A review on belt and chain continuously variable tr

MechanismandMachineTheory44(2009)19–41ContentslistsavailableatScienceDirectMechanismandMachineTheoryjournalhomepage:/locate/mechmtReviewAreviewonbeltandchaincontinuouslyvariabletransmissions(CVT):DynamicsandcontrolNilabhSrivastavaa,*,ImtiazHaquebaDepartmentofMechanicalEngineeringandEngineeringScience,UniversityofNorthCarolinaatCharlotte,9201UniversityCityBlvd.,DCH355,Charlotte,NC28223,USAbDepartmentofMechanicalEngineering,EIB106,ClemsonUniversity,Clemson,SC29634,USAarticleinfoabstractOverthelasttwodecades,significantresearchefforthasbeendirectedtowardsdevelopingvforthasbeenadirectconsequenceofthegrowingenvironmentalconcernimposingthedirectivesofreducedexhaustemissionsandincreasedvehicleeffinuouslyvariabletransmission(CVT)offersacontinuumofgearratiosbetweendesiredlimits,whichconsequentlyenhancesthefueleconomyanddynamicper-formanceofavehiclebybettermaghaCVTplaysacrucialroleintheplantoimprovevehiclefueleconomy,rentpaperreviewsthestate-of-the-artresearchondynamicmodeliicconcepts,mathematicalmodels,ngesandcriticalissuesforfutureresearchonmodelingandcontrolofsuchCVTsarealsodiscussed.Óehistory:Received14March2008Receivedinrevisedforl..................uctionWithgrowingsocioeconomicandenvironmentalconcern,automobileenergycepastfewdecades,vehicleshavebeenincreasinglyfacingstringent*.:+;fax:+.E-mailaddresses:nsrivast@,nilsri77@(tava),@().0094-114X/$-seefrontmatterÓ:10.1016/chtheory.2008.06.007

tava,/MechanismandMachineTheory44(2009)19–41performance,emissions,andfueleconomystandardsdrivenbyregulatoryandmarketforces(andardsintheUS,ACEAstandardsinEurope,emissionstradingpolicies,etc.).Emissionsofcarbondioxide(CO2),theprincipalgreenhousegasproducedbytransportationsector,havesteadilyincreasedalongwithtravel,energyuse,efuelecon-omyplaysacrucirethreefundamentalwaystoreducegreenhousegasemissionsfromthetransportationsector[1]:(a)increasetheenergyefficiencyoftranspor-tationvehicles,(b)substituteenergysourcesthatarelowincarbonforcarbon-intensivesources(ofalternativefueltechnologies),and(c)emendousgrowthinconsumerismandurbanization,thereislittlescopeforeghalternativeenergytechnologieslikefuelcells,electricmotors,batteries,solarpower,windpower,ignificantlylowergreenhousegasemissions,theycurrentlyfacemajorchallengesintermsofimplementationstrategies,infrastructure,cost,anddrivability[1].Ontheotherhand,significantresearchopportunityliesinrefiningthecurrentengineandtransmissiontechnologiestomeettheearlywaveofemissionsregul,continuouslyvariabletransmissions(CVT)havearousedagreatdealofinterestintheautomoanemergingautomotivetransmissiontechnologythatoffersacontnsequentlyenhancesthefueleconomyandaccelerationperformanceofavehiclebyallowingbettermatc,CVTsareaggressivelycompetingwithautomatictransmissionsandseveralvehiclemanufacturerssuchasHonda,Toyota,Ford,Nissan,etc.,arealreadynuouslyvariabletransmisr,inspiteoftheseveraladvantagesproposedbyaCVTsystem,thegoalsofhigherfueleconomyandbetterperformancehavenotbeenrealizedsignifirtoachieveloweremissionsandbetterperformance,itisnec-essarytocaptureandunderstandthedetaileddynamicinteractionsinaCVTsystemsothatefficientcontrollerscouldbedesignremanykindsofCVTs,eachhavingtheirowncharacteristics,calCVT[2],HydrostaticCVT[3,4],E-CVT[5,6],ToroidalCVT[7–9],Power-splitCVT[10–12],BeltCVT,ChainCVT,Ball-typetoroidalCVT[13],MilnerCVT[14],r,beltandchaintypesarethemostcommonlyusedCVTs,amongall,,thecurrentworkreviewsthestate-of-the-artre-search,inthecontextofdynamicsandcontrols,ofbeltandchainCVTsforachievierlyingtheories,mathematicalmodels,perwillnotonlygiveprofoundinsightintodynamicmodelingofsuchCVTsystems,butalsoevaluatecurrentandfutureresearchstrategiesneededtodesignefficientCVTcontrollers,identifypossiblelossmechanisms,characterizeoperatingregimes,tisdifficulttoreportacom-prehensive(-inclusive)surveyontheliteratureofthisrapidlyemergingfield,wehopethepublicationsreferencedhereareatleastrepresentativeoftheliteratureandwouldprovideagoodresearchplatformtopeopleinterestedindynamihorssincerelyapologizefornotbeicconfigurationofaCVTcomprisestwovariablediameterpulleyskeptatafixeddistt/chaincanundergobothradialandtangentialmotionsdep-beltCVTdrive:(a)basicconfiguration;(b)beltstructure[15].

