1、08FTM04AGMA Technical PaperThe Effect ofManufacturingMicrogeometryVariations on the LoadDistribution Factor andon Gear Contact andRoot StressesBy Dr. D.R. Houser, The OhioState UniversityThe Effect of Manufacturing Microgeometry Variations on theLoad Distribution Factor and on Gear Contact and RootS
2、tressesBy Dr. Donald R. Houser, The Ohio State 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.AbstractTraditionally, gear rating procedures on directly consider
3、manufacturing accuracy in the application of thedynamic factor, but only indirectly through the load distribution consider such errors in the calculation ofstresses used in the durability and gear strength equations. This paper discusses how accuracy affects thecalculation of stresses and then uses
4、both statistical design of experiments and Monte Carlo simulationtechniquestoquantifytheeffectsofdifferentmanufacturingandassemblyerrorsonrootandcontactstresses.Manufacturing deviations to be considered include profile and lead slopes and curvatures as well asmisalignment. The effects of spacing err
5、ors, runout and center distance variation will also be discussed.Copyright 2008American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2008ISBN: 978-1-55589-934-93The Effect of Manufacturing Microgeometry Variations on the Load DistributionFactor an
6、d on Gear Contact and Root StressesDr. Donald R. Houser, The Ohio State UniversityIntroductionGearratingformulashavenumerousdesignfactorsthat are intended to create realistic evaluations ofthe stress levels encountered by a gear pair. Thedynamic factor, however, is the only factor that hasmanufactur
7、ing accuracy directly included in itsevaluation. This paper discusses most of the otherfactors in the AGMA rating procedure 1 that mightbe influenced by manufacturing accuracy and thenuses load distribution analysis to assess the effectsof profile, lead and spacing deviations on rootstresses, contac
8、t stresses and load distributionfactors. The dynamic factor has received amplestudy in thepast 2-7,so itwill notbe furtherinves-tigated here. As part of the presented analysis, aprocedure is provided for obtaining an acceptablemicrogeometry design that is relatively insensitiveto manufacturing devia
9、tions and misalignment.Manufacturing accuracy definitionsPrior to looking at the factors that affect toothstresses,whatismeantinthispaperasmanufactur-ing accuracy will be defined. In this context,manufacturingaccuracyencompassesallfactorsinmanufacturing or assembly that change themicrogeometries and
10、 hence the load sharing of atooth pair. The AGMA accuracy classificationstandard 8 uses its quality number system todefine quality levels for profiles, leads, runout andspacing. The accuracy of housings, bearings andsupport shafting, geometry changes to the surfaceand root geometries and variables s
11、uch as centerdistance, backlash, outside diameter, tooththickness are not included, but are occasionallydiscussed. Alsonotincludedaredeviationsthataremeasuredthroughcompositeandsingleflanktests.However, these deviations result from theelemental variations that are considered.Provided below is a brie
12、f discussion of factorsaffecting the tooth microgeometry that are studiedin this paper.Lead deviations and misalignment: These arediscussed together since they have similar effectson the load distribution across the tooth face width.The effects of lead deviations, which essentiallyshift the load to
13、one end of the tooth, have beendiscussed in many papers 9-14. Misalignmentthat is atright anglesto thenormal contactingplaneis additive to lead slope deviation so in this paperthese effects for both the gear and pinion will belumped together into a single variation. The AGMAaccuracy standard 8 recog
14、nizes that there arepotentially two types of lead deviations, one of thelinear type and the second being a curvature devi-ation. The linear slope deviation may be added tomisalignment, while the curvature deviation istreated as a deviation in the specified lead crown.Profile deviations: Profile devi
15、ations are oftenthoughtofasdeviationsofthetoothformfromatrueinvolute, but, in loaded teeth operating at the gearpairs rated load, profile modifications in the form oftip and root relief are desirable variations from aperfect involute. Hence, profile deviations arethought of as deviations from the sp
16、ecified profileshape. Again,theAGMAaccuracy standardallowsone to specify deviations in terms of slope andcurvature. Profile deviations tend to affect tooth-to-tooth load sharing across the profile of the toothpair.Bias deviation: Although not spelled out in AGMAstandards, this typeof deviation,which
17、 isessential-ly a twisting of the tooth form, is identified by per-forming multiple profile and/or lead measurementson each measured tooth. This type of deviation,whichcommonlyoccurswhengearsare finishedbyscrew type generation grinding, also affects loadsharing.Spacing deviations: Tooth to tooth spa
