10FTM07

AGMA Technical Paper

A New Statistical Modelfor Predicting ToothEngagement and LoadSharing in InvoluteSplinesBy J. Silvers, C.D. Sorensen andK.W. Chase, Brigham YoungUniversity

A New Statistical Model for Predicting Tooth Engagementand Load Sharing in Involute Splines

Janene Silvers, Carl D. Sorensen and Kenneth W. Chase, Brigham Young University

[The 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 understood. 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 presented 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, 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 load 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 and 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 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 2010

American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314

October 2010

ISBN: 978--1--55589--982--0

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A New Statistical Model for Predicting Tooth Engagementand Load Sharing in Involute Splines

Janene Silvers, Carl D. Sorensen and Kenneth W. Chase, Brigham Young University

Introduction

Splined shafts are preferred 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 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 andbegin 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 notsufficient 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 behind this research is to permitdesigners to accurately predict the tooth loadingand avoid spline failure.

Previous work

Tooth 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 deflection 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, 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 teeth

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Combined load for multiple teeth

Figure 2. Force vs. deflection curve demonstrating sequential tooth engagement

The 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 valuecorresponding 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 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 variation

Multiple sources of error are present in toothmanufacturing for 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 clustered

near the middle of the 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 engage

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Figure 4. Normal distribution of tooth clearances

Mapping model

One method of predicting the clearance variation isthe mapping model, shown in Figure 5 for a10-toothspline. A uniform distribution is plotted on the y-axisand divided into 10 equal intervals. The center pointof each interval is projected 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.

Clearances of teeth determined fromnormal cumulative distribution function

Figure 5. Mapping model to predict toothclearance for a 10--tooth spline

The horizontal axis has units of standard deviation.If the several process errors are known from inspec-tion data, or estimated from previous experience,their 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 center 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 iscomplex statistically. It is dealt with in the nextsection.

Probabilistic tooth engagement model

If 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-ances, 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 by

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Tooth 1 vary for each spline assembly and for eachalternate 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 difficult 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 called 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 spline inFigure 6. Each tooth in the sequence has its owndistribution which is different from the normaldistribution.

Objective

The 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 generate a distri-bution of the tooth clearance for each tooth in thesequence. The results were compared to themapping model and to ProTEM.

Methods

Overview of ProTEM

ProTEM combines three related probabilities fortooth clearance. To calculate the probability ofTooth j on an N-tooth spline having clearance 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, and

S 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 for the jth tooth to engage on anN-tooth spline, h (x, j, N), is defined in Equation 1.

Predicted clearances on a 10--tooth spline

Figure 6. Tooth clearance predicted by ProTEM for a 10--tooth spline

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× (1− φ(x))N−jg(x)

h x, j, N =N !

(j− 1)! (N− j) !φ(x)j−1

(1)

where

g(x) is the probability density function for an indi-vidual tooth clearance;

φ (x) is the cumulative density function, CDF, foran individual tooth clearance.

Raising φ(x) to the power of j-1 gives the probabilitythat j-1 teeth have a clearance smaller than x. Like-wise, raising 1--φ(x) to the power of N-j gives theprobability that N-j teeth have a larger clearancevalue than tooth j’s clearance, x. The first term in the

PDF accounts for equivalent permutations, be-cause tooth j, the jth tooth 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-tributed. 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)= 1σ 2 π

e−(x−m)2

2 σ2 (2)

Normal probability density function, PDF

Normal cumulative distribution function, CDF

Figure 7. Manufacturing variation PDF and CDF, as used to determine probabilities in ProTEM

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The 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 areaun-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 curve asa function of x. At a given value of x, the height of theCDF curve, φ(x), gives the area integral from --1 tox, and represents the probability of a tooth clear-ance smaller than x. The height above the curve is1--φ(x), and represents the probability of a toothclearance larger than x. The total area under thePDF is equal to 1.0, which corresponds to the sumof φ(x) and 1--φ(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, thevariance, 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 kth moment for the distribution of

tooth j on an N-tooth spline can be found using themoment-generating function, which integrates theproduct of xk and the PDF, h (x, j, N), as shown inEquation 3 [3].

E xkj =

∞

−∞

xk h(x, j, N) dx (3)

Using this function, the first moment, or mean, isdefined in Equation 4, with k=1.

× (1− φ(x))N−jg(x) dx

E x j = ∞

−∞

x N !(j− 1)!(N− j) !

φ(x)j−1

(4)

The 2nd, 3rd, and 4th moments are calculated usingcorresponding 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 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 spline

Figure 8. tooth clearance predicted by ProTEM for Tooth 1 on a 10--tooth spline

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In 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. 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.

Monte Carlo simulation

Monte 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 engagement sequence withall the data for tooth j in a set, resulting in N sets of100,000 clearances each. Each set revealed askewed distribution, with its mean value shifted to

the left or right of the global mean clearance, form-ing a distribution of distributions, for comparison tothe ProTEM solution, as shown in Figure 6.

Distribution from Monte Carlo data

Two different methods were used to generate aCDF from the MCS data. The first method plottedthe clearance data as a discrete CDF, from whichthe PDF was obtained by numerical differentiation.The second method calculated the first fourmoments of the MCS data, which were fitted to ageneral skewed distribution.

MCS-Generated Discrete CDF

To generate a discrete CDF from MCS data, it isnecessary to assign a percentile (or cumulativeprobability value) to each data point. This is done byfirst sorting the data in increasing order. A percent-ile value is then calculated for point i in a set of Npoints from equation 5.

P i=i− 0.5

N(5)

This formula places the percentile of the smallestclearance in the sample at 1/(2N), and thepercentilefor the largest clearance in the distribution at1-1/(2N).

A discrete CDF is then determined by plotting thepercentile values Pi as a function of clearance. Thismethod is shown for 20 sample assemblies inFigure 9. The horizontal axis is in standarddeviation units of the global clearance distribution.

CDF of first tooth for 20 samples

Figure 9. CDF for Tooth 1 developed from MCS using 20 sample assemblies

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It is clear from this figure that the discrete CDF is notsmooth. As more samples are taken, the smooth-ness will increase. Nevertheless, due to thediscrete nature of this distribution, it will never beperfectly smooth.

Figure 10 shows the CDF plotted in the samemanner using 100,000 samples. This provides asmoother curve. Also plotted on the same graph isthe CDF of a standard normal distribution. Fromthis we can see that the clearance distribution ofTooth 1 is not normal.

In order to generate a PDF, the numerical derivativeof the CDF was taken. Initially, a central-differencemethod was used to take the derivative, as shown inFigure 11. Although the CDF in Figure 10 appears

smooth, taking the derivative amplifies the noisethat is present. In order to reduce the noise, a fivepoint-moving average was taken to smooth theCDF, and then the PDF was generated by fitting aparabola to a local segment of the CDF and takingthe analytical derivative of the parabolaat thecenterof the segment. This creates less noise than a finitedifference approach, as shown in Figure 11.

This procedure--creating a CDF from the MCS data,smoothing the curve, and taking the derivativeusinga parabolic fit--was repeated for each tooth on a10-tooth spline. The results are shown in Figure 12.The similarity to ProTEM (see Figure 6) is evident:both sets of PDF are skewed away from the overallclearance mean, and the means and spreadsappear comparable.

CDF of first tooth for 100,000 samples

Figure 10. CDF for Tooth 1, developed from MCS using 100,000 samples compared to a normalCDF with the same mean and standard deviation as in the MCS

Figure 11. Finite difference approach to taking numerical derivative compared to a parabolic fit

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Probability density functions based on Monte Carlo data

Figure 12. Tooth clearance found through MCS for a 10-tooth spline

Fitting a distribution to calculated moments

Generating a CDF, smoothing it, and taking thederivative to obtain the PDF, required significantcomputation, so another method was explored.Rather than directly computing a PDF, an analyticalprobability distribution was fit to the moments of thedistribution. Three distributions were compared: anormal distribution, a generalized gamma distribu-tion, and a lambda distribution.

