Chapter7DesignofGearMechanisms1.ParallelShaftGears(1)SpurgearsTheteetharestraightandparalleltothegearaxis.Thisgearisspurgear.Ifthegearshaveexternalteethontheoutersurfaceofthecylinders,theyrotateintheoppositedirection,asshowninFig7-1a.Aninternalgearcanmeshwithaexternalpiniononlyandtheyrotateinthesamedirection,asshowninFig7-1b.Thepinionandrackcombinationconvertsrotarymotionintotranslationalmotionorviceversa,asshowninFig7-1c.

7.1ClassificationofGearMechanismsFig.7-1Spurgears(直齒圓柱齒輪)Fig.7-2Helicalgears(斜齒圓柱齒輪)(2)HelicalgearsLikespurgears,helicalgearscanalsobeusedtoconnectparallelshafts,buttheirteetharenotparalleltotheirshaftsaxes,eachbeinghelicalinshape.Twomeshinggearshavethesamehelixangle,buthaveteethofoppositehands,asshowninFig7-2.Helicalgearcanalsobeclassifiedasexternalcontact,internalcontact,helicalpinionandrack.Fig.7-3Herringbonegears(人字齒輪)(3)HerringbonegearsHerringbonegearsareequivalenttoapairofhelicalgearswithoppositehelixanglesmountedsidebyside,asshowninFig7-3.Theaxialthrustforcesofthetworowsofteethcanceleachother.Sotheycanbeusedathighspeedswithlessnoiseandvibrations.2.SpatialGears(1)IntersectingshaftgearsIftheteethareformedontheconesurface,thisgeariscalledabevelgear.Iftheteetharestraightandcoincidentwiththeirconeelements,thisgeariscalledstraightteethbevelgearshowninFig7-4a.Whentheteethofbevelgearareinclinedatanangletothefaceofbevel,thisgearisknownashelicalbevelgearorspiralbevelgear,asshowninFig7-4b.Thecurvedteethbevelgearhasteeththatarecurved,butwithazerodegreespiralangle,asshowninFig7-4c.

Fig.7-4Bevelgears(錐齒輪)Fig.7-6Wormgears(蝸輪蝸桿)(2)SkewshaftgearsTotransmitmotionbetweentheskewshafts,thespiralgearsorcrossedhelicalgears,wormgearsandHypoidgearsareoftenusedinmachines.Thespiralgearsorcrossedhelicalgearsarelimitedtohighload,andtheirshaftscanbesetatanyangle,asshowninFig7-5.

(3)WormgearsFig7-6showswormgearsinwhichafewteethofthesmallergeararewrappedarounditscircumferenceanumberoftimesandformscrewthreads,andthelargerwheelhasconcaveshapeinitsdiameterdirectionsothattoenvelopetheportionofthesmallergear.Thesmallergeariscalledworm;thelargergeariscalledwormgear.Fig.7-5Crossedhelical

gears(交錯軸斜齒輪)1.LawofGearingFig.7-7Fundamentallawofgearing(齒廓嚙合的基本定律)7.2FundamentalLawofGearing[Duringmeshingofgears,theprofileofatoothongear1ismeshingatpointKwithanotherprofileonthatgear2toproducearotarymotion.Fig7-7showstwocontactingprofileswithcommonnormaln—n,andthisnormalintersectsthelineofthetwogearcentersO1O2atpointP.

Ifthepitchpointvariesforallphaseofthegearing,theratioisnotaconstant.Thistypeofgeariscallednoncirculargears.Fig7-8showsapairofnoncirculargears.Fig.7-8Noncirculargears(非圓齒輪)2.ConjugateProfilesWhenallthecommonnormallinesforeveryinstantaneouspointofcontactcanpassthroughthepitchpoint,thisisthefundamentallawofgearing;anymeshingprofileswhichsatisfythelawofgearingcanbecalledtheconjugateprofiles.

