1、91 FTM $1AWFinite Element Stress Analysis of aGenetic Spur Gear Toothby: Eugene A. Tennyson, The University of TennesseeAmerican Gear Manufacturers AssociationI I ITECHNICAL PAPERFinite Element Stress Analysis of a Generic Spur GearEugene A. TennysonThe University of Tennessee, Space InstituteTheSta
2、tements andopinions containedhereinare thoseof theauthor andshould notbe construedasan official actionoropinion of the American Gear ManufacturersAssociation.ABSTRACT:The prediction of bending stresses in a gear tooth, resulting fxom an externally applied torque, requires specialconsideration when d
3、esigning spur gear systems. The tooth geometry is such that excess risers exist which must beaccounted for. In addition, variables affecting the exact loadpoint on the tooth and the directionof the applied load arecritical. An interactive preprocessoris developed which generatesall the information,
4、includinga detailed tooth profile,necessary to perform a finite element bending stressanalysisofthe gear system. To validatetheprocedure, a test groupof spur gears is identified and analyzed. The results are then compared to those obtained via the American GearManufacturersAssociation (AGMA) standar
5、ds. The comparisonrevealed the finite element stressesto be slightlymoreconservative than the corresponding AGMA standard stresses. A generalized stress equationand geometry factor,based on the finite element approach, are also introduced. This paper is intended only as a proof of concept.Copyright
6、1991American Gear Manufacturers Association1500King Street, Suite 201Alexandria, Virginia, 22314October, 1991ISBN: 1-55589-615-4IAFinite Element Stress Analysis of a Generic Spur Gear ToothEugene A. Tennyson“The University of Tennessee Space Institute, Tullahoma, Tennessee 37388The predictionof bend
7、ingstresses in a gear tooth, resultingfroman externallyapplied torque,requiresspecialconsiderationwhendesigningspurgearsystems.The toothgeometryis suchthatstressrisers existwhichmustbe accountedfor.In addition,variablesaffectingthe exactload pointon thetoothand thedirectionof theappliedloadarecritic
8、al.Aninteractivepreprocessoris developedwhichgeneratesall theinformation,includingadetailedtooth profile,necessaryto performa finiteelementbendingstress analysisof the gearsystem.To validatethe procedure,a test groupof spur gears isidentifiedand analyzed.The results are then comparedto those obtaine
9、dvia the AmericanGearManufacturersAssociation(AGMA)standards.The comparisonrevealedthefiniteelementstressestobe slightlymoreconservativethan the correspondingAGMAstandard stresses.A generalizedstressequationand geometryfactor,basedon the finiteelementapproach,are also introduced.This paperis intende
10、donlyas a proofof concept.Introduction represents the transmitted tangential load and F theThe selection and design of spur gear systems is tooth face width. Although Eq. (1) representedprimarily guided by the anticipated bending stresses significant progress at the time, it is not very accu-in the
11、loaded tooth. Indeed, excessive bending stress rate in view of todays computational capabilities.in the fillet region is often the cause of gear failure.It is therefore important to have the means to _ereliably predict these stresses, w_Wilfred Lewis1, in 1893, proposed a bending stressformula, whic
12、h for the first time, took into account LoodPof_4“_nthe form of the tooth. To this day, this formula i I f- eh c_-e._remains the basis for most gear design. The geartooth is assumed to be a cantilever beam of uniformcross section, rigidly fixed at the base as shown inFigure 1. The “theoretical weake
13、st section“ AB isthen located by inscribing a parabola within thetooth outline. The parabola should be tangent to thefillets on either side, and its vertex is at the point Fig. 1 lewis Parabola of UniformStrengthwhere the line of action crosses the center line. Thebending stress a/: at AB is then de
14、termined by the In 1942, Dolan and Broghamer 2 conducted afollowing Lewis equation photoelastic study of stresses in gear teeth whichprovided much detailed information about the6Wt nature of stress distributions in the neighborhood ofL = hFt (1) fillets. This study also yielded a new stress concen-t
15、ration factor K which, when applied to the Lewisin which t is the length of AB and h is the distance equation, results in a more realistic value for thefrom C to AB in Figure 1. Furthermore, Wt bending stress at the tooth fillet. The factor Kreflects the combined effect of the tangential compo-nent
16、Wt and the radial component Wr of theexternal load W on the tooth. As an example, the*GraduateStudent,Dept. of MechanicalEngineering. stress concentration factor K for a 20 pressure anglestub tooth is given byIt_ o.15 f t1 0.45 Model DefinitionK = 0.18 + I.rfJ I.hJ (2) ToothProfileOne of the most wi
