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CHINESEJ0URNALOFMECHANICALENGINEERINGVo123,No3,2010367DOI:103901CJME201003367,availableonlineatwwwcjmenettom;wwwcjmenettomcnDynamicModelingofaRollerChainDriveSystemofInputShaftConsideringtheFlexibilityxuLixin,YANGYuhu一,CHANGZongyu,andLIUJianping1SchoolofMechanicalEngineering,TianjinUniversity,23anjin300072,China2CollegeofEngineering,OceanUniversityofChina,Qingdao266100,ChinaReceivedJune16,2009;revisedFebruary5,2010;acceptedFebruary10,2010;publishedelectronicallyFebruary25,2010Abstract:Rollerchaindrivesarewidelyusedinvarioushigh-speed,high-loadandpowertransmissionapplications,buttheircomplexdynamicbehaviorisnotwellresearchedMoststudieswereonlyfocusedOlltheanalysisofthevibrationofchaintightspan,andinthesemodels,manyfactorsareneglectedInthispaper,amathematicalmodelisdevelopedtocalculatethedynamicresponseofarollerchaindriveworkingatconstantorvariablespeedconditionInthemodel,thecompletechaintransmissionwithtwosprocketsandthenecessarytightandslackspansisusedTheeffectoftheflexibilityofinputshaftondynamicresponseofthechainsystemistakenintoaccount,aswellastheelasticdeformationinthechain,theinertialforces,thegravityandthetorqueondrivenshaftThenonlinearequationsofmovementarederivedfromusingLagrangeequationsandsolvednumericallyGiventhecenterdistanceandthetwoinitialpositionanglesofteethondrivinganddrivensprocketscorrespondingtothefirstseatingrolleroneachsideofthetightspan,dynamicsofanyrollerchaindrivewithtwosprocketsandtwospanscallbeanalyzedbytheprocedureFinally,anumericalexampleisvenandthevalidityoftheproceduredevelopedisdemonstratedbyanalyzingthedynamicbehaviorofatypicalrollerchaindriveThemodelcanwellsimulatethetransverseandlongitudinalvibrationofthechainspansandthetorsionalvibrationofthesprocketsThisstudycanprovideaneffectivemethodfortheanalysisofthedynamiccharacteristicsofallthechaindrivesystemsKeywords:rollerchain,dynamicmodeling,Lagrangeequations1IntroduetioRAsaneffectivewayoftransmittingpower,chaindrivesimpact,polygonalactionandexternalperiodicloadwereconsideredThen,arefineddynamicmodelofrollerchaindrivewasdevelopedbyKIM,etal,thecollisionsandreboundsoftheengagingrollerwerestudiedTheauthorscanbefoundextensivelyinnearlyallmechanicalaboveallusedthesimplifiedmodelandhavestudiedaengineeringareasThemajoradvantagesofchaindrivesaretheflexibilityinselectingshaftcenter,theabilitytodrivemorethanoneshaftthehighreliabilityanddurabilityHowever,theapplicationofchaindrivesintroducesproblemssuchasnoiseandvibration,whichmotivatedresearcherstoinvestigatetl1edynamicbehaviorofthemforvariousengineeringpurposesForacomprehensivereviewonthissubject,thereaderisreferredtoWANGetaltForrecentwork,VEIKOS,etal【。devisedarealisticanalyticalprocedureforthedynamicanalysisofrollerchaindrivesManytestcaseshadbeenperformedusingthisprocedurewithvarioustypesofchains,operatingatdirentspeeds,sprocketratiosandloadsCHOI,etal4developedamodelbasedonaxiallymovingmaterialtostudytransversevibrationofrollerchaindrivesThechainspanwasapproximatedasrigidbaratlowspeedsandelasticbaratmoderateorhighspeedsInthismodel,theCorrespondingauthorE-mail:ThisprojectissupportedbyNationalNaturalScienceFoundationofChina(GrantNo50605060),andTiinMunicipalScienceFoundationofChina(GrantNo06YFJMJC03300)sprocketandthetightsideofthechaindriveWhelltheemphasisisonthesystemsgloba1vibrationanddynamicbehavior,acompletechain仃ansmissionwhichregardsthechainspansastwoaxiallymovingstringswimtheirendsfixedtotworigidsprocketswasdevelopedbyWANG【oJInthesubsequentworkLIU。