A normal distribution was fit using the first twomoments: the mean and standard deviation.However, it does not include skewness, which isvisible in the numerically determined PDF. The gen-eralized gamma distribution uses the first fourmoments to fit the distribution, but the accuracy ofthe fit varied based on the overall clearance meanthat was chosen. (The mean had to be shifted fromzero because the generalized gamma distribution isonly valid for x-values greater than zero.)

The lambda distribution was also fit using the firstfour moments. It provides a close fit when com-pared to the numerically-generated PDF, as shownfor Tooth 3 in Figure 13 and was the best distributionfound to model the MCS data [4].

When comparing the MCS distribution to theProTEM distribution, the lambda fit to the MCS datawas used.

Tooth 3

Figure 13. Comparison of lambda fit to MCSdata and numerically generated PDF for Tooth

3 on a 10-tooth spline

Results

Raw data

The mean clearances were found using eachmethod of predicting and simulating. The resultsare compared for a 10-tooth spline in Figure 14.The MCS results agree almost exactly withProTEM. The mapping model is close, but themean value deviates slightly higher for the first toothand the last tooth to engage. The deviation in thefirst tooth is the most likely to be significant,because the first tooth is the most highly-loadedtooth on the spline.

The mean clearance of the Tooth 1, the most signi-ficant value, was compared between the threemethods as the number of teeth on the spline was

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increased. The results are shown in Figure 15.MCS and ProTEM agree almost exactly, while themapping model gives consistently higher results,underestimating the clearance of the first tooth.

The standard deviation of the tooth clearances fromProTEM and MCS are shown in Figure 16. The res-ults agree favorably. The mapping model does notpredict the standard deviation.

Table 1 compares numerical results for the first fourmoments predicted by ProTEM to the momentsfound through MCS for each tooth on a 10-toothspline. The results closely match each other. Tooth1 and Tooth 7 are singled out as typical examples.

Table 2 compares the same moments for the first 10teeth to engage on a 100-tooth spline. Again, theresults compare favorably. Table 3 demonstratesthat for the mean clearance, the difference betweenthe ProTEM model and the MCS model is less than0.5%. For the standard deviation, the difference isless than 1.5%. For the third moment, most of thedifferences are less than 3%, although one variesby 13% (but it is worth noting that all of the third mo-ments are very small). For the fourth moment, thedifferences are less than 5%. Given the randomnature of the MCS process, these results serve toconfirm the validity of the ProTEM model.

Predicted mean clearances for 10 teeth

Figure 14. Comparison of mean clearances from the mapping model, ProTEM, and MCS for eachtooth on a 10-tooth spline

Predicted mean clearance of first tooth

Figure 15. Comparison of mean clearances for Tooth 1 from the mapping model, ProTEM, andMCS as the number of teeth in the spline is increased

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Standard deviation of 1st tooth clearance

Figure 16. Comparison of standard deviation of the clearance for Tooth 1 from ProTEM, and MCSas the number of teeth in the spline is increased

Table 1. First four moments compared for a 10 tooth spline

Table 2. First four moments compared for 10 teeth of a 100 tooth spline

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Table 3. Quantitative differences of moments between ProTEM and MCS distributions.

