1、11FTM06AGMA Technical PaperReversed Gear ToothBending Stress and LifeEvaluationBy J. Chen, SAIC MotorReversed Gear Tooth Bending Stress and Life EvaluationJoe Chen, SAIC Motor (retired General Motors)The statements and opinions contained herein are those of the author and should not be construed as
2、anofficial action or opinion of the American Gear Manufacturers Association.AbstractThere is wealth of literature on the subject of single (or uni-) directional gear tooth bending stress and liferelationships(i.e.,S-Ncurves)thathasbeenpublishedinvariousjournalsandhandbooksoverpastdecades.Several of
3、them were adopted as industrial standards by different gear societies worldwide. However, theyhave limited information regarding the fluctuating bi-directional (reversed) gear tooth bending fatigue lifeprediction. To fill in this gap for practical applications, the author first intended to apply tra
4、ditional fatiguetheories such as modified Goodman, Gerber and Morrow to derive a series of S-N equations. Uponcorrelation of these equations with the regressed test results, significant deviations were found. From theobservation of test results, it was found that the slopes and endurance limits on t
5、he fitted S-N curves fromseveral different tested conditions were reasonably similar, if the test gears had been made from the samebatch of material and manufacturing process. Based on the above observation, the author proposed a newapproach to reduce the deviation from the above theories.Copyright
6、2011American Gear Manufacturers Association1001 N. Fairfax Street, 5thFloorAlexandria, Virginia, 22314October 2011ISBN: 978-1-61481-005-63Reversed Gear Tooth Bending Stress and Life EvaluationJoe Chen, SAIC Motor (retired General Motors)IntroductionOne of the major concerns that have been raised by
7、many gear designers regarding the gear tooth bendingstressis,“Whatisthedegradingfactor?”, whendealing withthe fullyor partiallyreversed loadsexerted onanidler gear compare to the uni-directional loads. Many related studies have been published over the yearsabout the cumulative damage and the associa
8、ted load/stress and life relationships (S-N curve) on mechan-ical components such as shafts, hubs and springs. However, limited information in this area on the geardesignandanalysishasbeenrevealed. Mostofthegearhandbooksandgearstandardsprovidefixedvaluesfor fully reversing load, such as 0.66 or 0.70
9、 as the deratingfactor forrough estimation. Amethod toaccountfor fluctuating or partially reversing load has not been fully covered and needed to be fulfilled. The authorproposes a new calculation approach, derived from the well-established fatigue theories, and enhanced bythe correlated test result
10、s.Analytical methodologyBackground reviewToquicklyaccessthecalculationprocessfor thegear reversalbending issue,the authorhad initiallyselecteda well known and popular stress-life fatigue rule, the Modified Goodman Method 1, to formulate theanalytical calculation procedure, then applied it for gear t
11、ooth bending evaluation. It will be reviewed anddescribed as follows.Fundamental of the Modified Goodman diagramWhen an element is subjected to a series of fluctuating loads that are either uni-directional or bi-directional,thecorrespondingstress-life(S-N)relationshipsasillustratedonFigure 1,canbeex
12、pressedbytheModifiedGoodman equation, the equivalent stress, ar, from the combined of mean stress, mean, and alternativestress, alt, can be expressed by the following equations.altar+meanSu= 1, or ar=alt1 meanSu(1)wheremean=max+ min2(2)alt=max min2(3)Suis ultimate tensile strength of material, and m
13、axand minare the principle maximum and minimumstresses.When both stress and life are in log scale, the corresponding stress life (S-N) relationship can be expressedas follows:ar=2 N(4)whereais constant; is slope of the S-N curve.4Figure 1. Modified Goodman diagram with the corresponding S-N curveIn
14、addition, a commonly used value, “R-ratio”, is used by many fatigue test engineers for fluctuating loadingcalculation, is introduced as follows:R =minmax(5)For example, when the load is in uni-directional R =0,sincemin= 0. In addition, max= min; or when arotational shaft is subjected to a constant b
15、ending load, R =-1.0,sinceminis a negative value.Other fatigue bending stress related theoriesThere are many other fatigue bending stress theories available, the author intended to use the Goodman-Morrows diagram instead of Modified Goodmans diagram, because it is easier for designers to obtain theu
16、ltimate strength SFthan the Suvalue from the actual static bending fracture test. Also because most of thetest gears are made from case hardened alloy steels, but not the basic non-heat-treated steel. SFobtainedfromtheactualtestalsoincludedthestressrisecausedbythesmallerrootfilletradius,potentialund