tava,/MechanismandMachineTheory44(2009)19–VTdrive:(a)basicconfiguration;(b)chainstructure[16,72].leyontheenginesideiscalledthedriverpulleyandtheoneonthefi.1[15]andFig.2[16,72]alV-beltCVT,torqueistransmittehereisfrictionbetweenbandsandbeltelements,thebands,likeflatrubberbelts,,thereisacombinedpush–pullactitherhand,inachainCVTsystem,theplatesandrockerpins,asdepictedinFig.2b,abeltCVT,thecontactforcesbetwe,excitationmechanismsexist,usesvibrationsintheentirechainCVTsystem,ltandchainCVTsystemsfallintothecategoryoffriction-limiteddrivesastheirdynamicperformanceandtorquecapacityrelysignificantlyonthefrictioncharacteristicofthecontactpatchbetweenthebelt/yofresearchhasbeenconductedondifferentaspectsofCVTs,mance,slipbehavior,efficiency,configurationdesign,lossmechanisms,vibrations,operatingregime,tercomprehensibility,cmodelingofbeltCVTManypapershavebeenpublishedovetcommonlyusedpower-transrelevantworkonmetalandrubberV-beltsissubsequentlycitedassuchCVTsareextensivelyresearchedbytoday’istingmodelsofbeltCVTs,withafewexceptions,aresteady-t[17,18]extensivelyworkonunderstandingthemechanicsoftractionbelts,horusedquasi-staticequilibriumanalysistodevelopasetofequathebeltiscapableofmovingbothradiallyandtangentially,variableslidinganglericaldescriptionofabeltCVTdrive.

tava,/MechanismandMachineTheory44(2009)19–usly,manyresearcherslikeKimmich[19],Gutyar[20],Amijima[21],etc.,assumedconstantiableslidingangleapproachrequiredthattheequilibrium,compatibility,andconstitutiveequationsbesolvedsimultaneouslyforpredictingthedynamicsofabelt–ntripetaleffectsweremodeledtoaccountfortheinfl.3and4illustratethegeometricconfiguration,slidingplane,andsimplifiedkinematicsofaV-beltCVTdrivewithnegligiblebeltfleslidingoractivearcs,representedinFig.3as(1,10),contributeactivelytotorquetransmission.(a,a0)representthewrapanglesofthebelt–pulleycontactingarcs,whereas(sin,sl)representtheinputandloadtorquesonthedriveranddrivenpulleys,t[22]sclassifiedonthebasisofcreep,compliance,sheardeflection,andflhoralsodiscussedslipduringwedgingduetopoorfielementanalysiswasusedtocalculatesheardeflectionsinthebeltandtodeterminestick–r,theworkdidnotaccountfortheinfluenceofbeltinertia,loadingconditions,andbeltradialvariations(duetopulleyaxialforces)t[17]alsostudiedtheinfluenceoffle,oflexure,tultoftheflexuraleffects,thecontactarcbecomessmallehoruflexuralrigidityhastremen-dousinflariationofcurvaturemaychangethedirectionoffrictionalforces,whichconsequentlyaffectsthetorquecapacityofthebelt–t[23]alsostudiedtheinfluenceofpulleyskewnessandflfldeformationmodel:(a)axialdeformation;(b)pulleyskewness[44].

tava,/MechanismandMachineTheory44(2009)19–4123orshortensthegroovewidth,therebyinflplacementofpulleysheavescanbeattributedtothephenomenaoflocaldeflection,platedeflection,.5[44]illustratesthevar-iationinpulleygroovewidthduetoelasticaldeflheorywasusedtoobtaintheglobaldefl,theplateequationsandthebeltequationsweresolvedsimultaneouslytoobtainthedynamicperformanceindicatorsofthebelt–ssofthhorreportedtheexistenceofsingularsolutions(alsoknownasorthogonalpoints,‘‘orthogonalpunkts”)wherethefric-tionalforcesareradiaracterofasolutionatanorthogonalpointtoagreatextentdkconcludedthattheaxialforcesforpulleyswithsmalltomediumskewnessdidnotdiffermuchfromtheonesinidealcase(rallelpulleyhalves).Ontheotherhand,largeskewnessconsiderabandDolan[24]extendedGerbert’swork[17]toformulateclosed-formsolutionsusingapproximatemathematicalfunolicandtrigonometricformsweredevelopedtodescribethetensiondistributionoverthemainactivearcofcontact(excludingtheseatingandunseatingre-gions).TheauthorswereabletogetresultsincloseagreementtoGerbert’snumericalresults;however,[25]studiedtransientmechanicsofrubberV-bultsobtaineddifferedsignifierateshiftspeedandhighangularspeed,partialadheclosed-formapproximationswereproposndGerbert[26]developedathirdorderrubberV-beltmodelconsideringnoinertialandfltroducedtheconceptof‘‘adhesive-like”oposedthatV-beltsdonotsticktopulleywallsalongtheentirecontactarc,butpassthroughanij[27]alsoanalyzedthemechanicsofametalV-beltCVTundertheinflhorusedtheconceptsofvirtualdisplacementtoobtainapproximationstobelttrajectories,tensiondistri-butions,axialthrusts,horreportedthattheinfluenceofpulleydeflfluencedecreasesnore,theuseofamixedmodelwassuggestedatsmallerradiiinordertoapt[28]alsousedtheconcehor’dyinvolvedthetheoreticalproblemofdetdeformationwascorrelatedonlytotheaxialforces,horincor-poratedtransverseelasticityofthebeltandusedforce-deformationequationstoobtainslidingconditionsandapproximateclosed-formsolutionstothebelt–metal.[15]incorporatedelastohydrodynamiclubricationtheorytomodelfrictionbetweenametalV-beltandthepulleysheaves,horsalsodiscussedtheinflr,certainunrealisticassumptions,suchasuniformbandtension,uniformloadingandunloadingarcsoverthepulleywraps,negligibleinertialeffects,etc.,ariatndPlay[29]edquasi-staticequilibriumanalysisandnumericaltechniqtelementswerealwaysassumedtobeincompression,soobservedthatonlypartofthecontactarc(iveorslidingarc)elbandsaidedpowertransmissionatlowtransmissionratios,terincludedanapproximatemodelofpulleydeformationsintheirCVTmodelandobserveaetal.[30]developedatheoreticalmodel-basedonquasi-staticequilibriumanalysistodescribevariationsinbandtensionandsegmentcompressionfosoincludedrelativemotionbetweenbandsandsegments(sincetheyhavedifferentradiiofrotation)r,microslipbehaviorofth,theauthorsalsoincorporatedanapproximatepulleydeformationmodeltostudytheinfluenceofpulleyflghtheresultsfromthenumericalmodelwereabletoconformtotheexperimentalobservationsforbandtensileforce,co-workers[31,32]oposedaspeedratio–torqueload–tainedtheequationsofmotionusingquasi-staticequilibriumconditionsandreportedthatthegrosscalresultsshowedthatthebeltradialdisplacementincreasedintheradialinwarddirectionforthedrivenpulley,whilethatofthectsofinertiaandflros[33]investigatedetcreepvelocitydepends

