1、08FTM06AGMA Technical PaperTooth Fillet ProfileOptimization for Gearswith Symmetric andAsymmetric TeethBy A. Kapelevich, AKGears, LLCand Y. Shekhtman, One StepMolding SolutionTooth Fillet Profile Optimization for Gears with Symmetricand Asymmetric TeethAlex Kapelevich, AKGears, LLC and Yuriy Shekhtm
2、an, One Step Molding SolutionThe 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.AbstractThegeartoothfilletisanareaofmaximumbendingstressconcentration.However,itsprofileistypic
3、allylessspecified in the gear drawing and hardly controlled during gear inspection in comparison with the gear toothflanks. This paper presents the fillet profile optimization technique for gears with symmetric and asymmetricteethbasedontheFEAandrandomsearchmethod.Itallowsachievingsubstantialbending
4、stressreductionin comparison with traditionally designed gears. This bending stress reduction can be traded for higher loadcapacity, longer lifetime, lower noise and vibration, and cost reduction.Copyright 2008American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virgini
5、a, 22314October, 2008ISBN: 978-1-55589-936-31Tooth Fillet Profile Optimization for Gears with Symmetric and Asymmetric TeethAlex Kapelevich, AKGears, LLC and Yuriy Shekhtman, One Step Molding SolutionIntroductionHistorically, gear geometry improvement effortswere concentrated on the working involute
6、 flanks.They are nominally well described and classified bydifferent standard accuracy grades, depending ongear application and defining their tolerance limitsfor such parameters as runout, profile, lead, pitchvariation, and others. Working involute flanks arealso modified to localize a bearing cont
7、act and pro-vide required performance at different tolerancecombinationsandpossiblemisalignmentasaresultof operating conditions (temperature, loads, etc.).Their accuracy is thoroughly controlled by gearinspection machines.The gear tooth fillet is an area of maximum bendingstressconcentration. Howeve
8、r,itsprofileandaccu-racy are marginally defined on the gear drawing bytypically very generous root diameter toleranceand, in some cases, by the minimum fillet radius,which is difficult to inspect. In fact, tooth bendingstrength improvement is usually provided by geartechnology (case hardening and sh
9、ot peening tocreate compressive residual stress layer, forexample) rather than gear geometry.Thegeartoothfilletprofileistypicallydeterminedbythe generating cutting tool (gear hob or shapercutter) tooth tip trajectory (Figure 1), also called thetrochoid. If the cutter parameters are chosen ordesigned
10、 to generate the involute flank profile,which must work for the specific gear applicationand satisfy certain operation conditions, the filletprofile is just a byproduct of the cutter motion. Thefillet profile and, as a result, bending stress are alsodependent on the cutter radial clearance and tipra
11、dius. The standard radial clearance usually is0.25/P or 0.20/P+0.002”, where P is the standarddiametralpitch. Thestandardcuttertoothradiusforthe coarse pitch gears is 0.3/P. For the fine pitchgears the standard cutter tooth radius is notstandardized and can be as low as zero 1.Unlike the contact Her
12、tz stress, the bending stressdoes not define the major dimensions or the gears,such as pitch diameters or center distance. If thecalculated bending stress is too high, in manycases, the number or teeth can be reduced and thecoarser diametral pitch (larger module) can beapplied to keep the same pitch
13、 diameters, centerdistance, and the same (or close) gear ratio. Thismakes the gear tooth physically larger and reducesbending stress to an acceptable level. Of course,this increases specific sliding and reduces contactratioandgearmeshefficiency,butthisisbetterthanthe broken teeth.Key1 Cutter tooth t
14、ip2 Gear tooth fillet as a trajectoryof the cutter tooth tip Rack profile (pressure) angleA AddendumC Radial clearanceR Cutter tip radiusFigure 1. Gear tooth fillet generation by the rack cutter (gear hob)2There are two general approaches to reducingbendingstressforthegiventoothsize. Oneofthemistoal
15、terthegeneratingcuttertoothtip- mostcom-monapplicationofthis approachis therack withthefull tip radius. Another approach is to alter the geartooth fillet profile - the most common applicationhere is the circular (instead of trochoid) fillet. Fur-ther development of both these approaches isbased on a
16、 mathematical function fitting techniquewherethecuttertipradiusor thegear toothtrochoidfillet profile is replaced by a parabola, ellipsis, chaincurve, or other curve, reducing the bending stress(see for example 2, 3). Bending stress reductionachievedbysuchfilletprofileimprovement isvariedand greatly
17、 depends on the cutter or gear toothparameters. Theresultingtoothfilletprofilemustbechecked for interference with the mating gear atvarious gear (and center distance) tolerancecombinations.This paper presents the Direct Gear Design filletprofile optimization technique, which allows for asubstantial
18、bending stress reduction in comparisonto traditionally designed gears. It also describeshow bending stress reduction can produce othergear performance benefits.Optimization methodDirectGearDesign4definesallgeargeometrypa-rameters without using the pre-selected basic orgenerating rack. It is applied
19、for custom gears andallows for the separation of the active involute flankand tooth fillet design.Theflankprofilesaredesignedfirsttosatisfyprima-ry performance requirements, such as maximumload capacity with acceptable contact stress level,maximum gear mesh efficiency (minimum specificsliding), etc.
