1、10FTM14AGMA Technical PaperAnalysis and Testing ofGears with AsymmetricInvolute Tooth Formand Optimized FilletForm for PotentialApplication inHelicopter Main DrivesBy F.W. Brown, S.R. Davidson,D.B. Hanes and D.J. Weires,The Boeing Company, andA. Kapelevich, AK Gears, LLCAnalysis and Testing of Gears
2、 with Asymmetric InvoluteTooth Form and Optimized Fillet Form for PotentialApplication in Helicopter Main DrivesFrederick W. Brown, Scott R. Davidson, David B. Hanes and Dale J. Weires,The Boeing Company, and Alex Kapelevich, AK Gears, LLCThe statements and opinions contained herein are those of the
3、 author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractGears with an asymmetric involute gear tooth form were analyzed to determine their bending and contactstresses relative to symmetric involute gear tooth designs which are repres
4、entative of helicopter main drivegears. Asymmetric and baseline (symmetric) toothed gear test specimens were designed, fabricated andtested to experimentally determine their single-tooth bending fatigue strength and scuffing resistance. Also,gears with an analytically optimized root fillet form were
5、 tested to determine their single-tooth bending fatiguecharacteristics relative to baseline specimens with a circular root fillet form. Test results demonstrated higherbending fatigue strength for both the asymmetric tooth form and optimized fillet form compared to baselinedesigns. Scuffing resistan
6、ce was significantly increased for the asymmetric tooth form compared to aconventional symmetric involute tooth design.Copyright 2010American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October 2010ISBN: 978-1-55589-989-93Analysis and Testing of Gears wit
7、h Asymmetric Involute Tooth Form andOptimized Fillet Form for Potential Application in Helicopter Main DrivesFrederick W. Brown, Scott R. Davidson, David B. Hanes and Dale J. Weires,The Boeing Company, and Alex Kapelevich, AK Gears, LLCIntroductionThe objective of the work described in this article
8、isto begin the process of evaluating the potentialbenefits of asymmetric involute gear teeth andoptimized root-fillet geometry for helicopter maintransmission applications. This involves not onlyquantifying performance improvements achievedby these concepts, but evaluating the practicality ofmanufac
9、turing gears with asymmetric teeth andoptimized root-fillet geometry for aerospace applic-ations. This work was performed under thesponsorship of the Center for Rotorcraft Innovation,CRI. The authors are grateful to CRI for theopportunity to investigate these technologies.In helicopter main drive ap
10、plications, minimizinggear weight while maintaining the necessarybalance of tooth bending strength, pitting resistanceand scuffing resistance is given high priority duringdesign of the gears. In many helicopter applica-tions, gears are required to transmit high continuoustorque loads in one directio
11、n but are lowly loaded inthe opposite direction. Traditionally, spur geardesigns in helicopter gear boxes utilize convention-al (symmetric) involute teeth which provide thesame torque capability in both the drive and coastloading directions. An overall weight reduction maybe realized by using gears
12、with higher capabilitythan conventional gears in the primary drivedirection, even if some capacity is sacrificed in thesecondary coast direction.The design intent of asymmetric gear teeth is toimprove performance of the primary drive profiles atthe expense of the performance for the oppositecoast pr
13、ofiles. In many cases the coast profiles aremore lightly loaded, and are loaded only for a relat-ively short duration. Asymmetric tooth profilesmake it possible to simultaneously increase thecontact ratio and operating pressure angle in theprimary drive direction beyond the conventionalgears limits.
