AGMA-91FTM16-1991.pdf

91 FTM 16 A V Contact Analysisof Gears Using a Combined FiniteElementand Surface IntegralMethod by: S. M. Vijayakar, Advanced Numerical Solutions and D. R. Houser, Ohio State University ,L AmericanGearManufacturersAssociation III TECHNICALPAPER Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- Contact Analysisof Gears Using a Combined Finite Element and Surface IntegralMethod S. M. Vijayakar, Advanced Numerical Solutions and D. R. Houser, Ohio State University The Statementsandopinionscontainedhereinarethoseof theauthorandshouldnotbeconstrued asanofficial actionor opinion of theAmerican Gear Manufacturers Association. ABSTRACT: Anewmethodis describedfor thesolutionofthecontactproblemingears. Themethodusesacombinationofthefinite element method anda surfaceintegralformof theBousinesqand Cermtisolutions. Numericalexamplesarepresented of contacting hypoid gears, helical gears andcrossed axis helicalgears. Copyright © 1991 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 October,1991 ISBN: 1-55589-614-6 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- Contact Analysisof Gears using a CombinedFinite Elementand Surface Integral Method SandeepM. Vijayakar AdvancedNumericalSolutions 2085 Pine Grove Lane, Columbus OH 43232, and DonaldR. Houser Professor, Dept.of Mech. Eng. The Ohio State University ColumbusOH 43210 INTRODUCTION Researchin the mid and late eightiesshowedthat The completeandaccuratesolutionof thecontactthe gear contactproblemwas not unsurmountable,but problemof three-dimensionalgears has been,fortherequiredan approachthat combinedthe strengthsof pastseveraldecades,oneof themore soughtafter,thefiniteelementmethodwiththoseofother albeitelusivesolutionsin the engineeringcommunity,techniquessuchasboundaryelementandsurface Eventhe arrivalof finiteelementtechniqueson theintegralmethods.Conceptsfrommathematical sceneinthemidseventiesfailedtoproducetheprogrammingcouldbe usedto advantagein solving solutiontoanybutthemostsimplegearcontactthe contact equations.An innovativeapproachtowards problems,theformulationofthefiniteelementsthemselves couldgoalongwaytowardssolvingthemesh The reasonsfor this are manyfold.When gearsaregenerationand geometricaccuracy problems.With the broughtin contact,the widthof the contactzoneisideaof incorporatingthebestoftheseandother typicallyan order of magnitudesmallerthan the othertechnologiesinmind,developmentof whatis now dimensionsof the gears.This gives rise to the need forCAPP(ContactAnalysisProgramPackage)was begun a veryhighlyrefinedfiniteelementmeshnearthefour years ago. It has evolvedinto a powerfulcollection contactzone.But giventhe fact that the contactzoneof computerprogramsthat providethe geardesigner movesoverthe surfaceof the gear, one wouldneedawith an insightinto the state of stressin gearsthat has veryhighlyrefinedmeshalloverthecontactingthus far neverbeenpossible.Some of the featuresthat surface.Finiteelementmodelsrefinedto this extentCAPPsupportsare:friction,sub-surfacestress cannot be accommodatedon eventhe largest of today'scalculation,stresscontours,transmissionerror,contact computers.Compoundingthis difficultyis the fact thatpressuredistributionsand load distributioncalculation. thecontactconditionsareverysensitivetothe geometryof the contactingsurfaces.General purposeFigures1 to 5 showexamplesof gear sets for which finiteelementmodelscannotprovidethe requiredthis process has been successfullyused. levelof geometricaccuracy.Finally,the difficultiesof generatinganoptimalthree-dimensionalmeshthatCONTACT ANALYSIS can accuratelymodelthe stressgradientsin the critical regionswhileminimizingthe numberof degreesofIn earlierstudiesVijayakar1988,1989;Bathe 1985, freedomof themodelhavekeptthe finiteelementChowdhury1986of contactmodeling,a purefinite methodfrom beingwidelyusedto solve the completeelementapproachwas used to obtain complianceterms gear contact problem,relatingtraction at one locationof a body to the normal displacementat anotherlocationonthecontacting 1 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- A v Figure 4- Contactanalysisof a 90° crossedaxis external helicalgear set. Figure1- Contactanalysisof helical gears. Figure 5- Contact an_YSciSl ga9_tcrossedaxis external gearbody.Itbecameapparentthatinordertoobtain sufficientresolutionin the contactarea,the size of the Figure2-Contactanalysis of hypoidgears,finiteelementmodelwouldhaveto be inordinately large.Afiniteelementmeshthatis locallyrefined aroundthecontactregioncannotbe