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Effectsofmisalignmenterror,toothmodificationsand
transmittedtorqueontoothengagementsofapairofspurgears
ShutingLi⁎
DepartmentofMechanical,ElectricalandElectronicEngineering,InterdisciplinaryFacultyofScienceandEngineering,ShimaneUniversity,Matsue690-8504,Japan
articleinfoabstract
Articlehistory:Received11November2013Receivedinrevisedform3September2014Accepted16September2014Availableonline7October2014Inthelastresearch[S.Li,Effectsofmachiningerrors,assemblyerrorsandtoothmodificationsonload-carryingcapacity,load-sharingrateandtransmissionerrorofapairofspurgear,Mech.Mach.Theory42(2007)698–726],effectsofmachiningerrors,assemblyerrorsandleadcrowningontoothsurfacecontactstresses(CS),rootbendingstresses,load-sharingratios(LSR)andtransmis-sionerrorsofapairofspurgearswereinvestigatedthroughperformingloadedtoothcontactanal-ysis(LTCA)withdevelopedfiniteelementmethod(FEM)programs.Butthisresearchcouldn'tinvestigatetheeffectsoftoothprofilemodificationandleadrelievingontoothengagements.Also,theeffectsofmachiningerrors,assemblyerrorsandtoothmodificationsontoothmeshstiff-ness(MS)couldn'tbeinvestigated.So,asacontinuousstudyofthelastresearch,thispaperinves-tigatestheeffectsoftoothprofilemodificationandleadrelievingontoothengagementsofapairofspurgearsandtheeffectsofmisalignmenterrorofgearshaftsontheplaneofaction,toothleadcrowingandtransmittedtorqueontoothMS.Anarccurveisusedtomodifytoothprofilesofapairofspurgearsinthispaper.Thisisbecausethismethodisusedverypopularlyforthespurgears.Methodsusedinthelastresearcharealsousedheretoinvestigatetheeffectsofthetoothprofilemodification,leadrelievingandtransmittedtorqueontoothengagements.Basedontheresults,itisfoundthatthetoothprofilemodificationandleadrelievinghavesignificanteffectsontoothCS,LSRandMS.ItisalsofoundthattransmittedtorquehasalittleeffectontoothMS,buthasnoeffectonLSRofthegears.Forthelead-relievedgears,calculationresultsshowthatedge-loadshappenedatthejointpartsoftherelievedpartandthenon-relievedpartoftoothleadwhentheleadisre-lievedwithstraightlines.Sincetheedge-loadsresultedingreatercontactstressesatthejointpartsandweakenedtoothcontactstrength,attentionmustbepaidtotheleadrelieving.Itisnec-essarytoreducetheedge-loadsassmallaspossiblethroughmakingthejointpartsmoothwhentheleadrelievingismade.©2014ElsevierLtd.Allrightsreserved.Keywords:SpurgearsAssemblyerrorsToothmodificationsMeshstiffnessLoad-sharingratiosContactstresses
1.Introduction
Itiswell-knownthatmachiningerrors,assemblyerrorsandtoothmodificationsofapairofgearshavesignificanteffectson
vibration,noiseandstrengthofthepairofgears.Thoughthisproblemhasbeenwellinvestigatedbymanyresearchers[1–20],
manyproblems,suchashowtoestimatevibration,noiseandstrengthlevelsofapairofgearswiththemachiningerrors,assembly
errorsandtoothmodificationsexactlyintheory,havenotbeensolvedcompletely.
Umezawaetal.[3–6]conductedresearchesontoothMScalculationsofapairofgearswithtoothsurfacedeviationsusingan
analyticalmethod.InUmezawa'sresearch,toothdeformationwasdividedintotoothdeflectionandtoothapproach(resulted
fromMechanismandMachineTheory83(2015)125–136
⁎Tel./fax:+810852328908.E-mailaddress:shutingli@ecs.shimane-u.ac.jp
.
http://dx.doi.org/10.1016/j.mechmachtheory.2014.09.0110094-114X/©2014ElsevierLtd.Allrights
reserved.ContentslistsavailableatScienceDirect
MechanismandMachineTheory
journalhomepage:www.elsevier.com/locate/mechmttheHertziancontactofcontactteeth).Thetoothdeflectionwascalculatedbyacantilevertheoryandthetoothapproachwas
calculatedbytheSneddonequation.Effectofthetoothsurfacedeviationsonrotationalvibrationofapairofhelicalgearswasalso
investigated[7].
SinceitisadifficultthingtouseUmezawa'smethodtosolvetoothcontactproblemsofapairofgearswithcomplicatedmachining
errors,assemblyerrors,toothprofilemodificationsandthinrims,Li[11,12]developedFEMsoftwaretodoLTCAofapairofgearswith
machiningerrors,assemblyerrorsandtoothmodificationsbasedontheprincipleofthemathematicalprogrammingmethodpresent-
edbyConryandSeireg[13,14].Inthelastresearchoftheauthor[12],effectsofmachiningerrors,assemblyerrorsandleadcrowning
ontoothsurfaceCS,rootbendingstresses,LSRandtransmissionerrorsofapairofspurgearswereinvestigatedusingthedeveloped
FEMsoftware.Butthisresearchcouldn'tinvestigatetheeffectsoftoothprofilemodificationandleadrelievingontoothengagements.
Also,theeffectsofthemachiningerrors,assemblyerrorsandtoothmodificationsontoothMSofapairofspurgearscouldn'tbein-
vestigated.So,asacontinuousstudyofthelastresearch,thispaperinvestigatestheproblemsthathavenotbeensolvedinthelast
research.ThesamemethodsandFEMprogramsareusedtodoLTCAofapairofspurgearswithtoothprofilemodificationandlead
relieving.Theeffectsofmisalignmenterrorontheplaneofaction,toothleadcrowingandtransmittedtorqueontoothMSofthe
pairofspurgearsarealsoinvestigated.
Basedoncalculationresults,itisfoundthattoothprofilemodificationandleadrelievinghavesignificanteffectsontoothCS,LSR
andMS.ItisalsofoundthattransmittedtorquehasalittleeffectontoothMS,buthasnoeffectonLSRofthegears.Forthelead-