14、 The main advantage of asymmetricgears is contact stress reduction on the drive flanks,resulting in reduced gear weight and higher torquedensity.Many traditional helicopter spur gear designs utilizecircular root-fillet geometries that are form groundalong with the gear flanks. Gear specimens wereana
15、lyzed, designed and manufactured to comparethe single tooth bending fatigue strength of gearswith optimized root-fillets with gear specimenshaving circular root-fillets.The items addressed in this article include analysis,design, manufacture and test of gear testspecimens with asymmetric teeth. Conv
16、entionalsymmetric tooth specimens were also producedand tested to provide a baseline for comparison.The design and manufacturing of the gear speci-mens is representative of helicopter main drivegears. Testing included single tooth bending fatigueand scuffing tests of both the asymmetric andbaseline
17、gears. Additionally, symmetric toothedgear specimens with optimized root fillet geometrywere analyzed, designed, manufactured and testedin single tooth bending fatigue and compared toconventional specimens with circular root filletgeometry as a baseline.BackgroundMinimizing gear weight while maintai
18、ning anacceptable balance of tooth bending strength,pitting resistance and scuffing resistance is givenhigh priority when designing gears for helicopterapplications. Typically helicopter main drive gearsare required to transmit high continuous torqueloads in their primary or drive direction. Torquel
19、oads in the opposite (secondary or coast) directionare lower in magnitude and of shorter duration thantorques in the primary direction. In simple planetarygear arrangements often used in helicopter gearboxes, the planet gears are required to transmit loadon both sides of their teeth due to contact w
20、ith thesun gear at one mesh and contact with the internalring gear at the other mesh. In these cases the sizeof the planet gear teeth are usually dictated by therequirements of the sun-planet mesh and stresses4are lower at the planet-internal ring gear mesh dueto the more conforming contact. Traditi
21、onally spurgear designs in helicopter gear boxes are symmet-ric involute teeth which provide essentially the sametorque capability in both the drive and coast loadingdirections. There may be an overall weight benefitfrom using gears with higher capability in theprimary drive direction, even if some
22、capacity issacrificed in the secondary direction.The design intent of asymmetric gear teeth is toimprove performance of the primary drive profiles atthe expense of the performance for the oppositecoast profiles. The coast profiles are unloaded orlightly loaded during a relatively short work period.A
23、symmetric tooth profiles make it possible to simul-taneously increase the contact ratio and operatingpressure angle beyond the conventional gearslimits. The main advantage of asymmetric gears iscontact stress reduction on the drive flanks,resulting in higher torque density.Gears with asymmetric teet
24、h have been known formany years. Prof. Willis wrote about buttress gearsin 1838 1. Since then many articles on the subjectof asymmetric gears have been published.However, there are very few practical applicationsof such gears. One of them was an application ofthe asymmetric teeth in the planetary ge
25、arbox in aturboprop engine 2.The work described herein presents the design,manufacture and test of:S Asymmetric involute gears with circular root-fillet geometryS Symmetric gears with optimized root-filletgeometryThe asymmetric tooth geometry and optimized filletgeometry was developed by the Direct
26、GearDesignmethod 3. Unlike traditional gear design,this method does not use a preselected basic orgenerating gear rack to increase the gear toothprofile, but defines the gear tooth formed by twoinvolutes of two different base circles (in case ofasymmetric teeth) with the arc distance betweenthem and
27、 tooth tip circle to avoid the sharp pointedtooth tip. If these base circles are identical, the gearhas symmetric teeth. The fillet between teeth is notin contact with the mating gear teeth. However, thisportion of the tooth profile is critical because this isthe area of the maximum bending stress c
28、oncentra-tion. The fillet profile is designed independently anda subject of optimization providing minimumbending stress concentration and sufficientclearance with the mating gear tooth tip in mesh 4.Test specimen design and analysisTest specimen gears designed for this program arerepresentative of
29、helicopter main drive gears indiametral pitch, pressure angle, material and pro-cessing. Standardized conventional tootheddesigns have been developed for bending fatigueand scuffing test rigs that Boeing Rotorcraft uses forgear research. The standardized test specimendesigns were modified to incorpo
30、rate the asymmet-ric tooth configuration and another for the optimizedfillet configuration. Specimens of each type weremanufactured using aerospace productiontechniques and requirements. A manufacturingapproach was developed with a goal of reducingmaterial and processing variability.The single tooth
31、 bending fatigue tooth gears are 32tooth gears with groups of four teeth removed perquadrant to allow for assembly into the single toothbending fatigue, STBF, test fixture. Table 1 andTable 2 summarize the basic design parameters forthe single tooth bending fatigue test gears, and alsoshow these des