Inordertomaintainthefundamentallawofgearingtobetrue,thegeartoothprofilesonmeshinggearsmustbeconjugateofoneanother.Thereareinfinitenumbersofpossibleconjugatepairsthatcanbeused,butonlyafewcurveshavebeenpracticallyappliedasgearteeth.Thecycloidprofileisstillusedinwatchesandclocksasatoothform,butmostothergearsusetheinvolutecurvefortheirshape.1.Involute(1)DevelopmentoftheinvoluteFig7-9showsaninvolutegeneratedbyalinerollingonthecircumferenceofacirclewithcenteratO;Whenthelinerolls,thepathgeneratedbythepointKonthelineistheinvolutecurve.

7.3InvolutePropertiesandInvoluteToothProfilesFig.7-9Developmentofthe

involute(漸開線的形成)(2)Involuteproperties1)Thelengthofthegeneratinglineisequaltothearclengthwhichthegeneratinglinerollswithoutslippingonthebasecircle.2)Thegeneratinglineisalwaystangenttothebasecircle,anditisalwaysanormaloftheinvoluteatapointK.3)ThelengthNKisthecurvatureradiusoftheinvoluteatpointK.PointNisthecenterofthecurvatureoftheinvoluteatpointK.

4)Theshapeoftheinvolutedependsupontheradiusofthebasecircle.Thesmallertheradiusofthebasecircleis,thesteepertheinvoluteis.Iftheradiusofthebasecircleisinfinite,theinvolutecurvebecomesastraightline.ThisisshowninFig7-10.Fig.7-10Shapeofinvolutes

andradiusofbasecircles

(漸開線的形狀與基圓半徑)2.MeshingofInvoluteProfiles(1)TheinstantaneousangularvelocityratioisconstantFig7-11showstwobasecircleswithcentersatO1,O2andradiirb1,rb2.

(2)SeparabilityofthecenterdistanceAnychangeincenterdistancewillhavenoeffectupontheinvoluteprofiles;thegearratiocannotbevaried.

(3)StationaryactionforceThelineofactionincaseofinvoluteteethisalongthecommonnormalatthecontactpoint,andthecommonnormalisthecommontangentofthebasecircles.Fig.7-11Gearingofinvoluteprofiles

(漸開線齒廓的嚙合)Fig.7-12Spurgearnomenclatures(漸開線直齒圓柱齒輪名詞術語)7.4NomenclaturesofStandardSpurGearandGearSizes1.GearTeethNomenclatures(1)AddendumcircleItisacirclepassingthroughthetopesoftheteeth.Theradiusanddiameteraredenotedasraandda.(2)DedendumcircleItisacirclepassingthroughtherootsoftheteeth.Theradiusanddiameteraredenotedasrfanddf.

(3)ReferencecircleItisadatumcircleingeardesignandmeasurement.Theradiusanddiameteraredenotedasrandd.Notethatthesubscriptwillbeomittedwhenexpressingthedimensionsofagear,suchasr,d,p,etc.(4)BasecircleItisacirclegeneratingtheinvolutecurves.Theradiusanddiameteraredenotedasrbanddb.(5)Tooththickness,spacewidthandcircularpitchThetooththicknessisthethicknessofthetoothmeasuredalongthecircumference,suchassK.Thespacewidthisthespacebetweentheadjacentteethmeasuredalongthecircumference,suchaseK(6)Addendum,dedendumandfulldepthTheaddendumistheradialheightfromthereferencecircletotheaddendumcircle,denotedasha.Thededendumistheradialheightfromthereferencecircletotherootcircle,denotedashf.Thefulldepthistheradialdistancebetweentheaddendumcircleandtherootcircle,denotedashf.(7)NormalpitchItisthecircularpitchmeasuredalongtheirnormal,denotedaspb.Itisequaltothecorrespondingpitchonthebasecircle.(8)FacewidthItisthelengthofthetoothparalleltothegearaxis,denotedasB.2.ParametersofInvoluteGear(1)NumberofteethItisthetotalnumberoftheteeththegearpossesses,anditisalwaysaninteger,denotedasz.

(2)ModuleThelengthofcircumferenceofthereferencecircleisequaltothesumofthenumberofthecircularpitchesonthereferencecircle,sowehave:πd=pz.(3)PressureangleThedefinitionofpressureanglehasbeendemonstratedandherewe

Fig.7-13Gearsizewithsamenumber

ofteethanddifferentmoduleofteeth

(同齒數、不同模數齒輪尺寸)emphasizethattherearedifferentanglesonthedifferentcircumferencesofthegear.

