1、10FTM07AGMA Technical PaperA New Statistical Modelfor Predicting ToothEngagement and LoadSharing in InvoluteSplinesBy J. Silvers, C.D. Sorensen andK.W. Chase, Brigham YoungUniversityA New Statistical Model for Predicting Tooth Engagementand Load Sharing in Involute SplinesJanene Silvers, Carl D. Sor
2、ensen and Kenneth W. Chase, Brigham Young UniversityThe 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.AbstractLoad-sharing among the teeth of involute splines is little under
3、stood. Designers typically assume only afraction of the teeth are engaged and distribute the load uniformly over the assumed number of engaged teeth.This procedure can widely over- or underestimate tooth loads.A new statistical model for involute spline tooth engagement has been developed and presen
4、ted earlier, whichtakes into account the random variation of gear manufacturing processes. It predicts the number of teethengaged and percent of load carried by each tooth pair. Tooth-to-tooth variations cause the clearancebetween each pair of mating teeth to vary randomly, resulting in a sequential
5、, rather than simultaneous toothengagement. The sequence begins with the tooth pair with the smallest clearance and proceeds to pick upadditional teeth as the load is increased to the maximum applied load. The new model can predict the numberof teeth in contact and the load share for each at any loa
6、d increment.This report presents an extension of the new sequential engagement model, which more completely predictsthe variations in the engagement sequence for a set of spline assemblies. A statistical distribution is derivedfor each tooth in the sequence, along with its mean, standard deviation a
7、nd skewness. Innovative techniquesfor determining the resulting statistical distributions are described. The results of an in-depth study are alsopresented, which verify the new statistical model. Monte Carlo Simulation of spline assemblies with randomerrors was performed and the results compared to
8、 the closed-form solution. Extremely close agreement wasfound. The new approach shows promise for providing keener insights into the performance of splinecouplings and will serve as an effective tool in the design of power transmission systems.Copyright 2010American Gear Manufacturers Association500
9、 Montgomery Street, Suite 350Alexandria, Virginia, 22314October 2010ISBN: 978-1-55589-982-03A New Statistical Model for Predicting Tooth Engagementand Load Sharing in Involute SplinesJanene Silvers, Carl D. Sorensen and Kenneth W. Chase, Brigham Young UniversityIntroductionSplined shafts are preferr
10、ed over keyed shafts fortransmitting heavy torque in industrial andautomotive applications. The splined shaft andmating hub have matching sets of teeth over the fullcircumference, as shown in Figure 1. If the toothloads were distributed uniformly around the circum-ference, each tooth would carry an
11、equal share.However, due to manufacturing variations, the toothclearance between each pair of teeth varies, so theteeth do not engage all at the same time. Thus, theload is not shared uniformly.In practice, as the shaft is turned, the tooth pair withsmallest clearance gap will make contact first and
12、begin to carry the load. As the torque increases, thefirst tooth deflects enough for a second pair, with thenext smaller clearance, to engage and begin toshare the load. This process of sequential engage-ment continues with increasing load until the fullload is applied. The full load is generally no
13、tsufficient to engage all of the teeth, so some teethwill not carry any load.As a result of this sequential engagement, the firstpair of teeth to engage will carry more of the load,causing the first tooth to be most likely to fail. Eachtooth in succession will carry a smaller share. Themotivation be
14、hind this research is to permitdesigners to accurately predict the tooth loadingand avoid spline failure.Previous workTooth engagement is driven by deflection: as theforce increases and the engaged teeth deflect,clearance gaps between other tooth pairs close andadditional tooth pairs engage. This de
15、flection canbe described using a strength of materials deflec-tion model, as was done by DeCaires 1. His modelencompasses deflection due to shear, bending, andcontact forces, and was verified by FEA. Figure 2shows a force-deflection curve calculated using thismodel.Each tooth is modeled as a spring,
16、 which acts inparallel with the other teeth. When multiple pairs ofteeth are engaged, the stiffness, Ki, of each add to-gether, which can be seen in the figure. At eachdata point, another pair of teeth engages, changingthe slope incrementally.Figure 1. External and internal spline teeth4Combined loa
17、d for multiple teethFigure 2. Force vs. deflection curve demonstrating sequential tooth engagementThe amount of load carried by a given tooth can befound by extending the slope of each segment of thegraph, as shown, and then measuring the verticaldistance between segments at the deflection valuecorr
