1、99FTM15 e I ITheoretical Model for Load Distribution on Cylindric Gears: Application to Contact Stress Analysis by: J.I. Pedrero, M. Arts, M. Pleguezuelos, UNED and C. Garca-Masi and A. Fuentes, Universidad Politcnica de Cartagena 1 I TECHNICAL PAPER COPYRIGHT American Gear Manufacturers Association
2、, Inc.Licensed by Information Handling ServicesTheoretical Model for Load Distribution on Cylindric Gears: Application to Contact Stress Analysis J.I. Pedrero, M. Arts, M. Pleguezuelos UNED, and C. Garca-Masi and A. Fuentes Universidad Politcnica de Cartagena The statements and opinions contained he
3、rein are those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract In this paper, the load sharing along the line of contact on spur and helical gear teeth is determined from the hypothesis of minimum elastic potential.
4、From this non-uniform load distribution and Hertzs equation, a method for determining both the value and the location of the critical contact stress is described. Obtained results are compared with those given by IS0 and AGMA standards. Copyright O 1999 American Gear Manufacturers Association 1500 K
5、ing Street, Suite 201 Alexandria, Virginia, 22314 October, 1999 ISBN: 1-55589-753-3 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesTHEORETICAL MODEL FOR LOAD DISTRIBUTION ON CYLINDRIC GEARS: APPLICATION TO CONTACT STRESS ANALYSIS J. I. Pedrero, Profes
6、sor ; M. Arts, Professor ; M. Pleguezuelos, Assistant Professor . r) C. Garca-Masi, Associate Professor ; A. Fuentes, Associate Professor (“) (*) r) UNED, Departamento de Mecnica Apdo. 60149,28080 Madrid, Spain Universidad Politcnica de Cartagena. Departamento de Ingeniera Mecnica y Energtica. Po Al
7、fonso XII1 44, 30203 Cartagena, Murcia, Spain Nomenclature C Ei, Ez F Fn a Fnt operating center distance modulus of elasticity compressive load normal load component of the load contained in the transverse plane height of both cylinders transmitted power radii of both elastic cylinders deformation p
8、otential elastic coefficient numbers of teeth on pinion and wheel fa ce width fractional parts of E, and E/r load per unit of length linear coordinate along the line of contact length of contact minimum length of contact effective length of contact for IS0 model normal module base pitch pinion and w
9、heel outside radii pinion and wheel base radii pinion and wheel contact radii Dinion mean radius axial contact ratio ES transverse contact ratio EU pl, Poissons ratios relative radius of curvature P pl, p2 pinion and wheel curvature radii at the tip relative curvature radius at the lowest point Ps o
10、f single tooth contact of the pinion relative curvature radius at the lowest point P of contact of the pinion p, p pinion and wheel curvature radii at the transverse section pl, p2 pinion and wheel curvature radii at the normal section OH contact stress 01 pinion rotational velocity 5 F dimensionles
11、s parameter for the rotational position Introduction Both IS0 i J and AGMA 2,3 models for the contact stress between two cylindric gear teeth are based on Hertzs equation for the contact between two elastic cylinders 4 I 11 I - U uni ta ry potent ia1 F R, +% o, = - W inverse unitary potential i;. *(
12、!$+!A) at standard transverse pressure angle at operating transverse pressure angle standard helix angle base helix angle where F is the compressive load, i the height of both cylinders, and R, p and E the radius, the COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Han
13、dling ServicesPoissons ratio and the modulus of elasticity, respectively, for each cylinder. Replacing f by the normal load F, L by the length of contact I, Ri and Rz by the curvature radii of both normal profiles of pinion and wheel at the contact point p1 and pi2, and doing the elastic coefficient
14、 ZE Subscript 1 and 2 in equation (4) denote the pinion and the wheel, respectively. Fni can be computed from the transmitted power P, the pinion base radius fb1 and the pinion rotational velocity o1 as while the curvature radii at the transverse section, as seen in Figure 1, may be derived from we
15、have (3) Finally, taking into account that F, = Fn cos, and p = p, cos, , where Fnf is the component of the load contained in the transverse plane, p the radius of curvature of the transverse profile and pb the base helix angle, we obtain f 1 / r Figure 1. Curvature radii where rc denotes the radius
16、 of the contact point. Figure 1 shows that and p are related by where alt is the transverse operating pressure angle. 444 II To simplify the notation, if we do - = - + -L , we have P Pl P2 (4) I ni 6, =z, I- , P (7) and, according to Equations (6), we can express p as a function of the radius of the
17、 pinion contact point rc1. Equation (i) gives the values for the nominal contact stress at any contact point and it is involved in both IS0 and AGMA models for pitting calculations. From a theoretical point of view, the critical value, or the determinant value, of CRI will be located at the contact
