Int.J.Mach.ToolsManufact.Vol.31,No.2,pp.167-181,1991.0890-6955/9153.00+.00PrintedinGreatBritainPergamonPressplcGENERATIONANDGEOMETRYOFHYPOIDGEAR-MEMBERWITHFACE-HOBBEDTEETHOFUNIFORMDEPTHF.L.LITVlN,*W.S.CHAING,*C.KUAN,*M.LUNDYtandW.J.TSUNGt(Received23August1989；infinalform15August1990)Abstract-Theauthorshavedescribedthemethodandkinematicsforgenerationofhypoidgear-memberswithface-hobbedteethofuniformdepthandderivedequationsforthetoolgeometryandthetoothsurface.Thedevelopedequationsarenecessaryforthecomputerizedinspectionofgeartoothsurfacebycoordinatemeasuringmachines,forgenerationofdiestoforgethegears,andtorepresentpartiallytheinputdataforcomputerizedsimulationofmeshingandcontactofhypoidgeardrives.1.INTRODUCTIONHYVOlDgearshavefoundwidespreadapplicationintheautomotiveindustry.TheyaremanufacturedusingcuttingmachinesdesignedbytheGleasonWork(U.S.A.)andOerlikon(Switzerland).Ahypoidgeardriveisdesignedfortransformationofmotionsbetweencrossedaxes.ThereferencesurfacesofthepinionandgeararepitchconesthatareintangencyatpointP(pitchpoint)choseninthefixedcoordinatesystem(Fig.1).Usually,theaxesofrotationofhypoidgearsformanangleof90.TheshortestdistancebetweentheaxesofrotationisdesignatedbyEandtheapicesofpitchconesarenotifiedbyO1and02.Theplanethatistangenttothepitchcones(pitchplane)passesthroughtheapicesofthepitchconesandpitchpointP.Figure2showspitchpointP,theshortestdistanceE,theangularvelocitiestoo)andto(2)ofthepinionandthegear；dlandd2determinethelocationofapicesofpitchconesO1and02withrespecttogearshortestdistance.Therearetwotypesofhypoidgears:(i)withtaperedteeth,and(ii)withteethofuniformdepth,andtheyaregeneratedrespectivelyby:(i)theface-milledmethod,and(ii)theface-hobbedone.Theface-milledmethodisbasedontheapplicationofatoolwhosegeneratingsurfaceisacone.Theteetharetapered.Eachtooth(sometimeseveneachsideofthetooth)isgeneratedseparatelyandindexingisrequiredforthegenerationofthenexttooth.Thegeartoothsurfaceisanenvelopeofthefamilyoftoolsurfaces.Theexceptionistheformate-cutgearwhosesurfacecoincideswiththetoolsurface.Theface-hobbedmethodisbasedongenerationofthetoothsurfacebytwofinishingblades.Thetoothsurfacerepresentsafamilyofcurves(straightlines)thatisgeneratedbytheshapeoftheblade.Theteethareofuniformdepth.Theprocessforgenerationisacontinuousoneandindexingisnotrequired.ThetheoryofgenerationandmathematicaldescriptionofhypoidgearswithtaperedtoothsurfaceshasbeenthesubjectofintensiveresearchbyGleasonengineers1,andLitvinetal.2.MethodsfordeterminationofanenvelopeofafamilyofsurfaceshavebeendiscussedbyLitvin3.Thetheoryofgenerationandmathematicalrepresentationofsurfacesforface-hobbedhypoidgearshasnotbeenpublishedintheliterature.Thereareseveralreasonswhymathematicalrepresentationofhypoidgeartoothsurfacesisimportantformanufacturingthemwithhighprecision:*DepartmentofMechanicalEngineering,UniversityofIllinoisatChicago,Illinois,U.S.A.tOearEngineeringDepartment,DanaCorporation,Indiana,U.S.A.167168F.L.LtrvtNetal.02/.7_,L.FI.1,Pitchconesofhypoidgears.d2!r,(2),f/fIPitchplaneFt.2.Pitchplaneofhypoidgears.(1)Itisnecessarytooptimizethemachine-toolsettingsinordertoreducetheleveloftransmissionerrors(theycausevibrations)andprovideaproperbearingcontact.Thiscanbeachievedbycomputerizedsimulationofmeshingandcontactofgeartoothsurfacesincontinuoustangency.Analytically,thismeansthatthegeartoothsurfacesmusthaveacommonpositionvectorandeollinearnormalatanyinstantaneouspointofcontact3.