiresaremoreprevalent.Onedifficultyoftheanalysisofthetire/roadinteractionisitsstrongnonlinearity.Thesolvingandmodelingcomplexityoftheseproblemsaredictatedbythesein-terestingandchallengingnonlinearlities.DanielsonandNoor[1]developedaFEMtiremodelinthecylindricalcoordinatesbyusingexactcircularshapeofthetire.Theexactmodelingofcirculargeometriesprovidesbetterfiniteelementpredictionsandreducesthenumberofelementsalongthecircumferencedirection.BurkeandOlatunbosun[2]simulatedthecontactoftire/roadinteractionapplyinggapelementswiththeapplicationofMSC/NASTRANV67.WiththehelpofMSC/MARC,Liuetal.[3]obtainedthepressuredistributionwithapneumatictireof12.5/75R20whenconsideringthechangeofdrivingforceinthecontactoftireandrealroad,suchassandroad.Wanjiietal.[4]predictedthecharacteristicsoftireapplyingthetangentforcesinthecontactarea.Asemi-analyticaltiremodelwasproposedbyHiromaetal.[5]tosimulatethestableandtransientresponses.Mastinuetal.[6]analyzedthestressdistributioninthetirefootprintvalidatingwithexperimentaldata.Anotherdifficultyinmodelingthetire/roadcontactishowtoapplytheuncertainboundaryconditions.Traditionally,theLagrangianMultiplierMethodorthePenaltyFunctionMethod[7,8]areused.AlthoughtheLagrangianMultiplierMethodcouldprovidetheaccuratemathematicaldescriptionbyintroducingthecontactconstraintsintogoverningequations,itincreasesthenumberofthedegreeoffreedomsandconsequently,thecostofcomputation.Ontheotherhand,thenumberofDOFsisnotincreasedinthePenaltyFunctionMethod,butthemethodisnotrobustthestabilityofthesolutionisdependentontheselectionofthepenaltynumber.Allofthosemethodshaveinfluenceonthemodelingofthetire/roadcontact.Basedonthegeometryofaradialtire,Bezierpolynomialsareemployedtorep-resentthespatialsurfaceandtheRitzprocedurewithparametrictrialfunctionsisappliedtoobtaingoverningequationsinthispaper.Thecontactconstraintsareintegratedintothegoverningequationsdirectlyasboundaryconditions,allowinganeasiersolutiontothegoverningequations.Throughanexample,theaccuracyandeffectivenessofthepresentmethodtomodelthecontactinteractionofthetire/roadiscomparedtothepredictionoftheFEMresultsproducedbythecommercialFEMcodeMSC/MARC.2.Basicformulation2.1.Strain-displacementrelationsRadialtirestypicallyhaveavariablethicknessstructureasshowninFig.1.Itiscomposedofsteelbelts,nylonbelts,rubberandcarcass.Thesteelandnylonbelts100Y.Xuetal./JournalofTerramechanics42(2005)99–113orientateindifferentanglewiththeradialdirectionaccordingtothedesignandperformanceofthetire.Duetothedifficultyofrepresentingthecomplexpatternofaradialtire,theassumptionthattheradialtireiswithoutpatternisemployedinthepresentresearchandthecomputationalaccuracyhasbeenaffectedverysmallbytheassumptionfromtheviewofSection4.Althoughitisdifficulttodefinethestructureofaradialtire,itcanbeapproximatedasanenclosureoftheouterandinnercontoursurfaces.Theouterandinnercontoursurfacesareeasilyrepresentedbyftopðs;tÞandfbotðs;tÞ,respectively.Themiddlesurfacecanbedefinedasashellofvariablethicknesswiththefollowingform:h0¼ðftopðs;tÞþfbotðs;tÞÞ=2:ð1ÞTheheightsfromtheoutercontoursurfaceandinnercontoursurfacetothemiddlesurfacear