17、dely accepted gear tooth pro-where .rf represents the fillet radius of the tooth as files is defined by the involute curve which providesshown in Figure 1. a natural line of contact for two mating gears.The American Gear Manufacturers Association Before a finite element grid can be generated, it is(
18、AGMA) incorporated both the Lewis equation and necessary to derive a set of cartesian coordinatesthe Dolan and Broghamer stress concentration factor (z;i, YIi ) for a generic point on the tooth profile.K into its standard. The AGMA bending stress fora statically loaded standard addendum spur gear wi
19、th z8500 helix angle is given by “_ .2.80 _votuteAGMA= FJ (3) “g zTs_ oid/_ 2.70where rAGMA is the bending stress number, Rp thepitch radius of the gear, and Pd the diametral pitch. 2.65The Geometry Factor Y depends on the Lewis Form z60Factor Y as well as on the stress concentration factorI I I/_ I
20、n practice then, Eq. (3) is often used to deter- 0.o437o._75 o.m2mine the maximum static bending stress in a spur 1/2ToothThickness(In)gear tooth subjected to the usual operating loads.An alternative to the AGMA standard approach to Fig. 2 Involute and Trochoid Curvescomputing bending stresses in ge
21、ars is provided bythe finite element method. The possibility of using One of the more popular methods for cuttingthe finite element method is very appealing, espe- involute teeth is the hobbing process 4-5.This processcially to the designer who is only occasionally consists of traversing a straight
22、sided rack across aresponsible for gear selection and application. In rotating gear billet. As the rack moves across theaddition, it is expected that, when properly imple- rotating gear an involute is generated on the gearmented, the finite element approach will yield highly tooth blank. In addition
23、, a fillet is cut in the root ofaccurate results, the tooth in the form of a trochoid curve as depictedIn the course of the present study an interactive in Figure 2. Equations for both these curves can bepreprocessor was developed which queries the user derived. Assuming a base radius R b is given,
24、it canfor minimal information concerning the spur gear be seen from Figure 3a that the profile angle _bpisystem at hand. The preprocessor outputs all the corresponding to a generic point on the involute atinformation needed by a standard finite element radius ri is given byprogram to perform a stres
25、s analysis of the geartooth. In order to validate the procedure, a test _pi=sin -1 (R_/ri) (4)group of spur gear systems was identified andanalyzed using the finite element method. Theresulting bending stresses are compared with those The involute angle 0i is then defined asobtained via the AGMA sta
26、ndard Eq. (3). In addi- Oi = t_ntbpi- dppi (5)tion, a new generalized bending stress equation andgeometry factor are proposed, based on the finiteelement results. In the next section, the model Also, from Figure 3b it can be seen thatdefinition for a generic spur gear tooth will be tidiscussed. Oa=
27、0i +_r i (6)2AYli = ri cos Obi (1O)The cutting edge of the hob rack generates theshape of the root fillet. This shape, which is calledthe trochoid, is bounded by the dedendum circle andthe involute curve as shown in Figure 4a.Fig. 3a The Involute Curve mF?4a The Tr_hOidTiUl_eUsing Eqs. (4-6), it is
28、now possible to determinethecrdinates(xliYli)fagenericpintnthe _ _/_ Ibzinvolute. First, for a given tooth thickness tp at the pitch radius, the tooth thickness ti at any otherradius ri can be found asti=2ri_Rp+ Oa-Oil (7) Fig. 4b Hob Rack GeometryIn order to define the cartesian coordinates (XTi, Y
29、Ti)where Oa is defined by Eq. (6) for ri = Rf. In order of a generic point on the trochoid curve, it is neces-to obtain the coordinates (xli,Yfi) about the tooth sary to define the distance L in Figure 4b. L is thecenterline the angle 0bi is defined distance from the point Z to the hob tooth centerl
30、ine. The point Z is the center of the hob tip radiusR h-Obi= Oa- 0i = Op+ tp _ 0i (8) Defining T h as the hob tooth thickness, and Tp as2Rp the tooth thickness at the pitch radius, it is seen thatso thatx,ii = ri sin Obi (9) Th = _r/Pd - Tp (11)bz= b- R h (12) whereand tan/3i -dzZi = -RP_iein_Ti+bzc
31、s_i (18)-dYzi bzsila_Ti + RprlicosrliL=2T“ -bztand_ff r! (13)eos_pp Finally, transferring the trochoid coordinates to thegear tooth coordinate system via the angle A inwhere _pp is referred to as the standard pressure Figure 4a, it follows thatangle, or the tooth pressure angle, and is defined byEx.