tava,/MechanismandMachineTheory44(2009)19–41notonlyonthestructuralcharacteristicsofthebelt,powertransmissionwithabelt–pulleysystem,dualchangeinvelocityfromthelargervaluetothelowervalue,andinversely,tcreepoccursinthedirectionofmotiononteportedthatthebeltcreepvelocityateachpointofthearcofcreepisalinearfunshietal.[34]investigatedthetorquetransmitationprocedurewasoutlinedtopredicttheslipratioandslip-limittorqueofametalV-esearchfocusedonthemicroslipcharacteristicofaV-beltCVT,phypothesiswasbasedontheassumptionthatsliponlyoccurredonthepulleywherethegapswerepresentandthosegapswerhorsobservedtheexistenceofactiveandidlearcsduringdifferentphasesoftrans-mission,asdepictedinFig.6[34].Theactivearcsarethoseregionsofbelt–pulleywrapthatcontributeeffectivelytotorquetransmission,hecompressiveforceinbeltelementsdecreasesintheidlearcregion,thereisredistributionofele-mentalgaps,[35]didperformance-basedanalysisofametalV-beltdrive,usingquasi-staticequilibriumconcepts,onbetweenindividualbandsinabandpacetal.[36]analyzedslipandeffiampingforcelevelsreducetheeffir,highclampillamountofslipisallowed,roposedthattheamountofradialslipdependsonthepulleydeformationeffects,shiftingspeed,tialslipwasdefinedbasedontheredistributionofelementalgaps(asin[34]).Itwasreportedthatthetractioncoefficient,ameasureofthetorquecapacityofaCVT,r,oncethemaximumtorquecapacityofaCVTisattained,thesliprisesdramaticallyandthetractioncoeffictionbetweenthebeltandthepulleywasmodeledaccordingtoStribeck’cedistributionswereobtainedusingAsayama’s[30]horsalsoconcludedthatthetractioncoefficientislargelydependenr,theefficienaetal.[37]analyzedtheaxialforchorsproposedauniquerelationtorelatethationsofmotionwerederivedusingquasi-staticequi-libriumconceptsandarelationshipwasdevelopedtorelatetheaxialforcestobelttension,erimentsindicatethatthethrustforceprofilevariesoverthepulleywrap;itisthhorsreportedthatevenwhenonlyinitialtensionisap-plied,thethrustpressureisnotconlsoreportedthatthethrustetal.[38]developedaunteandcontinuousshiftingbehaviorsweresimulatedinordertoana-lyzeefficiultsshowedhighlossofeffi-chanismduetoelementalgapredistributioninametalbeltCVT[34].

tava,/MechanismandMachineTheory44(2009)19––pulleycontactforcedistributionatnoloadfordifferentclamping(axial)forces[39].Tanaka[39]experimentallymeasuredthecontactforcesbeativechangeintheshapeofbeltwrappingarcdultsshowedthattheshapeofbeltwrappingarcwasnotco-axialwiththepulleyaxis,enconditionanddrivingwithasmalltorque,thecontactforcedistri-butionexhibitedpeaksattheinletandoutletofthepulleys,asdepictedbyarepresentativeexampleplotinFig.7[39].Thepeakontheoutletwashigher,roposedthatasymmetricalpulleydeformationandself-lockhorsalsoob-servedthatthehighpeakosattributedtotheincreaseinbeltcoml.[40]alsousedsimplifieddynamicmodelsofametalV-beltCVTtoansoexperimentallyinvestigated[41]thecreepmodewhenthebeltclamp-ingforcewasnotsmall,sonotedthattheshiftspeedincreepmodeincreasedinproportiontothepulleyrotatingspeed,,alargeradialslipoccurredoverthgerandPfeiffer[42,43]developedadetailedelasticmodelofmetalV-beltCVTsy,shaft,c-titdynamicswasspecifigitudr[44]analyzedthemechanicsofametalchainandV-beltconsideringlongitudinalandtransversestiffnessofthechain/belt,leydeformationwasmodeledusingastandardfileywasassumedtode-formintwoways,elwasprimarilyusedtostudyeffii[45]alsodevelopedamultibodydynamicmodelofametalV-beltCVTtofindthemetalblocktrajectoriesandtocalculatetheforcesactingontheblockandthering/g/r,intheirstudy,thesimulationwasconductedundersteady-stateceetal.[46]developedatheoreticalmodelofametalpushingV-belttounderstandthetwasmodeledasaone-dimensionalcontinuousbodywithzeroradialthicknessandinfi-dimensionalequationsweredefinedtoencompassdifferentloadingscenarios;however,theeetal.[47]alsostudiedtheinfluenceofclearancebetweehorsreportedthatpowertransmissionisassuredonlyifanactivearcexistswheretheelementsarepressedagaearc,wherethesteelelementsareseparatedandnocompressionexists,riverpulley,a‘‘shock”rivenpulley,therewasnoshocksection,butan‘‘expansionwave”kindofpheeetal.[48]usedtwofrictionmodels,namelyaCoulombfrictionmodelandavisco-plasticfrictionmodel,to