18、point in which (il, (b) d,+dp 1. Also, as seen in Figure 7, the value of ggiving the local maximum value of v(g)p-(;) is always placed at the interval of single tooth contact of the transverse section C, 5 6 5 ths , or what is the same, d, 2 A 5 1. 200 I , I 1 178 156 134 112 90 J I l l I A B CD E 2
19、00 181 168 152 136 120 A B C D E l I I 176 - 164 - 152 - 140 - A B C D E Figure 7. Typical aspects of function v(;)p- (;): (a) without local maximum, (b) with local maximum different from the absolute one, ,(c) with local maximum being the absolute one. Both intervals define a rectangle in the diagr
20、ams takes values form O t l-du . This result is similar to the interpolated values considered by IS0 l and AGMA 3, though the interpolation is made in terms of 2 instead of p- . If c, -+ O the left side of the rectangle in Figure 8.b is placed at A;, = du + c, + da, , whose intersection with the lin
21、e At = A& defines the critical point Ag = du, or what is the same f = 4, +da = C, . This point is the critical point obtained for the case of spur gears above, so there is ”continuity” in the model. Similarly, when the maximum of v(if)p”(-f) is located at col, the critical stress is always located a
22、t the intersection point of the line 6 = lo -E and the e left side of the rectangle in Figure 8.b, , = &, + eP. From both equations, the critical point is = 6, , which means that the critical contact stress is always placed at the lowest point of single tooth contact, without any influence of the ax
23、ial contact ratio. At any rate, the continuity” of the model is also ensured, as this point is the same as that of spur gears. Results and discussion For spur gears, both the location of the point of critical contact stress and the value of the load per unit of length given by the model presented he
24、rein, are the same as those in l-31, so there is nothing to discuss. Only some special cases may present the critical contact stress at the lowest point of contact, but these cases are less probable as the load on this point is the third part of the total load, as discussed above. For helical gears
25、with axial contact ratio greater than 1, IS0 l and AGMA (31 standards suggest to compute the determinant contact stress at the pitch point and the pinion mean radius, respectively. From the model presented herein the critical point is the maximum of v()p-(t), denoted by E,”, which is always containe
26、d in the interval of single tooth contact C,S, neglecting the special cases in which it is located at 6. A comparative study of the obtained values of &,with respect to those given by i and 3, CI= and &MA, has been performed by considering more than 600 different cases, with different values of the
27、pinion tooth number, gear ratio, shift x-factor, pressure angle and helix angle. Table 1 shows the results of some of these cases. Typically, when the pinion mean radius coincides with the operating pitch point, and consequently c, =CAGMA, &, is quite similar to both for small gear ratios, while dif
28、ferences increase as the gear ratio increases, decrease with the pressure angle and remain uniform with variations of the helix angle. On the contrary, when C,co # SAGMA, cVp is very similar to AGM MA for small gear ratios and to cis for high ones. Comparisons among ISO. AGMA and minimum-potential s
29、tresses, given by Equations (12), (15) and (37), respectively, are not suitable as 10 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesTable i. Results of the comparative study among ISO, AGMA and minimum-potential models for contact stress. these nomin
30、al values should be corrected by several influence factors, some load distribution factors among them, which will be obviously different for each case. More interesting may be the comparison between the nominal contact stresses at tiso, (EAGMA and &, all of them computed by Equation -(37). Table 1 s
31、hows some results. Tendencies are the same: greater differences for high gear ratios and small pressure angles, for riso = SAGMA, and values close to AGMA or &O, according to the gear ratio is small or high, when riso t AGMA. However, values of the stresses are much more similar among them than thos
32、e of the corresponding 4 parameters. This means that IS0 and AGMA predictions on the load for pitting occurring and its location are in good agreement with the models of load distribution and contact stress presented herein. For helical gears with axial contact ratio less than 1, IS0 and AGMA sugges
33、t to compute the contact stress at an interpolated value of 4 between the lowest point of single tooth contact, corresponding to E, = O, and the pitch point (EO) or the pinion mean point (AGMA), corresponding to cg = 1. In our case, three different possibilities should be distinguished: O o d, +E, .