(4)CoefficientofaddendumAstandardvalueoftheaddendumisha=h*am.h*aiscalledthecoefficientofaddendum,usually,h*a=1,fornormalteeth;h*a=0.8,forshorterteeth.

(5)Coefficientofclearance3.GeometricalSizesofStandardSpurGears

Ifagearhasthestandardmoduleandpressureangleonthereferencecircle,standardaddendumanddedendum,alsothetooththicknessisequaltothespacewidthonthereferencecircle,itiscalledthestandardgear.

Tab7-2showstheformulaofcalculatingthesizesofstandardgears.4.ToothThicknessAlonganArbitraryCircle

Ingeardesignandmanufacturing,thetooththicknessonthearbitrarycircle,suchastooththicknessonthegeartop,sometimesmaybenecessary.Fig7-14showsageartooththathasthicknesssKatradicallocationrK.Fig.7-14Tooththicknessalonganarbitrarycircle(任意圓齒厚)5.TerminologyforInternalGears

Aninternalgearhasitsteethcutontheinsideoftherimratherthanontheoutside,andithasconcavetoothprofiles,whilethetoothprofilesoftheexternalgearareconvex.Fig7-15showsatypicalinternalgear,andthefollowingsaredifferentfromtheexternalgear.Fig.7-15Terminologyforinternal

gears(漸開線內齒圓柱齒輪術語)6.TerminologyforaRackArackisaportionofagearhavinganinfinitebasediameter,thusitsreferencecircle,addendumcircle,dedendumcircleareallstraightlines.Theinvoluteprofileoftherackbecomesastraightlineandisperpendiculartothelineofaction,seetheFig7-16.

Fig.7-16Terminologyforarack(齒條術語)Thecharacteristicsofarackareasfollows.1)Theprofilesareallskewlines,andtheyareparallelonthesamesideoftheteeth.Thepressureanglesatdifferentpointontheprofileareallthesameandtheyareequaltothenominalpressureangleof20°.2)Thepitchremainsunchangeableonthereferenceline,addendumline,andsoon.Itsvalueisp=πm;thebasepitchispb=πmcosα.3)Theaddendumanddedendumarethesamewiththeexternalgears.

1.ConditionsofCorrectlyMeshingforInvoluteGears7.5MeshingDriveofStandardSpurGearsFig.7-17Meshingofteeth(齒輪嚙合)Gearstransmitmotionbymeansofsuccessivelyengagingteeth,butnotbothgearsaretobemeshedtogethercorrectly.Fig7-17showsapairofmeshinggearsinwhichallthecontactpointsbetweenthetwogearswiththeinvoluteprofilesmustlieonthelineofaction,sothatthepitchpointremainfixed.2.ConditionsofContinuousTransmissionofGears(1)MeshingprocessofapairofgearsInFig-7-18,twogears1and2withrotatingcentersatO1andO2respectivelyareincontactatpointB2andK.

(2)Conditionsofcontinuoustransmissionofgears

Fig7-18showsthatthelaterpairofteethisjustcomingintocontactattheinitialpointB2andthepreviouspairofteethisincontactatthepointK,andthecontactwillnotyethavereachedfinalpointB1.Thus,forashorttimetherewillbetwopairsofteethincontact,thecontinuoustransmissionissatisfied.

Fig.7-18Meshingprocessofteeth(輪齒的嚙合過程)1)Contactratioforexternalspurgears.Fig7-19showsapairofmeshinggearshavinginvoluteteeth.ThelengthB2B1canbecalculatedfromthefollowingrelationship.Fig.7-19Contactratioforexternal

gears(外嚙合齒輪重合度計算)(3)ValueofcontactratioRearrangingtheaboveequations,weobtain:2)Contactratioforinternalspurgears.ThecontactratioforinternalgearisillustratedinFig7-20.Inthesamemethod,wecanobtainthefollowingformula.Fig.7-20Contactratioforinternalgear