tava,/MechanismandMachineTheory44(2009)19–41modelfrictionbetweenthebeltandthepulleyforacchorsreportedthattheCoulombfrictionmodelisunabletocorrectlypredicttheshiftdynamicsofaCVTduringslowmaneuvers(-modephases),butitcouldwellpredictthelimitingtractioncapacityanddynamicbehaviorofaCVTinslip-mode(rapidshifting)r,avisco-plasticmodelisnotonlyabletoaccuratelypredictCVTbehaviorincreep-modephases,horsalsoproposedsimplerelationstocorrelateshiftspeedtoclampingforceratio,driverpulleyspeed,,Carboneetal.[49,50]extendedtheirpreviouswork[46]toinvestigatetheinflbfaleffectsofthebeltwereneglected;however,pulleybendingwasconsideredbasedonSattler’smodel[44].Theauthorsassumedequalpulleydeformationsandalsothatthebelt–pulleywrapanglesdidnotdeviateconsiderablyfrom180°.Theonlyinertialeffeer,horspredictedthattheCoulombfrictionmodelwasabletoaccuratelydescribeshiftingbehaviorofaChorssuggestedthatinsteadystate,thepressureandtensiondistributir,pulleybendingplayedasignifihownthatincreepmode,therateofchangeoelwasalsoabletopredicttheinflhorsreportedthatduringcreep-modephases,theshiftspeedmentsdonebyanumberofresearchers,especiallythosedonebyFujiiandKanehara,haveshownthatbothtensileforceinthebandorksuggeststhattheseforcesvarynon-uniformlyoverthepulleywrap,whichinvatal.[51]experimentallyinvestihorsobservedtheexistenceofactiveandidlearcsonthedrivingpulleyandsuggestedthatundersomeoperatingconditions,itisplausibleforthebandstocontributeupto40–45%horsalsoreportedthatthebandtensiondistributiodpacktensileforcewasobservedtodeuently,thebandtensionimpededtorquuldhappentosuchanextentthatinlowratio,forarangeoflowertorquelevels,thendco-workers[52]usedanumberofstraingaugedbeltelemerimentswereconductedonastoriesofpulleynormalforce,beltcompressiveforce,transmittingforce(-gentialfrictionforcebetweenabeltelementandthepulley),radialfrictionforce,elementshoulderforce,twasdifficulttomeasurecompressioninthefreespanofthebeltCVTsystem,astraight-linefilmentnormalforceexhibitedpeaksattheentryandexitofthepulleys(similartotheobservationsofIdeandTanaka[39]).Also,itwasobservedthatthhorsalsoreialfrictionforcerestrictedthepenetrationofabeltelementintothepulley,exceptattietal.[53]developedanumericalmodeltocalculatethesteady-stateforcedistribualV-beltismodeledusingalumped-parameterapproachwiththreekindsofspringsandtwokindsofinterfacialelementsfortheblockandthering,-parameterdiscretemodelofametalV-beltCVT[53].

tava,/MechanismandMachineTheory44(2009)19–4127asshowninFig.8[53].Theinterfacialelementswereusedtocapturefrictioneffectsbetweentheblockandthepulley,ndco-workers[54]laterextendedtheirpreviousexperimentalsetuptorecordtheforcesinstoriesofvariousforcesactinginabelt–r,er,sincetheexperimentswereconductedatlowspeedsandlowpressures,,FujiiandKaneharaco-authoredapaperwithKuwabara[55]wheretheyproposedanadvancednumericalmocesactinginthesystemwereestimatednotonlyatsteadystaeltakesintoaccountthevariousdynamicinteractionsoccurringamongbands,bservedthatbandstransmittednegativepowerunderover-driveconditions(epitchradiusondrivenpulleyissmallerthanthedriverpulleyradius).So,inordertomeetload,eaterthrustwasneehorsalsoobservedthatthethrustratio(ratioofthedriveraxialforcetothedrivenaxialforce)slightlyincreasedwithincreasingthecoeffifluenceoffle.9[55]depictssomesteady-stateresultsfromFujiiandco-workers[55]thathighlightthecombin,theauthors(tal.)definedspeedratioasthendKaneharaalsoco-authoredapaperwithFujimura[56,57]wheretheyusedmeancoefficientsoffrictionannggradientisanon-dimensionalparameterwhichisdefinedastheradialincerimentalresultsshowedthatthrustforceandslidingspeedinfluencedthemeancoeffihorsalsoobservedthattheshiftinggradientwasinflratioisdefineaseinshiftinggradientwasobservedwithincreasingthethrustofapulleyinwhichthebeltpitchradiuswasincreasing,whereasitdecreasedwithincr,theauthorsdidnotincludetheinfluenceofbelndKaneharalaterco-authoredapaperwithKataoka[58]wheretheyanalyzedshiftmechanismsofthorsconcludedthattheshiftinggradientisgovernednotonlybyblockelasticdeformations(duetothepulleythrusts),horsdevelopedquasi-staticequilibriumequaivinganddrivenpulleythrustsmentalconformitywasalsoreportedfornotonlysteadystate,ctionalperformanceofCVTfluidsandthefrictionalcharacteristicsofblock–pulleycontactinterfaceswereevaluatedbyapplyingthemeancoeffioundfromexperimentsthattheesti-matedcoefficientoffrictionofCVTfledwiththerel-ativeslidingspeedoftheblockswithrespecttothepulleys,slidingdirection,storiesofbelttensileandcompressiveforcesatsteadystatefordifferentspeedratios[55].

