外文翻译--凸轮速度对凸轮系统影响的实验研究 英文版.pdf

PergamonMech.Mach.TheoryVol.31,No.4,pp.397-412,1996Copyright©1996ElsevierScienceLtd0094-114X(95)00087-9PrintedinGreatBritain.Allfightsreserved0094-114X/96$15.00+0.00ANEXPERIMENTALSTUDYOFTHEEFFECTSOFCAMSPEEDSONCAM-FOLLOWERSYSTEMSH.S.YANandM.C.TSAIDepartmentofMechanicalEngineering,NationalChengKungUniversity,Tainan70101,Taiwan,RepublicofChinaM.H.HSUDepartmentofMechanicalEngineering,KungShahInstituteofTechnologyandCommerce,Yungkang,Tainan71016,Taiwan,RepublicofChina(Received9September1994；receivedforpublication26October1995)Altraet-Traditionally,inacam-followersystem,thecamisoftenoperatedataconstantspeedandthemotioncharacteristicsofthefolloweraredeterminedoncethecamdisplacementcurveisdesigned.Fromthekinematicpointofview,theapproachbyvaryingcaminputdrivingspeedisanalternativewayforimprovingthefollowermotioncharacteristics.Hereweshowhowtofindapolynomialspeedtrajectoryforreducingthepeakvaluesofthemotioncharacteristics.Furthermore,constraintsandsystematicdesignproceduresforgeneratinganappropriatetrajectoryofthecamangularvelocitiesaredeveloped.Designexamplesaregiventoillustratetheprocedureforgettinganappropriatespeedtrajectoryasvariablespeedcam-followersystems.Furthermore,anexperimentalsetupwithaservocontrollerisdevelopedtostudythefeasibilityofthisapproach.Experimentaldatashowthattheresultsareveryclosetothoseoftheory.NOMENCLATUREa-accelerationofthefollowerA,At-normalizedaccelerationofthefollowerc,d,e,n,Ta,Tb,x,y-constantparametersh-maximumdisplacementofthefollowerj-jerkofthefollowerJ,Jc-normalizedjerkofthefollowers-displacementofthefollowerS-normalizeddisplacementofthefollowert-timeforthecamtorotatethroughangle0T,Tp,Tpj,Tpv-normalizedtimev-velocityofthefollowerV,Vc-normalizedvelocityofthefollowert-camanglerotationfortotalrisehtim,t2,f13,t4-camrotationangle?-normalizedcamangleofrotationO-camangleofrotation-timeofcamrotationfortotalrisehT,%,%,r4-timeofcamrotationca-camangluarvelocityCOave-averagecamangularvelocityofacompletecycletOsm,cos2,co,co4-averagecamangularvelocityinafollowermotionperiodoh-the1stderivativeofco6b-the2ndderivativeoftot-normalizedcamangularvelocitytF-the1stderivativeoffthe2ndderivativeoffINTRODUCTIONInacam-followersystem,theloadproducedbyinertiaforcesispronetodeflectionandcreatesvibrations；andtheloadintroducedbyjerksmaycausevibrationsaswell,Thesewillaffecttheoperatinglifeofthecam.Therefore,thedesignofmotioncurvestominimizedynamicloadingisofimportanceforhighspeedcammechanisms.Itiswellknownthatthevelocityandaccelerationcurvesarerequiredtobecontinuousandtohavesmallerpeakvalues.Inaddition,thejerkcurveshouldbefinite.397398H.S.Yanetal.Acamisoftenassumedtobeoperatedataconstantspeedindesigningacam-followersystem.However,themotioncharacteristicsofthefollowerarechangedasthecamspeedvaries,Traditionally,toachievethedesiredmotionisanapplicationofsynthesisforobtainingnewdisplacementcurveswhichhavebetterdynamiccharacteristics.Inthispaper,weproposeanalternativemethodbyvaryingthecamspeeds.Theconceptofusingvariablespeedsinacam-followersystemdesignwasseldomstudiedintheliterature.Rothbart1designedavariablespeedcammechanisminwhichtheinputtothecamistheoutputofaWithworthquick-returnmechanism.TesarandMatthew2derivedthemotionequationsofthefollowerbyconsideringthecaseofvariablespeedcams.ThecriteriaforselectingproperangularvelocitieswhichwilleliminatethediscontinuityinmotioncharacteristicsofthefollowerareinvestigatedbyYanetal.3.Fromthekinematicpointofview,theobjectiveofthisworkistofindcamspeedtrajectoryforreducingthepeakvaluesofthefollower-outputmotion.Furthermore,constraintsandsystemdesignproceduresforgeneratingapropertrajectoryofthecamangularvelocitiesaredeve!oped.Designexamplesaregiventoillustratetheprocedureforgettingaproperangularspeedforagivenfollowersystem.Anexperimentalcam-followersystemissetupinwhichaservomotoriscontrolledtogeneratethedesiredspeedtrajectoryforperformanceevaluations.MOTIONEQUATIONSForacam-followersystem,thefollowerdisplacement,s(t),isafunctionofcamrotationangleO(t).Mathematically,itcanbeexpressedas:s(t)=f(O(t)(1)whereO(t)isthecamrotationangleattimet.Thefollowervelocity,v(t),ofthefolloweristhengivenby:v(t)=f(O)o(t)(2)wheref(O)=df(O)/dO,andco(t)=dO(t)/dtisthecamangularvelocity.Furthermore,thecorrespondingfolloweracceleration,a(t),andjerk,j(t),are:a(t)=f"(O)co2(t)+f(O)dg(t)(3)j(t)=f"(O)co3(t)+3f"(O)co(t)cb(t)+f"(O)6J(t)(4)wheref"(O)=df2(O)/dO2,f"(O)=df3(O)/dO3,(t)=dco(t)/dt,andc3(t)=dcoZ(t)/dt2.Equations(1)-(4)presenttherelationshipbetweencaminputangularvelocityco(t)andfollower-outputmotionss(t),v(t),a(t),andj(t).Obviouslyifco(t)isaconstant,theycanbegreatlysimplified.Lethbethetotaldisplacementofthefollowerasthecamrotatesanangleintimeperiod3.Furthermore,denoteT=t/r,=0/8,andS-s/h.NowwehaveT0,1,Vt0,r,0,1,V00,/andSe0,1,Vs0,h.Then,equations(1)-(4)canberewrittenintermsoftheirnormalizedformsasfollows:s(t)S(T)=g(),)=h(5)V(T)=g(y)t2(T)(6)A(T)=g"(y)fl2(T)-I-g(y)il(T)(7)J(T)=g"(y)3(T)+3g"(7)t2(T)tI(T)+g(y)(T)(8)wheret2(T)=dy(T)/dTisthenormalizedcamangularvelocityandV(T),A(T),andJ(T)arethenormalizedvelocity,acceleration,andjerkofthefollower,respectively.Therelationshipbetweenequations(1)to(8)canbefoundas:s=hs(9)hv=-v(10)TEffectsofcamspeedsoncam-followersystems399ha=A(ll)_hjj-.(12)Whenthecamoperatesataconstantspeed,i.e.f(T)=1,thenormalizedvelocity,Vc(T),acceleration,At(T),andjerk,J¢(T),ofthefollowercanbeexpressedas:V(T)=g(7)(13)Ac(T)=g"(7)(14)J(T)=g"()(15)where7(T)=T.CRITERIAFORDESIGN(T)Foragivencam-followersystem,thepeakvaluesofthenormalizedvelocity,acceleration,andjerkresultingfromaconstantdrivingspeedmaypossiblybereducedifweproperlycontrolitsinputspeedtrajectoryf(T).Forexample,toreducethepeakvaluesofnormalizedvelocity,f(T)canbechosensothatIV(Tpv)l<II/o(Tpv)lwhereVchaspeakvaluesatnormalizedtimeTpv.Then,fromequations(6)and(13),weselectthat12(T)mustsatisfythecondition:-1<f(Tpv)<1.