(內嚙合齒輪重合度計算)3)Contactratioforapinionandarack.ThecontactratioforapinionandarackisillustratedinFig7-21.WherePB1isasthesameasbeforeandPB2dependsupontherelativepositionofthepinionandtherack.Ifthepinionmesheswitharackwithoutchangingthedistanceofthecenter,thatistosay,thereferencelineoftherackistangenttothereferencecircleofthepinion,thelengthofPB2isasfollows:Fig.7-21Contactratioforapinionand

arack(齒輪齒條嚙合的重合度計算)Fig.7-22Natureofteethaction(重合度的意義)Thecontactratiomeanstheaveragenumberofpairsofteethwhichareincontactusuallyisnotaninteger.Iftheratiois12,asshowninFig7-22,itdoesnotmeanthatthereare12pairsofteethincontact.Itmeansthattherearealternatelyonepairandtwopairsofteethincontact,oronepairofteethisalwaysincontact,andtwopairsofgearsareincontact20percentoftimes.FromFig7-22,weknowthattherearetwopairsofteethincontactonthesegmentsofB2K′andKB1,andonthesegmentKK′onlyonepairofteethisincontact.3.RelativeSlideBetweenContactTeeth

FromFig7-22weknowthattheworkingprofileofgear1isfrompointB2toitstoothtop,andtheworkingprofileofgear2isfrompointB1toitstoothtop.Whenthepairofteethcontactsatanarbitrarypoint,suchaspointK′,theslidewilloccurbetweentheteethsurfacesalongtheirtangentdirection.ThisisbecausethatthevelocityofpointK′isnotthesameonthetwogears.4.CenterDistanceofGearsWhentheshaftsofapairofgearsaremountedcorrectly,thefollowingconditionsmustbesatisfied.1)Theradicalclearancebetweentheaddendumcircleofagearandthededendumcircleofthemeshinggearmustbethestandardvalue,thatisc=c*m;seetheFig7-23.

Fig.7-23Normalcenterdistance(無側隙嚙合的徑向間隙)2)Thebacklashmustbezerotheoretically;thetooththicknessonthepitchcircleisequaltothespacewidthonthepitchcircleofthemeshinggear.5.PinionandRackExample7-1Twoinvolutegearsinmeshhaveamoduleof2.5mmandapressureangleof20°.Thenumbersofteetharez1=22,z2=33.Thecoefficientofaddendumish*a=1,andc*=0.25.Findthefollowings:1)Sizesofthetwogears;2)Contactratio;3)Ifthecenterdistanceisincreased1mm,findthecontactratio.

Fig.7-24Meshingofapinionandarack(齒輪與齒條嚙合)1.GearTeethForming(1)FormingCuttingProbablytheoldestmethodofcuttinggearteethismilling.Aformmillingcuttercorrespondingtotheshapeofthetoothspaceisusedtocutonetoothspaceatatime,afterthegearisindexedthroughonecircularpitchtothenextposition.Therearetwokindsofmillingcutter:oneisthedisccuttershowninFig7-25a;theotheristhefingercuttershowninFig7-25b.

7.6FormingandUndercuttingofGearTeethFig.7-25Formingcutting(仿形加工)(2)GeneratingCuttingIngenerating,atoolhavingashapedifferentfromthetoothprofileismovedrelativetothegearblanktoobtainthepropertoothshape.Themostcommonmethodsofgeneratinggearteethareshapingmethodandhobbingmethod.1)Shapingteeth.Shapingisahighlyfavoredmethodofgeneratingteethofgear.Thecuttingtoolusedintheshapingmethodiseitherarackcutterorapinioncutter.Fig7-26showsshapingteethwithapinioncutter;Fig7-27showsshapingteethwitharackcutter.Fig.7-26Shapingteethwith

apinioncutter(齒輪插刀)Fig.7-27Shapingteethwith

arackcutter(齒條插刀)Fig.7-28Hobbingteeth(滾齒加工)2)Hobbingteeth.Fig7-28illustratesthegeneratingprocesswithahob.Ahobisacylindricalcutterwithoneormorehelicalthreadsquitelikeascrewthreadtap,andhasstraightsideslikearack.Thehobandtheblankarerotatecontinuouslyattheproperangularvelocityratio,andthehobisthenfedslowlyacrossthefaceoftheblankfromoneendofteethtotheother.Allteethhavebeencut.Fig.7-29Undercutting(根切現象)2.Undercutting