tava,/MechanismandMachineTheory44(2009)19–41SrivastavaandHaque[59–61]developedalwastounderstandthetransientdynamicbehaviorofabeltertialcouer-actionbetweenthebandpackandthebeltelement(whichhasbeenneglectedinalotofpreviousmodels)aleffectsduetobeltmotionwereneglected;however,pulleyflexibilitywastakenintoaccountusingSferra’scorrelations[38].Also,thecontactbetweenthebeltandthepulleyswasmodelednotonlybyclassicalcontinuousCoulombfrictionlaw,butalsobycertainmathematical/analyticalmodelsoffric-tionthatcapturedifferentloading/operatingconditions(on,lubrication,etc.).Itisevidentfromtheirworkthatnotonlytheconfigurationandloadingconditions,butalsotheinertialforces,influencethedynamicperformanceofaCVT,horsalsoreportedthattnsequentlyledtotheformationofshocksectionsattheelement–bandinterr,.10[61]depictsthefree-bodydiagramsofthemetalV-beltCVTdrivereportedbytheauthorstocatavaetal.[59,62]alsohighlightedthesignificanceofafeasiblesetofinitialoperahorsproposedthataCVT,beingahighlynonlinearsystem,needsaspecificsetofoperatingconditions,whichcanbefoundusinganefficientsearchmechanism,horsusedgeneticalgorithms(GA)tocapturethisfeasiblesetandalsohighlighteditsefficiencyincapturingthissetbycomparingittotheresultsgeneratedfromthedesignofexperiments(DOE).TheoptimizationobjectivefunctionwasatavaandHaque[63]developedametalpushingV-beltCVTmodelatsteadystatetostudyitsmicroslipbehaviorandtodefiscussedtheinflsbasedontheredistributionofelementalgapsandformationofinactivearcs(asproposedbyKobayashietal.[34]).-bodydiagramsofametalV-beltCVTsystem:(a)bandpackforces;(b)beltelementforces;(c)forcesonthepulleysheave;(d)torquesonthedriverpulley[61].

tava,/MechanismandMachineTheory44(2009)19–4129topredhorsagainobservedthattheCVToperatedinadefiniteregimeofaxialforcesanedictedtheminimumaxialforcenecessarytoinitiatetorquetransmissionandthemaximumaxialforcethattheCVTcouldsustain(basedonslipbehaviorandnotonstress,wear,orfatigueeffects).Theoperatingregimesforaxialforcesandtorqueswereob.11[61]illustratesrep-resentativetimehistoriesfromtheauthors’beltCVTmodelforanoperatingconditionof200Nminputtorqueand100NmloadtorqueataconstantdriverultsemphasizetheinfluenceofinertialeffectsonthedynamicperformanceandtorquecapacityoftheCVTsystem(referto[61]fordetails).Akehurst[64]theoreticallyandexperimentallyinvesthordevelopedfrictioneffectswerenntorquelosseventisproposedtooccurduetobandsslidingrelativetoeachotherandthesegment,therlossmodelsweredosseswereduetopulleydeflectionscausingthesegmentstocontactthepulleypastidealexitandentrypoints(wedgeloss)andtopenetratefurtherintothepulleythantheidealbeltpitchradius(penetrationloss).ThewedgelossproposedwassimilartothepreviouslydevelopedempiricalmodelofMick-lemetal.[65]andwasbasedontheradialdefltialbeltslipwasagaininvestigatedonthebasisofredistributionofgapsbetweenthebeltelements(asin[34]).Inadditiontothetorquelossassociatedwiththebelt,theauthorstudiedotherparasiticlossesinthetransmissionsystem,likesealandbear-ingdraglosses,pumplosses,clutchdraglosses,meshinglosses,dystate,itwasobservedthatthelossesinhighratioatlowspeedswerelowerthanlowratiolosses,r,asspeedsincreased,storiesfrombeltCVTmodel:(a)elementandpulleyspeeds;(b)elementandbandspeeds;(c)beltelementacceleration[61].

tava,/MechanismandMachineTheory44(2009)19–41Source of Energy

Spiral motion of

the belt

edself-excitationmechanisminametalbeltCVT[70].fi,itwasnotr,inlowratioconditions,thetorquelossundernandLong[66]presentedanoverviewofcurrentmanual,automatic,andcontinuouslyvariabletransmissioneffi-horsqualitativelydiscussedlossmechanismsinthesetransmissionsandsuggesteddesignimprovementstoenhancetransmissioneffielt-typeCVTsrequirepressuresofupto3MPa,thepumpingpowehorssuggestedtheuseofaradialballpumpohorsreportedthatthemanualtransmissionisthemosteffir,sincetheirtechnologyisquitemature,thereisverylittleroomforimprovementintermsofeffiandFussner[67]alsodiscussedthemechanisms,forces,andefficienciesofdifferenttypesofCVT,especiallypushbeltCVT,elastomerbelt-typeCVT,toroidalCVT,nutatingtypeCVT,untofpowerwhichcanbetransmittedbyapushbelt-typeCVTisdepen-dentoneitherthetensilestramicswithinthesteelbandsplhorssuggestedthatthetorquetransmissioninametalbeltCVTsystemisduetobothpushingforcesintndNair[68]developedmathematicalmodelsofdifferentCVTdesignsbyusingexistingliteratureandlaternormalizedtheinformationtoallowcomparisonofdifferentCVTconcepts(al,chain,belt,hydrostatictypes)elswereusedtocomputeefficienchorsreportedthattherubberbeltCVT,ingeneral,isthemosteffichainCVTisinefficientforlowtorquesandthetractionCVTislesseffindSung[69]analyticallyandexperimasisofmotioncharacteristicsoftheCVT,themajorcomponentsweremodeledasaninstantaneousflespanoftheaxiallymovingbeltwasmodeledasaflexiblecouplerlink,whilethevariablediameterdrivinganddrivensheavesweremodeledasthecrankandrockerlinks,horsusedthemixed-variationdaryandinitialconditionsweredeterminedfromthephysicalconfietricstudyonnhorsobservedthatthenaturalfrequencieser,insteadystate,,thenaturalfrequenerandco-workers[70]analyzedself-inducedvibrationsinapushingV-beltCVTusinghighlysimplifihownthatcertainfrictioncharacteristics,especiallythosehavingneg-ativegradientwithrespecttorelativevelocity,ctioncharacteristicandtheelasticityofhorsalsoobservedthatincreasingpulleystiffnessdecreasedbeltvibration,.12[70]cmodelingofchainCVTInadditiontobeltCVTs,considerableresearcheffortisalsobeingdirectedtowardsunderstdriveCVTconsistsoftwovariat[71]licationsfromdifferentsourcesweregroupedintothreesections:dynamicsofaxiallymovingmaterials,chaindrivedynamics,theliteraturediscussedbytheauthorwasrelatedtothedynamicsofrollerchaindrives,rubberV-andflatbelts,arollerchaindrive,aCVTchaindrivetransmitspowerexclusivelythroughfrictionalforcesintheconontacthastwopossiblestates,stickorslip,sibletransitionsb[72]devel-opedafiniteelementmethod(FEM)basedstaticmodelofaelgivesagoodtheoreticaldescriptionofthetorquetransferbehaviorofCVTchaindrives.

