Design,generationandstressanalysisofface-gear

Design,generationandstressanalysisofface-geardrivewithhelicalpinionFaydorL.Litvina,IgnacioGonzalez-Perezb,AlfonsoFuentesb,*,DanieleVecchiatoa,BruceD.Hansenc,DavidBinneycaGearResearchCenter,DepartmentofMechanicalandIndustrialEngineering,UniversityofIllinoisatChicago,Chicago,IL60607-7022,USAbDepartmentofMechanicalEngineering,PolytechnicUniversityofCartagena,CampusMuralladelMar,Dr.Fleming,s/n,30202,Cartagena,Murcia,SpaincSikorskyAircraftCorporation,6900MainSt.,Stratford,CT06615,USAReceived11September2003;receivedinrevisedform23September2004;accepted30September2004AbstractTwoversionsofface-geardrivegeometrywithahelicalpinionareconsidered.Oneversionisbasedonascrewinvo-lutehelicoid.Thesecondversionisanewgeometrydevelopedasenvelopesoftwomismatchedparabolicracksofthepinionandtheshaper.Aface-geardrivewithaspurpinionisconsideredasaparticularcaseofthedevelopedtheory.Anewmethodofgrindingorcuttingofface-gearsbyawormofspecialshapehasbeendeveloped.Thefollowingprob-lemsareconsidered:(i)generationofagrindingwormwithsurfacesfreeofsingularitiesand(ii)generationofface-gearbyashaperwithsurfacesfreeofundercuttingandpointing.Toothcontactanalysisandstressanalysisareappliedcom-binedwithinvestigationoftoothbearingcontactonbothsides.Thefollowingadvantageshavebeenachievedforthegeometryproposed:(a)existenceofalongitudinalbearingcontact,(b)avoidanceofedgecontact,and(c)reductionofcontactstresses.Aphenomenonofasymmetryoftoothbearingcontactforthedrivingandcoastsideshasbeendiscov-ered.Acomputerizeddesign,generationandstressanalysisforthenewtypesofface-geardriveshavebeendeveloped.
2004ElsevierB.V.Allrightsreserved.Keywords:Design;Stressanalysis;Face-geardrive;Helicalpinion0045-7825/$-seefrontmatter
2004ElsevierB.V.Allrightsreserved.doi:10.1016/j.cma.2004.09.006*Correspondingauthor.Tel.:[phone];fax:[phone].E-mailaddress:[e-mail](A.Fuentes).Comput.MethodsAppl.Mech.Engrg.194(2005)3870–3901[6–9,16,17].OneofthemainachievementsisgrindingoffacegearsbyawormofaspecialshapeproposedinRefs.[9,18].Grindingallowsapplicationofhardenedmaterialsforface-gearsandimprovesthecurrenttechnologyofmanufacturingofface-geardrives.Newdevelopmentsofthetheoryofgearinghavebeenproposedrecently[2,6–8,10,11,19–22].Enhancedmethodsofstressanalysisbasedonthefiniteelementmethod[23]havebeenalsodevelopedandapplied[1,5,12].Nomenclatureanpressureangleinnormalsectionacpressureangleincross-sectionbHelixangles0,w0spaceandwidthoftheteethoftherack-cutterkccoefficientoftooththickness(ui,li)(i=1,s)surfaceparametersofskewrack-cutter(hi,
i)(i=1,s)surfaceparametersofhelicalpinion(i=1)andshaper(i=s)cmshaftangleformedbyshaperandface-gearDcmshaftangleerrorDEshortestoffsetdistanceformedbycrossedaxesofthepinionandtheface-gearDqaxialdisplacementofface-gearkwleadangleofthewormcwscrossinganglebetweenaxesofshaperandwormgshalfofthewidthofthespaceonthebasecylinderRi(i=s,1,2,w)toothsurfaceoftheshaper(i=s),pinion(i=1),face-gear(i=2),andgeneratingworm(i=w)wi(i=s,1,2,w)anglesofrotationoftheshaper(i=s),pinion(i=1),face-gear(i=2)andworm(i=w)formedduringtheprocessofgeneration/i(i=1,2)rotationanglesofthepinion(i=1)andtheface-gear(i=2)formedduringthemeshingprocessai(i=1,s)rack-cutterparabolacoefficientforthepinion(i=1)andtheshaper(i=s)aplparabolacoefficientofparabolicfunctionthatrepresentstoolplungingafaddendumofface-gearEwsshortestdistancebetweenaxesoftheshaperandthewormuoparameterofparabolaapexLi(i=1,2)inner(i=1)andouter(i=2)limitingdimensionsoftheface-gearMji,Ljimatrices4·4and3·3fortransformationfromSitoSjofpointcoordinatesandvectorsmnmoduleNi(i=s,1,2,w)numberofshaperteeth(i=s),pinion(i=1),face-gear(i=2),andgeneratingworm(i=w)Pddiametralpitchrpi(i=1,s)pitchcircleradiusofthepinion(shaper)rbi(i=1,s)basecylinderradiusofthepinion(shaper)rpwpitchcircleradiusofthewormF.L.Litvinetal./Comput.MethodsAppl.Mech.Engrg.194(2005)3870–39013871Recentstudiesofface-gearshavebeenlimitedtoface-geardriveswithspurinvolutepinion.Forfurtherextensionofapplicationofface-geardrives,theauthorsofthispaperconsider:(1)Face-geardriveswithhelicalpinions(Fig.1)withtwotypesofgeometry:(a)screwinvolutesurfaceand(b)asurfacedeterminedastheenvelopetoaskewparaboliccutter.(2)Thegenerationoftheface-gearofthedrivemaybeaccomplished:(i)byashaperand(ii)agrindingorcuttingwormprovidedbyaspecialthreadsurface(seeSection8).Thecontentsofthispapercover:(i)Avoidanceofsingularities,undercutting,andpointingofgeneratedsurfaces.(ii)Simulationofmeshingandcontactfordeterminationoftransmissionerrorsandshiftofbearingcontact.(iii)Enhancedstressanalysis.Applicationofface-geardriveswithhelicalpinionsisillustratedwithexamplesofgeardrives(seeSection2).Thedevelopedtheoryisillustratedwithnumericalexamples.2.Examplesofapplicationofface-geardriveswithhelicalpinion2.1.Geartrain1Thetrainisformed(Fig.2(a)):(i)byco-axialface-gears1and4and(ii)carriercatwhicharemountedhelicalpinions2and3.Pinions2and3arerigidlyconnectedandperform:(a)transfermotionwi