etal【integratedthelocalmeshingphenomenawiththeglobalchainsprocketsystemdynamicbehaviorInordertoanalyzethediscretenatureofthechainspansinglobalchainsprocketsystem,thechainstrandwasmodeledasaseriesofindividuallumpedmassconnectedtogetherbymasslessspringsdamperlThefirsttousethecompletegeometrywithadynamicmodelfordeterminingtheloaddistributionsalongthesprocketsandtheforcesalongthechainwasTROEDSSON,etaltjInthemode1thechaintransmissionwasdividedintofourseparateparts:thetightspan,slackspan,drivingsprocketanddrivensprocket,andeachpartwasmodeledandanalyzedseparatelyBecausethecompatibilitybetweenthepartsmustbeconsideredinacompletemodel,theboundaryconditioncheckingandsolvingprocedurearecomplicatedAmodelofaroller:!:竺里坐!蘭!里!erChainDriveSystemConsideringtheFlexibilityofInputShaftchaindriveincludingtheimpactbetweenthechainandguidebarswasdevelopedbyPEDERSENetalJJJInthemodel,arealtoothprofileisreplacedbyacirculartoothprofileandthecontactsbetweenrollersandsprocketteetharedetectedbykinematicconstraintsofusingunilateralconstraintmethodologyFiniteelementtechniquesandnumericalsimulationsoftwarewereusedbvZHENGetal,forpredictingthemeshingnoiseduetotheimpactofchainrollersagainstthesprocketofchaindrivesInthemodel,thecompletestandardgeometryofsprocketsandal1componentsofchainlinkswereusedwithminorgeometrysimplificationTheobjectiveinthisstudyistodescribeanautomatedgenerallyapplicablecomputerproceduretowardspredictivedesignandanalysisofthecompleterollerchaindrivesunderdynamicconditionsAmathematicalmodelisdevelopedandthenonlinearequationsofmovementarederivedfro而usingLagrangeequationsTheeffectoftheflexibilityofinputshaftondynamicresponseofthechainsystemistakenintoaccountaswellastheelasticdeformationinthechaintheinertialforcesthegravityandthetorqueondrivenshaftArollersprockettoothmotionneglected(7)Allpartsofthemodel,bothchainandsprocketsareregardedastwodimensionalbodiesandallvariationsinthedirectionalongtherotationaxisofthesprocketsareneglectedFig1showsamodelofatypicaldrivingsprocketwithchainlinksarounditAcoordinatesystemisdefinedinthecentreofthedrivingsprocketThisistheglobalcoordinatesystem,usedtodescribethegeometry,whencalculatingthepositionoftherollersandthesprocketsThepositionofanylumpedmasspointfcanbeexpressedinxandYidirectionsThestretchingforcesinthetwoconnectinglinksarecalled一1andThecontactforcebetweenthesprocketandtherolleriscalledandthegravitationforceismgInertialforcesinthechainoneintheradialdirection:mR1andoneinthetangentialdirectiont=mR。l1,areconsideredsincetherotationofsprocketsisnotatanearlyconstantangularvelocityIntheseforcesthestretchingforcesandthecontactforcecanbeexpressedasthespring-dampingforcesAdditionally,itissupposedthatalltheforcesactatmasscenterInthemodel,theinputshaftissimplifiedasatorsionspringwithconstantandcontactdetectionanalysisisdescribedindetailinthistorsionalstiffnessanddampingTheconstantorvariablestudyComparedwiththeworkmadebyTR0EDSS0Netal【JwhatisthesimplestnodivisionisneededinthecompletechainmodelandtheboundaryconditioncheckingiseasDynamicsofanyrollerchaindrivewithtwosprocketsandtwospanscanbeanalyzedbytheprocedure,ifthreenecessaryparameters,thecenterdistance5andthetwoinitialpositionanglesofteethand07ondrivinganddrivensprocketscorrespondingtothefirstseatingrolleroneachsideofthetightspan,aregivenFinally,anumericalexampleisgivenandtheproceduredevelopedhereisdemonstratedbyanalyzingthedynamicbehaviorofatypicalchaindrive。2DynamicModel-Forpurposesofanalysis,thechainstrandwillbeapproximatedasaseriesofpointmasseslumpedattherollercenterswhichareconnectedtogetherbymasslessspringsdamperasusedintheoreticalworks”Forsimplifyingcomplexityofthemodel,thefollowingassumptionsaremadeinthederivationofthechainequationsofmotion(1、A11thelinkshaveequalmass,stiffnessandundeformedpitchOnehalfofthetotal1inkmassislumpedateachroller(2、Therotationalinertiaoftherollersabouttheircenterofgravityisneglected(3)Thechainpitchisequaltothesprocketpitch(4、Allmechanicalclearancesareneglected(5)ArealtoothprofileisreplacedbyacirculartoothprofiletllJ(6、ThefrictionbetweentherollerandthetoothflankinthepartofthechainincontactwiththesprocketisspeedsboundaryconditioncanbeimposedontherotationanglefofinputshaftWhentherollerengagesintothedrivensprocket,itsforceconditionisalmostthesameasitondrivingsprocketAtorqueMisimposedonthedrivensprocketshaftinordertosimulatetheworkingloadmR。