usedwhenthe contactzonetravelsover the surfacesof the two bodies. Otherresearchersworkingin the tribologyareade Mul1985,Seabra1987, Lubrecht1987 haveobtained compliancerelationshipsin surfaceintegralformby integratingthe Greensfunctionfor a pointloadon the surfaceof a half space(the Bousinesqsolution)overthe areasofindividualcellsdemarcatedon thecontact zone. Thismethodworkswell as longas the extentof thecontactingbodiesismuchlargerthanthe dimensionsof the contactzone,and the contactzoneis far enoughfromthe othersurfaceboundariesso that the twocontactingbodiesmaybe treatedas elastic halfspaces.Theseconditionsare, however,not satisfied by gears The approachthatis describedhere is basedon the assumptionthatbeyonda certaindistancefromthe contactzone,thefiniteelementmodelpredicts deformationswell.Theelastichalfspacemodelis Figure3- Contactanalysis of worm gears,accuratein predictingrelativedisplacementsof points Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- J=-6 nearthe contactzone.Undertheseassumptions,it is Apossibleto makepredictionsof surfacedisplacements_0 thatmakeuseof the advantagesof both,the finite velementmethod,aswellasthesurfaceintegral approach. Thismethodisrelatedto asymptoticmatching methodsthatarecommonlyusedto solvesingular perturbationproblems.Schwartz and HarperSchwartz 1971 haveusedsuch an asymptoticmatchingmethod cylinderspressedagainstan elasticcylinderin plane strain. In orderto combinethesurfaceintegralsolution withthefiniteelementsolution,areferenceor 'matching'interfaceembeddedin the contactingbody isFigure6- Computationalgrid in the contact zone of the used.Thismatchingsurfaceis farenoughremovedgears. fromthe principalpointof contact so thatthe finite elementpredictionof displacementsalongthis surface is accurateenough.At the same time, it is close enoughforce appliedat the locationp which is on the surface of to the principalpointof contact so that the effect of thethe gear.The superscripts(si) and(fe) on a term will finiteextentof the body does not significantlyaffect themeanthatthe termhas beencalculatedusingsurface relativedisplacementsof pointson thissurfacewithintegralformulaeandafiniteelementmodel, respectto pointsin the region of contact,respectively.Subscripts1 and2 willdenotegears number1 and2, respectively.Whenthissubscriptis Contactanalysisis carriedout in severalsteps.Theomittedinanequation,theequationwillbe first step is to lay outa grid at each contactzone.Thenunderstoodto applyto both the gears crosscomplianceterms betweenthe variousgrid points are calculatedusinga combinationof a surfaceintegralLetu(p;q)= -u(p;q).nbetheinwardnormal formof theBousinesqandCerrutisolutionsandthecomponentof the displacementvectoru(p;q), wheren finiteelementmodelof thecontactingbodies.Finallyis the outwardunit normalvectorat the point p. loaddistributionsandrigidbodymovementsare calculatedusinganalgorithmbasedon theSimplexThedisplacementu(rij;r)of a field pointr due to a methodVijayakar1988.load at the surface grid pointr can be expressedas: ll Inordertodiscretizethecontactpressure distributionthatis appliedonthesurfacesoftwou(rij;r) = (u(rij;r)-u(rij;q ) )+ u(rij;q ) contactinggearteeth,a computationalgridis set up. Figure6 showssucha computationalgridthat has beenwhereq is somelocationin the interiorof the body, set up in the contactzone of the gears.The entirefacesufficientlyremovedfromthe surface(Figure7). If the widthof one of the gears (gear 1) which is mappedontofirsttwo termsare evaluatedusingthe surfaceintegral _:_-1,+1 is dividedinto2N +1 slices.Nis a userformulaeandthethirdtermis computedfromthe selectablequantity.The thicknessof eachslice in the _finiteelementmodel,thenwe obtainthe displacement parameterspace isA_ = 2/(2N+1). For each slice j=-Ntoestimate: +N, a cross sectionof gear 1 is taken at the middleof the slice, and a pointis locatedon this slicethat approachesu(rij;r)(q ) = (u(Si)(rij;r)_u(Si)(rij;q) + u(fe)(rij;q) the surfaceof the matinggear (gear 2) the closest.This selectionis carriedout usingthe undeformedgeometry. If the separationbetweenthe two gears at this closestThe term in parenthesesis the deflectionof r with pointislargerthana userselectableseparationrespecttothe'referencepoint'q.Thisrelative tolerance,thenthe entiregear slice is eliminatedfromcomponentis betterestimatedby a localdeformation furtherconsideration.Otherwise,a setof gridcellsfieldbasedon the BousinesqandCerrutihalfspace identifiedby the gridcell locationindices(i,j), i = - Msolutionsthanby the