机械原理Chapter-6--Gear-Mechanisms-4

Chapter6GearMechanisms§6.9HelicalGearsforParallelShafts§6.10WormGearing§6.11BevelGearsGenerationandCharacteristicsofHelicalTeeth•gearsalwayshaveacertainfacewidth.•TheprofileofaplanargearisformedbyastraightlineKKonageneratingplaneasthatplanerollswithoutslippingonthebasecylinder.GenerationandCharacteristicsofHelicalTeeth•Inthecaseofaspurgear,thestraightlineKKisparalleltotheaxisofthegear.•Thetoothsurfaceofthespurgearisthereforeaninvolutecylinder.GenerationandCharacteristicsofHelicalTeeth•Thetoothsurfacesofspurgearscontactonastraightlineparalleltotheaxesofthegears.•Thisimpliesthattoothprofilesgointoandoutofcontactalongthewholefacewidthatthesametime.GenerationandCharacteristicsofHelicalTeeth•Thiswillthereforeresultinthesuddenloadingandsuddenunloadingonteethastoothprofilesgointoandoutofcontact.•Asaresult,vibrationandnoiseareproduced.•Thatiswhytransmissionbyspurgearsisnotcompletelysmooth.GenerationandCharacteristicsofHelicalTeeth•Helicalgearsareproducedtoovercometheshortcomingsofspurgears.•ThestraightlineKKonthegeneratingplaneisnolongerparalleltotheaxisofthegear.GenerationandCharacteristicsofHelicalTeeth•Asthegeneratingplanerollswithoutslippingonthebasecylinder,everypointonthestraightlineKKwillproduceaninvolute.•Thecurveconnectingthestartingpointsoftheinvolutesonthebasecylinderisahelix.•Thesurfaceprofileofahelicalgearisthereforecalledaninvolutehelicoid.GenerationandCharacteristicsofHelicalTeeth•Thetoothsurfacesoftwohelicalgearsforparallelshafts•aregeneratedbythesamestraightlineKKinclinedtotheaxesofthegears•asthegeneratingplanerollsontwobasecylinderswithparallelaxes.GenerationandCharacteristicsofHelicalTeeth•Thetoothsurfacesoftwoengaginghelicalgearscontactonastraightlineinclinedtotheaxesofthegears.•Thelengthofthecontactlinechangesgraduallyfromzerotomaximumandthenfrommaximumtozero.Ob12b2GenerationandCharacteristicsofHelicalTeeth•Theloadingandunloadingoftheteethbecomegradualandsmooth.•Thatiswhyhelicalgearscanoperateatahigherspeed.Ob12b2ParametersofHelicalGears•Therearetwosetsofparametersforahelicalgear.•Onesetisonthetransverseplaneandtheothersetonthenormalplane.ParametersofHelicalGears•Gearteetharecutbymovingorfeedingthestandardtoolperpendiculartothenormalplane.•Thecuttersarethesameasthoseforspurgearswithstandardparameters.ParametersofHelicalGears•Therefore,parametersonthenormalplane(withsubscript“n”)arethestandardvalues.•Ontheotherhand,thetransverseprofile(withsubscript“t”)ofahelicalgearisaninvolute,thesameasinthecaseofaspurgear.ParametersofHelicalGears•Tomakeuseoftheformulaeforspurgears,theparametersintheequationsforspurgearsshouldbereplacedbythoseonthetransverseplaneofhelicalgears.•Therefore,itisnecessarytosetuprelationshipsbetweenbothsetsofparameters.ParametersofHelicalGears•Byunwrappingthereferencecylinder,itcanbeseenclearlythatdBπdπdbλββbβptnpπdlppntcoscostnmm•βisthehelixangleonthereferencecylinder.Itcanbepositiveornegative,referringtorighthandedorlefthanded.ParametersofHelicalGears•Therelationshipbetweenαnandαtcanbeseenclearlyinahelicalrack.•△abcisonthetransverseplaneand△a′b′conthenormalplanewithab=a′b′•anda′c=accosβ.bb'aca'cb(a)αβαtnb'(a')costancos'''tantnabacbacan=20t20ParametersofHelicalGears•Theaddendumanddedendumofastandardhelicalgearcanbecalculatedonthenormalplaneasbb'aca'cb'(a)b(a)βαtαn•Sincethecuttersforhelicalgearsarethesameasforspurgears,mn,•ha*nandc*ntakethesamestandardvaluesasforspurgears.ParametersofHelicalGears•Thereferencediameterandcentredistanceshouldbe•calculatedonthetransverseplaneascosntzmzmdcos2)()(212121zzmddan•theadjustmentofthecentredistancecanbedonebynotonlyaddendummodificationbutalsochangingthehelixangleofthegears.ProperMeshingConditionsforHelicalGears•Astransverseprofilesofhelicalgearsareinvolutes,apairofhelicalgearsmountedonparallelshaftsshouldhavethesamemoduleandpressureangleonthetransverseplane•accordingtothepropermeshingconditionofspurgears,•i.e.mtl=mt2,αt1=αt2.ProperMeshingConditionsforHelicalGears•Inaddition,teethonbothgearsshouldslantatthesameangle.•Thismeansβ1=-β2foranexternalgearpairorβ1=β2foraninternalgearpair.•Sincecosβ1=cosβ2,mt1=mt2andαt1=αt2•implymnl=mn2andαn1=αn2ProperMeshingConditionsforHelicalGears•Therefore,themodulesandpressureanglesofbothgearsonthenormalplanearealsoequaltoeachother,respectively.•Bothαnandmnshouldbestandardvalues.ContactRatioforaHelicalGearPair•Bythedefinitionofcontactratiointhecaseofaspurgearset,contactratio•whereB1B2isthelengthoftheactuallineofaction,orthedistancemovedbythetoothpairalongthebasecircleduringengagement;pbisthebasepitch.OOpbBB1212PKN2N1ωωαααα11a''a22bpBB21ContactRatioforaHelicalGearPair•unfoldingbasecylindersforaspurgearandhelicalgear.•LineB2B2indicatesthestartpositionofmeshingandB1B1theendpositionforthespurgearpair.LLBBBBBB'B111112222B1B'BBβbbβ△ContactRatioforaHelicalGearPair•ThecorrespondinghelicalgearscomeintocontactalsoatB2B2,butgradually.•OneoftheendfacesofthehelicalgearbeginstocomeoutofcontactatB1B1,buttheteethdonotgetoutofcontactfullyuntilthetootharrivesatB1′B1′.LLBBBBBB'B111112222B1B'BBβbbβ△Cont