外文翻译--一种滚子链传动的动力学建模 中文版.pdf
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外文翻译--一种滚子链传动的动力学建模 中文版.pdf
CHINESEJ0URNALOFMECHANICALENGINEERINGVo123,No3,2010·367·DOI: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:yangyuhutju。edu。cnThisprojectissupportedbyNationalNaturalScienceFoundationofChina(GrantNo50605060),andTiinMunicipalScienceFoundationofChina(GrantNo06YFJMJC03300)sprocketandthetightsideofthechaindriveWhelltheemphasisisonthesystemsgloba1vibrationanddynamicbehavior,acompletechain仃ansmissionwhichregardsthechainspansastwoaxiallymovingstringswimtheirendsfixedtotworigidsprocketswasdevelopedbyWANG【oJInthesubsequentworkLIU。etal【integratedthelocalmeshingphenomenawiththeglobalchainsprocketsystemdynamicbehaviorInordertoanalyzethediscretenatureofthechainspansinglobalchainsprocketsystem,thechainstrandwasmodeledasaseriesofindividuallumpedmassconnectedtogetherbymasslessspringsdamperlThefirsttousethecompletegeometrywithadynamicmodelfordeterminingtheloaddistributionsalongthesprocketsandtheforcesalongthechainwasTROEDSSON,etaltjInthemode1thechaintransmissionwasdividedintofourseparateparts:thetightspan,slackspan,drivingsprocketanddrivensprocket,andeachpartwasmodeledandanalyzedseparatelyBecausethecompatibilitybetweenthepartsmustbeconsideredinacompletemodel,theboundaryconditioncheckingandsolvingprocedurearecomplicatedAmodelofaroller:!:竺里坐!兰!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,onlyCHrNESEJOURNALoFMECHANICALENGINEEIUNG·369·onesinglepointatthetoothsurfacecansimultaneouslybeincontactwiththesameroller(a)Realtoothprofile(b)CirculartoothprofileFig2SimplificationofthetoothprofileThechainspanisfinallymodeledasaseriesofmasspointsconnectedwithmasslessspringelementsasshowninFig3Bothofthevibrationsinandydirectionscanbereflectedbythemode1Thelinksinboththetightandtheslackspanssufferthee佰:ctofgravityInthemovementtheengagementanddisengagementphenomenonbetweenrollersandsprocketteethwilloccurperiodicallyThecontactdetectionbetweenthemwillbespecifiedinnextpart:motionandcontactdetectionanalysisFig3Modelofthechainspan3MotionandContactDetectionAnalysisAnexampleofageneralchaintransmissionisshowninFig4ItrepresentsthesimplestkindoftransmissionsinceitonlyincludestwosprocketsandonechainOnesprocketisdriving,theotheroneisdrivenandthechaintransmitsthepowerbetweenthemThetwosprocketcentersareassumedtobelocatedatthesameheightwiththegravitationalforceactinginthenegativevdirection辨N-1)rNrlr2r3llgntslae-xi鼍。iideA一:sprocke一f:,slacks囝一Fig4CompletechaintransmissioninglobalcoordinatesystemTodescribethemodelafewparametershavetobedefinedAsshowninFig4,thereareNlinksinthechainThefirstlinkseatedonthedrivingsprockettoothwillbeindexed订,alongthearrowpointingdirectionthenumberincreaseItisassumedthatthefirstlinkseatedonthedrivensprockettoothwillbeindexedrp,andthelastoneintheehainisindexedrN(1<p<)Theinitialpositionanglesofteethmeshedwithrollerr1andrpondrivinganddrivensprocketsaredefinedandrespectivelyIntheanticlockwiserotationofdrivingsprocketthepositionanglesofsprocketteethcorrespondingtorollerrlandrpcanbeexpressedasfollows:=+,0pl=02+2,(1)(2)where1and2aretherotationalangleofdrivinganddrivensprockets,respectivelyThen,thepositionanglesofthecorrectseatingteethcorrespondingtoroller订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,【:!:坚!璺里竺!苎!垦!里!SystemConsideringtheFlexibility。fInputShRRtR,(10)fk(yHR。2Generally,thenumberoflinksinthechainisnotanintegralmultipleofthenumberofsprocketteethTherefore,whenthechainlinkrlengagesintothedrivingsprocketagain,theseatingtoothiSnotthepreviousoneAtthispoint,inordertokeepthecorrectmeshingeffects,thekinematicconstraintsbetweenrollersandsprocketteethneedtoberedefinedWiththesamecasewhenthechainlinkrpengagesintothedrivensprocketagain,thekinematicconstraintsbetweenrollersandsprocketteethalsoneedtoberedefined4DerivationofEquationsofMotionk(yf卜hik(。1sin0sfYf)×RtR(Rlcos0sfXi)+(。1sin0sfYi)qik。(。2sin0pfYi)×RtR+Theequ砒i。fm。i。n酊ederVedmusingc(j2i_1-f)+c(一川)一c。fblR。1sin(+90Lagran。)一geequationsasz,、lI十lICSl一S。,dI_+:Qi)a口fJagfga口Thederivationofthekineticenergy,deformationenergy,dissipationfunctionandgeneralizedforceofthesystemareshowninAppendixThentheapplicationofLagrangeequationsleadsto2Jv+2equationsofmotionTheequationsoflinksinXandydirectionsarelistedasfollows:m2fk(xf一1一Xi)L1k(xfXi+1)11(一lXi)+(一Yi)hikc(1cos0s一Xf)×RtR(1cOS0sfXi)+(1sin0sfqik。(+R2cos0pff)×RtR+f一qic。×。2sin(Opf+90。)一f+g=Qs+Qp(13)Theequationsofmotionofdrivinganddrivensprocketsareasfollows:Xf)sinf十Rs1(Rs1sin0sfYf)cos0sfIxRtR(lcos0s一Xi)+(R1sin0sfYi)+v一RRcos(0s+90。)一毫×i=lsin(0sf+90。)+lR。1ARlsin(0sf+90。)一×cos(0+90。)一-rp1)一c一)=0,(14)22+gfko一R。2(fi+R。2cOS0pff)sin0pf+i=1R2(2sinOpYi)cosOpfIxRt(b2R22R2cos(0f+90。)一毫Ix+c(Xi_l-Xi)+c(Xi-2i+1卜cc1Rslcosf+90。)一sif+90o):n(”9oof×毫卜gfcc【s2×c。(+90。)一戈f=“+Qpr】2、c。s(pi+90o):(15)r121。,JJSOCR一+=:厂Cg