20、The tooth fillet design is based on completelydefined involute flank parameters. The initial filletprofile is a trajectory of the mating gear tooth tip inthe tight (zero backlash) mesh. For practical pur-poses, this trajectory is defined at the minimumcenter distance (including both gears runout),ma
21、ximum tooth thickness, and maximum outerdiameter of the external mating gear(for aninternalmating gear the minimum inner diameter is used).This allows the exclusion of interference with themating gear tooth.The fillet optimization consists of three majorcomponents 5:S trigonometric functions for fil
22、let profileapproximation;S FEA for stress calculation;S a random search method to define the optimalset of the trigonometric functions coefficients,which allows them to reach the minimumbending stress.The trigonometric functions are selected in such away that the first and the last FE nodes of the i
23、nitialfillet profile are placed on the form diameter circle(Figure 2) and cannot be moved during theoptimization process. The rest of the initial fillet FEnodes are moved along the beams that passthroughthefilletcenter. Thecenterofthefilletisthecenter of the best-fitted circle. The bendingstresses a
24、re calculated for every new fillet profileconfiguration. The adjustment of the optimizingvariable parameters is defined based on thesuccessful (leading to stress reduction) and unsuc-cessful (leading to stress increase) iteration stepsand some random vector. The number of iterationsteps (or optimiza
25、tion time) and minimal iterationstepsarelimited. Therandomnatureofthismethoddoes not yield absolutely identical results for thesame set of gear parameters and number of itera-tion steps. The program was adjusted so that themaximum bending stress difference betweenrepeated calculations does not excee
26、d 2%. The fil-letshapesforthesecasesarealsoslightlydifferent.Optimization ResultsAsanexampleofthefilletprofileoptimization,differ-entfillets wereconstructed forthe gearpair withthestandard involute tooth profile and the followingparameters (see Figure 3):S Number of teeth of both mating gears 24;S D
27、iametral pitch 12;S Generating rack profile (pressure) angle 20;S Addendum coefficient (also known as normal-ized addendum coefficient) 1.0;S Face width of both mating gears 1.0”;S Operating torque 200 in-lb.Table 1 presents the FEA of the different filletprofiles. Italsoindicatesthattheoptimizedfil
28、lethasthe largest curvature radius at the maximum stress3pointandtheshortestradialdistancefrom thispointto the load application point. The isogram chartsillustrate the bending stress distribution.Figure 4 also shows the bending stress distributionalong the different fillet profiles. It clearly indic
29、atesthat the optimized fillet has the lowest maximumbendingstress,whichisevenlydistributedalongthelarge portion of the fillet profile. Other fillet profileshave significantly greater maximum stresses thatare sharply concentrated.Key1 Involute tooth flanks2 Form circle diameter3 Initial fillet profil
30、e4 Fillet center5 Optimized fillet profile;Figure 2. Fillet profile optimizationKeyF Applied loadH Radial distance between load applicationand max. stress pointsC Actual radial clearance1 Fillet profile (black) generated by thestandard (for coarse diametral pitch) rackwith the tip radius 0.3/P2 Fill
31、et profile (pink) generated by thestandard (for fine diametral pitch) rackwith the tip radius equal zero3 Fillet profile (dark blue)generated by thefull tip radius rack4 Circular fillet profile (light blue)5 Optimized fillet profile (green)6 Trajectory of the mating gear tooth tip intight mesh (red)
32、Figure 3. Standard gear tooth with different fillets; X and Y coordinates at the center of the gear4Table 1. FEA of different fillet profilesRack cutterwith tipradiusR=0Rack cutterwith tipradiusR=0.3/PRack cutterwith full tipradiusCircularfillet profileOptimizedfillet profileFillet profile numberat
33、Figure 32 1 3 4 5Bending stress iso-gramsX-coordinate ofload applicationpoint, in-.0593 -.0593 -.0593 -.0593 -.0593y-coordinate ofload applicationpoint, in1.0167 1.0167 1.0167 1.0167 1.0167X-coordinate ofmaximum stresspoint, in-.0825 -.0858 -.0868 -.0822 -.0813Y-coordinate ofmaximum stresspoint, in.
34、899 .9026 .9026 .9158 .9234Fillet curvature ra-dius at the maxi-mum stress point,in.0231 .0399 .047 .0483 .1093Radial distance be-tween load applica-tion and maximumstress points, in0.1157 0.1118 0.1117 0.0989 0.0915Radial clearance, in .0208 .0208 .0246 .0165 .0159Maximum bendingstress, psi8686 728
35、7 6602 6412 5731Relative stress dif-ference, %+19.0 0 - 9.4 -12.0 -21.45Key1 Fillet profile (black) generated by thestandard (for coarse diametral pitch)rack with the tip radius 0.3/P2 Fillet profile (pink) generated by thestandard (for fine diametral pitch)rack with the tip radius equal zero3 Fille
36、t profile (dark blue) generated bythe full tip radius rack4 Circular fillet profile (light blue)5 Optimized fillet profile (green).Figure 4. Bending stress distribution chart along the fillet profilesBenefits of fillet optimizationIf load capacity of the gears with conventional (tro-choidal or circu
37、lar) fillet profiles is limited by themaximum bending stress, the fillet optimizationincreases gear load capacity proportionally to thebending stress reduction. However, very often,gear load capacity, and consequently gear drivesize and weight reduction, is limited by the toothsurface durability, wh
38、ich greatly depends on thecontact stress. In this case, the bending stress re-duction provided by the fillet optimization can beconverted into the contact stress reduction. Figure5a presents the charts of the bending (black) andcontact(blue)stresses,calculatedforthegearpairswith the standard involut
39、e profiles. These gearshave gear ratio 1:1, the constant center distanceaw=60mm,thefacewidthofbothgearsb=10mm,and the driving torque T = 50 Nm. The number ofteeth varies from 12 to 75 and module varies ac-cordinglyfrom5mmto0.8mmtokeeptheconstantcenterdistance. Thebendingstressesarepresent-edintwocha
40、rts;one forthe gearswith thestandard(generated by 20pressure angle rack) trochoidalfillet profile and another one for the gears with theoptimized fillet profile. For example, the bendingstress level of 180 MPa is considered acceptable.This level is achievable for the 20-tooth gears withthestandardfi
41、lletorforthe28toothgears(withfinermodule). However,the28toothgearshaveahighercontact ratio and as a result lower contact stress.The fillet optimization allowed converting potentialbending stress into the 6% contact stress reductionby using the gears with greater number of teeth.This 6% contact stres
42、s reduction doubles life of thesteel case hardened gears with a high number ofload cycles.Similarly to the conversion of the bending stress re-ductionintocontactstress reductionand longerlife,the fillet optimization allows achieving higher gearmesh efficiency. Figure 5b presents the charts ofthe ben
43、ding stresses (black) and gear mesh effi-ciency (blue) for the same gear pairs. In thisexample, the finer module gears with greaternumber of teeth and the optimizedfillet, whichhavethe same maximum bending stress level, provideless specific profile sliding and, as a result, 0.6%higher gear mesh effi
44、ciency in one gear pair. Thiscan be very beneficial for high power multistagegear transmission, because this will reduce heatgeneration, required lubrication system, etc.6(a)(b)Figure 5. a contact stress reduction;b increased mesh efficiency.The potential benefits of the bending stress con-centratio
45、n reduction by the tooth fillet profile opti-mizationcan beextended. This allowsusing agearwithagreaternumberofteethandfinermodulethatgenerates less noise and vibration. It is likely pos-sibletoincreasethehydrodynamicoilfilmthicknessand reduce the flash temperature, because of thereduced profile sli
46、ding.Application of fillet optimization forsymmetric and asymmetric gearsThefillet profile optimization is for custom gears. Inprevious paragraphs, the optimized fillets wereconstructed to the standard involute tooth flanksonly to compare them with the fillet profiles of thestandard gear teeth and d
47、emonstrate possiblebending stress concentration reduction. The au-thorshavenointentionofrecommending usingfilletoptimizationforstandardgears. Thebenefits ofthestandard gears include their universality and suit-ability to the majority of non-critical gear applica-tions. Theyareavailableoff-the-shelf;
48、 theirdesignvalidation is simple and typically does not requirespecial testing.Incustomgearsthenonstandardgeargeometry,in-cluding the optimized fillet profile, is necessary toguarantee required performance. Custom gearsare used for extreme and highly competitiveapplications likeaerospace andracing d
49、rives,auto-motive gear transmissions, etc. Forming geartechnology, like plastic and metal injection molding,powder metal processing, precision gear forging,extrusion, and die casting allow to extend theimplementation of the nonstandard gears with theoptimized fillet profile for many custom gearapplication. Examples,ofsuchgearsarepresentedin Figure 6.a) Polyurethane die cast gearfor industrial applicationb) Metal machined gearsfor automotive applicationFigure 6. Examples of the custom gears with the optimized fillet profile7Many gea
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