(16)Forthecasethatwewanttoreducethepeakvaluesofthenormalizedacceleration,i.e.IA(r)l<IAc(r)lwereAohaspeakvaluesatnormalizedtimeTp.Basedonequations(7)and(14),f(T)shouldbechosensuchthatT.-1-F2(Tpa)<g(y(Tpa)t)(p)<tl_f2(Tp).(17)"g"(Y(Tpa)-""-tNotethatg"(7(Tp)mustbenonzero.Similarly,ifitrequiresJ(Tpj)lIJc(Tpj)lwhereJ¢haspeakvaluesatnormalizedtimeTpj,thenfromequations(8)and(15),weneedsomeII(T)whichsatisfies3g"(7(Tpj)tl(Tpj)t)(Tpj)+g(Tpj)(Tpj)<.1-S(Tpj)(18)-1+f3(Tpj)<g"(y(Tpj)asg"(r(rpj)#0.Foravoidingexcessivevibrationinthefollower,theharmonicsoftl(T)shouldbechosenaslowaspossible.Here,wechooseanappropriatespeedtrajectory.Sincethevelocityandaccelerationcurves,equations(6)and(7),arerequiredtobecontinuousandthejerkcurve,equation(9),shouldbefiniteaswell,(T)mustbeatleastsecondorderdifferentiable.Consideringthethecontinuityoft(T),theslopeoft(T)maybezeroatT=0andl,i.e.t)(0)-0and(1)-0.Furthermore,duetotheboundaryconditionsofthenormalizedcamrotationangle,(0)=0and),(1)=l,theintegrationoff(T)mustsatisfythefollowingcondition:ff2(T)dT+=(T)¢(19)forsomeconstantc.Inavariable-speedcam-followersystem,thecamoperatesatanangularvelocityco(t)androtatesanangleflinatimeperiod3,wehave:fco(t)dt=ft.(20)Sinceco(t)=fl/ztl(T),equation(20)isinfactequalto:f01(T)dT=1.(21)400H.S.Yanetal.Here,weonlyconsiderthecaseoft(T)t>0,i.e.thedirectionofcamspeedisnotchanged.Asaresult,designcriteriaforselectingfZ(T)toreducethepeakvaluesoffollower-outputsare:(a)(I)forthecaseofreducingthepeakvalueofthenormalizedvelocity:-l<(Tpv)<1(II)forthecaseofreducingthepeakvalueofthenormalizedacceleration:g(?(TPa)(T)<1-fZ2(T)-1+f2(rpa)<g"(?(T)(III)forthecaseofreducingthepeakvalueofthenormalizedjerk:-1+fZ3(Tp)<3g"(?(Tpj)fl(Tpj)(Tp)+g(T(rpj)(Tpj)g"(y(r.j)<1-f3(Tpj)1.(b)fZ(T)isatleastsecondorderdifferentiable.(c)fi(0)=fi(1)=0.(d)ConstantcsatisfiesSfl(T)dT+c=(T)subjecttotheboundaryconditions(0)=0,(1)=1.(e)£Z(T)hastheharmonicsaslowaspossible.(f)SI)(T)dT=1.(g)n(r)/>0.Letequations(5)-(8)representthenormalizedmotioncharacteristicsofthefollowerintherisingperiod.Then,themotioncharacteristicsinthefallingperiodare:S(T)=1-g()(22)V(T)=-g(,)fZ(T)(23)A(T)=-g"(?)fZ2(T)-g()(T)(24)J(T)=-g"(,)n3(T)-3g"(r)n(T)t(T)-g(r)t)(T).(25)Itiseasytofindthattheabsolutevaluesofthenormalizedvelocity,acceleration,andjerkinthefallingperiodareequaltothoseintherisingperiod,respectively.Hence,wehavethefollowingfact:Ifthesamedisplacementcurveisusedintherisingandfallingperiodsofafollower,thefunctionsoffl(T)inthesetwoperiodsareidentical.ANGULARVELOCITYf(T)Consideracam-followersystemwhichhasacamprovidingacycloidalmotioncurvewherecam-inputfZ(t)isapolynomial.Intherising(orfalling)periodandapplyingcriteria(a)and(g)toreducethepeakvaluesofmotioncurves,wechoosethefollowingpolynomialfZ(T),Fig.1:.t_.3.dtitiT0TaTb1Fig.1.Polynomialangularvelocityinrisingorfallingperiod.