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Onthelobeprofiledesigninacycloidreducerusingbymeansoftheprincipleoftheinstantvelocitycenterinthegeneralcontactmechanismandthehomoge-*Correspondingauthor.Tel.:+82552797579；fax:+82552635221.E-mailaddress:smkwonchangwon.ac.kr(S.-M.Kwon)./locate/mechmtMechanismandMachineTheory41(2006)596616MechanismandMachineTheory0094-114X/$-seefrontmatterC2112005ElsevierLtd.Allrightsreserved.neouscoordinatetransformation.Itisconsideredthefourtypesofthecycloidreducersinthisstudy:thestationaryringgeartypeepicycloidreducer,therotatingringgeartypeepicycloidreducer,thestationaryringgeartypehypocycloidreducerandtherotatingringgeartypehypocycloidreducer.Designexamplesforthefourtypesofthecycloidreducersarepresentedtosimulatetheoperationandtodemonstratethefeasibilityofthisapproachusingacomputer-aidedprogramdevelopedonC+language.C2112005ElsevierLtd.Allrightsreserved.Keywords:Cycloidreducer；Epicycloidplategear；Hypocycloidplategear；Instantvelocitycenter；HomogeneouscoordinatetransformationinstantvelocitycenterJoong-HoShin,Soon-ManKwon*DepartmentofMechanicalDesignandManufacturing,ChangwonNationalUniversity,9Sarim-dong,Changwon,Kyongnam641-773,SouthKoreaReceived31January2005；receivedinrevisedform17May2005；accepted28July2005Availableonline21September2005AbstractAcycloidspeedreducerisoneoftherotationalspeedregulationdevicesofthemachinery.Ithasadvan-tagesofthehigherreductionratio,thehigheraccuracy,theeasieradjustmentofthetransmissionratioandthesmallerworkspacethananyotherkindsofthereducer.Thispaperproposesasimpleandexactapproachforthelobeprofiledesignofthecycloidplategear,whichisamainpartofthecycloidreducer,doi:10.1016/j.mechmachtheory.2005.08.0011.IntroductionSpeedreducersareusedwidelyinvariousapplicationsforspeedandtorqueconversionpur-poses.Amongthem,acycloidreducerhasbeenusedfordecadesowingtotheirsmoothandhighperformance,highreliability,longservicelife,compactness,exceptionaloverloadcapacity,lowtozerobacklashthroughrollingtoothengagementinthecontactmechanism,andotheradvantages.Thereforeitmakesanattractivecandidateforlimitedspaceapplicationstoday.Acycloidplategear,whichisamainpartofthecycloidreducer,meshesinallteethorlobesatanyonetimewiththerollergear(orringgear)consistedofseveralrollersonthecircularpitchline.Generally,itisclassifiedintofourtypesofthecycloiddrivesbythelobeprofileofthecycloidplategearandtherollergearC213smotion:thestationaryringgeartypeepicycloidreducer,therotat-ingringgeartypeepicycloidreducer,thestationaryringgeartypehypocycloidreducerandtherotatingringgeartypehypocycloidreducer.Foranexample,thestationaryringgeartypeepicycloidreducer(seeFig.1)basicallyhasonlythreemajormovingparts:highspeedinputshaftwithintegrallymountedeccentriccamandrollerbearingassemblycorrespondingtothedistanceofcentersbetweenrollergearandcycloidalplategear,cycloidalplategear,andslowspeedoutputshaftassembly.Astheeccentriccamrotates,itrollsthecycloidplategearsaroundtheinternalcircumferenceofthestationaryringgear.TheJ.-H.Shin,S.-M.Kwon/MechanismandMachineTheory41(2006)596616597resultingactionissimilartothatofawheelrollingaroundtheinsideofaring.Asthewheel(cycloidalplate)travelsaroundtheringgear,thewheelitselfturnsslowlyonitsownaxisinanoppositedirection.Thatis,foreachcompleterevolutionofthehighspeedshaftthecycloidalplategearturnsonelobepitchintheoppositedirection.Ingeneral,thereisonelesscycloidaltootharoundtheplategearthantherearerollersinthestationaryringgearhousing,whichresultsinreductionratiosbeingnumericallyequaltothenumberoflobesontheplategear.Thereducedrotationoftheplategearsistransmittedtotheslowspeedoutputshaft,notdepictedinFig.1,Fig.1.Shapeofastationaryringgeartypeepicycloidreducer.bymeansofdrivepinsandrollerswhichengagewithholeslocatedaroundthemiddleofeachplategear.TotheauthorsC213bestknowledge,littlepublishedinformationisavailableonanalysisanddesignofthecycloidreducer.BotsiberandKingston1introduced,withlittleanalyticalwork,thetheoryofoperationofthecycloiddrivemechanism.MalhotraandParameswaran2studiedtheeectsofdesignparametersonforcesforvariouselementsofthecycloidspeedreduceraswellasthetheoreticaleciency.BlancheandYang3developedananalyticalmodelofthecycloiddriveswithmachiningtolerancesandinvestigatedtheeectsofmachiningtolerancesonbacklashandtorqueripple；andthey4alsopresentedacomputer-aidedanalysisproceduretoverifytheperformanceofcycloiddrives.LitvinandFeng5useddierentialgeometrytogeneratethecon-jugatesurfacesofcycloidalgearing.Recently,YanandLai6havepresentedageometricdesignrotatingringgeartypeepicycloidreducerinSection3,thestationaryringgeartypehypocycloidreducerinSection4,andlastlyinSection5therotatingringgeartypehypocycloidreducer.Based598J.-H.Shin,S.-M.Kwon/MechanismandMachineTheory41(2006)596616upontheproposedapproach,aprogramforshapedesignautomationhasbeendevelopedwithC+language.Finally,designexamplesarepresentedtodemonstratethefeasibilityofthisapproach.2.StationaryringgeartypeepicycloidreducerAccordingtoKennedyC213stheorem810,thethreeinstantvelocitycenterssharedbythreerigidbodiesinrelativemotiontooneanother(whetherornotconnected)alllieonthesamestraightline.Fig.2showstheconstructionnecessarytofindinstantvelocitycenters.InFig.2links2and3areindirectcontact.Allpinjoints(IC12,IC13)arepermanentinstantcenters.IfthepointofcontactdoesnotlieonthelineofcentersIC12IC13,thesetangentialcomponentswillnotbeContactPointCommonTangentLineCommonNormalLineIC23IC13IC12Link1Link2Link3Link1conceptofahypocycloidalreducerusingthetheoryofconjugatesurfaces.Mostrecently,Lietal.7haveintroducedadoublecrankring-plate-typecycloiddriveandpresenteditsworkingprin-ciples,advantagesanddesignissues.Inthispaper,weproposeanewapproachfortheexactgeometricdesignofthecycloidalplategearswithoutinterferenceinthecycloiddrivesusingtheprincipleofinstantvelocitycenterandthehomogeneouscoordinatetransformationtechnique.Itisconsideredthefourtypesofthecycloidreducersinthisstudy；thestationaryringgeartypeepicycloidreducerinSection2,theFig.2.Instantvelocitycentersofacontactmechanism.equal,andslidingexists.Hencetheonlyrelativemotionwhichlinks2and3canhaveattheirpointofcontactisinthedirectionofthecommontangent,andtheircenterofrelativerotation,instantvelocitycenterIC23,mustthenliealongthecommonnormal.However,byKennedyC213sthe-oreminstantvelocitycenterIC23mustliealonglineIC12IC13.HenceinstantvelocitycenterIC23liesatthepointoftheintersectionofthecommonnormalandthelineofcentersIC12IC13.Fig.3isaschematicofastationaryringgeartypeepicycloidreducer.ThismechanismemploysacrankO1OCtodevotetheepicycloidalplategearthatorbitsaboutthecenter(O1)oftheinputshaftduetotheeccentricityoftheshaft.Atthesametime,thecycloidalplategearrotatesaboutitsowncenter(OC)intheoppositedirectionoftheinputshaft,duetotheengagementwiththestationaryringgear.Theresultingmotionofthecycloidalplategearisacompoundmotion.Wecanconsiderthatitconsistsofthreelinksinkinematics:theframecorrespondingtoO1OR(hererollersbeingattachedtothestationaryringgear)asLink1,theeccentricdistanceO1OCasLink2,andthecycloidplategearasLink3.ByKennedyC213stheorem,wecaneasilydeterminethethreeinstantvelocitycenters,i.e.apointO1asIC12,apointOCasIC23andapointMasIC13,respectively,asshowninFig.3.HerewewilldenotetheeccentricityO1OCcorrespondingtotheFromvelocityJ.-H.Shin,S.-M.Kwon/MechanismandMachineTheory41(2006)5966165992velocityx3oftheoutputcycloidplategear(Link3)areillustratedinthesamedirection(coun-terclockwise).ThemagnitudeofthevelocityV23ofthepointIC23asshowninFig.4canbedeter-mine