18、cingdeviations may affect dynamics, but have a greatereffect on tooth to tooth load sharing 15. In thispaper, AGMA quality number values are used andthese load sharing effects are analyzed.4Runout deviations: Runout results from eccentrici-ties both in the manufacture and the assembly ofgears. The m
19、ost common form of runout is radialrunout, which manifests itself in terms of cyclicspacing deviations, cyclic changes in the profileslope, and cyclic changes in the operating centerdistance and/or the effective outside diameter. Thelatter two effects slightly change the profile contactratio of the
20、gear pair. In the analysis of this paper,only the spacing deviation effect of profile runout isconsidered. Another form of runout commonly re-ferredtoasleadrunout orlead wobbleoccurs intheface width direction. It is assumedthat leadwobbleeffects are already included in the tolerance usedfor lead dev
21、iations so no special analysis of leadwobble is performed in this paper.AGMA rating equationsSince this study concentrates on the effects ofmanufacturing deviations on stresses, we first lookat the current AGMA method for computing thesestresses and discuss in general how each factor inthesestresseq
22、uationsisaffectedbymanufacturingdeviations.The AGMA stress formulas 1 for bending and du-rability are respectively given below:Contact stress equation:sc= CpWtKoKvKsKmdFCfITensile bending stress equation:st= WtKoKvKsPdFKmKBJwhere,sccontact stress number, lb/in2;Cpelastic coefficient, lb/in20.5;Wttra
23、nsmitted tangential load, lb;Kmload distribution factor;Kooverload factor;Kvdynamic factor;Kssize factor;Cfsurface condition factor for pittingresistance;F net face width of narrowest member, in;I geometry factor for pitting resistance;d operating pitch diameter of pinion, in;sttensile stress number
24、, lb/in2;KBrim thickness factor;J geometry factor for bending strength;Pdtransverse diametral pitch, in- 1.The factors that are unlikely to be affected much bymanufacturing (deviations in materials are not con-sidered in this paper) include the elastic coefficient,the overload factor, and the size f
25、actor. As men-tioned earlier, the dynamic factor has been studiedextensively 2-7 and is the only factor used in theaboveformulasthatiscurrentlybasedongearqual-ity. The load distribution factor certainly is affectedby misalignment and profile and leadmodifications.In a way, the current use of this fa
26、ctor does dependuponaccuracy,butthisuseisnotquantifiedintermsof the accuracy numbers. In the simulations of thispaper, the load distribution factor is evaluatedbased on quality numbers. The surface conditionfactor is certainly affected by manufacturing, butalsoisnotconsideredbyAGMAinitsaccuracydefi-
27、nitionsandwillnotbestudiedinthispaper. Bothge-ometry factors are subject to manufacturing toler-ances, but again they are not quantified in theAGMA tolerancenumbers sowill onlybe brieflydis-cussed. The bending strength geometry factor issubject to changes in surface finish and shape im-perfections i
28、n the root fillet region that could affectroot stresses. For instance, this author once hadsome gears made in which the hob feed rate wasvaried and the teeth with the coarserfeed ratewerefound to have lower lives based on single tooth fa-tigue testing 16. With regard to the surface dura-bility geome
29、try factor, it has been shown that thecontact stress may increase significantly at loca-tions on the tooth flank where there are abruptchanges in thetooth form,an examplebeing thera-dius of curvature change in the tip relief “break” re-gion 17. The rim thickness factor is subject tomanufacturing to
30、the extent that there are toler-ances on the thickness value that could affect rootstresses.Baseline microgeometry selectionWhenstudyingmicrogeometryvariations,onemustfirst start with a baseline microgeometry. This, inessence, means to define the baseline profiles andleads of the design. Every gear
31、designer has theirown approach to establishing these parameters.Some designers might choose their profile so as to5avoid corner tooth contact and tip interference; oth-ers might minimize transmission error and othersmay wish to minimize the effects of spacing devi-ations. When selecting lead variati
32、on, some de-signers useno leadcrown, othersselect astandardamount of lead crown based on experience and stillothers might prefer end relief ratherthan leadcrow-n. In this study, we shalluse aload distributionsim-ulation 18-20 to select both the profile and leadmodifications that will avoid corner co
33、ntact and tipinterference and at the same time provide reason-able insensitivity to misalignment.The basic gear geometry to be used in this paper isgiveninTable1:Thegeometryissimilartooneusedin a previous study 21 except that the root diame-ters have been adjusted slightly.Table 1. Helical gear geom
34、etryPinion GearNumber of teeth 25 31Normal diametral pitch(1/in)8.598Normal pressure angle(deg)23.45Helix angle (deg) 21.50Center distance (in) 3.50Face width (in) 1.25 1.25Outside diameter (in) 3.360 4.110Root diameter (in) 2.811 3.561Standard pitch diameter, dp(in)3.125 3.875Transverse tooth thick