32、 (4) for ri =Rp. Furthermore, b is the deden- ZFi=YTisinA-zTiCOSA (19)alum radius and be is shown in Figure 4b.As the gear blank rotates over an angle ni, the _lFi=YTiCOsA+ZTisinA (20)hob traverses a distance niRp. The exact trochoidalcurve can then be generated by letting ,7i vary from where0 to 7r
33、/2. The coordinates (Zgi,7Zi) of the tro-choid center Z are found as A= _Pd L (21)2Rp Rp_gi=(Rpr_i)eos_i-(Rp-bz)ein_i (14) Equations (9-10) and (19-20) can now be used todetermine an exact profile for a generic spur gearYzi = (RP - bz) cos _i + (R/ _i ) ein_i (15) tooth. Equation (18) must be numeri
34、calty solved for _ibetween 0 and _r/2. In addition, the involute coordi-Next, the coordinates (gTi_YTi) of the correspon- nates must be evaluated from the base curve where 0iding point on the trochoid can be obtained by is 0, to the top of the land radius where ri is equaladding to (zg i, Ygi ) the
35、coordinates of the hob tip to Rp. And finally, the intersection of the involuteradius R h . and the trochoid must be found in order to deter-vr mine the exact transition point. The analyst is nowin a position to generate a finite element grid for axr generic tooth. To apply a given load to the tooth
36、, iti is necessary to determine the coordinates of theapplication point of the load.x7Tooth Load, Spur gears develop critical bending stress whenloaded at the highest point where a pair of teeth arestill in contact. This point is called the Highest Pointof Single Tooth Contact (I-IPSTC). This point
37、isYz used as the load application point only for highquality gears; the tooth tip is used for standardquality gears. Referring to Figure 6, the normaloperating pressure angle n_,for a gear set withFig. 5 TroehoidCoordinates pitch radii Rpa and Rpp is given byLetting fl be the angle formed by a line
38、normal to _n_= (Rpc +Rpe)sind,pp- (Rpp+a)2_(Rpe_b)2 1/2the trochoid generated by the point Z and the YTaxis, as shown in Figure 5, it is seen that (22)To determine the HPSTC, consider again Figure 6,ZTi = zgi +.R h cc_s _i (16) from which it follows thatYTi -Ygi - Rh sinl_i (17) R l= Rp + a (23)A_o
39、= ee-1 (R_/RI) (24) finite element model are taken as shown in Figure 7.The boundary between the tooth and the rest of theEquations (23) and (24) are valid for both the gear gear is pinned. This allows for the necessary freedomG and the pinion P. Furthermore, of movement when bending loads are appli
40、ed.In this study the tooth stress problem is consideredAO=ER_p+R_p_2Rt_pRtpeOS(,oG_eb1_p)l/2 to be a plane stress problem. This presumes that theload is uniformly distributed along the width of the(25) tooth and that the front and back faces are allowedto freely expand.2 _rRbpAC = (26) ResultsNpStre
41、ssesOC=AC-AO (27) The principle stresses for each statically loadedtooth were determined using a quadratic finiteOp element mesh as shown in Figure 7. This mesh_ _ prved t be adequati_ _:_:Fig.6 GeometryofMeshingTeethTherefore, the location Re of the HPSTC for the cgear is found to be Fig. 7 Finite
42、Element Mesh and Boundary ConditionsR=ttp, cTU=ER_G+OU2_2R_,aOCeos(Cjoa_tbp_,) 1/2 for all cases in terms of convergence requirements.(28) A total of twenty gear sets were used in this study.Finite Element Model The sets consisted of five specific gears, eachWhen the tooth model accurately reflects
43、the matched with three pinions and a hypothetical tipgeometry, a finite element stress analysis can yield loading. For each gear and pinion set the HPSTChighly accurate results. By using the analytical equa- was determined and incorporated into the finitetions for the involute and the trochoid, it i
44、s possible element model in order to produce more accurateto generate an accurate profile of the tooth. From results. Table 1 lists the specifications for the partic-this, a finite element grid for the tooth can be ular set of gears used.establish as shown in Figure 7. The actual maximum principle s
45、tress occurs at theNext, the HPSTC is located analytically. If the tooth fillet as expected. Due to the geometry of theexact location of the HPSTC does not cooincide tooth, the maximum compressive stress occurs at thewith node location, a close approximation can be side of the tooth opposite the loa
46、d, and is slightlyobtained by applying the load as a pressure between higher than the maximum tensile stress which occursthe nodes surrounding the load point, on the loaded side of the tooth. The location of theThe boundary conditions for a single tooth type maximum stress is extremely close to that
47、 found via5the method of the theoretical weakest section pressure angle of 20* and helix angle of 0% It waspresented by l.zwis. Figure 8 shows a typical tooth observed that the diametral pitch of the gears hadmodel with the maximum principle stress contour no effect on the geometry factor JF“ Theref
48、ore, thelines as generated by the finite element program, factor is applicable for any diametral pitch. Becausethe stresses in the gears are functions of the appliedforce, face width, diametral pitch and pitch radius,the following number a0 is defined_il/_“_._ WPd(29)(:_ aO=In ( W,F,P,I, Rp ) = F R
49、pBy using the results of the finite element analysis,Appli_ai_ JF is given asJ,= a 0 = WP,_ (30)unesof.-%_“_,fb,_._:_: _ FEA FRpaFE AThe stress equation then becomesWPd (31)aFEA = FRp J-_R.9.HPSTC I I i 4, + “where JF is defined for different mating gear sets asa function of N 1 and N 2. N 1 and N 2 represent theFig. $ MaximumPrinciple Stress ContourLinesnumber of teeth on the gear and the pinion respec-The stresses obtained from the finite element tively. Evaluation of JF for various values ofN 2analysis are higher than those predicted by th