tava,/MechanismandMachineTheory44(2009)19–4131SrnikandPfeiffer[72–74]velopedaplanarmodelofchorkdealtwithmulti-bodyformalismsandfirtoaccountforthepolygonalexcitationsduetothediscretenatureofchain,inlinkswerethusmodeledaskinematicallydecoupledrigidbodiesinter-connectedbyspring-damperforceelements,asproposedbyFritzandPfeiffer[75].tingitsowninternaldynamics,thepairofrockerpinswasmoelwassimulatedunderthecondition.13[74]horsusedclassicalCoulombfrictionlawaswellasitstimeefficientcontinuousapproxpstoriesfromchainCVTmodel:(a)pulleynormalforce:(b)chaintensileforce[74].-bodydiagramsforchainCVTmodel:(a)forcesonachainlink;(b)chainlinkinteraction;(c)link-pulleycontactdescription[16].

tava,/MechanismandMachineTheory44(2009)19–41linksandthepulleywasrhorsusedtheTheoryofUnilateralContacts(developedbyPfeifferandGlocker[76])toformulatethesystemequationsasalinearcomple-mentarityproblem(LCP)ulationresultsshowedtheinfluenceofpuhorsreportedthatpulleydeformationhasadecisiveinfluencebservedthattheefficiencyofachainCVTdrivewithflsattributedtosubstantialradialmovementsofthechainlinksinthepulleygroovesforthecasewithhigherpulleyflhorsalsodiscussedtheinfluenceofchain’roposedthatanunin-tegratedchainpit.14[16]depictsyrandPfeiffer[77–80]horsmodeledthelinksanvenasmallpulleymisalignmentcouldyieldsignificanttensileforcesinthechain,leydeformationwasmodeledusingastaticfilleyshadonerotational,twotransla-tional(in-plane),andoneaxial(out-of-plane)leymainlinkwasmodeleksalsohadafewmoredegrsticityandthetranslationaldampingofthejointweretakenintoaccountbythelinkforceelement,whereastherotationaldampingandtheaxialfrictionbinkforceelement,ectofmovingcontactsrelativetotheplatespringreferencepointsbetweenarockerpinandaaringeffectsbetweenthepinendsandthesheaveswereobservedduetotheinflu-enceofshear,torsional,horsreportedthatowingtobendingeffects,thringthepulley,theshhorsalsosuggestedthataCVTchainwithoutclaspplatesismoreflroposedthatthetensileforcesinchar,thiswasnotagoodsolutionbecausewiththesamecrosssectionofpins,r,thetensileforcesinhorsalsodidacomparativestudyontheefficienciesofmetalV-beltsandchains,andconcludedthatCVTchaindriveshavehighereffirativestudyonthedynamicsandperformanceofametalV-beltCVTandachainCVTwasdonebyLebrechtandco-workers[81].Theauthorsobservedthatforthesametorque-run-upconditions,theclamper,sincebeltstendtoslipmorethanchains,theefficiencyofabeltCVTsystemwasobservelsoreportedthatthebeltCVTdriveexhibitslowercontactforcesatthepandPfeiffer[82]analyzedthenndPfeiffer[83]developedamethodtoassessnoiseodymodelsofgearbox,chainCVT,horsusedcom-plexnonlinearstiffnessfunctionsintheanalysistocaptureexcitationmechanismsrelatedtdtocalculatetheA-ratedequivalentcontinuousforcelevelofthebear-ingforces,whichisproportionaltothesound,enkoandSankar[84]investcorporatedanonlineardynamicmodelofclearanceandana-lyzeditsinflge[85]developedafastcomputationalalgorithmtocomputedynamicindicatorsofachainCVTfromamathematicalmodelwhichincludeddeformations,loadings,ghthefrictioncharacteristicofcontactingsurfacesinevitablyplaysacrucialroleinCVT’sperformance,literaturepertainingtotheinfluenceoffrictiononCVTdynamicsisscarce[48,57,58,60,61,70,86–89].Almostallthemodels,exceptafew,mentionedinliteratureuseCoulomb-ever,dependingondifferentoperating(orloading)conditionsanddesignconfigurations,tance,inafullylubricatedCVT,thefrictioncharacteristicofthecontactzonemaybearresemblancetotheStribeckcurve[90]er,veryhighforcesinthecontactzonefurtherleadtotheconditionsofelasto-plastic-hydrodynamiclubrication,htetal.[70]analyzedself-inducedvibrationsinametalpushingV-beltCVTbyusinghighlysimplifieportedthatcertainfrictioncharacteristics,especiallythosehavingnegativegra-dientwithrespecttorelativevelocity,ctioncharacteristicandtheelastir,itisnotclearwhethersuchphenomusnecessarytostudytheinflu-enceofdifferentfrictioncharacteristicsontheperformanceofaCVTinordertobetteridentifyvariouslossmechanismsanddesigneffitavaandHaque[16,91–94]developedaplanarmultibodymodelofchainCVTandinves-

