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英文资料翻译英文原文:DesignandmachiningofcylindricalcamswithtranslatingconicalfollowersByDerMinTsayandHsienMinWeiAsimpleapproachtotheprofiledeterminationandmachiningofcylindricalcamswithtranslatingconicalfollowersispresented.Onthebasisofthetheoryofenvelopesfora1-parameterfamilyofsurfaces,acamprofilewithatranslatingconicalfollowercanbeeasilydesignedoncethefollower-motionprogramhasbeengiven.Intheinvestigationofgeometriccharacteristics,itenablesthecontactlineandthepressureangletobeanalysedusingtheobtainedanalyticalprofileexpressions.Intheprocessofmachining,therequiredcutterpathisprovidedforataperedendmillcutter,whosesizemaybeidenticaltoorsmallerthanthatoftheconicalfollower.Anumericalexampleisgiventoillustratetheapplicationoftheprocedure.Keywords:cylindricalcams,envelopes,CAD/CAMAcylindricalcamisa3Dcamwhichdrivesitsfollowerinagroovecutontheperipheryofacylinder.Thefollower,whichiseithercylindricalorconical,maytranslateoroscillate.Thecamrotatesaboutitslongitudinalaxis,andtransmitsatransmitsatranslationoroscillationdisplacementtothefolloweratthesametime.Mechanismsofthistypehavelongbeenusedinmanydevices,suchaselevators,knittingmachines,packingmachines,andindexingrotarytables.Inderivingtheprofileofa3Dcam,variousmethodshaveused.Dhandeetal.1andChakrabortyanddhande2developedamethodtofindtheprofilesofplanarandspatialcams.Themethodusedisbasedontheconceptthatthecommonnormalvectorandtherelativevelocityvectorareorthogonaltoeachotheratthepointofcontactbetweenthecamandthefollowersurfaces.Borisov3proposedanapproachtotheproblemofdesigningcylindrical-cammechanismsbyacomputeralgorithm.Bythismethod,thecontourofacylindricalcamcanbeconsideredasadevelopedlinearsurface,andthereforethedesignproblemreducestooneoffindingthecentreandsideprofilesofthecamtrackonadevelopmentoftheeffectivecylinder.Instantaneousscrew-motiontheory4hasbeenappliedtothedesignofcammechanisms.Gonzalez-Palaciosetal.4usedthetheorytogeneratesurfacesofplanar,spherical,andspatialindexingcammechanismsinaunifiedframework.Gonzalez-PalaciosandAngeles5againusedthetheorytodeterminethesurfacegeometryofsphericalcam-oscillatingroller-followermechanisms.Consideringmachiningforcylindricalcamsbycylindricalcutterswhosesizesareidenticaltothoseofthefollowers,PapaioannouandKiritsis6proposedaprocedureforselectingthecutterstepbysolvingaconstrainedoptimizationproblem.Theresearchpresentedinthispapershowsqnew,easyprocedurefordeterminingthecylindrical-camprofileequationsandprovidingthecutterpathrequiredinthemachiningprocess.Thisisaccomplishedbythesueofthetheoryofenvelopesfora1-parameterfamilyofsurfacesdescribedinparametricform7todefinethecamprofiles.HansonandChurchill8introducedthetheoryofenvelopesfora1-parameterfamilyofplanecurvesinimplicitformtodeterminetheequationsofplate-camprofilesChanandPisano9extendedtheenvelopetheoryforthegeometryofplatecamstoirregular-surfacefollowersystems.Theyderivedananalyticaldescriptionofcamprofilesforgeneralcam-followersystems,andgaveanexampletodemonstratethemethodinnumericalform.Usingthetheoryofenvelopesfora2-parameterfamilyofsurfacesinimplicitform,TsayandHwang10obtainedtheprofileequationsofcamoids.Accordingtothemethod,theprofileofacamisregardedasanenvelopeforthefamilyofthefollowershapesindifferentcam-followerpositionswhenthecamrotatesforacompletecycle.THEORYOFENVELPOESFOR1-PARAMETERFAMILYOFSURFACESINPARAMETRICFORMIn3DxyzCartesianspace,a1-parameterfamilyofsurfacescanbegiveninparametricformas(1)whereζistheparameterofthefamily,andu1,u2,aretheparametersforaparticularsurfaceofthefamily.Then,theenvelopeforthefamilydescribedinEquation1satisfiesequation1andthefollowingEquation:(2)wheretheright-handsideisaconstantzero7.Litvinshowedtheprovingprocessofthetheoremindetail.IfwecansolveEquation2andsubstituteintoequation1toeliminateoneofthethreeparametersu1,u2,andζ,wemayobtaintheenvelopeinparametricform.However,oneimportantthingshouldbepointedouthere.Equations1and2canalsobesatisfiedbythesingularpointsofsurfacesdescribedbelowIthefamily,eveniftheydonotbelongtotheenvelope.PointswhichareregularpointsofsurfacesofthefamilyandsatisfyEquation2lieontheenvelope.Theconditionforthesingularpointsofasurfaceisdiscussedhere..aparametricrepresentationofasurfaceis(3)whereu1andu2aretheparametersofthesurface.Apointofthesurfacethatcorrespondstoinagivenparameterizationiscalledasingularpointoftheparameterization.Apointofasurfaceiscalledsingularifitissingularforeveryparameterizationofthesurface7.Apointthatissingularinoneparameterizationofasurfacemaynotbesingularinotherparameterizations.Forafixedvalueofζ,equations1and2represent,ingeneral,acurveonthesurfacewhichcorrespondstothisvalueoftheparameter.Ifthisisnotalineofsingularpoints,thecurveslsoliesontheenvelope.Thesurfaceandtheenvelopearetangenttoeachotheralongthiscurve.Suchcurvesarecalledcharacteristiclinesofthefamily7.theycanbeusedtofindthecontactlinesbetweenthesurfacesofthecylindricalcamandthefollower.THEORYOFENVELOPESFORDETERMINATIONOFCYLINDRICAL-CAMPROFILESOnthebasisofthetheoryofenvelopes,theprofileofacylindricalcamcanberegardedastheenvelopeofthefamilyoffollowersurfacesinrelativepositionsbetweenthecylindricalcamandthefollowerwhilethemotionofcamproceeds.Insuchacondition,theinputparametersofthecylindricalcamserveasthefamilyparameters.Becausethecylindricalorconicalfollowersurfacecanbeexpressedinparametricformwithoutdifficulty,thetheoryofenvelopesfora1-parameterofsurfacesrepresentedinparametricform(seeequations1and2)isusedindeterminingtheanalyticalequationsofcylindrical-camprofiles.Asstatedinthelastsection,acheckforsingularpointsonthefollowersurfaceisalwaysneeded.Figure1ashowsacylindrical-cammechanismwithatranslatingconicalfollower.Theaxiswhichthefollowertranslatesalongisparalleltotheaxisofrotationofthecylindricalcam.aistheoffset,thatis,thenormaldistancebetweenthelongitudinalaxisofthecamandthatofthefollower.RandLaretheradiusandtheaxiallengthofthecam,respectively.TherotationangleofthecylindricalcamisФ2aboutitsaxis.Thedistancetraveledbythefolloweriss1,whichisafunctionofparameterФ2,asfollows:(4)Thedisplacementrelationship(seeequation4)forthetranslatingfollowerisassumedtobegiven.Infigure1b,therelativepositionofthefollowerwhenthefollowermovesisshown.Thefollowerisintheformofafrustumofacone.Thesemiconeangleisα,andthesmallestradiusisr.δ1istheheight,andμisthenormaldistancefromthexzplanetothebaseofthecone.ThefixedcoordinatesystemOxyzislocatedinsuchawaythatthezaxisisalongtherotationaxisofthecam,andtheyaxisisparalleltothelongitudinalaxisoftheconicalfollower.theunitvectorsofthexaxis,yaxisandzaxisarei,jandk,respectively.Bytheuseoftheenvelopetechniquetogeneratethecylindrical-camprofile,thecamisassumedtobestationary.Thefollowerrotatesaboutthedamaxisintheoppositedirection.ItisassumedthatthefollowerrotatesthroughanangleФ2abouttheaxis.Atthesametime,thefolloweristransmittedalineardisplacements1bythecam,asshowninFigure1b.Consequentlyusingthetechnique,ifweintroduceθandδastwoparametersforthefollowersurface,thefamilyofthefollowersurfacescanbedescribedas(5)where0≤θ<2piAndф2istheindependentparameterofthecammotion.Referringtotheoryofenvelopesforsurfacesrepresentedinparametricform(seeequations1and2),weproceedwiththesolvingprocessbyfinding(6)Therearenosingularpointsonthefamilyofsurfaces,since(r+δtanα)>0inactualapplications.Theprofileequationsatisfiesequation5andthefollowingequation:(7)Whereor(8)WhereSubstitutingequation8intoequation5,andeliminatingθ,weobtaintheprofileequationofthecylindricalcamwithatranslatingconicalfollower,anddenoteitas(9)Asshowninequation8,θisafunctionoftheselectedfollower-motionprogramandthedimensionalparameters.Asaconsequence,thecylindrical-comprofilecanbecontrolledbythechosenfollower-motioncurvesandthedimensionalparameters.Twovaluesofθcorrespondtothetwogroovewallsofthecylindricalcam.Nowtheprofileofthecylindricalcamwithatranslatingconicalfollowerisderivedbythenewproposedmethod.Asstatedabove,Dhandeetal.1andChakrabortyandDhande2havederivedtheprofileequationofthesametypeofcambythemethodofcontactpoints.Acomparisonoftheresultiscarriedouthere.Sincethesamefixedcoordinatesystemandsymbolsareused,onecaneasilyseethattheprofileequationisidenticalalthoughthemethodsusedaredifferent.Moreover,wefindthattheprocessoffindingthecamprofileissignificantlyreducedbythismethod.CONTACTLINEAteverymoment,thecylindricalcamtouchesthefolloweralongspacelines.Thecontactlinesbetweenthecylindricalcamanditsfollowerarediscussedinthissection.Theconceptofcharacteristiclinesinthetheoryofenvelopesfora1-parameterfamilyofsurfacesmentionedabovecouldbeappliedtofindingthecontactlinesinacylindricalcom.TheprofileofacylindricalcamwithatranslatingconicalfollowerisgivenbyEquation9.Then,thecontactlineataspecificvalueofф2,sayф20,is(10)Where,inEquation10,thevalueofθisafunctionofδdefinedbyEquation8.ThecontactlinesbetweenthesurfacesofthecamandthefollowerateachmomentisdeterminedbyEquation10.weseethattherelationshipbetweenthetwoparametersθandδofthefollowersurfaceisgivenbyEquation8,anonlinearfunction.Thus,onecaneasilyfindthatthecontactlineisnotalwaysastraightlineontheconicalfollowersurface.PRESSUREANGLETheanglethatthecommonnormalvectorofthecamandthefollowermakeswiththepathofthefolloweriscalledthepressureangle12.thepressureanglemustbeconsideredwhendesigningacam,anditisameasureoftheinstantaneousforce-transmissionpropertiesofthemechanism13.Themagnitudeofthepressureangleinsuchacam-followersystemaffectstheefficiencyofthecam.Thesmallerthepressureangleis,thehigheritsefficiencybecause14.Infigure2,theunitnormalvectorwhichpassesthroughthepointofcontactbetweenthecylindricalcamandthetranslatingconicalfollowerintheinversionposition,i.e.pointC,isdenotedbyn.Thepathofthefollowerlabeledastheunitvectorpisparalleltotheaxisofthefollower.fromthedefinition,thepressureangleΨistheanglebetweentheunitvectorsnandp.Since,atthepointofcontact,theenvelopeandthesurfaceofthefamilypossessesthesametangentplane,theunitnormalofthecylindrical-camsurfaceisthesameasthatofthefollowersurface.ReferringtothefamilyequationEquation5andFigure2,wecanobtaintheunitvectoras(11)wherethevalueofθisgivenbyequation8,andtheunitvectorofthefollowerpathis(12)Bytheuseoftheirinnerproduct,thepressureangleΨcanbeobtainedbythefollowingequation:(13)ThepressureanglederivedhereisidenticaltothatusedintheearlyworkcarriedoutbyChakrabortyandDhande2.CUTTERPATHInthissection,thecutterpathrequiredformachiningthecylindricalcamwithatranslatingconicalfollowerisfoundbyapplyingtheproceduredescribedbelow.Usually,withtheconsiderationsofdimensionalaccuracyandsurfacefinish,themostconvenientwaytomachineacylindricalcamistouseacutterwhosesizeisidenticaltothatoftheconicalroller.Intheprocessofmachining,thecylindricalblankisheldonarotarytableofa4-axismillingmachine.Asthetablerotates,thecutter,simulatingthegivenfollower-motionprogram,movesparalleltotheaxisofthecylindricalblank.Thusthecuttermovesalongtheruledsurfacegeneratedbythefolloweraxis,andthecamsurfaceisthenmachinedalongthecontactlinesstepbystep.Ifwehavenocutterofthesameshape,anavailablecutterofasmallersizecouldalsobesuedtogeneratethecamsurface.Underthecircumstances,thecutterpathmustbefoundforageneralendmillcutter.Figure3showsataperedendmillcuttermachiningacurvedsurface.Thefrontportionofthetoolisintheformofacone.ThesmallestradiusisR,andthesemiconeangleisβ.Ifthecuttermovesalongacurveδ=δ0onthesurfaceX=X(δ,ф2),theangleσbetweentheunitvectorofthecutteraxisaxandtheunitcommonnormalvectornatcontactpointCisdeterminedby(14)Thusthepathofthepointόonthecutteraxisthatthevectornpassesthroughis(15)andthetipcentreTfollowsthepath(16)Figure4showsataperedendmillcuttermachiningthegroovewallofacylindricalcam.Theaxisofthetaperedendmillisparalleltotheyaxis.Notethatthetwoconditions (17)(18)forthegeometricparametersofthecutterandtherollerfollowermusthold,orotherwisethecutterwouldnotfitthegroove.Theunitvectorofthecutteraxisis(19)Fortheprofileofthecylindricalcamwithatranslatingconicalfollowergivenbyequation9,theangleσisdeterminedbytheinnerproduct:(20)Thus,byusingtheresultsobtainedearlier,thepositionofthetipcentreofthecuttercanbederivedas(21)whereNUMERICALEXAMPLETheproceduresdevelopedareappliedinthissectiontodeterminethecylindrical-camprofile,andtoanalyseitscharacteristics.Themotionprogramofthefollowerforthecylindricalcamwithatranslatingcylindricalcamisgivenas(22)wherehandλaretwoconstants.Andh=20unitsandλ=60℃.Themotionprogramisadwell-rise-dwell-return-dwellcurve,andtheriseandreturnportionsarecycloidalcurves15.Figure5showsthemotionprogram.Thedimensionalparametersusedforthecylindricalcamandthefollowerareasfollows:semiconeangleoffollowerα=0℃heightoffollowerδ1=15unitsdistancefrombottomoffollowertoxzplaneμ=55unitssmallestradiusoffollowerr=7.5unitsoffseta=20unitsradiusofcamR=73unitsaxiallengthofcamL=100unitsTheprofileofthecylindricalcamobtainedbyapplyingEquation9isshowninFigure6.InFigure6,thegroovewallwiththesmallerzcoordinatesissideⅠ,andtheotherissideⅡ.ThevariationsofthepressureanglesfortheriseandreturnportionsareshowninFigures7and8forsideⅠandⅡ,respectively.Itcanbeseenthatthepressureanglesforbothsideshappentobeidentical.CONCLUSIONSAshasbeenshownabove,theapplicationofthetheoryofenvelopesaffordsaconvenientandversatiletoolfordeterminingthecylinder-camprofileswithtranslatingconicalfollowers.Bymeansoftheanalyticalcamprofileequations,itcanbeeasilyextendedtoaccomplishthetaskfortheanalysisofthecontactlineandthepressureangle.Further,thecutterpathrequiredintheprocessofmachiningisgeneratedfortaperedendmillcutters.Sincethesamefixedcoordinatesystemandsymbolsareusedinthisstudy,onecanseethattheresultsforcamprofilesandpressureanglesareidenticaltothoseobtainedinpreviousresearch1,2.Onlyonecoordinatesystemisusedinthisapproach.Asaresult,theprocessofderivationissimple.Workiscurrentlyunderwaytofacilitatetheimplementationofthetoolpathforthemachiningofthecylindricalcamonanumericallycontrolledmillingmachine.翻译:MACROBUTTONMTEditEquationSection2SEQMTEqn\r\hSEQMTSec\r1\hSEQMTChap\r1\h圆柱凸轮的设计和加工有人提出了具有平移圆锥传动件的圆柱凸轮的轮廓确定及其机加工的简单方法.在单参数曲面族的包络线理论的基础上,给定从动件运动规律的具有平移圆锥传动件的圆柱凸轮的轮廓的设计是很简单的.通过这种设计方法得到的轮廓曲线可以进行凸轮切线和压力角等几何特征的分析研究.在机加工过程中,可以使用锥形端铣刀,它的尺寸小于等于圆锥传动件的尺寸.很多实例证明该方法的实用性.关键词:圆柱凸轮,包络线,计算机辅助设计和计算机辅助制造.圆柱凸轮是利用其圆周上的沟槽来驱动传动件的空间凸轮.传动件是圆柱或者圆锥形状的,可以做平行移动也可以做摆动.凸轮绕着它的纵向轴线旋转,同时将平移或摆动运动传递给传动件.