FiniteElementsinAnalysisandDesign35(2000)213225StressanalysisoftheringincontinuouslyvariabletransmissionmechanismSerdarTumkorDesignandManufacturingInstitute,StevensInstituteofTechnology,CastlePointonHudson,Hoboken,NJ07030,USAAbstractDuringtheserviceperformanceofafrictionringinacontinuouslyvariabletransmission(CVT)mecha-nism,themostcommonlyobservedfailuremodeisfatiguefailure.Themainpurposeofthisstudyistodeterminethereasonofthisfailureandtoimprovetheshapeofthering.Theringismodeledbytheniteelementmethod(FEM)andalsocheckedbyanalyticalformulasgeneratedforcircularrings.Thecalculationsshowthathighinteriorstressesarepresentonthecontactsurfaces.Theshapeoftheringismodiedtondtheoptimumshapeanddimensionsofthering.Byturningagrooveinsideofthering,thesurfacestressisdecreased,butthestressesnearthegrooveisincreasedbasedonaFEManalysisdependingonthegrooveradii.Afteranalysisandoptimizationofthegrooveradius,thedimensionR1.95mmiscalculatedandtested.Fatiguefailureisobservedtodecreasebytestingthemodiedring.(2000ElsevierScienceB.V.Allrightsreserved.Keywords:Continuouslyvariabletransmission；Shapeoptimization；Parametricniteelementmodel1.IntroductionInIndustrycontinuouslyvaryingspeedsareneededformanyapplicationsconcerningmecha-nisms.Thesetypesofmechanisms,whichhaveavariablevelocityandtorqueratio,arecalledcontinuouslyvariabletransmission(CVT)mechanisms.Di!erenttypesofCVT-driveshavebeensuccessfullyusedindi!erentindustrialapplicationsformanyyears.Variablemetalbeltandchaindrivesareverywellknown.Inthescopeofthisstudy,aconicalandsteelringassembledtractiondriveisinvestigated.ACVT-drivewithasteelringisinexpensivebecauseofthelow-costsimpleringpartused.Buthightorquecannotbetransmittedbyfrictionaldrives.Dependingontime,atypeoffatiguefailurecalledpittingcanbeobservedduetothenormalforce,whichisneededforthefrictionalboundbetweentheconesandring.TheliteratureonCVTisveryscarce.AperformancecharacteristicofavariablemetalV-beltdrive,whichhasasimilarassemblyandperformancecharacteristic,isgivenbySun1.Speed0168-874X/00/$-seefrontmatter(2000ElsevierScienceB.V.Allrightsreserved.PII:S0168-874X(99)00065-7NomenclatureAareaofthecrosssection(mm2)b1width(mm)ddiameter(mm)Emodulusofelasticity(Youngsmodulus)(Pa)FNnormalforce(N)hdistancefromthecentroidal(mm)kconstantMbendingmoment(Nmm)Rgrooveradius(mm)rradius(mm),radialpositionofthestressrnneutralaxisradius(mm)Rrcentroidalaxisradius(mm)x,y,zcoordinateskcoe$cientoffrictionlpoissonratiop.!9maximumstress(MPa)phcircumferentialnormalstress(MPa)ratio,transmittedtorque,andfrictioncoe$ciente!ectforthevariablemetalV-beltdriveareanalyzedbyKaram2.TheCVTmechanismofpowertransmittinghasbeenstudiedbyKuwabaraetal.3,4.Dynamicofthefrictionalchaindrivesissimulatedtogetrealisticpredictionsaboutthesystembehavior5,6.ThenonlineardynamicsofthechaindriveCVTarealsostudiedbyPauschandPfei!er7inautomotivedrivetrainsystems.AnotherstudyonCVTisperformedbyTakemotoetal.8aboutthenoiseproblemcausedbymetal-to-metalcontactscheme.Basedontheniteelementmethod(FEM),newoptimizationprocedureshavebeendevelopedtotransferbiologicaloptimizationmechanismstomechanicalengineering9.Itmeansthatanybiologicalloadcarriertendstoachieveaconstantstressatleastinatimeaverage.Unacceptablehighstressesduetostressconcentrationswouldcausefailure,whiletheunderloadedzonesarewastedmaterials.Therefore,theprincipleoflightweightdesignisthemaincriterionfortheshapeofnaturalstructures.Insomecasesthegeometricalconstraintsshouldbeconsideredforthefunctioningofthemechanisms,thereforethegeometrybecomesmorecriticalthantheweightconsiderations.Inthosecasesprinciplesofthecomputeraidedoptimization(CAO)Methodcanbeusedhowever,certaingeometricconstraintsmustbeconsidered.ThismethodwasdevelopedbyMattheckandBurkhardt10toimprovetheshapeofhighlyloadedcomponentswithrespecttoareductionandhomogenizationofstressesonthesurface.IntherstFEMrunthestressesarecalculatedwithaniteelementmodel,inwhichactualloadcasesandboundaryconditionsareused.Thisstressdistributionwillleadtoimprovetheshape.AfteranotherFEMrunofimprovedstructurethestresspeakswillbereducedandhomogenized.Kasperdiscussessomeoptimizationmethodsincombina-tionwithgeneralnumericeldcomputationmethodslikeFEM11.214S.Tumkor/FiniteElementsinAnalysisandDesign35(2000)213225Fig.1.TheassemblyofCVTmechanism.InthispaperaCVTmechanismisreported,whichhasafatiguedamageproblembecauseofthehighstressonitsmetalring.Anapproximateanalyticalsolutionforthecircumferentialstressiscalculated.Tondoutthestressdistribution,FEManalysisisdone.ModicationsoftheringdesignareachievedusingahybridoftheCAOandcurvettingprocedurestogether.Themodiedcasesaretestedandthefatiguefailureisseentodecrease.2.DescriptionofthemechanismInFig.1therearetwofrictionconesonthemotorshaftofthemechanismformingapulley.b1conesarexedandb2conesarefreeintheaxialdirection.b2conesareallowedtomovewithintheprescribedrange.Bothoftheseconesmovethesamedistancesincetheyareconnectedtoeachother.AnIsometricviewandthepictureoftheCVT-mechanismareshowninFigs.2and3,respectively.Velocityratiovariesbetweentheminimumandmaximumvaluesdependingonthelocationoftheringrelativetothepulley.Torqueistransmittedthroughlinecontactsbetweenthepulleyformedbytheconesandthesteelring.Frictioninterfaceissatisedbytheappliednormalforceduetogivenprestresstoconnectionpointsofthesteelringandcones.Forrollingwithoutsliding,thenormalforceandthefrictionforcearerelatedbytheCoulombsmodelasfollows:FN*F&/k,(1)wherethecoe$cientoffrictionl0.10.13isassumed12.OtherfrictionmodelsbesidetheCoulombsarereviewedbyOlssonetal.13.Forasimilarmechanism,themetalV-beltbehaviorofaV-beltCVTwasinvestigatedanalyticallyandexperimentallybyKima