1、09FTM17AGMA Technical PaperVariation Analysis ofTooth Engagement andLoad-Sharing inInvolute Splinesby K.W. Chase, C.D. Sorensenand B. DeCaires, Brigham YoungUniversityVariation Analysis of Tooth Engagement and Load-Sharing inInvolute SplinesKennethW.Chase,CarlD.SorensenandBrianDeCaires,BrighamYoungU
2、niversityThe statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractInvolute spline couplings are used to transmit torque from a shaft to a gear hub orother rotatingcomponent.
3、Externalgearteethontheshaftengageanequalnumberofinternalteethinthehub. Becausemultipleteethengage simultaneously, they can transmit much larger torques than a simple key and keyway assembly.However,duetomanufacturingvariations,theclearancebetweeneachpairofmatingteethvaries,resultinginonly partial en
4、gagement.A new model for tooth engagement, based on statistics, predicts that the teeth engage in a sequence,determinedbytheindividualclearances. Astheshaftloadisapplied,thetoothpairwiththesmallestclearanceengagesfirst,thendeflectsastheloadincreases,untilthesecondpairengage. Thetwoengagedpairsdeflec
5、ttogetheruntilthethirdpairengage,andsoon,untilthefullloadisreached. Thus,onlyasubsetofteethcarrytheload. Inaddition,theloadisnon-uniformlydistributed,withthefirsttoothcarryingthebiggestshare. Asaconsequence, the load capacity of spline couplings is greatly reduced, though still greater than a single
6、keyway.Thestatisticalmodelpredictstheaveragenumberofteethwhichwillengageforaspecifiedload,plusorminustheexpectedvariation. Italsoquantitativelypredictstheloadandstressineachengagedpair. Criticalfactorsin the model are the stiffness and deflection of a single tooth pair and the characterization of th
7、e clearance.Detailed finite element analyses were conducted to verify the tooth deflections and engagement sequence.The closed form statistical results were verified with intensive Monte Carlo simulations.Themoreaccuratemodelhasleadtoincreasedunderstandingofthemechanicsofinvolutesplinecouplingsand s
8、hould permit better prediction tools for designers and improved performance of their designs.Copyright 2009American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314September 2009ISBN: 978-1-55589-970-73Variation Analysis of Tooth Engagement and Load-Sharing i
9、n Involute SplinesKenneth W. Chase, Carl D. Sorensen and Brian DeCaires, Brigham Young UniversityIntroductionAn involute spline coupling consists of a shaft withmachined gear teeth on its exterior, mated to a hubwith a matching set of interior teeth, as shown inFigure 1. They are found in gear train
10、s, transmis-sions, pumps, and many other rotating machines.The involute profile makes them self-centering.Thetransmittedtorqueis distributedover theteeth,decreasingtheloadonanyonetooth. Itisthereforesuperior to a single key and keyway assembly, andleads to lighter, more efficient shaft designs.In th
11、eory, the full ring of gear teeth on the shaftengage with an equal number of teeth in the hub,resulting in the load equally distributed over all theteeth. In practice, only a fraction actually transmitthe load. Due to manufacturing variations, evenwith precision gear hobbing processes, only afractio
12、n of spline teeth engage. Splines, therefore,perform far below their theoretical capacity.Designers commonly assume only 1/4 to 1/2 of theteethcarry thefullload. They also approximatetheload as uniformly distributed among the load-bearingpairsofteeth. Theseassumptionsareoftensatisfactory, but can le
13、ad to early failures whenapplied to high load applications.Consider an extreme application the multi-diskbrakesonanindustrialminingdumptruck,showninFigure 2. The tires on this behemoth are 13 ft. indiameter. Thedriverclimbstwostoriestothecab. Ithauls loads up to 380 tons over challengingterrain.Stop
14、ping at any speed punishes the brakes.a) External tooth splineb) Internal tooth splineFigure 1. Involute splinesFigure 2. Industrial mining truck (courtesy Caterpillar, Inc.)4The multi-disk brake assembly consists of a set offriction plates, separated by pressure plates, asshown in Figure 3a. Fricti
15、on plates are splined onthe inside circumference and engage the shaftspline. Pressure plates are splined on the outsidecircumference and engage the non-rotating hub.Thebrakeis actuatedby aringof hydraulicpistons,which clamp the pressure plates to provide brakingforce. The piston cylinders can be see
16、n on thebrake assembly in Figure 3b.Thesplinecouplingsmust transmitthehighbrakingforces to the hub. Examination of failed platesrevealed cracks at the base of the teeth, anindication of bending fatigue failure. Failed toothfragmentscanjambetweentheplatesorblockcool-ing passages, which may lead to co
17、mplete failure.Also, uneven wear was observed, suggestingunequal tooth loads.A desire for a better understanding of themechanics of tooth engagement led to the currentstudy. Concerns focus on the effect of variation onthe load distribution within an involute spline joint.Insights leading to more rel