18、esponding to the applied force. The first tooth toengage, hereafter referred to as Tooth 1, carries thelargest load.DeCaires model can be used to determine thepercentage of the total load carried by Tooth 1. Thepercentage is compared to the number of teethengaged in Figure 3. Tooth 1 always carries
19、a largerpercentage of the total load than any other teeth.Although the total load on Tooth 1 continues toincrease, the percent of the total decreases due toload sharing by an increasing number of teeth.Models for tooth clearance variationMultiple sources of error are present in toothmanufacturing fo
20、r both the internal and externalteeth, so the resulting tooth clearance is a combina-tion of several random variables. Therefore, anormal distribution of tooth clearances is a reason-able assumption. This distribution is shown inFigure 4. Note that more clearances are clusterednear the middle of the
21、 distribution and spread outnear the tails. The teeth are self-sorting in order ofincreasing clearance-teeth will engage in order,from the smallest clearance to the largestclearance, regardless of their location in theassembly.Figure 3. Percent of load carried by Tooth 1as subsequent teeth engage5Fi
22、gure 4. Normal distribution of tooth clearancesMapping modelOne method of predicting the clearance variation isthe mapping model, shown in Figure 5 for a 10-toothspline. A uniform distribution is plotted on the y-axisand divided into 10 equal intervals. The center pointof each interval is projected
23、horizontally across tointersect the normal cumulative distribution function(CDF), then vertically down to the x-axis. Theresulting distribution on the x-axis is normal. Thismodel predicts the mean, or most likely, clearancevalues for the first tooth to engage, the second toothto engage, and so on.Cl
24、earances of teeth determined fromnormal cumulative distribution functionFigure 5. Mapping model to predict toothclearance for a 10-tooth splineThe horizontal axis has units of standard deviation.If the several process errors are known from inspec-tion data, or estimated from previous experience,thei
25、r standard deviations may be added byroot-sum-squares to estimate the resultant clear-ance standard deviation. The horizontal axis maythus be scaled to the corresponding dimensionalclearance.Using the CDF to transform one distribution into an-other is an established procedure 2. In this case,the cen
26、ter point of each increment, when mappedacross and down to the horizontal axis, locates themost likely clearance for each tooth pair. In reality,the clearance of Tooth 1, Tooth 2, etc. is subject torandom variation about the most likely value. This isan important aspect of tooth engagement, which is
27、complex statistically. It is dealt with in the nextsection.Probabilistic tooth engagement modelIf two splined shafts are assembled to two hubs, theresulting clearances will be similar, but not exactlythe same. Since the process variations are random,each assembly will have slightly different clear-a
28、nces, just as no two snowflakes are exactly alike.Furthermore, if a spline is disassembled, the shaftis rotated a couple of teeth, and reassembled, all theclearances are rearranged. This means that boththe force-deflection curve and the load carried by6Tooth 1 vary for each spline assembly and for e
29、achalternate meshing of the same spline-hub pair.Consequently, neither can be defined for all cases,but they can be predicted by statistically modelingthe expected clearance variation.For a given spline assembly, the clearance for everytooth pair could be measured in theory, althoughthis would be di
30、fficult in practice. As a large numberof spline assemblies is measured, a histogram forthe clearance of each tooth pair can be developed,and will approach a continuous distribution.In this paper, a more compete statistical model topredict tooth clearances is developed and valid-ated. This model is c
31、alled the Probabilistic ToothEngagement Model, or ProTEM. While themapping model predicts the mean or averageclearance for each tooth pair, ProTEM also predictsthe complete probability distribution of clearancesfor each pair of teeth. The ranges of predictedclearance values are shown for a 10-tooth
32、spline inFigure 6. Each tooth in the sequence has its owndistribution which is different from the normaldistribution.ObjectiveThe objective of this paper is to evaluate the Prob-abilistic Tooth Engagement Model (ProTEM). AMonte Carlo Simulation was used to model severalthousand spline assemblies and
33、 generate a distri-bution of the tooth clearance for each tooth in thesequence. The results were compared to themapping model and to ProTEM.MethodsOverview of ProTEMProTEM combines three related probabilities fortooth clearance. To calculate the probability ofTooth j on an N-tooth spline having clea