finiteelementmodel.The gross to Mandthe positionvectorsris set upcentereddeformationof the bodydueto the fact that it is not a 1jhalf spacewill not significantlyaffect this term. On the aroundthisclosestpointof slice, j. The numberMiscontrary,theremainingtermu(fe)(rij;q)is notuserselectable.The dimensionof the gridcells in the profiledirectionAs is also user selectable,significantlyaffectedby local stressesat the surface.This is becauseq is chosento be far enoughbeneaththe Let u(p;q)denotethedisplacementvectoratthesurface.Thistermis thereforebestcomputedusinga locationq on a gear dueto a unit normalcompressivefiniteelementmodelof the body.The valueu(rij;r)(q) Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- O Jo 0.4of the finiteelementwerecreatedsemi-automatically. Only a sectorcontainingthree teeth of each gear was °2o0_1 0_2 053o._o.55 0.5_ 0_0._o.69 0_modelled,witheachtoothbeingidentical.The gear _p_c_c_,_(gear no. 1) and the pinion(gear no. 2, the smaller gear) werethenorientedinspaceaspertheassembly Figure14: Variationof sub-surfaceshear stress withdrawings,andthe analysiswascarriedoutfor each individualtime step. Figure 2 showsthe six tooth gear depth underthe pointof maximumcontact pressure,andpinionmodel.Sectoralsymmetryisusedto generatestiffnessmatricesfrom the stiffnessmatrix of manufacturethesegearshave many kinematicsettings,onetooth.Forthisparticulargear set, a threetooth Thesettingsarechosensuchthatthecontactzonemodel sufficesbecauseat the most two teeth contact at a remainsin the centerof the toothsurfaces as the gearstime.Figure15 showsthe surfacesof the threetooth rollagainsteachother.Aheuristicprocedureisgear.Figures16, 17 and18 showthe contactpattern availableto selectthe settings,but in practice,these(which is the locusof the contact zone as the gears roll settingshaveto be selectedafter a tediousiterativeagainst each other), for a gear torque of 240, 480 and 960 processinvolvingcuttingand testingactual gears. Evenin-lbs, respectively.Figures19 and 21 showviews of the so,it isverydifficultto predictthe actualcontactcontact zonewith contactpressurecontourson the gear stresses,fatiguelife, kinematicerrorsand other designfor two particularangular positions.Figures20 and 22 criteria,especiallywhennotinstalledinidealshowmagnifiedviewsof the contactzone for these two conditions.The contactstressesare so sensitiveto thepositions.Theyshowcontoursofnormalcontact actualsurfaceprofilethatconventional3-Dcontactpressureson the surfaces.Computationalgridsof 11x25 analysisis not feasible,cells were used on these surfaces to obtain the pressure distributions.Finally,thepositionof the pinionwas A sample90° hypoidgear set from the rear axle of aperturbedslightlyfromthedesignlocation,and the commercialvehiclewas selected.The gear ratio of thisFigures 23 (a) to (d) showthe contact patterns that were set was41:1I andthe axialoffsetwas 1.5 inches.Theobtained.When comparedto the contactpattern for the gear surfaceshad been experimentallyshownto be idealunperturbedpositionin Figure 16, it showsthat the best for this particulargear ratioandaxial offset.In othercontactpatterndoesindeedoccuratthedesigned words,the contactzonewasfoundto remainin theposition,lendingcredencetothenotionthatan centralportionofthegearteethin theoperationalanalysisof the kinddescribedin this paperhas the potentialto be used in the design process itself. Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- v Figure 16: The locus of the contact zone at a gear torque of 240 in-lbs.Figure 19: Contact pressurecontours for position1. Figure17: The locus of the contact zone at a gear torque of 480 in-lbs.Figure 20: Contactpressurecontours for position1. normal stresscalculatedat varioussectionsin a pair of contactinghelicalgearteeth. Figure 25 showscontour curvesofmaximumprincipalnormalstressdrawn along the surfaceof the gear tooth, and Figure 26 shows a contoursurfaceof maximumprincipalnormalstress withina gear tooth.It is also possibleto draw contour curvesand surfacesfor the minimumprincipalnormal stress and the Von Mises' octahedralshear stress.Figure 27 is an exampleof an arrowdiagramthat can be used to showboth the magnitudeas well as directionof the principalnormalstresses.Stressesare depictedby arrowspointinginthe principaldirections.Tensile _stressesare depictedby outwardpointingarrowsand compressivestressesare depictedby inwardpointing arrows.The lengthof anarrow is proportionalto the Figure18: The locus of the contact zone at a gear torque o