tava,/MechanismandMachineTheory44(2009)19–4133tigatedtheinfluenceofclearanceandportedthatclearanceandfrictionparametersdrasticallyinfluencetheperformanceofaCVTsystem,reduceitstorquecarryingcapacity,andmayevenrenderthr,theirworkdidnotincludeanymathemati.15and16[16]illustraterepresentativetimehistoriesfromtheauthors’chainCVTmodelemphasizingtheinfluenceofcontact-zonefrictionconditionsandclearancesbetweenchainlinksonthetorquecapacityanddynamicperformanceofthechainCVTsystem(referto[16]fordetails).Theauthorsreportedthatcertaincontact-zonefrictioncharacteristics(lubri-cationandstiction)couldcausereductioninthetorquecapacityoftheCVTsystemandalsoitavaandHaque[60,61]developedadetailedtransientdynamicmodelofametalpushingV-beltCVTtoevaluateitsdynamicperformanceindicesundertheinfluenceofpulleyflexibilityandvary-ingfrictioncharacteristicsofthebelt–ferentmathematicalmodelsoffrictrictionmodelswereabletocaptureeffectsduetokineticfriction,stiction,horsperformedacomparativeanalysisonthedynamicperformanceofametalV-beltCVTundertheinflr,chaosrelate,SrivastavaandHaque[95]exploitedtheirpreviousplanarmultibodychainCVTmodel[16]toidentifyinterestingfriction-induccperformanceindicesofchainCVTemphasizingreducedtorquecarryingcapacity[16].cperformanceindicesofchainCVT:tensileforceinlinksshowingirregularperiodicity[16].

tava,/MechanismandMachineTheory44(2009)19–romnonlineardynamicssuchasLyapunovexponents,recurrenceplotting,phase-spacereconstruction,ployedtocharacterizetheobservedchaoslwasnotonlytounderstandfriction-inducednonlineardynamicbehaviorofachainCVTdriveasthechainlinkstraversedthecontactingarcsofdriveranddrivenpulleys,butalsotoevaluatesystemperformanceundertheinflhorsreportedthatcertaincontact-zonefrictionconditionsnotonlyre-ducetorquetransmittingcapacityofaCVT,butalsoinduceeatrolThecontrolaspectofachievingadesiredgearratioprofilebypulleyactuationforcelopmentofanoptimumCVTcontrolstrategyisnotaneasytaskowingtotwopartiallyoppositefeaturesthathavetobesatisfied:thereductionoffuelconsumptionandtherequirementofappro-priatedrivability/accelerationperformance(whichisdependentonthetorquecapacityoftheCVTsystem).Anaccurateandfastcontroloancedcontrolstrategymustimplementanaccuratemodeloftransmissionshiftingdynamicsinordertoforeseetheactualclampingforcesn,thechallengesforanefficientCVTcontrollerprimar-ilyaretoincreasethetorquecapacityofaCVTsystem,minimizebeltsliplosses,rtoachieveminimumfuelconsumptionrelativetothevariouslevelsofdesireddrivetorque,bothenginespeedandenginetorqueneedstobecontrolledsimultaneously,therebyreqposeofanintegratedCVT-enginecontrolistoachieveoptimalengineoperationforminimumfuelconsumptionwhilesatisfyingthedriver’imalengineoperation,theengineshouldbeoperatedonopti-maloperatingline(OOL).InFig.17[96],anforminimumfuelconsumptioncanbeobtainedfromthespecifiimalengineoperationpointisdefinmfuelconsumptioncanbeachievedbyoperatingtheengineattheoptimaloperationpointbysimultaneousTVO(throttlevalveopening)andCVTratiocontrol,ne-CVTintegratedcontrolwassuggestedbyTakiyamaandMorita[97].Theauthorsdevelopedanalgorithmtosimultaneouslycontrolthevehiclevelocityfromthedifferencebetweenthedesiredandactualvelocityandthr,theauthorsneglectedthetransientcharacter-isticsofpowertrain,whichresultedinpoorvehicle/,Takiyama[98]modifiedthepreviouscontrolalgorithm[97]byincorporatingair–fuelratioeffectsinthemodel,espeedcontrolsubsystem,fueloptimizingcontrolsubsystem,andA/Fcontrolsubsystemweretreatedassingleinput–outputsubsystems(SISO)-aguchietal.[99]developedanintegrathorssuggestedthatinanintegratedengine-CVTcontrol,itisnecessarytocaetal.[100]developedanintegratedengine-CVTcontrhorsusedtheenginetorquetocohorscalculatedthetargettorquebyassumingthattheacceleratorpedaltravelrepreseloperatingcharacteristicsofaCVTsystem[96].

tava,/MechanismandMachineTheory44(2009)19–4135KimandKim[96]developedanintegratedengine-CVTcontrolalgorithmbyconsideringthepowertrsoproposedcompensationahorsconductedexperimentstoconcludethattheoptimalenginespeedcompensationalgorithmgivesbetterengineoperationaroundtheOOL,comparedtotheoptimaltorquecompensationalgo-rithm,co-workers[101]experimentallyinvestigatedthere-sponsecharacteristicsofaparallelhybridelectricvehicle(HEV)CVTanddevelntheratiocontrolalgorithm,theeffectofCVTsystemdynamicsontheHEVengineoper-ationisinvestigatedbya‘‘hardware-in-the-loop”ulationshowedthattheengineperformanceimprovedbyusingaclosedloopcontrol,wherevariablecontrolgainswereuseddependingontheshiftdirectionandtheCVTspeedratio,consideringthenonlinearcharacteristicsofratio-controlhydraulicvalvesandCVTbelt–rtomaintainasteady-statespeedratio,singclampingforcesinthevariatorimprovestheefficiencyofaCVT;however,etal.[102,103]developedarobustgain-schedulingPIcontrollerbasedonalinearizedslipmodeltomeasureandcontrolslipinaCnswerescheduledbasedonprimaryspeed,ratio,pointalsovariedwiththeratio,sincethemaximumtractioncoeffir,theslipcontrolsystemwasdesignedforquasi-staticratiocontrol,ndLewis[104]developedasimulation-bodyformalismswereusedtomodelthebelt,andamodifiedPIcontStefanopoulou[105]consideredthewheelspeedproblemofanautomotivepowertrainwithaconventiontrolstrategies:(a)speedenvelopestrategy;(b)singletrackstrategy;(c)offthebeatentrackstrategy[108].