34、 1 For the first two cases we had obtained that the critical contact stress was located at the same point as in the case of 21, .e., point ,gvp in which v(4)p-l (c) is maximum. But as seen in Table 1, this value of cvp is smaller than tiso and (AGMA for cg z 1, and now ciCo and &GMA decrease, so dif
35、ferences will be smaller. And differences in terms of contact stresses will be even smaller. For the last case, the critical contact stress was located at the point of = tb +E , corresponding to an interpolated value between tVp and &. Since the critical ,g of the three models tend to &, differences
36、 will also tend to be smaller. Conclusions A model for the load distribution along the line of contact on spur and helical gears has been developed based on the minimum deformation potential principle. This distribution and Hertzs equation have been used to establish a model for the contact stress.
37、11 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicese For spur gears, the determinant contact stress is usually located at the lowest limit of the interval of single tooth contact. Only in special cases, for small pinion tooth number and high gear ratio
38、s, the critical point may be placed at the inner limit of the interval of contact, but this possibility is less probable as the load at this point is three times smaller than that at the other one. For helical gears with axial contact ratio greater than 1, the contact stress is maximum at a point in
39、 which the inverse unitary potential multiplied by the relative curvature radius is maximum. This point is always inside the interval of single tooth contact, except in special cases similar to the previous ones. The same point is the critical one for axial contact ratio smaller than 1 but greater t
40、han 1 minus the fractional part of the transverse pressure angle, and also for axial contact ratio smaller than 1 minus the fractional part of the transverse pressure angle if this point belongs to the line of contact at the moment in which the last transverse section of the pinion is loaded at the
41、lowest point of single tooth contact. If the point doesnt belong to the contact interval, the critical point is the contact point of the first transverse section at the same moment. This point corresponds to one obtained from a linear interpolation, in terms of the 6 parameter, between the highest p
42、oint of single tooth contact, for axial contact ratio equal to 1, and the lowest point of single tooth contact, for axial contact ratio equal to O. Since the last one is the point obtained for spur gears, the continuity of the model is achieved. All the above results are in good agreement with calcu
43、lation methods proposed by IS0 and AGMA standards. Ac knowiedgement Thanks are expressed to the Spanish Council for Technical and Scientific Research for the financial support for the research project PB95-0876, “Generation of Conjugate Profiles for Gear Teeth. Development of the Behavior Models for
44、 Bending and Wear“, and also to the Vicerrectorate of Research of the UNED for the additional support. I11 I21 131 91 References IS0 International Standard 6336-2:1996, “Calculation of Load Capacity of Spur and Helical Gears - Part 2: Calculation of Surface Durability (Pitting)“ , International Orga
45、nization for Standardization, Genve (1996). AGMA Standard 2001-C95, “ Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth“, American Gear Manufacturen Association, Alexandria, VA (1 995). AGMA Information Sheet 908-889, “ Geometry factors for Determining the P
46、itting Resistance and Bending Strength of Spur, Helical and Hemngbone Gear Teeth“, American Gear Manufacturers Association, Alexandna, VA (1989). J. Hertz, “On the Contact of Elastic Solids“, ,Miscellaneous Papers, Macmillan (1896). H.Winter and T.Placzek, “Load Distribution and Topological Flank Mo
47、dification of Helical and Double Helical Gears“, European Journal of Mechanical Engineering. Vol. 36, No. 3 (1991). Y. Zhang and Z. Fang, “Analysis of Tooth Contact and Load Distribution of Helical Gears with Crossed Axes“, Mechanism and Machine Theory, Vol. 34, no. 1 (1999). J. Boemer, “Very Effici
48、ent Calculation of the Load Distribution on External Gear Sets - the Method and Applications of the Program LVR“, Proceedings of the Yh International Power Transmission and Gearing Conference, San Diego (1 996). J. I. Pedrero, M. Estrems and A. Fuentes, “ Determination of the Efficiency of Cylindric
49、 Gear Sets“, Proceedings of the 4I World Congress on Gearing and Power Transmissions, Vol. 1, Paris (1 999). J. I. Pedrero, M. Artes and A. Fuentes, “Modelo de Distribucin de Carga en Engranajes Cilndricos de Perfil de Evolvente“ , Revista iberoamericana de 12 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesIngenieria Mecnica, Vol. 3, No. 1 (1999). lo A. Fuentes, “Modelo de Clculo a Flexin de Engranajes Cilndricos de Perfil de Evolvente“, P
