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STUDYONTHELUBRICATIONFACTOROFINVOLUTESPURGEARSAbstract:Basedonalotofnumericalsolutionstotheproblemsofthethermalnon-Newtonianelasto-hydrodynamiclubricationandsomefatiguetestswithrollers,thelubricationfactorofinvolutespurgears(calledgearforshort)isinvestigated.TheresultssuggestthatgearlubricationeffectsbearcloserelationstoadimensionlessparameterDwhichissyntheticallydeterminedbygearingrotationalspeed,load,material,dimensionandlubricantviscosity.WhenD8,thegearfatiguelifeincreasesasthelubricantviscosityisincreased；WhenD8,however,thelifedecreaseswiththeviscosityaddition,whichisinmarkedcontrasttothelubricationfactorZLrecommendedbytheInternationalStandardforComputingCylindricalGearStrength.Attheend,asetofformulaeforcalculatinggearlubricationfactorssuitablefordifferentworkingconditionsareadvanced.Keywords:InvolutespurgearElastohydrodynamiclubrication(EHL)Non-NewtonianfluidLubricationfactorFatiguetestsNOTATIONSA,B-Pressure-densitycoefficientsb-Hertzhalfwidthc1,c2-Specificheatofpinionandgearef-SpecificheatoflubricantD0-Temperature-densitycoefficientE-EquivalentYoungmodulusEl,E2-YoungmoduliofpinionandgearF-Frictionalforceperunitwidthu-Coefficientoffriction-Lubricantfilmthickness0-Filmthicknessconstantkf-Thermalconductivityoflubricantk1,k2-Thermalconductivitiesofpinionandgearp-PressurepH-MaximumHertzianpressureP0-Appliedloadperunittoothlength.R-Equivalentradius,1/R=1/R1+1/R2R1,R2-RadiiofcurvatureofpinionandgearS0-Temperature-viscosityindexT-TemperatureT0-InletlubricanttemperatureT1,T2-Temperaturesofpinionandgearu-X-componentoflubricantvelocityul,u2-Pinionandgearteethsurfacevelocitiesur-Contactrollingvelocity,ur=(u1+u2)/2U-DimensionlessvelocityparameterU=(0ur)/(ER)W-DimensionlessloadparameterW=P0/(ER)x-Coordinatedirectionalongsurfacemotiony-Coordinatedirectionacrosslubricantz-ViscosityindexofRoelandsmodel0-Ambientviscosityoflubricant*-Equivalentviscositye-Cross-filmeffectiveviscosityoflubricant0-Ambientdensityoflubricante-Cross-filmeffectivedensityoflubricant1,2-Densitiesofpinionandgear-Shearstresswithinlubricantfilm0-Eyringstressxy,-Shearstressonpinionsubsurfacex,y-Normalstressesonpinionsubsurfaceeff-Contactstressontoothsurface1,2-Possionratioesofpinionandgear0INTRODUCTIONThecurrentstandardforcomputingcontactfatiguestrengthofgearisbasedonHertziantheorywhichisonlysuitablefordrycontact.Inordertotakeintoaccountthelubricationeffect,lubricationfactorZLisraisedbytheInternationalStandardforComputingCylindricalGearStrength(ISO/423E).Also,variousfactorsrelatedtogearlubricationaregiveninsomeotherGearStandards,suchasD.B,AGMA,JAMA,ThomasandDIN3990andsoon.Althoughthewaystodefinethelubricationfactorarevariousindifferentgearstandards,theprincipleisidenticalthatthehighertheviscosity,thegreaterthevalueofthelubricationfactorandthelongerthegearfatiguelife.Sofarastheauthorsknow,however,itisinconsistentwithsomepracticalconditionsoccurredintheindustrialgeardrives.Therefore,itisessentialtodofurtherresearches.Theproblemsofgearlubricationhavebeeninvestigatedbyquiteafewpeople.Forexample,R.ErrichellomadeacomprehensiveEHLanalysisofgeartransmissionsandgavearecommendationforselectinglubricanttype,viscosityandacasehistorywhichdemonstratedmanyoftheprinciplesofgearlubrication.L.ThibaultandJ.Kalonkostudiedtheeffectofappropriateselectionoflubricantongeardrivelife.Someguidelinesweregivenonhowtoselecttheproperlubricantintheirpaper.R.Rakiclhlinvestigatedtheinfluenceoftribologicalpropertiesoflubricantonthereliabilityofgearandfoundthatreliabilityofgearwasgreatlyaffectedbyboththetypeoflubricantoilandtheviscositygrade.Morerecently,alotofexperimentalresearchesweredonebyO.JarvioandA.Lehtovaara.Basedontheirexperimentalresults,theinfluenceoflubricanttypeanditsadditivesonmeancoefficientoffrictioningearcontactwasevaluated.Inthepresentanalysis,morethanfiftynumericalcalculationsofthermalnon-NewtonianEHLarecompleted.Sixsetsofsimulatedfatiguetestswithrollersaredone.Threeformulaeforcomputinggearlubricationfactorsarepresented.1BASICEQUATIONS1.1Reynoldsequationer()ph=uxxx(1)Anequivalentviscosity*hasbeenintroducedinEq.(1)todescribethenon-Newtonianlubricantproperties.IftheEyringsfluidmodelisadoptedandassumingthevariationsofboththeviscosityanddensityacrossthefilmthickness(alongthey-direction)canbeignored,whichisacceptableaccordingtoL.Chang,theeffectiveviscosity*couldbecalculatedasfollows0011=sinh()e(2)ThepressureboundaryconditionsofEq.(1)aregivenbyi(x)=0pandeex=xp(x)=|=0xpwheretheinletlocation,xi,ischosenaccordingto|xi|4bandtheoutletlocation,xe,isdeterminedinthesolutionprocess.1.2Filmthicknessequation2202x-s(x)=+0ln()2seixxxpdsRE(3)1.3Energyequation22=+ffTTTppuuckuxyTTy(4)Theboundaryconditionsofaboveequationcanbeexpressedasf=001111kT(x,0)=|+kxyTdsTycuxsf=02222kT(x,)=|+kxyTdsTycuxs(5)1.4Viscosity-pressure-temperaturerelationThecross-filmeffectiveviscosityoflubricantasafunctionofpressureandtemperaturecanbecalculatedusingtheRoelandsrelation0-900138=exp(ln+9.67)1+5.1101138szeTpT(6)1.5Density-pressure-temperaturerelationThecross-filmeffectivedensitycanbedeterminedbytheDowson-Higginsonrelation000=1+1+eAPDTTBP(7)Theaveragecross-filmtemperatureisusedinthecalculationsofboththeeande.1.6Loadequation0=eixxppsds(8)1.7FrictionalforceequationAsismentionedabovethattheEyringsnon-Newtonianfluidmodelisusedinthisanalysis.Forthismodel,theconstitutiveequationforthestress-strainrelationis00=sinhuyor00=sinhuaryThus,thefrictionalforceonthesurfaceofthepiniontoothisobtainedas00=sinheixxyouFdxardxy(9)then,thecoefficientoffrictionwillthusbe=F/p0.1.8StressequationsWhenpressureandsurfaceshearstressareknownitispossibletocomputethesubsurfacestresscomponentsatanypointm(x,y)accordingtothefollowingexpressions232222=+eixxxyxxpdxy22222+2=+eixxyxyxyxpdxy(10)Inthepresentanalysisthemaximumshearstressischosenasstresslevelmeasure.2COMPUTATIONSCHEMETheproblemissolvedindimensionlessformevenifallequationsaregivenindimensionhere.Tosolvethethermalnon-Newtonianlubricantproblem,theaboveEqs.(1)(8)aresolvedsimultaneouslytoobtainthepressure,filmt