Whengearteethareproducedbyageneratingprocess,thetopofcuttingtoolremovestheportionoftheinvoluteprofileneartherootteeth.Thisiscalledundercutting.TheundercuttingweakensthetoothbyremovingmaterialatitsrootshowninFig7-29.Severeundercuttingwillpromoteearlytoothfailure.Itmayalsoreducethelengthofcontactandresultinrougherandnoisiergearaction.Inmachinedesign,theundercuttingmustbeavoidedoreliminatedbythedesigner.Fig.7-30Rackcutterprofile

(齒條插刀的齒廓)(1)CausationofundercuttingThedifferencebetweentherackcutterandtherackisthattheaddendumoftherackcutterisgreaterthanthatoftherack,adistancec*m,asshowninFig7-30.Fig.7-31Undercuttingprocess(根切的形成)Fig7-31illustratesageneratingprocessofastandardgearwiththerackcutter.TheinitialpointofcuttingisatpointB1whichistheintersectionbetweenthelineofactionandtherightedgeoftherackcutter.Whentherackcutterismovedfromposition1toposition2,theinvoluteprofileofthegearwillcompletelybecut.BecausetheaddendumlineoftherackexceedstheextremepointN,thegeneratingprocesscannotbestopped,suchasatposition3,obviously,theinvoluteprofileoftheroottoothontheleftedgeofthecutterwillbecutaway.(2)MinimumnumberofteethtoavoidundercuttingThemethodstoeliminateundercuttingareasfollows:1)Reducetheheightofthetoothofthecutter.ThesegearstobecutarecalledshorttoothgearsFig.7-32Minimumnumberofteethtoavoid

undercutting(避免根切的最少齒數);theyhaveasmallcontactratio.2)Increasethepressureangle.Thisresultsinasmallerbasecircle,sothatmoreofthetoothprofilebecomesinvolute.3)Reducethenumberofteeth.Whenthenumberofteethofageartobecutisreduced,theradiusofbasecircleisreducedtoo,andtheextremepointNislikelytofallontherightofthepointB2wheretheaddendumlineoftherackcutterintersectsthelineofaction.SeetheFig7-32.1.ConceptofNonstandardGears7.7NonstandardSpurGearsIngeneratingastandardgearwitharackcutter,iftheaddendumlineoftherackcutterexcessestheextremepointNwheretheaddendumlineofthecutterintersectsthelineofaction,theundercuttingwilloccur.Tosolveit,therackcuttercanbemovedadistancexmoutwardsuntiltheaddendumlineofthecutterfallsdowntheextremepointN,asshowninFig7-33.Inthiscase,thereferencelineisnolongertangenttothereferencecircleofthegear,thelinewhichistangenttothereferencecircleofthegearisthepitchline,andthetooththicknessandtoothspaceonthepitchlineoftherackisnotequal.Thetooththicknessandtoothspaceonthereferencecircleofthegeartobecutisnotequaltoo.Thisgeariscalledthenonstandardgearormodifiedgear.

2.MinimumCoefficientofOffsetFig.7-33Smallestcoefficientofoffset(最小變位系數)WhentheaddendumlinepassesthroughtheextremepointN,thereisjustnoundercuttingtooccur,asshowninFig7-33.Therefor,wehave:3.ComparisonofStandardGearandNonstandardGear(1)NovariedsizesThebasecircle,referencecircle,circularpitchandbasepitcharenotvaried.Theinvolutecurveisnotchanged;itisusedonthedifferentportionoftheinvolute.