MomentDisagreement between ProTEM and MCS10 tooth spline 100 tooth splineTooth 1 Tooth 7 Tooth 1 Tooth 7

m --0.05% --0.36% --0.05% --0.03%m2 --1.30% --0.20% 0.21% 0.04%m3 --2.20% 0.17% 2.78% 13.33%m4 --3.43% 0.16% 1.64% 1.03%

Distributions compared

In addition to comparing the raw data, the distribu-tions were compared. (The mapping model doesnot produce a distribution of tooth clearance, and isnot shown here.) Figure 17 shows graphical com-parisons of ProTEM to MCS for a 10-tooth spline,and for the first 10 teeth to engage on a 20-, 40-, and100-tooth spline. The MCS plots shown are thelambda fits to the MCS data (which were determ-ined to be equivalent to the numerically generatedMCS distributions). The plots correlate very closelyfor a 10-tooth and 20-tooth spline. The distributionsappear to diverge more for a 40-tooth and 100-toothspline; however, they still show a very similar rangeof probable clearances for each tooth. Additionally,the first tooth clearance, which is the most critical inpredicting tooth failure, is still very closely correlatedbetween ProTEM and MCS.

Conclusion

The results from ProTEM compare favorably bothquantitatively (looking at the moments) andqualitatively (comparing the distributions) with MCSusing a large number of samples. This comparisonvalidates ProTEM as a predictor of probable toothclearances.

Methods compared

MCS and ProTEM give equivalent results for clear-ance variation in splines; however, ProTEM is moreefficient to use. Table 4 compares the tasks in-volved in generating the data of interest. It is clearfrom this table that ProTEM does not require asmany tasks or as much computation to gather thedesired information about the clearances as MCSdoes.

Figure 17. Comparison of distributions produced using ProTEM and using MCS for a 10-toothspline and for the first 10 teeth on a 20-, 40-, and 100-tooth spline

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Table 5 compares the accuracy, necessarycomputation, and the process of determining qualitylevel for each method. From this we see thatProTEM requires the least computation, while stillproviding high accuracy. Determining the qualitylevel in production requires less computation usingProTEM than using either approach with MCS.

An important difference between MCS and ProTEMis that MCS requires the whole spline to bemodeled, using several thousand sample assem-blies, to be able to gather pertinent information,while ProTEM analyzes only the first few teeth in asingle assembly.

Applications

This study is a continuation of a statisticalinvestigation of involute spline performance. Theprevious work developed a strength of materialsmodel for tooth stiffness, which is an essentialelement in predicting tooth engagement and loadsharing. This model was verified by finite element

analysis. The results were used to develop aspreadsheet (called STEM), based on the mappingconcept, to statistically predict the most likely, oraverage tooth engagement.

ProTEM is a more sophisticated statistical model. Itprovides a tool to predict the range of tooth clear-ances statistically for each tooth on an N-toothspline. This information may be used to determinethe probable values (mean value and range aboutthe mean) for the following:

S Sequence of tooth engagement

S Number of teeth carrying a specified load

S Load distribution among teeth

S Loads and stresses in each tooth

By having this information, engineers will have anincreased understanding of the mechanics of splineengagement. Designers are provided with a tool toaccurately predict performance of splines. This hasthe potential to be used to optimize the design andproduce higher performance splines.

Table 4. Comparison of computational efficiency

Tasks involved in generating probability distribution and finding mean and rangeMCS: Direct numerical

computation MCS: Lambda fit to data ProTEM

S Perform simulation,generating over a millionvalues used in computation

S Compute mean and rangedirectly from data

S Plot and smooth CDFS Fit parabola segments using

least squaresS Take numerical derivative to

get distribution

S Perform simulation,generating over a millionvalues used in computation

S Compute mean, range, andother moments directly fromdata

S Use moments to fit Lambdadistribution

S Get distribution directly fromformula, for the given toothnumber and number of teethon spline

S Calculate range fromdistribution

S Calculate mean frommoment-generating function

Table 5. Comparison of accuracy, computation, and accessibility

Method MCS: Direct numericalcomputation MCS: Lambda fit to data ProTEM

Accuracy ofmethod

Dependent on sample size: requires more than 10,000samples, generating a complete distribution for eachtooth

Good: agrees well withMCS. Equivalent to alarge sample size

Computationrequired Most computation Much computation Least computation

Process ofdeterminingquality level

Perform entire simulation;count the number ofsplines that fail to meetdesign requirements

