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通过控制接触的路径和传输误差来为螺旋锥轮设计齿轮机床设置外文文献翻译中英文翻译.docx
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通过控制接触的路径和传输误差来为螺旋锥轮设计齿轮机床设置外文文献翻译中英文翻译
附录外文文献
DesignofPinionMachineTool-settingsfoSpiralBevelGearsbyControllingContactPathandTransmissionErrors
CaoXuemeia,*,FangZongdea,XuHaob,SuJinzhana
a:
SchoolofMechatronics,NorthwesternPolytechnicUniversity,Xi’an710072,China
b:
ZhongnanTransmissionMachineryWorksofChangshaAviationIndustries,Changsha410200,China
Received16September2007;accepted20February2008
Abstract:
Thispaperproposesanewapproachtodesignpinionmachinetool-settingsforspiralbevelgearsbycontrollingcontactpathandtransmissionerrors.Itisbasedonthesatisfactionofcontactconditionofthreegivencontrolpointsonthetoothsurface.Thethreemeshingpointsarecontrolledtobeonapredesignedstraightcontactpaththatmeetsthepre-designedparabolicfunctionoftransmissionerrors.Designedseparately,themagnitudeoftransmissionerrorsandtheorientationofthecontactpatharesubjectedtoprecisecontrol.Inaddition,inordertomeetthemanufacturingrequirements,wesuggesttomodifythevaluesofblankoffset,oneofthepinionmachinetool-settings,andredesignpinionmachinetool-settingstoensurethatthemagnitudeandthegeometryoftransmissionerrorsshouldnotbeinfluencedapartfromminoreffectsonthepredesignedstraightcontactpath.Theproposedapproachtogetherwithitsideashasbeenprovenbyanumericalexampleandthemanufacturingpracticeofapairofspiralbevelgears.
Keywords:
spiralbevelgear;contactpath;transmissionerror;blankoffset;toothcontactanalysis
1Introduction
Spiralbevelgearsareamongthekeycomponentsofaerospacepowerplantsingeneral,helicoptergeardrivesinparticular,whichregardthemeshingperformance,enduranceandreliabilityascriticalsafetyfactors.Therefore,designingspiralbevelgearshasallthetimebeendrawingcloseattentionofresearchersinmanycompaniesandinstitutions.Therequirementsofreducingnoiselevelandincreasingenduranceofspiralbevelgearshaveraiseformidablechallengetodesignerstoo.
Thebasesfordesigninglow-noisespiralbevelgearswithlocalizedbearingcontactswerepresentedinrelatedRefs..Inordertoabsorblargererrorsinalignmentandhavebetterstability,thecontactpathshouldbedesignedtobeastraightline.Astransmissionerrorsaremostlyblamedfornoiseandvibrationingearingsystems,thetransmissionerrorsofparabolictypeareconsideredtobeabletoabsorblineardiscontinuouseffectscausedbymisalignmentreferredtoasthemainsourceofnoise.
Insomecases,themachinetool-settingsdesignedbywayoftheexistinglocalsyntheticmethodarewellbeyondtheappropriateapplicablerangeofthemachine.
Inthispaper,anewintegratedapproachisproposedonthebaseofmeetingthecontactconditionsinclusiveofthepredesignedparabolicfunctionoftransmissionerrorsandthespecificallyorientedstraightcontactpaththroughthreegivencontrolpointsonthetoothsurface.Asaresult,asearlyasinthedesigningstage,theoperatingperformancecouldbecontrolled.Inaddition,thevaluesofblankoffsetcanbesomodifiedastobewithintheappropriaterangeofthemachinewithoutanyinfluence
onthemagnitudeoftransmissionerrorsapartfromtherebeingminoreffectsonthepredesignedlinearcontactpath.
Theproposedapproachisbasedonthefollowingassumptions:
(1)Thegearmachinetool-settingsarepredeterminedandcanbeadopted.
(2)Themaininputparametersareη2andm'21.Ofthem,η2determinestheorientationangleofthepredesignedstraightcontactpath,andm'21thesec-
ondderivativeofthetransmissionfunction,whichdeterminesthepredesignedmagnitudeoftheparabolicfunctionoftransmissionerror.Inthispaper,thevaluesofη2andm'21aregiveninadvance,but,inpractices,theycanbeoptimizeddependingontheappliedloadstoobtainthefavorablemeshingperformancethroughtheloadedtoothcontactanalysis(LTCA).
2ActiveToothSurfaceDesignbyThreeGivenMeshingPoints
Afterthethreecontrolmeshingpointshavebeendeterminedonthegearsurfacebythepredesignedstraightcontactpathandtheparabolicfunctionoftransmissionerrors,thepinionmachinetool-settingscanbedetermined.
2.1Determinationofthreecontactpoints
Fig.1showsthethreecontactpointsonthegeartoothsurface.Γ2isthepitchangleofthegear.Thecontactpathisdesignedtobeastraightlineandη2istheorientationofcontactpath.
Fig.1Threecontrolmeshingpointsandcontactpath.
Whenthegearsurface∑2rotatesbyφ02,φ12andφ22,thepinionsurface∑1contactswithitatmeshingpointsM0,M1andM2withrotationalanglesφ02,φ12andφ22respectively.Letthecycleofpinionmeshingbe2π/Z1,whereZ1isthetoothnumberofpinion,andthenφ11=π/Z1+φ01andφ21=−π/Z1+φ01canbedetermined.