35、nessat dp(in)0.1934 0.1934Profile / face contact ratio 1.383 / 1.254Total contact ratio 2.637Pinion torque, lb-in 5000Theprocedureforcomingupwithanacceptablemi-crogeometry is as follows:Step1. Identify the rough torque rating for the gearpair. For the sample gear set, we used an AGMArating formula w
36、ith approximate constants to comeup with a torque rating of roughly 5000 lb-in.Step2. Usingthisrating,aloaddistributionanalysiswith perfect involutes (shafts not included in theanalysis)isperformed. Figure1showstheloaddis-tribution at one contact position for the perfectinvo-lute analysis. Oneobserv
37、es thatthere issignificantcontact at the tooth tips (tip interference) and at theenteringcornerofthetooth(cornercontact). Figure2 shows a composite plot of contact stresses thatresult from the analysis of many positions of con-tact. The stresses at the tip and root andat thecor-ners are abnormally h
38、igh due to the tip interferenceand corner contact and due to the fact that the ra-dius used to compute the contact stress at the tip ismuch smaller than that along the tooth flank. Peakcontact stresses are about 240 ksi in the corners,200 ksi on the tip edge and 170-180 ksi in the toothcenter. FromF
39、igure3,whichshowsrootstressesat5 equally spaced locations along the root of thetooth, the peak pinion root stress is about 44 ksi.The root stresses of the gear were quite similar, soin all subsequent analyses, we shall only observethe pinion root stresses.Figure 1. Load distribution of a perfectinvo
40、lute with 5000 lb-in torqueFigure 2. Contact stress distribution of aperfect involute with 5000 lb-in torque6Figure 3. Pinion root stresses at 5 locationsacross the face width for a perfect involutewith 5000 lb-in torqueStep 3. Identify the maximum tooth deflection, thevalue of which is then used as
41、 a guideline in select-ing the values of tip and root relief. This deflection,whichmaybetakenfromthetransmissionerrorplotof Figure 4, is about 0.001 inch.Figure 4. Transmission deviation of a perfectinvolute with 5000 lb-in torqueStep4. For reference,at thedesign loaddeterminetheeffectofamplitudeoft
42、ipreliefandleadcrowningon the major design parameters, namely, contactstressandrootstress. Fornarrowfacewidthhelicalgears it has been found that both circular lead andcircular profile modifications perform well in distrib-uting the load and in reducing transmission error.Figure 5 shows the results o
43、f such an analysis forour gear pair when operating at a pinion torque of5000 in-lb.Figure 5 is quite interesting since it shows thethreshold modifications that are required in order tominimize both tip interference and corner contactstresses. The stresses at the lower left corner areabnormally high
44、due to the corner contact. As oneincreases the lead crown, these stresses drop, butsoonthetipinterferencestressesdominateandanyfurther increase of lead crown amplitude causesthese stresses to increase. In order to totally mini-mize the tip interference stresses, one must applytip relief. In this cas
45、e, applying about 0.0005 in. ofprofile crown and about 0.0002 in. of lead crownprovides a minimum contact stress of 201 ksi. Aseither the profile crown or the lead crown is furtherincreased, one observes that the contact stress in-creases. Thisincreaseisessentiallyduetoafocus-ing of the load closer
46、to the center of the tooth.Figure 5. Effect of profile and lead crown onthe peak contact stress at 5000 lb-in torqueA - 240 ksi, B - 201 ksi, C - 213 ksi, D - 230 ksiFigure 6 shows that, after a small initial amount ofleadcrownisapplied,rootstressesincreasewithin-creasing profileand leadcrown. One o
47、bservesthatthereisaslightconflictbetweenthe respectiveopti-mal microgeometries desired for root stresses andcontact stresses. However, our final microgeome-try is selected based on, first, insensitivity to mis-alignment, and second, insensitivity to all manufac-turing deviations so at this time this
48、 is not a bigissue.Asamatterofinterest,Figure7showshowtheloaddistribution factor, Kmchanges with microgeometryvariation. The “optimum” microgeometry now hasabout 0.0002 in of lead crown and no profile crownand at this condition, the load distribution factor isabout 1.19. Any further misalignment or
49、load shiftdue to shaft deflections is likely to increase thisvalue. Also,notethatthisperfectlyalignedloaddis-tribution factor increases as we increase either thelead crown or the profile crown.7Figure 6. Effect of profile crown and leadcrown on pinion root stress at 5000 lb-intorqueA - 43 ksi, B - 48 ksi, C - 53 ksi, D - 60 ksiFigure 7. Effect of lead crown and profilecrown on load distribution factor at 5000 lb-intorqueA - 1.19, B - 1.39, C - 1.58, D - 1.71Step 5. Determine candidate profile crowns andperformmicrogeometrysimula
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