这种机械原理长期广泛应用在各种设备中,比如,运输机,纺织机,包装机,旋转分度盘等等.为获得三维凸轮的轮廓曲线,曾用过各种方法.DHANDE和CHAKRABORTY和DHANDE发明了确定平面和立体凸轮轮廓的一种方法.这种方法是在一个前提下使用的,即认为在主动轮和从动件交点处,凸轮的径向矢量和速度矢量二者相互垂直.BROISOV提出了借助计算机辅助计算的方法来解决圆柱凸轮机构设计上的问题.通过这种方法,可以把圆柱凸轮的轮廓考虑成为展开的线性曲面.这样,设计时就只需在实际圆柱上找到凸轮轨迹的中心和轮廓边缘.瞬间螺旋运动理论已经应用到凸轮机构的设计中.GONZALEZ-PALACIOS在统一标准下应用这种理论得到了平面,球面和柱面凸轮机构.GONZALEZ-PALACIOS和ANGLES又应用这个理论确定了球面摆动辊子凸轮机构的几何形状.考虑到用与传动件同样尺寸的圆柱刀具加工圆柱凸轮,PAPAIOANNOU和KIRITSIS提出了通过解决最优化受限问题来选择刀具步距的程序.在这份研究报告提出了一个新的简单的程序来确定圆柱凸轮的轮廓方程并提供机加工过程中所要求的刀具路径.它是通过应用以参数形式描述的单参数曲面族的包络线理论来完成的.Hanson和Churchill引用隐函数形式的单参数平面曲线族包络线理论确定盘形凸轮的轮廓曲线方程。Chan和Pisano将这种盘形凸轮几何轮廓包络线理论扩展运用到非规则曲面的传动零件系统中,他们创建了普通凸轮伟动系统中凸轮轮廓的解析法描述,并举例证明了该方法适用于数字形式。Tsay和Hwang将这种包络线理论应用到隐函数形式的双参数曲面族上,建立了它们的轮廓线方程。根据这种方法,凸轮的轮廓曲线被看作是,当凸轮作圆周回转时传动件在不同位置,其轮廓的包络线。单参数曲面族的包络线理论在三维笛卡尔坐标系中,单参数曲面族可以用下面的公式来表示: 其中是曲面族的参数,是曲面族中特定曲面的参数。这样,方程(1.1)所描述的曲面族的包络线即满足方程1.1又满足下面的方程: 其中,方程右边是常数0,Litvin对这个定理进行了详细的验证。如果我们能够解出方程1.2,并把结果代入方程中就能消去,中的一个参数,我们就可以得到参数形式的包络线方程。然而还有重要的一点要指出,方程1.1和方程1.2也能够被曲面族外的异常点面非包络线上的点所满足,只有曲面族上有规律的,满足方程的点才位于包络线上。现在讨论奇点出现的条件。曲面的参数形式表达式为 其中,是曲面的参数。满足下面方程的点即为异常点:。如果一个奇点在曲面各种参数形式下都是奇点则称该奇点为曲面的奇点。在一种曲面参数形式下为奇点,在其它形式下却不一定是。为了确定的值,一般地,方程1.1和方程1.2代表曲面上的一条曲线,这些曲面对应于参数的值。如果曲面不包含奇点,那么该曲线就在包络线上。曲面和包络线与这条曲线相切。这些曲线称为曲面族的特征线。利用它们可以找到圆柱凸轮曲面与传动件的接触线。确定圆柱凸轮轮廓的包络线理论。在包络线理论的基础上,圆柱凸轮的轮廓可以被看作是在凸轮运动过程中,传动件曲面在圆柱凸轮表面与传动件之间的位置时的传动件曲面的包络线。在这种条件下,输入的圆柱凸轮参数作为曲面族的参数,因为圆柱或圆锥传动件曲面可以很容易地以参数形式表示出来,以参数形式表示(见方程1.1和1.2)的单参数曲面族的包络线理论用来确定圆柱凸轮轮廓的解析方程。正如上面最后一部分所述,传动件曲面上奇点的检验总是必要的。图a表示的是带有平移圆锥传动件的圆柱凸轮机构。传动件移动轨迹的轴线与圆柱凸轮轴线重合,a是凸传动件纵向轴线间距离的偏移值。R和L是凸轮的半径和长度,是绕轴旋转的角度,传动行程,是的函数,它们之间关系为 它们之间的替代关系通常是给定的。图b表示传动件移动时与凸轮二者的位置关系。传动件是圆锥截体圆锥顶角一半为,最小半径为,是高,是从面到圆锥顶点的距离。以凸轮旋转轴线为z轴,y轴平行于圆锥传动件轴线建立0xyz坐标系。Xyz轴的单位方向矢量分别为i,j,k.通常在静止状态下,利用包络线技术来得到圆柱凸轮轮廓。传动件反方向绕凸轮轴线旋转。传动件绕z轴转过角。同时,传动件距凸轮线性偏移s,如图b所示。因此,应用这种理论,如果我们引进和两个参数,传动件曲面族可以用下面公式来描述: 其中,,是与凸轮运动有关的独立参数。参考如方程1.1和1.2所描述的参数形式的曲面的包络线理论通过下面这个方程我们继续解决这个问题。 实际应用中,如果,在这些曲面族中就不会出现奇点。轮廓方程满足方程.5和下面的方程: 其中,或 其中将方程1.8代入方程1.5中,消去,我们得到带有平移圆锥凸轮的圆柱凸轮的轮廓方程, 如方程1.8所示,的功能是用来选定传动件运动过程和作为尺寸参数,结果,圆柱凸轮轮廓可以通过选择传动件运动曲线和尺寸参数来控制,的两个值同圆柱凸轮的两个螺旋角一样。现在,带有平移圆锥传动件的圆锥传动件的圆柱凸轮的轮廓可以利用这种新的方法设计出来。如上面所述,Dhandeetal.和Chakraborty,Dhande通过这种相关联通点的方法推导出了同类型凸轮轮廓的方程,在这,对这一结果进行了毕较。自从这种方法应用以来,尽管方法不同,但大家却可以很容易得到同样的轮廓方程,而且,我们发现使用这种方法来确定轮廓方程的过程大简化了。相交线:任何时候,圆柱凸轮与它的传动件相交线为空间曲线时它们之间的相交线是必需要讨论的。上文提到的单参数曲面族的包络线理论中的特征线观点可以应用到这里来确定相交线。凸轮方程向方程1.9一样,那么,当取具体值时,相交线方程为: 其中,方程1.10中,的值和方程1.8中定义的的功能是一样的。凸轮和传动件在任意时刻的表面交线是同方程1.10来确定的。我们知道传动件表面参数和之间的关系是由非线性方程1.8给出的。因此,大家可以很容易的发现圆锥传动件表面的交线并不总是一条直线。压力角凸轮和其传动件的公法线与传动件的轨迹所成的角叫压力角。设计时必须考虑压力角,它们衡量机构是否恰当进行连续力传递的参数,在这种凸轮传动系统中压力角的大小影响系统的效率,压力角越小,效率越高。如图2,圆柱凸轮和它的圆锥传动件的单位法向量通过二者的交线上的一点,例如C点,以n来表示。传动件的路径用单位向量p表示,p平行于传动件的轴线,从压力角定义可知,矢量n和p所夹的角就是压力角。在交线处,既然系包络线和曲面有共切面,那么,圆柱凸轮表面的法线和传动件表面法线是共线的。参照方程1.5和图2,可以得到单位法矢量如下: 其中,的值由方程1.8给定,传动件的单位速度矢量为: MACROBUTTONMTPlaceRefSEQMTEqn\h(SEQMTSec\c

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