17、ercutandimperfections on the root surface during the process.Gear tooth root bending stress calculation methodsTwo major branches of gear root bending stress calculations that have been widely adopted and used byvariousgearindustriesaroundtheworld. TheAGMAmethod2thatappliesthe Lewisformula withparab
18、olagearloaddiagramhasbeendominatedintheU.S.,whilethe30degreetriangleloaddiagram34andcalcu-lation procedures established by ISO, DIN and JSME, are widely used in Europe and Japan The authorintendedtousetheAGMAmethodwithsomeminorsymbolmodifications(Figure 2)inthispaper,wherebothtangential force, P cos
19、 L, and separating force, P sin L, that exerted on the gear tooth are taken intoconsideration5Figure 2. Gear bending stress comparison on tensile versus compressive sidesGear root stress at tensile, “C” side (i.e., where the gear load is exerted upon)CTen= sb Tensile sC=6 hPcosLFWT2cP sinLFWTc(6)Gea
20、r root stress at compressive “D” side:DComp= sCb Comp sC= 6 h PcosLFWT2cP sinLFWTc(7)The net gear root stress difference between the tensile side to the compressive side is equal to two timestheroot normal compressive stress, scor2 PsinLFWTcAlthoughthecompressivesideofrootbendingstress,asshowninequa
21、tion 7was notexplicitly expressedbythe above mentioned gear standards, it could be easily derived from the general stress calculation that isbasedonthecantileverbeambendingstresstheorywhenthebeamissubjectedtobothnormalandtangentialloads as shown in Figure 2.Gear fillet root bending stress under bi-d
22、irectional (reversed) loadsBased on the above calculation procedure, it allows the user to put either equal (full reversed), or different(partiallyreversed)gearloadsonbothsidesofthegeartooth(Figure 3). For demonstrationpurpose, anidlergear is subjected to the full reversed load, P1= P2, on both toot
23、h sides is used.Therefore, if the stress concentration factor and other gear degrading factors such as misalignment, speedandshockload,arenottakenintoconsideration,themaximumfullreversedrootstressatP1sideoratP2sideof the gear tooth can be expressed by equations 8 through 11.P1sideWhen the gear is su
24、bjected to full reversed loads (loadP1released first, then followed byP2). Themaximumroottensilestress,max,inducedbythegearloadP1andthemaximumrootcompressivestress,min,inducedby the opposite side gear load P2can be calculated by equations 8 and 9.max=6 hP1cosLFWT2cP1sinLFWTc(8)min= 6 hP2cosLFWT2cP2s
25、inLFWTc(9)6Gear root stresses under fully reversed loadsFigure 3. Gear tooth subjected to bi-directional loadsAccording to Goodman rule, the mean stress, mean, is equal to;mean=max+ min2(10)And the alternative stress, alt, is equal to:alt=max min2(11)P2sideSince both sides of the gear tooth are symm
26、etric and subjected to the equal loads, P1= P2, the criticalstresspoints at P2side should have the same stress level as P1side. Therefore, the cumulative gear damage andlife on both sides should have the same value.Procedure to establish S-N curve for gear subjected to partially or fully reversed lo
27、ad,based on Goodman-Morrows ruleUsing the GoodmanMorrow rule, the equivalent alternative stress, ar, and the corresponding gear life canbe calculated accordingly. The following steps and corresponding figures illustrate the detailed constructionprocedure for reversed gear root stress and the corresp
28、onding life calculation.Step 1. Establish the S-N curve for a given gear material under the uni-direction-loadsIn order to build the bi-directional gear loading S-N curves, the baseline uni-directional gear root bendingS-N curve needed to be established first. Since based on Goodman-Morrows equation
29、s, the equivalentstress value (AR) applied to establish the S-N curve is different from the conventional calculated maximumgearrootstressobtainedfromFEAorgearstandardformula. Therefore,thegearfilletrootstress(S)andthecorrespondinglife(N)neededtobereconstructedbasedonthisequivalentstress,ar,sobothuni
30、-directionalas well as the bi-directional loads S-N curves can be constructed by using a common stress base.Once the material yield tensile strength, Sy, and the endurance limit strength, Se, have been defined, themaximum, max,and the minimum, min= 0, root stresses under a loading condition can be o
31、btained. Theassociated equivalent stress, ar, can thus be calculated using equation 4, and equations 8 through 11. Thecorresponding S-N curve of the uni-directional bending can be constructed accordingly. To facilitate the arcalculation for the uni-direction load (Figure 4), a 45 degree line can be
32、applied along the line, then meanequals alt.7ConventionalS-NGoodman-MorrowMethodFigure 4. Goodman-Morrow S-N curveStep 2. Establish a full reversed bending S-N curve/s with equivalent stressBased on the above stress and life (S-N) calculated from the Goodman-Morrow rule, the S-N curve of agiven gear
33、 material at a full reverse bi-direction condition (R= -1.2) can be constructed using the followingsteps (see Figure 5):1. Construct the S-N curve under the uni-direction load range as described in Step 1.2. Apply the Goodman-Morrows equations to calculate the equivalent stress, ar, at any given loa