AtthemeancontactpointM0,theinstantaneoustransmissionratioisequaltothegearratio.Usuallythispointischosentobeinthemiddleof
thetoothsurface,anditslocationcanbeadjustedaccordingtodesignrequirements.Therotationanglesofthegearφ02andthepinionatthispointφ01,canbedeterminedfromthelocationandthemeshingequationatM0.
2.2Determinationoftransmissionerrors
Thetransmissionerrorfunctionisrepresentedby
δφ2=(φ2−φ02)−(φ1−φ01)Z1/Z2
(1)
Whereφi(i=1,2)istherotationangleofthepinion(i=1)orthegear(i=2)intheprocessofmeshing,andZithetoothnumberofthepinion(i=1)orthegear(i=2).
Theparabolicfunctionoftransmissionerrorisrepresentedby
δφ2=
·m'21(φ1−φ01)2
(2)
wherem'21isthederivativeofthetransmissionratio.
φ2isdeterminedasfollows
φ2=Z1(φ1−φ01)/Z2+φ02+
·m'21(φ1−φ01)2(3)
Therotationanglesofgearφ12andφ22,atthemeshingpointsM1andM2,canbedeterminedfromEq.(3).
2.3Determinationoforientationofcontactpath
Asthecontactpathisdesignedtobeastraightline,thethreemeshingpointsM0,M1andM2areonit.Themeshingequationbetweenthegearsurface∑2andthepinionsurface∑1atthecontactpointM1shouldobserve
n·ν12=f(θg,øg,φ12)=0(4)
wherenandν12aretheunitnormalandtherelativevelocityatthemeshingpoint,θgandøgrepresentthesurfacecoordinatesofthegeartoothsurface.
BasedonthelocationofthecontactpointM0andthegivenorientationofcontactpathη2,thelocationofthecontactpointM1canbedetermined.ThelocationequationofcontactpointM1is
g(θg,øg,φ12)=0(5)
BysolvingEqs.(3)-(5),thepositionvectorandunitnormalofthepointM1atthegearsurface∑2canbeobtained.ThepositionvectorandtheunitnormalofthepointM2atthesurface∑2canalsobedeterminedthesameway.
2.4Determinationofø11andø21
Fig.2showsthecoordinatesystemsforpiniongeneration.ThecoordinatesystemsSm1,ScandSbarefixedonthemachine.Thepinionmachinerootangleγ1determinestheorientationofSbwithrespecttoSm1.XG1isthemachinecentertobackforgenerationofpinion;Em1theblankoffsetandXB1theslidingbase.ThecradlecoordinatesystemSprotatesabouttheZm1-axis.Theangleøpisthecurrentrotationangleofthecradleintheprocessofgeneration.ThecoordinatesystemSfisconnected
rigidtothepinionhead-cutter,whichintheprocessofgenerationperformsrotationwiththecradle(transfermotion)andrelativemotionwithrespectto
Fig.2Coordinatesystemsforpiniongeneration.
thecradleabouttheZf-axis.ThemovablecoordinatesystemS1isconnectedrigidtothegeneratedpinionandrotatesabouttheXb-axis.Theangleø1isthecurrentrotationangleofpinionintheprocessofgeneration.ø11andø21aretherotationanglesofpinionintheprocessofgenerationatpointsM1andM2.BytransformingthecoordinatefromS2toSc,theunitnormalncofcontactpointsM1andM2insystemSccanbedetermined.ThenbytransformingthecoordinatefromSftoSm1,theunitnormalnm1ofpinion-cutter-generatingsurfaceinSm1canbedetermined.SincetheaxisofSm1isparalleltotheaxisofSc,theunitnormalofcontactpointinthesystemScisequaltothatinthesystemSm1.Thefollowingequationholdstrue.
nc(ø1)=nm1(θp+øp)(6)
fromwhichø11andø21canbedetermined.
2.5DeterminationofXG1,Em1,ø1pandø2p
ThepositionvectorsrcfofpointsM1andM2inthesystemSccanbederivedfromtheirpositionvectorsonpinion-cutter-generatingsurfacesbytransformingthecoordinatefromSftoSc.ThepositionvectorsrccanbederivedfromthepositionvectorsofcontactpointsM1andM2onthesurface∑2bytransformingthecoordinatefromS2toSc.rcfandrccoincidewitheachotherattheinstantaneouspointofcontactM1(M2).Thepositionvectorequationis
rcf=(XG1,Em1,øp,sp)=rc(7)
AsforthepointsM1andM2,thepositionvectorequations,whichareequivalenttosixindependentscalarequationswithsixunknowns,canbe
usedtodeterminethesixunknowns:
XG1,Em1,ø1p,ø2p,s1pands2p.Hereø1p
andø2paretherotationanglesofcradleintheprocessofgeneratingcontactpointsM1andM2.
2.6Determinationofrp,sr1,q1,XB1andmp1
Fromthegivenangleη2onthecontactpath∑2,andthelengthofthemajoraxisoftheinstantaneouscontactellipse,theprincipalcurvaturesanddirectionsatmeancontactpointM0forthepinionhead-cuttercanbedeterminedwiththelocalsynthesis.Thenthecutterpointradiusrpcanbedetermined.BasedonthepositionvectorsforthepointM0inthesystemsScandSm1,themachinetool-settingssr1,q1andXB1canbedetermined.Likewise,basedontheunitnormalforthepointM0inthesystemScandtheequationofmeshingbetweenthepinionhead-cutterandthepiniontobegenerated,thecuttingratiomp1canbedetermined.
2.7Determinationofmodifiedroll
Themodifiedrollmeansthecuttingrationotbeingconstantintheprocessofgeneration.Therotationangleofth
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