Fig.7-34Tooththicknessofapositivemodifiedgear(正變位齒輪齒厚)(2)VariedsizesThetooththicknessandwidthspacearevaried.Fig7-34showsageneratingprocessofastandardgearandapositivemodifiedgear.Theaddendumanddedendumarevaried.FromtheFig7-35,wecanobservethattheaddendumofapositivemodifiedgearislargerthanthatofthestandardgear,anditsdedendumissmallerthanthatofthestandardgear.Inthesamemethod,wecanfindthedifferencebetweenthestandardgearandnegativemodifiedgearintheiraddendumanddedendum.Fig.7-35Toothprofilesofthestandardgearand

modifiedgears(變位齒輪與標準齒輪的齒廓)4.BriefIntroductionofNonstandardGears

Supposingthatthecoefficientsofoffsetofthetwogearsarex1andx2,thetypesofgeartransmissioncanbeclassifiedasfollows.1)Thecoefficientsofoffsetx1andx2areallzero,thatis:x1+x2=0，andx1=x2=0.Itisapairofstandardgears.2)Thecoefficientsofoffsetx1andx2areequalinmagnitudebutoppositeindirection;thatis:x1+x2=0，andx1=-x2≠0.

3)Thesumofthecoefficientsofoffsetx1andx2isnotequalzero,thatis:x1+x2≠01.ShapeoftheToothofHelicalGearFig.7-36Toothsurfaceofaspurgearandahelicalgear(漸開線圓柱齒輪和斜齒輪輪齒的齒面)7.8ParallelHelicalGears

IfaplaneSrollsonabasecylinder,alineKK′paralleltotheaxisofthecylinderintheplanewillgenerateaninvolutesurfaceofaspurgearonthecylinder,showninFig7-36a.Thus,whenapairofspurgearsisinmesh,contactbetweentheteethisalongthelineparalleltotheaxis.However,ifthelineintheplaneisinclinedanangleβbtotheaxisofthecylinder,itwillgeneratesaninvolutehelicoids,asshowninFig7-36b.2.BasicParametersofaHelicalGear(1)HelixangleFig7-37ashowsahelicalgearwithrighthandthread.Ifweextendthewidthofthehelicalgearuntilitisequaltotheleadofthehelix,thendevelopthebasecylinderandreferencecylinder;thehelixanglescanbedeterminedfromFig3-37b.

Fig.7-37Helixangles(螺旋角)(2)NormalmoduleandtransversemoduleThetransverseplaneofahelicalgearisperpendiculartoitsaxis,anditisacirclewhichisusedtocalculateitsgeometricalsizes.Thenormalplaneisperpendiculartoitstooththread;itisanellipsewhichisusedtoanalyzeforceaction.Therefore,theparametersofthehelicalgearcanbedividedintothatofthetransverseplaneandnormalplane.Fig.7-38Helicalgeartoothrelations(斜齒輪模數關系)Fig.7-39Normalpressureangle

andtransversepressureangle

(法向壓力角和端面壓力角)

(3)Normalandtransversecoefficientsofaddendum(4)Normalandtransversecoefficientsofclearance(5)NormalandtransversepressureanglesThespurgearsareidentifiedbymeansofonepressureangle;thegeometryofhelicalgearsrequiresthetwopressureangles.Fig7-39showsthetoothprofileofahelicalrackonthetransverseplaneandnormalplane.4.ConditionsofCorrectlyMeshingforHelicalGears

Theconditionsofcorrectlyengagingforapairofhelicalgearsarethatthenormalmoduleandnormalpressureangleofthetwomeshinggearsmustbethesamerespectively,orthetransversemoduleandtransversepressureangleofthetwomeshinggearsmustbethesamerespectively;theirhelixanglesmustbeequalandhaveoppositehelicaldirections.Theyare:mn1=mn2或mt1=mt2

αn1=αn2或αt1=αt2

β1=-β2（“-”代表旋向相反）5.ConditionsofContinuousTransmissionofGearsFig.7-40Contactratioforhelicalgears(斜齒輪傳動重合度)Thecontactratioofapairofhelicalgearsisgreaterthanthatofspurgears.ThiscanbeillustratedinFig7-40.Theupperfigureisadevelopedbasecylinderofaspurgearinwhichthetransverseplaneiscoincidentwiththenormalplane;thelineB1B2representsthecontactlengthofthegears.Thecontactratiois:ε=εa+εβTheFig7-whichhasthesamenumberofteethandthetransversemodulewiththeFig7-40aspurgear.6.EquivalentSpurGear