tava,/MechanismandMachineTheory44(2009)19–41ofthepowertrainusingsimplifiedCVTmodelstoimprovethevehicle/powertrainperformancewithoutsignifilaandSchmid[106]usedrobuststate-feedbacklinearizationapproachtokeeptheengine-CVTsystemontheoptimalfuel-efficiencycurveforacontinuouslyrunningengine(evehicle/carisinthehighpowerregime).LiuandPaden[107]andPfiffnerandGuzzella[108]surveyedthebasicshiftcontrolstrategiesofaCVTsystemforavehitegorizedshiftcontrolstrategiesintothreebroadphilosophies:(a)‘‘Speedenvelope”strategywherethedesiredoperatingareaoftheCVTsystemisformedbytwocurvesintheenginespeedversusvehiclespeedplane,asshowninFig.18a[108].Theimprovementsinthefueleconomyofavehiclearerealizedbysimplychoosingarelativelylowdesiredenginespeedatcruisingconditions;(b)‘‘Singletrack”strategywheretheenginetorqueisbroughtasquicklyaspossibletothe‘quasi-static’peakefficiencycurveX,asshowninFig.18b[108].Concurrently,thegearratioisadjustedtocorrespondtotheproposedfinalsteady-stateoperatingpoint;(c)‘‘Off-the-beatentrack”strategywheretwoormoretrajectoriesthatrepresentdifferentdrivingmodes(theeconomyandtheperformancemode)areusedtocontinuouslyadjustthegearratiotoreachthefinalr,sincethesethreestrategiesarebasedonheuristicsphilosophy,Pfiffneretal.[109]laterextendedtheirpreviousworktoexploreoptimalandsuboptimalcontrolstrategiesforfuel-efficientCVTpowertrainsbyusingsimplifiKim[110]developedanalgo-rithmtalbeltCVT,linepressuficientlinepressurecausesgrossslippageofthebelt,therhand,excesuently,tion,sincethemaximumprimarypressure(aquantitythatinfluencesshiftspeed)islimitedbythelinepressure,themetalbeltCVTcanalsopossiblysufferfrhorsfirstcreatedaCVTshiftspeedmapusingsimplifintheCVTshiftspeedmap,analgorithmtocalculateheshiftspeedcontrolalgorithmandsimplifieddynamicmodelsofthelinepressurecontrolvalveandtheratiocontrolvalve,per-formancesimulationsofaCVT-eetal.[111]developedanadaptivenonlinearcontrolalgorithmfortheasymptotictrackingofthedesiredwheelspeedbyensuringthattheCVTratiotracksadesiredgearratioprofiusedintheirmodeltocapturethegearratiodynamicswasassumedasapurefital.[112]developedacontinuous-timemodelandcontrollerforthedrivetraindynamicsandthehhorintroducedseparatenotionsofgeometricratioandspeedratioandtheirrelationtoeachotherthroughtheoutputpowereffirivenmodelwasthencreatedtortrollerwasdesignedusetal.[113]modeledtheCVTsystemasafirst-orderlagsystemwithanuncertaintimeconstantandtimedelay,andappliedl-synthesis,arobustcontrolmethod,rtofurtherimprovethecontrolperformance,,theirCVTl.[114]suggestedafuzzylogicbasedratiocontrolalgo-rithmforthemetalbeltCVTsystemconsideringtheon–offcharacteristicsoftmentalresultsshowedthatadesiredspeedratiocouldbeachievedatsteadystatebyfuzzylogicinspiteofthefltion,itwasfoundthatfasterresponseandbetterrobustnesscharacteristil.[115]developedamodel-basedcontrolalgorithmforthepressure-controlssure-controlCVTsystem,thedeshorsproposedthatlinearcontrolalgorithmssuchasPIDtypecontrolcouldbeusedforthepressure-controltypeCVTwhereasnonlinearoradaptivecontrollogicshouldbeimplementedfortheflsetal.[116]developedanewratiohedynamicmodelsofthevariatorandhydraulics,andcompensatorconstraints,asetpdbackcalratiocon-trollerguaranteedthatatleastoneofthepressuresetpointswasalwaysminimalwithrespecttoitsconstraints,nterestingCVTmodelingandcontrolstrategiespro-posedintheliteratureincludeKolmanovskyetal.[117],Pfiffneretal.[118],Shafaietal.[119],Laanetal.[120],Sorge[121,122],KongandParker[123],Frankandco-workers[124,125],Saito[126],Tanietal.[127],sionsAlthoughacontinuouslyvariabletransmissionplaysacrucialroleintheplantoimprovethefueleconomyanddynamicperformanceofavehicle,itscoectedin-creaseinfuetroeltandchaintypeCVTsarethemostcommonlyuseddrivetransmissions,thispaperreviewsthestate-of-the-artresearchthathasbeelsdiscussedinliteraturevaryintheirlevelofcomplexity,theirmodeofanalysis,

tava,/MechanismandMachineTheory44(2009)19–r,theliteraturereviewedrevealssignificantopportunitiesofresearchthatcouldbeneces-sarytogainbetterinsightintothedynamicsofsuchCVTsystemstomaximizethedynamicperformanceandfueleconomyofaCVT-equippedvehicle,designbetter/efficientcontrollers,identifylossmechanisms,andcharacterizeoperatingregimesformaximumtorquetransmissibilityoreffintheliteraturesurveyed,thefollowingconclusionscouldbedrawn: