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OnthelobeprofiledesigninacycloidreducerusingbymeansoftheprincipleoftheinstantvelocitycenterinthegeneralcontactmechanismandthehomogeCorrespondingauthor.Tel.82552797579fax82552635221.Emailaddresssmkwonchangwon.ac.krS.M.Kwon.www.elsevier.com/locate/mechmtMechanismandMachineTheory412006596–616MechanismandMachineTheory0094114X/seefrontmatterC2112005ElsevierLtd.Allrightsreserved.neouscoordinatetransformation.Itisconsideredthefourtypesofthecycloidreducersinthisstudythestationaryringgeartypeepicycloidreducer,therotatingringgeartypeepicycloidreducer,thestationaryringgeartypehypocycloidreducerandtherotatingringgeartypehypocycloidreducer.DesignexamplesforthefourtypesofthecycloidreducersarepresentedtosimulatetheoperationandtodemonstratethefeasibilityofthisapproachusingacomputeraidedprogramdevelopedonClanguage.C2112005ElsevierLtd.Allrightsreserved.KeywordsCycloidreducerEpicycloidplategearHypocycloidplategearInstantvelocitycenterHomogeneouscoordinatetransformationinstantvelocitycenterJoongHoShin,SoonManKwonDepartmentofMechanicalDesignandManufacturing,ChangwonNationalUniversity,9Sarimdong,Changwon,Kyongnam641773,SouthKoreaReceived31January2005receivedinrevisedform17May2005accepted28July2005Availableonline21September2005AbstractAcycloidspeedreducerisoneoftherotationalspeedregulationdevicesofthemachinery.Ithasadvantagesofthehigherreductionratio,thehigheraccuracy,theeasieradjustmentofthetransmissionratioandthesmallerworkspacethananyotherkindsofthereducer.Thispaperproposesasimpleandexactapproachforthelobeprofiledesignofthecycloidplategear,whichisamainpartofthecycloidreducer,doi10.1016/j.mechmachtheory.2005.08.0011.IntroductionSpeedreducersareusedwidelyinvariousapplicationsforspeedandtorqueconversionpurposes.Amongthem,acycloidreducerhasbeenusedfordecadesowingtotheirsmoothandhighperformance,highreliability,longservicelife,compactness,exceptionaloverloadcapacity,lowtozerobacklashthroughrollingtoothengagementinthecontactmechanism,andotheradvantages.Thereforeitmakesanattractivecandidateforlimitedspaceapplicationstoday.Acycloidplategear,whichisamainpartofthecycloidreducer,meshesinallteethorlobesatanyonetimewiththerollergearorringgearconsistedofseveralrollersonthecircularpitchline.Generally,itisclassifiedintofourtypesofthecycloiddrivesbythelobeprofileofthecycloidplategearandtherollergearC213smotionthestationaryringgeartypeepicycloidreducer,therotatingringgeartypeepicycloidreducer,thestationaryringgeartypehypocycloidreducerandtherotatingringgeartypehypocycloidreducer.Foranexample,thestationaryringgeartypeepicycloidreducerseeFig.1basicallyhasonlythreemajormovingpartshighspeedinputshaftwithintegrallymountedeccentriccamandrollerbearingassemblycorrespondingtothedistanceofcentersbetweenrollergearandcycloidalplategear,cycloidalplategear,andslowspeedoutputshaftassembly.Astheeccentriccamrotates,itrollsthecycloidplategearsaroundtheinternalcircumferenceofthestationaryringgear.TheJ.H.Shin,S.M.Kwon/MechanismandMachineTheory412006596–616597resultingactionissimilartothatofawheelrollingaroundtheinsideofaring.Asthewheelcycloidalplatetravelsaroundtheringgear,thewheelitselfturnsslowlyonitsownaxisinanoppositedirection.Thatis,foreachcompleterevolutionofthehighspeedshaftthecycloidalplategearturnsonelobepitchintheoppositedirection.Ingeneral,thereisonelesscycloidaltootharoundtheplategearthantherearerollersinthestationaryringgearhousing,whichresultsinreductionratiosbeingnumericallyequaltothenumberoflobesontheplategear.Thereducedrotationoftheplategearsistransmittedtotheslowspeedoutputshaft,notdepictedinFig.1,Fig.1.Shapeofastationaryringgeartypeepicycloidreducer.bymeansofdrivepinsandrollerswhichengagewithholeslocatedaroundthemiddleofeachplategear.TotheauthorsC213bestknowledge,littlepublishedinformationisavailableonanalysisanddesignofthecycloidreducer.BotsiberandKingston1introduced,withlittleanalyticalwork,thetheoryofoperationofthecycloiddrivemechanism.MalhotraandParameswaran2studiedtheeffectsofdesignparametersonforcesforvariouselementsofthecycloidspeedreduceraswellasthetheoreticalefficiency.BlancheandYang3developedananalyticalmodelofthecycloiddriveswithmachiningtolerancesandinvestigatedtheeffectsofmachiningtolerancesonbacklashandtorquerippleandthey4alsopresentedacomputeraidedanalysisproceduretoverifytheperformanceofcycloiddrives.LitvinandFeng5useddifferentialgeometrytogeneratetheconjugatesurfacesofcycloidalgearing.Recently,YanandLai6havepresentedageometricdesignrotatingringgeartypeepicycloidreducerinSection3,thestationaryringgeartypehypocycloidreducerinSection4,andlastlyinSection5therotatingringgeartypehypocycloidreducer.Based598J.H.Shin,S.M.Kwon/MechanismandMachineTheory412006596–616upontheproposedapproach,aprogramforshapedesignautomationhasbeendevelopedwithClanguage.Finally,designexamplesarepresentedtodemonstratethefeasibilityofthisapproach.2.StationaryringgeartypeepicycloidreducerAccordingtoKennedyC213stheorem8–10,thethreeinstantvelocitycenterssharedbythreerigidbodiesinrelativemotiontooneanotherwhetherornotconnectedalllieonthesamestraightline.Fig.2showstheconstructionnecessarytofindinstantvelocitycenters.InFig.2links2and3areindirectcontact.AllpinjointsIC12,IC13arepermanentinstantcenters.IfthepointofcontactdoesnotlieonthelineofcentersIC12–IC13,thesetangentialcomponentswillnotbeContactPointCommonTangentLineCommonNormalLineIC23IC13IC12Link1Link2Link3Link1conceptofahypocycloidalreducerusingthetheoryofconjugatesurfaces.Mostrecently,Lietal.7haveintroducedadoublecrankringplatetypecycloiddriveandpresenteditsworkingprinciples,advantagesanddesignissues.Inthispaper,weproposeanewapproachfortheexactgeometricdesignofthecycloidalplategearswithoutinterferenceinthecycloiddrivesusingtheprincipleofinstantvelocitycenterandthehomogeneouscoordinatetransformationtechnique.ItisconsideredthefourtypesofthecycloidreducersinthisstudythestationaryringgeartypeepicycloidreducerinSection2,theFig.2.Instantvelocitycentersofacontactmechanism.equal,andslidingexists.Hencetheonlyrelativemotionwhichlinks2and3canhaveattheirpointofcontactisinthedirectionofthecommontangent,andtheircenterofrelativerotation,instantvelocitycenterIC23,mustthenliealongthecommonnormal.However,byKennedyC213stheoreminstantvelocitycenterIC23mustliealonglineIC12–IC13.HenceinstantvelocitycenterIC23liesatthepointoftheintersectionofthecommonnormalandthelineofcentersIC12–IC13.Fig.3isaschematicofastationaryringgeartypeepicycloidreducer.ThismechanismemploysacrankðO1OCÞtodevotetheepicycloidalplategearthatorbitsaboutthecenterO1oftheinputshaftduetotheeccentricityoftheshaft.Atthesametime,thecycloidalplategearrotatesaboutitsowncenterOCintheoppositedirectionoftheinputshaft,duetotheengagementwiththestationaryringgear.Theresultingmotionofthecycloidalplategearisacompoundmotion.WecanconsiderthatitconsistsofthreelinksinkinematicstheframecorrespondingtoO1ORhererollersbeingattachedtothestationaryringgearasLink1,theeccentricdistanceO1OCasLink2,andthecycloidplategearasLink3.ByKennedyC213stheorem,wecaneasilydeterminethethreeinstantvelocitycenters,i.e.apointO1asIC12,apointOCasIC23andapointMasIC13,respectively,asshowninFig.3.HerewewilldenotetheeccentricityO1OCcorrespondingtotheFromvelocityJ.H.Shin,S.M.Kwon/MechanismandMachineTheory412006596–6165992velocityx3oftheoutputcycloidplategearLink3areillustratedinthesamedirectioncounterclockwise.ThemagnitudeofthevelocityV23ofthepointIC23asshowninFig.4canbedeterminedbyV23¼Ex2¼ðEC0QÞx3.ð1ÞItmeansthattheactualorientationofx2andx3isintheoppositedirectiontoeachother.Pollitt11showedhowtofindthepointofcontactbetweenthecycloidplategearplanetarygearandthecylindricalrollerswhichmakeuptheteethofthestationaryringgearsungearLink1RollerGearLink3CycloidalPlateGearLink2RollerGearProfileICICICXfYfContactPointOOORM1223131Cthedefinitionoftheinstantcenter,bothlinkssharingtheinstantcenterwillhaveidenticalatthatpoint.InFig.4,theangularvelocityxoftheinputshaftLink2andtheangulareccentricbearingoftheinputshaftasE,O1MasQwhichisanunknowntobedeterminedbelow,andO1ORasR,respectivelyFig.4.ThecenterdistanceorcranklengthE,thenumberofrollersN,therollerradiusRr,andtheradiusoftherollergearRareusuallyassigneddesignparameters.Fig.3.Instantvelocitycentersofastationaryringgeartypeepicycloidreducer.
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