Inordertounderstandthetoothshapeofthenormalplaneofahelicalgear,ahelicalgearcanbecutbytheobliqueplanewhichisperpendiculartothetoothdirection,asshowninFig7-41.Theintersectionoftheobliqueplaneandthereferencecylinderproducesanellipsewhosesemimajoraxisisa,andsemiminoraxisisb.Fig.7-41Equivalentspurgear

(當量齒輪)7.AdvantagesandDisadvantagesofHelicalGears

Helicalgearsaremoreexpensivethanspurgearsbutoffersomeadvantages.Theyoperatequieterthanspurgearsbecauseofsmoothandgradualcontactbetweentheiranglesurfacesastheteethcomeintomesh.Spurgearteethmeshalongtheirentirefacewidthatonce;thesuddenimpactoftoothontoothcausesvibration.Also,forthesamegeardiameterandmodule,ahelicalgearisstrongerduetotheslightlythickertoothforminaplaneperpendiculartothegearaxis.

Theoneofthedisadvantagesofhelicalgearsisthattheyproduceanaxialthrustforcewhichisharmfultothebearings.Therefore,thehelixangleislimitedfrom8°~15°.1.CharacteristicsofWormGearsFig.7-42Wormandwormgear(蝸輪蝸桿)7.9WormandWormGearsWormandwormgearareusedtotransmitmotionbetweentwoshaftswhicharenonparallel,nonintersecting,usuallyatashaftangleof90°,asshowninFig7-42.2.TypesofWormGears

Fig.7-43WormTypes(蝸桿傳動類型)3.ParametersandSizesofWormGears(1)ParametersofwormgearsInthemainsectionwhichisperpendiculartotheaxisofthewormgearandcontainstheaxisofthecylindricalworm,theconjugateactionofwormsisthesameasarackandagear,asshowninFig7-44.1)Module

2)Pressureangle

3)Numberofteethandratio

4)Quotientofwormdiameter

5)Centerdistance

(2)ConditionsofcorrectlyMeshingforwormgearsTheconditionsofcorrectlyengagingforwormgearsarethatthemoduleandpressureangleofthewormandwormgearmustbethesamerespectively;theleadangleofthewormmustbeequaltothehelixangleofthewormgear,andhavethesamehelicalhand.Fig.7-44Archimedeswormandwormgear

(阿基米德圓柱蝸桿傳動)Fig.7-45Unwrappedthreadsonaworm(蝸桿分度圓柱展開圖)1.PitchAngleandGearRatioFig.7-46Straightbevelgears(直齒錐齒輪傳動)7.10BevelGearsBevelgearsmustbemountedsothattheapexesofbothpitchconesarecoincidentbecausethepitchoftheteethdependsupontheradialdistancefromtheapex.Fig7-46showsapairofstraightbevelgearsinmesh.Theshaftangleis90°.Someofthenomenclaturesareasfollows.Thegearratioofbevelgearsmaybewrittenasbefore.2.BackConeandEquivalentGearFig.7-47Toothsurfaceofabevel

gear(錐齒輪的齒面)

ImaginethatageneratingcircularplaneSwithradiusRwhosecenterOiscoincidentwiththeapexoftheconerollsontheconewithoutslipping,thepointontheplane，suchaspointK,willgeneratesasphericalinvolutefrompointA,asshowninFig7-47.Fig.7-48Backconeandequivalent

spurgear(背錐與當量齒輪)

Theinvoluteprofileonasphereisdifficulttomanufacture,thusanapproximationinaplaneismade.Itiscustomarytomakeaconewhichiscalledasthebackcone,andtheelementsofthebackconeareperpendiculartothoseofthepitchcone.Thebackconeistangenttothesphereatthepitchcircle,andthentheteethofthebevelgearonthesphereareprojectedonthebackcone.Theyarealmostthesame,aslongasthebevelgearhaseightormoreteeth;itisaccurateenoughforpracticalpurposes.Afterthebackconeisdeveloped,asectorspurgearhavingaradiusequaltothebackconeelementwillbeobtained,asshowninFig7-48.3.Parameters,SizesandProperties(1)ParametersParametersarestandardizedatthelargeendofthebevelgears.

(2)Conditionsofc