34、rance x,itcombines the probabilities that there are:S j-1 teeth with a clearance less than x,S one tooth with a clearance equal to x, andS N-j teeth with a clearance greater than x.Using these probabilities, a probability densityfunction, PDF, is developed for each tooth as a func-tion of x. The PDF
35、 for the jthtooth to engage on anN-tooth spline, h (x, j, N), is defined in Equation 1.Predicted clearances on a 10-tooth splineFigure 6. Tooth clearance predicted by ProTEM for a 10-tooth spline7 (1 (x)Njg(x)hx, j, N=N!(j 1)! (N j)!(x)j1(1)whereg(x) is the probability density function for an indi-v
36、idual tooth clearance; (x) is the cumulative density function, CDF, foran individual tooth clearance.Raising (x) to the power of j-1 gives the probabilitythat j-1 teeth have a clearance smaller than x.Like-wise, raising 1-(x) to the power of N-j gives theprobability that N-j teeth have a larger clea
37、rancevalue than tooth js clearance, x. The first term in thePDF accounts for equivalent permutations, be-cause tooth j,thejthtooth to engage, does not haveto be in any specific position on the spline.In this study, the various tooth clearances areassumed to be independent and to be normally dis-trib
38、uted. The PDF and CDF for a normal distributionare shown in Figure 7 where the height of the PDFcurve at x is g(x). The probability of a tooth having aclearance between x and x + dx is given by g(x) dx.For a normal distribution g(x) is expressed inequation 2.g(x) =1 2 e (xm )22 2(2)Normal probabilit
39、y density function, PDFNormal cumulative distribution function, CDFFigure 7. Manufacturing variation PDF and CDF, as used to determine probabilities in ProTEM8The red area under the PDF to the left of theclearance value, x, gives the probability of a toothhaving a clearance smaller than x. The blue
40、area un-der the PDF to the right of x gives the probability thata tooth has a clearance larger than x. Calculatingthese probabilities requires integration of the PDF.To save repeatedly integrating for each probabilityvalue, the CDF, plotted below the PDF curve,evaluates the shaded area under the PDF
41、 curve asa function of x. At a given value of x, the height of theCDF curve, (x), gives the area integral from -1tox, and represents the probability of a tooth clear-ance smaller than x. The height above the curve is1-(x), and represents the probability of a toothclearance larger than x. The total a
42、rea under thePDF is equal to 1.0, which corresponds to the sumof (x) and 1-(x), as shown in Figure 7.Any PDF, whether a normal or other distribution,may be characterized in terms of its area moments.The first moment, or “centroid” of the area under thecurve, is the mean value. The second moment, the
43、variance, or the standard deviation squared,describes the range of the distribution. The thirdmoment describes the skewness, or asymmetry.The fourth moment describes the kurtosis,orpeakedness. The kthmoment for the distribution oftooth j on an N-tooth spline can be found using themoment-generating f
44、unction, which integrates theproduct of xkand the PDF, h (x, j, N), as shown inEquation 3 3.Exkj= xkh(x, j, N) dx (3)Using this function, the first moment, or mean, isdefined in Equation 4, with k=1. (1 (x)Njg(x)dxExj= x N!(j 1)!(N j)!(x)j1(4)The 2nd,3rd, and 4thmoments are calculated usingcorrespon
45、ding k values. The moment generatingfunction is not closed form, so each of the momentsmust be calculated numerically.Because it defines the PDF, ProTEM gives a rangeof clearance values rather than just a mean value.The PDF for Tooth 1 on a 10-tooth spline is shown inFigure 8. Looking at the PDF, it
46、 is evident that themost likely clearance, the peak value, is not themean clearance; the distribution is skewed to theleft (the direction of the longer tail).Probability density for clearance of first tooth on 10-tooth splineFigure 8. tooth clearance predicted by ProTEM for Tooth 1 on a 10-tooth spl
47、ine9In order to generalize the results, ProTEMcalculates probabilities for clearances in terms of astandard normal distribution. The results are pro-duced in standard deviation coordinates, meaningthat a value of -1.54 corresponds to a clearancevalue 1.54 standard deviations below the meanclearance.
48、 By substituting the manufacturing meanand standard deviation, the results can be scaled tofit each manufactured data set.In ProTEM, each tooth distribution is skewed awayfrom the global mean clearance, or the manufactur-ing clearance mean, as shown in Figure 6 for theexample of a 10-tooth spline.Mo
49、nte Carlo simulationMonte Carlo Simulation (MCS) was used to verifythe ProTEM model. In this simulation, sets of toothclearances were generated with normallydistributed random errors and the resulting distribu-tions for Tooth 1, Tooth 2, etc. were compared toProTEM.After generating 100,000 sets of N toothclearances, each set, representing an individualspline assembly, was sorted in order of increasingclearance to determine the tooth engagementorder. The clearance data was then groupedaccording to the tooth engagem
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