34、d underthe fully reversed loading condition. Since under the fully reversed load, R = 1.20, and also becauseR = min/max ,the corresponding stresses maxand mincan be expressed as:mean=max+ min2= 0.1 maxandalt=max min2= 1.1 maxSo the fully reversed equivalent stress, ar, of a given load range can be c
35、alculated as:ar=alt1 meanSF3. The arcould also be defined from the graphic method (Figure 5) where point “H” represents theintersectionofOR(=min= -0.1)andtheOJ(=alt=1.1max). DrawalinefromSuorSftopoint “H”,theintersection point “K” is the equivalent stress arof the fully reversed loading condition, a
36、s defined.4. From point “K” draw a straight line to meet with the uni-directional load S-N curve-CDG at point “D”.Draw another straight line from the arof the same test load level, but under uni-direction load conditionpoint “L”, the point “E” where the load line and life line intersected is one of
37、the S-N curve points for thatparticular fully reversed load condition.5. Repeat the same process on different loads with the same bending condition (i.e., R=-1.2 in this case),the S - N curve (B - C-E) thus can be constructed as shown on Figure 5.8Gear root stress under fully reverse loading, P1= P2
38、Figure 5. Gear root stress and corresponding life for full reversed loadStep 3.Using the same procedure as described earlier in steps 1 through 5, many partially reversed loading (atdifferent R-ratios) S-N curves thus can also be constructed.Apply single tooth bench test to verify the analytical res
39、ultsTo further demonstrate and validate the above analytical theory, a sample test gear (Figure 6) was designedand used to perform the test. The results will be discussed in the following sections.Test gear material and heat treatment specificationsThe associated material physicalproperty andheat tr
40、eatmentspecifications (Table 1)for theabove testgearis attached for FEA modeling stress calculations.External gear data MetricNumber of teeth 24Module 4.500Pressure angle 20.00Pitch diameter 108.00Base diameter 101.4868Major diameter 118.23/118.36Root diameter 97.96/98.34Start of active profile (rol
41、l angle) 12.82”Circular tooth thickness 7.465/7.501Measure over two 2.937 diameter pins 120.235/120.318Figure 6. Test gear configuration9Table 1. Gear material and heat treatment specificationGear material Case carburized low carbon alloy steelSteel grade AGMA Grade 2BasedonAGMAGrade2specificationin
42、formation:Elastic modulus E 206.5 GPPoissons ratio 0.3Heat treat to obtain:Surface hardnessCore hardness58 to 62 HRc32 to 34 HRcUltimate tensile yield strength 2700 MPa (ref)Tensile yield strength 1500 MPa (ref)Endurance limit 600 MPa (ref)The author intended to apply several popular analytical meth
43、ods to perform the gear root bending stresscomparison,soasuitablemethodcouldbeselectedandusedforcorrelationstudywiththeactualbenchtests.There were two primary purposes for conducting these analyses; one was to compare the root bendingstresses - on both drive and coast sides, under uni-direction load
44、; andthe otherwas tofind theappropriatedcompress versus tensile stress ratio or R-ratio when the gear is subjected to a bi-direction load at a givenloading position.AGMA calculation methodBased on the above-mentioned AGMA Standard guideline, a spreadsheet was generated (Table 2) tocalculatethesingle
45、geartoothfilletbendingstress,thecalculationonlyprovidedtheloadside bendingstress.Some of the inputs were obtained from method given in ANSI/AGMA 2001-D04.FEA MethodA3-DFEAmodelwasestablished,basedonthegearprofilecoordinatesgeneratedbyagivenhobcutterandthe final ground specifications. Three different
46、 loading conditions at the same given loading point wereperformed by the FEA analysis. The corresponding tensile bending stress on the loading and thecompress-ive bending stress at opposite unloaded were calculated.The attached FEA modeling (Figure 7) illustrated the stress distribution contours at
47、both sides of the geartooth when one of the tooth was subject to the single directional load. The summary tabulation of themaximum root stresses and the associated compressive versus tensile stress ratios (R-ratio, Table 3) arealso attached for comparison reference.Figure 7. Typical 3D FEA root stre
48、ss modeling under uni-directional load10Table 2. Single tooth bending stress calculationTable 3. FEA gear root stress on both sides of tooth under uni-direction loadTorque,Nm500 %Diff 1000 %Diff 1750 %DiffLocation Drive side Coast sideComp/TensileComp/TensileCoast sideComp/TensileDrive side Coast si
49、deComp/TensileStress,MPa555.7 -663.3119.4% orR = -1.19/1107 -1329120.1% orR = -1.20/1931 -2336121% or R= -1.21/Boundary element methodThegearboundaryelementmethod5wasalsoappliedforthegearrootstressstudy. Figure8illustratesthegear root stress and stress distribution between the tensile and compressive (R ratio) under a unit loading(load = 1.0) condition.Stress analysis results comparisonTa