18、iable designs weresought.The objectives of the research reported hereinclude:S Develop a statistical model to predict toothengagement and loads;S Investigate the effects of tooth clearancevariations on spline performance;S Estimate tooth load sharing and stresses;S Verify the tooth engagement model
19、with MonteCarlo simulations;S Verify the loads with finite element simulations;S Determine the effects of spline designparameter combinations;S Develop software for analysis and design ofspline couplings.Previous studies of splines have investigated:S Splinestandardsanddesign;1, 2,3, 4,6,8, 9, 10S P
20、rocess error sources and resultant tootherrors; 2, 8, 11, 12S Tooth stresses and deflections; 3, 7, 11,13, 17S Spline tooth engagement; 2, 8, 11S Tooth load distributions. 2, 16Many studies of deflection and stress in gear teethhave been published, with valuable results. Butspline couplings, althoug
21、h they share involutegeometry with gears, are a very different applica-tion. Gear designers do not seek simultaneouscontact between all the pairs of teeth. For splinedesign, this is the goal, but it is not possible due totoothvariations. Thereis,as yet,noquantitativeal-gorithm for predicting tooth e
22、ngagement in splinesresultingfrommanufacturingprocesserrors. Witharealistic analytical tool, which includes all criticalspline parameters, as well as realistic estimates oftooth errors, designers may be able to find anoptimum combination which significantly improvesspline performance.a) CAD modelb)
23、Actual brakeFigure 3. Multi-disk brake assembly (courtesy Caterpillar, Inc.)5The results of this study include:S The effects of tooth errors on mating toothclearances;S Statistical characterization of clearancevariations;S Prediction of tooth engagement;S Sequence of tooth engagement vs. load;S Stre
24、ss in each toothresulting from non-uniformtooth loads.Analytical modelTooth errors/tooth clearanceThe natural variation in the spline manufacturingprocess leads to non-uniform clearance betweenpairs of mating teeth. Three common sources ofvariation for this study were suggested by a gearmanufacturin
25、g expert: index error, profile error andtooth thickness error 12, 15. Both the internaland external teeth are subject to all three. With sixsources of error, it may be assumed that the clear-ance variation would approximate a normal orGaussian distribution, as shown in Figure 4. 14Notethatthemajorit
26、y ofteethareclusteredclosetothe mean clearance, and spread out in the tails.Arequirementforsmoothbrakingactionofthetruckbrakesisthattheclearancebesufficienttoallowthepressureplatestoslideaxially. Extraclearanceanda decreased pressure angle are provided for thisapplication.Leaderror is anothersourcef
27、orlongsplineteeth. Itdid not contribute in this application because thebrakeplatesarethinandadjustindependently. Theshaft and hub centerlines were also assumed toremain in alignment due to their stiffness and self-centering geometry.Tooth engagementThe realization that tooth clearance varies fromtoo
28、thtotooth,ledtoanewstatisticalmodelfortoothengagement 5. Consider a shaft and hubassembly. As the shaft rotates to engage the mat-ing internal teeth, the clearance is reduced to zero.However, clearance between all tooth pairs doesnotgotozerosimultaneously. Thepairofteethwiththe smallest clearance en
29、gages first and begins totransmit thetorqueload. As theloadincreases,thetoothpairdeflect,untilthesecondpairengage. Theload continues to increase, causing both to deflectuntil a third pair engage, and so on, until the fullapplied load is reached.Figure 4. Normal distribution of tooth clearance6Thus,
30、tooth engagement is a sequential process,sorted in order of increasing clearance. However,due to the random natureof theprocess errors, theteethdonot engageinnumericalorder. Rather,theengagement sequence will be as random as theclearances.Tooth stiffnessTheresultant toothloads dependonthestiffness o
31、fa tooth under load. Three modes of deflectioncon-tribute: shear, bending, and contact, as shown inFigure 5.Figure 5. 3 modes of tooth deflectionFor small deflections, spline teeth may be modeledas a linear springs, as shown in Figure 6. When atooth on the shaft engages a tooth on the hub, theycombi
32、neas springs inseries. Their stiffnesses addas reciprocals. When two pairs of engaged teethshare the load, their stiffnesses add linearly. Thus,as teeth engage sequentially, the combinedstiffness increases incrementally.a) Series springsb) Parallel springsFigure 6. Tooth stiffness modelsA plot of th
33、e resulting force-deflectioncurve for thespline assembly, Figure 7, shows a steepeningcurve,composedofstraight,linearsegments. Eachchange in slope is the result of another pair cominginto engagement to share the load. At some point,the full applied load is reached, and the number ofengaged pairs is
34、determined. Thus, the number ofpairsthatengage,andtheirengagementsequence,is a complex interaction between the applied load,the tooth stiffness, and the clearance magnitudeand variation.Figure 7. Force-deflection plot due to sequential tooth engagement7NotethatthedeflectionfromTooth1to2,2to3,3to4, e
35、tc., gets smaller as each new tooth engages.This is due to the normal distribution of theclearance,showninFigure4. Toothclearancesareclustered more closely near the mean of theclearance distribution and spread out in the tails.Of course, the tangential force on the toothproduces a torque, and the ta
36、ngential deflectionproducesarotation. Bothforceanddeflectionactatthe same radius, hence, the force vs. deflectioncurveisequivalenttoatorquevs.angulardeflectioncurve.Load-sharingAsignificantresultofsequentialengagementisthatload sharing between the teeth is not uniform. InFigure 8, the force-deflecti
37、on curves for severalpairsofteethareshownastheyengagesequential-ly. Thefirstpairtoengagestartsatzeroandtheloadincreases withslopeK1. As theloadincreases,thefirst pair deflects until the second engages. Bothpairssharetheloadanddeflecttogetherwithacom-bined stiffness K1+K2, as shown in Figure 7. Thene
38、xt pair to engage increases the stiffness toK1+K2+K3.A vertical line in Figure 8 intersects all three force-deflection curves. The total load is F1+F2+F3.The loads are unequal because the first pair de-flected1,thesecond2,andthird3. Thefirstpaircarries a bigger share, because it has deflectedfurther
39、.The load-sharing plot, Figure 9, shows the load isnot uniformlydistributed. Thefirsttoothpaircarries18%of the load, with eachsuccessive pair carryinga smaller percentage. The curve is not smooth,because of the random variation in clearance.Figure 8. Force-deflection plot for individual teethFigure
40、9. Load sharing plot for 10 engaged teeth8Strength of materials model (SMM)Tooth stiffness is the key factor in an analyticalmodel of tooth engagement. The strength ofmaterials modelrepresents each toothas astubby,cantilever beam, with a tapered cross section,subject to a load acting normal to the s
41、urface. Theinvolute profile is approximated by a trapezoid, asshowninFigure10. Filletsareneglectedinthestiff-ness calculations, but are included in the stresscalculations. Deflections due to contact stress arealso neglected in the stiffness calculations.Figure 10. Simplified tooth geometryTheloaddoe
42、snotmovefromroottotip,asoccursininvolute gearing, but remains stationary, near thepitch diameter. The normal force, Fn, is resolvedinto tangential and radial components, as shown inFigure 11. The tangential force, Ft, produces bothbending and shear deflections and stresses. Theradial force, Fr, prod
43、uces an radial force and areverse bending moment, as shown in Figure 11.The stress distribution at the base of the tooth isshown for each component of load, and the result-ant combined stress distribution, in Figure 12. Thestress and deflections are all calculated in closedform,forinstantresults,aso
44、pposedtoconductingafull finite element analysis.Map uniform tooth spacing to a normaldistributionAs stated previously, the combination of severalsources of process variation, for both the externalandinternalsplinesismodeledasanormaldistribu-tion. The resultant variation in clearance isclustered abou
45、t the mean, becoming sparseapproaching the tails.Figure 11. Equivalent loads acting on a toothFigure 12. Total stress at the base of the tooth9Figure 13 illustrates a procedure for mapping auniformly distributed clearance onto a normal dis-tributionforaseventoothspline. Theverticalaxisisdividedintos
46、evenequalintervals, withthecenterofeach interval located and projected horizontally,until it intersects the Cumulative DistributionFunction(CDF)forthestandardnormaldistribution.At the intersection point, project down to the hori-zontal axis to determine the clearance for eachtooth. The Probability D
47、ensity Function (PDF) isplotted below the (CDF) to illustrate the resultingnormal distribution.Notethat theCDF goes toinfinity at y = 0andy = 7,but, since Tooth 1 and Tooth 7 are one half intervalinboard, the extreme values of clearance arealways finite. Also, note that the clearance alwaysincreases
48、 going from Tooth 1 to 7, but the changegets smaller as you approach the mean, thenincreases again after the mean is passed.These values of clearance represent the average,ormeanvaluesforTooth1,Tooth2,andsoon,upto7. But,ifclearancesweremeasuredfor1000splineassemblies, there would be a variation abou
49、t themean value for Tooth 1 through 7, resulting in a setof distributions, clustered about the mean clear-ance. A follow-on paper on this topic is mentionedin Appendix A.Themappedmethodgivesanadequateestimateofthenumberof teethengagedforany specifiedload,as well as the load sharing.VerificationTheSMMmodelfortoothdeflectionsandstresshasbeen verified by Finite Element Analysis (FEA).Finely meshed models were analyzed for severalcases of differing tooth sizes and parameters.Single pairs of engaged teeth were anal