(2)Itisnecessarytocompensateforthedistortionofgeartoothsurfacesthatarecausedbyheat-treatmentandbyerrorsintheinstallmentofmachine-toolsettings.Thisgoalcanbeachievedbypropercorrectionsofmachine-toolsettings.Forthispurposethetoothsurfaceofthefirstgearfromthesetofgearstobegeneratedisinspectedbyacoordinatemeasuringmachine2,4,5.Thisinspectionprovidestheinformationondeviationsoftherealtoothsurfacesfromthetheoreticalonethatmustberepresentedanalytically.Theminimizationofdeviationsbycorrectionofalimitednumberparameters-themachine-toolsettings-isatypicalmathematicalproblemofoptimiz-ation.HypoidGear-member169(3)Forgingofgearsrequiresgenerationofdiesthathavetocopythegeartoothsurfaces.Thediesaregeneratedpointbypointandthiscanbedoneifthegeartoothsurfaceisrepresentedanalytically.Thethreereasonsaboveexplainwhyanalyticalrepresentationofface-hobbedgeartoothsurfaceswillbenefittheirmanufacturing.Thecontentsofthispapercoverthedescriptionofthemethodforgenerationofthegear-memberofface-bobbedgeardrives,andtheanalyticalrepresentationofthegeartoothsurface.Theobtainedresultsarethebasisfor:(i)determinationoftoothsurfacedeviationsbycoordinatemeasurements,(ii)generationofdiesforforging,and(iii)aspartoftheinputdataforcomputerizedsimulationmeshingandcontactofhypoidgeardrives2,6.2.PRINCIPLESOFGEARTOOTHSURFACEGENERATIONTheprincipleofgeartoothgenerationisshownschematicallyinFig.3.TheheadcutterisprovidedwithNwgroupsofblades.Eachgroupcontains3blades:oneforroughcuttingandtwofinishingblades-oneforeachsideofthetooth.Whiletheheadcutterisrotatedthroughtheangle2r/Nw,thebeinggeneratedgearisrotatedthroughtheangle2rr/N2,whereN2isthenumberofgearteeth.Thusthenextgroupofbladeswillstarttocutthenextgeartoothafterthecurrentgroupofbladeshasfinishedcuttingthecurrenttooth.Thetransmissionofthecuttingmachineprovidesrelatedrotationsoftheheadcutterandthegearandthisistheconditionofthecontinuousindexing.TheschematicofthecuttingmachineisshowninFig.4.Themachinecradlecarriestheheadcutter.Theaxisofthegearblankformswiththeaxisofthecradletheprescribedanglethatdependsonthepitchconeangleofthegear.Intheprocessforgenerationthecradle,theheadcutterandthegearperformrelatedrotationsabouttheiraxes.TheinstalmentoftheheadcutterandthegearonthecuttingmachineareshowninFig.5.Sc2isthecoordinatesystemthatisrigidlyconnectedtothecuttingmachine.AnauxiliarycoordinatesystemSrisalsorigidlyconnectedtothecuttingmachine.Planexc2=0istheso-calledmachineplaneandtheoriginOc2ofSc2isthemachinecenter.Wewilldifferentiatebetwennontiltedandtiltedsettingsoftheheadcutter(Figs3and5).Iftheheadcutterisnottilted,itsaxiswithunitvectorccoincideswiththeaxisofrotationXr(Fig.5).Thetiltangleoftheheadcutterisdesignated2(Fig.5).ThetiltoftheheadcutterisusedtoimprovetheconditionsofgearmeshingandbearingTiRedHead-CutterGearBladesIFTG,3.Principleofgeargeneration.170F.L.LITVINetal.CradleCutterShaftofblankFro.4.Schematicofcuttingmachine.Projectionoftheheed-cutteraxisV.6,Ic2lYc2FI.5.Cuttingmachinecoordinatesystem.contact.TheoriginO,ofS,isthepointofintersectionoftheaxisofrotationoftheheadcutterwiththemachineplane.P