1,clg,一一,)Fig1ModelofsprocketwiththecontactingchainpartFig1showsastandardgeometryofsprockettoothwiththreearcsandonestraightlineModelingthefu11sprocketgeometryanddetectingthecontactbetweenitandrollerinmovementisquitedifficultSointhisstudyarealtoothprofileisreplacedbyacirculartoothprofileasusedintheworkt“JasshowninFig2TheadvantageofthissimplificationisthatthecontactsbetweenrollersandteethcanbedeterminedeasilyTheradiusofthecirculartoothprofileisR+andtherollerradiusisR,yieldingaclearanceofAR=RtRIntheengagingprocess,onlyCHrNESEJOURNALoFMECHANICALENGINEEIUNG369onesinglepointatthetoothsurfacecansimultaneouslybeincontactwiththesameroller(a)Realtoothprofile(b)CirculartoothprofileFig2SimplificationofthetoothprofileThechainspanisfinallymodeledasaseriesofmasspointsconnectedwithmasslessspringelementsasshowninFig3Bothofthevibrationsinandydirectionscanbereflectedbythemode1Thelinksinboththetightandtheslackspanssufferthee佰:ctofgravityInthemovementtheengagementanddisengagementphenomenonbetweenrollersandsprocketteethwilloccurperiodicallyThecontactdetectionbetweenthemwillbespecifiedinnextpart:motionandcontactdetectionanalysisFig3Modelofthechainspan3MotionandContactDetectionAnalysisAnexampleofageneralchaintransmissionisshowninFig4ItrepresentsthesimplestkindoftransmissionsinceitonlyincludestwosprocketsandonechainOnesprocketisdriving,theotheroneisdrivenandthechaintransmitsthepowerbetweenthemThetwosprocketcentersareassumedtobelocatedatthesameheightwiththegravitationalforceactinginthenegativevdirection辨N-1)rNrlr2r3llgntslae-xi鼉。iideA一:sprocke一f:,slacks囝一Fig4CompletechaintransmissioninglobalcoordinatesystemTodescribethemodelafewparametershavetobedefinedAsshowninFig4,thereareNlinksinthechainThefirstlinkseatedonthedrivingsprockettoothwillbeindexed訂,alongthearrowpointingdirectionthenumberincreaseItisassumedthatthefirstlinkseatedonthedrivensprockettoothwillbeindexedrp,andthelastoneintheehainisindexedrN(1p)Theinitialpositionanglesofteethmeshedwithrollerr1andrpondrivinganddrivensprocketsaredefinedandrespectivelyIntheanticlockwiserotationofdrivingsprocketthepositionanglesofsprocketteethcorrespondingtorollerrlandrpcanbeexpressedasfollows:=+,0pl=02+2,(1)(2)where1and2aretherotationalangleofdrivinganddrivensprockets,respectivelyThen,thepositionanglesofthecorrectseatingteethcorrespondingtoroller訂ondrivinganddrivensprocketscanbederivedfrom=+一=O2+2+01)x360。(pf)360。Z2(3)(4)wherezlandz2arenumbersofteethofdrivinganddrivensprockets,respectivelyInthemovement,thedeviationofrollerrifromtheircorrectseatingpositionsondrivinganddrivensprocketsdenotedbysfandpfareexpressedasfollows:。f=R。f一,pf=Rpf一,whichcanbewrittenassf=pfR1cos一Rs1sinfYf(5)(6)(7)(8)whereR。1,Rs2arethepitchradiusforthedrivinganddrivensprockets;isthecenterdistancebetweenthetwosprocketsWhenthecontacthappens,twokinematicconstraintsarerequiredtomeetForthecontactbetweenrollerriandthecorrectseatingtoothondrivingsprocket,theconstraintis(9)Forthecontactbetweenrollerrianditscorrectseatingtoothondrivensprocket,theconstraintis量=R,【:!:堅(jiān)!璺里竺!苧!墾!里!SystemConsideringtheFlexibility。fInputShRRtR,(10)fk(yHR。2Generally,thenumberoflinksinthechainisnotanintegralmultipleofthenumberofsprocketteethTherefore,whenthechainlinkrlengagesintothedrivingsprocketagain,theseatingtoothiSnotthepreviousoneAtthispoint,inord
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