32、ign parameters for the baselinespecimen design.Table 1. Single tooth bending fatigue testgearsParametersSymmetricteeth withcircularfillets(baseline)Symmetricteeth withoptimizedfilletsAsymmetricteethwithcircularfilletsNumber ofteeth32 32 32Diametralpitch5.3333 5.3333 5.3333Pressureangle25 2535 drive1
33、5 coastTable 2. Scuffing test gearsParametersSymmetric teethwith circularfillets (baseline)Asymmetricteeth withcircular filletsNumber ofteeth30 30Diametral pitch 5.3333 5.3333Pressure angle 2535 Drive18 Coast5The optimized fillet gear is similar to the baselinedesign with the circular fillet replace
34、d with the optim-ized root fillet. A comparison of the geometrybetween the circular fillet and the optimized filletgeometries are shown in Figure 1.Figure 1. Coordinate plot of circular fillet andoptimized fillet design geometriesSimilarly, the scuffing test gears are within thedesign experience ran
35、ge of typical main transmis-sion helicopter power gears. The test gear design isof similar size to a first stage planetary sun-planetmesh that can be found in a medium to large sizehelicopter. Basic parameters for the scuffing testgears are summarized in Table 2.Test specimen analysisThe test specim
36、en gear designs were analyzed topredict their bending and contact stresses, andcompared to stresses predicted for the baseline testspecimens.Asymmetric tooth geometry single toothbending specimensSingle tooth bending fatigue specimens weredesigned employing asymmetric involute teeth. Forcomparison,
37、conventional symmetric involute gearswere designed and tested. Both asymmetrictoothed and conventional baseline specimensemploy ground circular root-fillets.The asymmetric gear tooth form for the STBF(Single tooth bending fatigue) test specimens isnominally based on the standard STBF gearspecimen. T
38、his enables the asymmetric toothedspecimen to fit the existing test fixture with onlyminor modifications for tooth load angle, andprovides a direct comparison between asymmetricand conventional gears of the same diameter andface width. Finite element analysis (FEA) meshesof the baseline symmetric to
39、othed gear specimenand the asymmetric toothed specimen are shown inFigure 2 and Figure 3.Figure 2. FEA mesh STBF baselinesymmetric toothFigure 3. FEA mesh STBF asymmetric toothspecimenThe gear parameters and calculated bendingstresses for the STBF test gears are presented intheTable3.Asymmetric toot
40、h geometry scuffing testspecimensFEA meshes of the baseline (symmetric) scuffinggear tooth and the asymmetric scuffing gear toothare shown in the Figure 4 and Figure 5 respectively.The gear parameters and comparison results arepresented in the Table 4.6Table 3. Comparison of data for baselinesymmetr
41、ic and asymmetric toothed STBFgear specimensParameters1)BaselinesymmetrictoothedspecimenAsymmetrictoothedspecimenNumber of teeth 32 32Diametral pitch 5.3333 5.3333Drive pressure angle,deg25 35Coast pressureangle, deg25 15Pitch diameter, Pd,in 6.0000 6.0000Drive base diameter,in5.4378 4.9149Coast bas
42、e diameter,in5.4378 5.7956Outside diameter, in 6.4000 6.3864Root diameter, in 5.571 5.558Drive TIF diameter, in 5.6939 5.6581Coast TIF diameter,in5.6939 5.8110Circular tooththickness, in0.2905 -0.28850.2905 -0.2885Fillet radius, in0.074 mincircular0.078 mincircularFace width, in 0.375 0.375Torque, i
43、n-lb 5000 5000Load applicationradius, in3.06 3.06Calculated maximumbending stress, psi57887 54703NOTE:1)Length dimensions in inches, angles in degrees.Figure 4. FEA mesh baseline symmetricscuffing gear specimenFigure 5. FEA mesh asymmetric toothscuffing gear specimenTable 4. Comparison of data for b
44、aselinesymmetric and asymmetric toothed scuffingtest specimen gearsParameter1)BaselinesymmetrictoothedgearAsymmetrictoothedgearNumber of teeth 30 30Diametral pitch 5.0000 5.0000Drive side pressureangle, deg25 35Coast side pressureangle, deg25 18Pitch diameter, Pd,in 6.0000 6.0000Drive base diameter,
45、in5.4378 4.9149Coast base diameter,in5.4378 5.7063Outside diameter, in 6.400 max 6.4034 maxRoot diameter, in 5.459 max 5.510 maxDrive TIF diameter, in 5.6864 5.6415Coast TIF diameter,in5.6864 5.7607Circular tooththickness at (pd), in0.3106 -0.30860.3106 -0.3086Fillet radius, in 0.059 min 0.081 minFa
46、ce width, in 0.50 0.50Contact ratio 1.417 1.25Torque, in-lb 6000 6000Calculated maximumcontact stress, psi193180 174100NOTE:1)Length dimensions in inches, angles in degrees.7Optimized root-fillet geometry single toothbending specimensSingle tooth bending fatigue specimens weredesigned employing trad
47、itional symmetric involuteteeth. This specimen tooth geometry fits the singletooth bending fatigue test rig at Boeing. Baselinespecimens employ a ground circular root-filletrepresentative of gears in the aircraft application.The optimized fillets specimens share the sametooth geometry as the baselin
48、e except for the formof the root fillet. The form of the optimized root-filletprofile was determined analytically. The fillet optim-ization technique is based on two dimensional finiteelement analysis (FEA) employing a randomsearch method 4. The FEA meshes of the circularfillet gear tooth and the ge
49、ar tooth with the optimizedfillet are shown in the Figure 6. Notice thatoptimization of root-fillet results in a non-constantradius that is slightly “pinched” compared to thetraditional circular fillet. Optimized fillet geometry isdefined as a series of coordinate points. Thesepoints are plotted in Figure 7 and can be comparedto the coordinates of the circular root-fillets, seeFigure 8.Figure 6. FEA mesh STBF symmetric toothcircular filletThe gear parameters and comparison of results arepresented in Table 5.Test specimen manufacturing