Perform entire simulation;use tabulated lambda datato predict how manysplines fail to meet designrequirements

Integrate PDF of Tooth 1to predict how manysplines fail to meet designrequirements

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Future work

As a result of this study, our sponsor was motivatedto test a spline assembly under load, to measure thetorque vs. rotation accurately and see if thepredicted piece-wise linear behavior, shown inFigure 2, could be verified. The results were highlysuccessful. They even demonstrated thatdisassembling the spline and reassembling it,rotated to a new position, produced a completelynew torque-rotation plot, as predicted, since all thetooth clearances would change. These results arepresented by DeCaires [1].

One surprising feature was discovered. Some ofthe increments in slope were much larger thanpredicted. Examination revealed that they ap-peared to be integer multiples of the single tooth in-crement expected, indicating that more than onetooth was engaging simultaneously. This behaviorwas not predicted by the tooth engagement model.

It has been postulated that perhaps there is aperiodic fluctuation in the hobbing or generatingprocess that leads to correlation between teeth, inviolation of the assumption of independence.Perhaps it is a numeric interaction between thenumber of teeth on the spline and the number ofteeth, or threads, on the cutter.

This opens up new possibilities. If the correlationmechanism can be found, it may be possible todesign the process to deliberately cause the teeth to

engage in groups favorable to load sharing. It couldlead to splines with higher load carrying capabilities.

A third research effort has recently been conductedto analyze measured spline data provided by oursponsor. The data was of sufficient resolution tocalculate the expected clearances between matingspline teeth and to predict the tooth engagementsequence. It also enabled demonstration of thechanges which occur when a spline is disas-sembled, rotated and reassembled. Additionally, apreliminary study was made to simulate, by meansof a CAD model, how errors in the hobbing processare imprinted on production splines.

Effective tooth engagement

When determining the tooth loading, it is commondesign practice to divide the total applied load by thenumber of teeth assumed to be engaged. Typicalrecommended values are ¼ or ½ of the total teeth, de-pending on how heavy the load. However, even ifthe number of teeth engaged were known, this un-derestimates the maximum tooth load, since theload is not distributed uniformly. As shown inFigure 18, Tooth 1 carries 18% of the total loadwhen 10 teeth are engaged. Common design prac-tice would assume uniform loading on the engagedteeth, where each tooth carries 10% of the load.This assumes a lower force on Tooth 1 than is actu-ally present, and the tooth will fail before a uniformloading would predict.

Figure 18. Effective tooth engagement and the percent of load carried by 10 engaged teeth [1]

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In order to remedy this, the total load can be dividedby an effective number of teeth engaged. Theeffective number of teeth engaged is determinedfrom the Statistical Tooth Engagement Model(STEM), by dividing the total load by the amountcarried by Tooth 1, which is estimated by modelingprobable clearances of a spline. The STEMspreadsheet was developed for this purpose [1]. Inthe case of Figure 18, it would take five teeth, at aload equal to Tooth 1, to carry the load. Dividing theload by the effective number of teeth engaged givesan accurate estimate of the maximum tooth load,which can then be used to determine at what loadthe spline will fail.

The Effective Tooth Engagement provides a designprocedure similar to that which engineers

commonly use, but which yields a much moreaccurate estimate of tooth loads and stress.

References

1. DeCaires, Brian J., “Variation Analysis of Invol-ute Spline Tooth Contact,” MS Thesis, BrighamYoung University, April, 2006.

2. Press, W. H., et al., Numerical Recipes in C,Cambridge, 2nd Ed, 1992.

3. Hines, W. W., and D. C. Montgomery, Probabil-ity and Statistics in Engineering andManagement Science, John Wiley, 2nd Ed.,1980.

4. Shapiro, S. S., and A. J. Gross, StatisticalModeling Techniques, Marcel Dekker, 1981.