翻譯部分英文原文Parametricstudyofdynamicsofwormandworm-gearsetundersuddenlyappliedrotatingangleM.Y.Chung,D.ShawDepartmentofPowerMechanicalEngineering,NationalTsing-HuaUniversity,No.101KungFuRoadSectionII,HsinChu,Taiwan,ROCReceived13September2006；receivedinrevisedform23January2007；accepted25February2007Availableonline17April2007AbstractThedynamicsofaworm-gearsetunderinstantlyappliedrotatingangleareaffectedbyseveralfactors(includingthefrictionforceandelasticdeformationofthesurfacebetweengearteeth).Itisfoundthatthosefactorscauseanonnegligiblerotationalpositioningerror.Thegoalsofthisstudyareto(1)setupamathematicmodel,(2)carryoutanumericalsimulation,and(3)carryoutanexperimentandcompareitwiththenumericalresults.Theexperimentsandnumericalresultshaveverygoodagreement.Inthisresearch,themoment-inertiaofflywheel,friction,rigidityofshaftandrigidityofgeartootharealsostudied.Theresultscanbeusedaserrorestimationsofrelevantangles,angularspeedandangularaccelerationunderasuddenlyappliedrotationalangle,andareusefulforestablishingtheerrorcompensationrequiredforpositioncontrol.1.IntroductionWhenradartracksanaircraftwhichturnsatasharpangle；theradarmaynotbeabletolockontothetargetduetothesmalloscillationoftheantenna.Sometimes,theoscillationevencausesafailureoftheradarsystem.Accuratepositioningisacrucialsubjectinatrackingsystem.However,duetothedesignofthetrackingtransmissionmechanismisrestrictedbybothspaceandweight,thevolumeandtheweightofthetransmissionmechanismmustbelimited.Theworm-gearsetisagoodchoiceduetoitssmallvolumeandhighreductionratio.Owingtotheoscillationoftheantennaisanimportantdynamicsbehaviorforhighprecisionpositioningofradarsystem,thedynamicsofaworm-gearsetunderinstantlyappliedlargetorqueisveryimportanttounderstandthisbehavior.Thisbehaviorisaffectedbyseveralfactorsincludingthefrictionforceandelasticdeformationofthesurfacebetweengearteeth.Tounderstandtheeffectofthosefactors,alotofresearchhasbeenconducted.YukselandKahraman1studiedthein.uenceofsurfacewearonthedynamicbehaviorofatypicalplanetarygearset.Thewearmodelemployedaquasi-staticgearcontactmodeltocomputecontactpressuresandArchardswearmodeltodeterminetheweardepthdistributions.Parkeretal.2analyzedthedynamicresponseofaspurpairofawiderangeofoperatingspeedsandtorques.Comparisonsweremadetootherresearcherspublishedexperimentsthatrevealcomplexnonlinearphenomena.Thedynamicresponseofaspurpairwasinvestigatedusinga.niteelement/contactmechanicsmodelthatofferssigni.cantadvantagesfordynamicgearanalyses.Maliha3createdanonlineardynamicmodelofaspurgearpair,whichwascoupledwithlinear.niteelementmodelsoftheshaftscarryingthem,andwithdiscretemodelsofbearingsanddisks.Theexcitationsconsideredinthemodelwereexternalstatictorqueandinternalexcitationcausedbymeshstiffnessvariation,gearerrorsandgeartoothpro.lemodi.cation.Britton4producedasuper.nishedgearteeth(withaapproximately0.05mmRaFilm)andthefrictiontractionintheexperimentsweresimulatedtheoreticallyusingathin.lmnon-Newtonianmicro-elastohydrodynamiclubricationsolverandencouragingagreementbetweenfrictionmeasurementsandtheoreticalpredictionswasobtained.Kong5predictedelasticcontactandelastohydrodynamic.lmthicknessinwormgears.Usingtheundeformedgeometryofthegapbetweengearteethincontactathree-dimensionalelasticcontactsimulationtechniquehadbeendevelopedforcalculationofthetrueareaofelasticcontactunderloadrelativethewheelandwormsurfaces.Tuttle6studiedtheharmonicdrives,whichexhibitedverynonlineardynamicbehavior,inhismodelnotonlydynamicmodelsincludeaccuraterepresentationoftransmissionfriction,complianceandkinematicalerrorwereunderstand,butalsoimportantfeaturesofharmonic-drivegear-toothgeometryandinteraction.Experimentalobservationswereusedtoguidethedevelopmentofamodeltodescribeharmonic-driveoperation.Itisimportanttobeawarethatworm-gearperformanceisin.uencedbythelubricantappliedandmaintained.Helouvry7discussedtheissueoftheservo-orientationcontrolbeingaffectedbyasurgeofbothmaximumstaticfrictionforceanddynamicfrictionforce.Theyreportedtheoccurrenceofastablepositioningerror,aswellasthestopping(orstoking)duringtrackingattheturningpointofalimitingloopofthetrackingsystem.Whenthesystemwasinaonedirectionallow-speedtracking,itwaspossibleforthestickslipphenomenontooccurduetothefactthatthefrictionalforcetovelocitycurveappearstobeanegativeslopeandin.uencesthestaticfrictionforce.Itwasalsofoundthatasimilarphenomenonoccursathigh-speedtracking.Doupont8proposedafrictionalforcemodelwithnegativeslope.Inthereference,relativemotionfromstatictohigh-speedslidingwasdividedintofourstages.Inthe.rststage,theexternalforcewaslessthanthemaximumstaticfrictionforce.Inthesecondstage,twocontactingsurfacesstartedtodevelopasmallrelativevelocity.Inthethirdstage,therelativevelocitystartedtoincreasebetweentwoobjects,andlubricantgotintothecontactareaanddecreasedthefrictionalforce.Inthefourthstage,thelubricanthadfully.lledthecontactarea,andtheviscouscharacteristicandthefrictionalforceincreasedwiththevelocity.Oguridesignhandbook9includedseveralempiricalchartsandcalculationformulaeofthefrictioncoef.cientvs.slidingvelocity,theallowableerrorofmanufacturing,therecommendedvalueforthegapsbetweentwoteeth,andtheallowablecontactpressure.Shigley10discussedthewormspitchcirclelinevelocityaswellasthetransmissionfrictionalcoef.cient.Inthisstudy,onlythefrictionforceandelasticdeformationofthesurfacebetweengearteethareinterested.TheZK-typeworm-gearsetisusedtoeliminatethebacklash.Previousresearchworksongearsandgearsetswerefocusedonthelimitingloading,pressuredistributionandallowablepressureloadingofgears,andontherelationshipbetweenfrictioncoef.cientandslidingvelocity.Mostinvestigationsarebaseontheassumptionthatthetoothisrigid.Thisstudyisfocusedontheeffectsoftheelasticdeformationofthegeartoothsurfaceandthenonlinearfrictioncoef.cientonthesystemdynamics.Tostudythedynamicsoftheworm-gearset,adynamicmodeloftheworm-gearsetisdevelopedandthecomparisonbetweentheexperimentalresultsandthenumericalresultsisalsostudied.Theparametersofthedynamicmodelareadjustedbythe.ndingsofthecomparisonoftheanalysisandtheexperiment.Finally,theeffectsofmoment-inertiaof.ywheel,friction,rigidityofshaftandrigidityofgeartoothonthenonlinearbehaviorofthetransmissionmechanismarealsostudied.2.Thedynamicmodelsoftheworm-gearset2.1.Theequilibriumequationsofthewormandworm-gearsetThegeometricofworm-gearsetisshowninFig.1.Thereareaworm,awormgear,twoshafts,fourbearingsanda.ywheel.Theshaftofwormisdrivenbyanangley1.Therotationalangleofthe.ywheelisy4.Therotationalangleofthewormsbodyisy2(duetotheelasticdeformationofthewormshaft；y16y2).Therotationangleofworm-gearbodyisy3.ThemomentofinertiaofthewormisJw.Thepitchradiusofthewormshelicaltoothisrw.ThewormsrotationalinertiaasarigidbodyisJg.Thepitchradiusofthewormgearisrg.Themoment-inertiaofthe.ywheelisJ4.Inthisstudy,toneglecttheeffectofbacklash,ZK-typewormandwormgearareusedbothinanalysisandexperiment.Tosimplifythetheoreticalmodel,thebasicassumptionsarelistedasfollows:(1)The.ywheelisrigid.(2)Thewormsandworm-gearteethareelastic.(3)Thebodyofthewormgearisrigid.(4)Wormandwormgearareperpendiculartoeachother.(5)Theclearancebetweenteethisignored.(6)Noerroronthegeartoothpro.