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FEM stress analysis in hypoid gears
Mechanism and Machine Theory 35 (2000) 1197±1220
www.elsevier.com/locate/mechmt
FEM stress analysis in hypoid gears Vilmos Simon Szent IstvaÂn Egyetem, GeÂpeÂszmeÂrnoÈki Kar, Mechanika eÂs MuÈszaki AÂbraÂzolaÂs TanszeÂk, 2103 GoÈdoÈlloÈ, PaÂter KaÂroly u.1, Hungary Received 9 July 1997; received in revised form 14 September 1998; accepted 15 September 1999
Abstract Stress analysis in hypoid gears by using ®nite element method is performed in order to develop simple equations for the calculation of tooth de¯ections and stresses. A displacement type ®nite element method is applied with curved, twenty-node isoparametric elements. A method is developed for the automatic ®nite element discretization of the pinion and the gear. The full theory of mismatched hypoid gears is applied for the determination of the nodal point coordinates on the teeth surfaces. The corresponding computer program is developed. By using this program the in¯uence of design parameters and load position on tooth de¯ections and ®llet stresses is investigated. On the basis of the results, obtained by performing a big number of computer runs, by using regression analysis and interpolation functions, equations for the calculation of tooth de¯ections and ®llet stresses are derived. The advantages of the use of these equations in load distribution calculation in hypoid gears are shown. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Hypoid gear; Finite element method; Displacements; Stresses; Approximation
1. Introduction The ®nite element method (FEM) is particularly well suited to study tooth de¯ections, contact stresses and tooth ®llet stresses in gears. Most of the research by FEM has been directed towards the spur and helical gears, mostly with two-dimensional geometry. Because of the diculty of correctly identifying the three-dimensional geometry, only a few researchers have investigated hypoid and spiral bevel gears, especially hypoid gears. For hypoid gears the ®rst attempt has been made by Wilcox [1], using the ¯exibility matrix in combination with the ®nite element method. Handschuh and Litvin [2] developed an analytical method to solve the gear tooth surface coordinates and provide input to a three-dimensional geometric modeling program that enables stress analysis in spiral bevel gears by FEM. A similar method was 0094-114X/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 9 9 ) 0 0 0 7 1 - 3
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Nomenclature a bf1 , bf 2 E p , E g fhgm fF e i g fFj g Fn fw , fs kw , ks m NF Ni N1 , N2 rc frFj g fri g st wg g w p n , wn a11 , a12 a2 xFj , ZFj , zFj cp0 g s p Rfil , sRfil
hypoid oset pinion and gear face widths, moduli of elasticity of the pinion and gear materials, respectively factor of gear mean working depth equivalent nodal force concentrated load normal load displacement and stress in¯uence factors displacement and stress factors module number of concentrated loads acting on the element interpolation function numbers of pinion and gear teeth nominal (mean) cutter radius position vector of the loaded point expressed by normalized coordinates position vector of the element nodal points in the global coordinate system minimal normal pinion topland width cutter point width for gear ®nishing de¯ections of pinion and gear teeth, respectively cutter blade angles for pinion ®nishing cutter blade angle for gear ®nishing normalized coordinates of the loaded point pinion mean spiral angle maximum ®llet stresses in pinion and gear teeth
applied by Bibel et al. [3] to make a multi-tooth model (four gear and three pinion teeth) for FEM stress analysis in spiral bevel gears. Gap elements are used in [4] for the evaluation of stresses by FEM in spiral bevel gears. In the paper published by Bibel and Handschuh [5] the data for a commercially available 3D solid modeler was generated by numerically evaluating the kinematic motion of the manufacturing process for spiral bevel gears and the contact between the deformable gear teeth was modeled with the automatic generation of nonpenetration constraints. Handschuh [6] uses the FEM for the thermal and for the structural/ contact analysis in spiral bevel gears. Huang et al. [7] presented a new method of tooth contact ®nite element analysis for spiral bevel and hypoid gears, which combines 3D ®nite element contact stress analysis, in which the friction was considered, with loaded tooth contact analysis. In the paper written by Chen et al. [8] a computerized simulation of meshing and contact of loaded gear drives was presented, determining the instantaneous contact ellipse and the contact force distributed over the contact ellipse, the de¯ection of contacting teeth under the load, the load share, the real contact ratio and the real function of transmission errors caused by gear misalignment and tooth de¯ection, and the maximum bending stresses by FEM. The method is
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applied to a hypoid gear pair. This method is further improved by Litvin et al. [9]. In the paper published by Gosselin et al. [10], the contact deformations are calculated by FEM for line contact (spur) and point contact (spiral bevel) gear pairs and compared with recognized analytical formulations. The use of the ®nite strip method for the determination of de¯ections in spur, straight bevel and helical gear teeth is introduced by Gagnon et al. [11]. By the author of this paper research has been ongoing in an attempt to predict stresses and to calculate the load distribution in hypoid gears by using ®nite element method. A displacement type ®nite element method is applied [12] for the stress analysis in the pinion and the gear. Curved, twenty-node isoparametric elements are used (see Fig. 1). A great deal of work has gone into ®nite element modeling of teeth and bodies of the pinion and the gear. A method has been developed for the automatic ®nite element discretization of the pinion and the gear by arbitrarily chosen number and sizes of elements in the main directions and in the dierent regions of the teeth in order to get a ®ner mesh where extreme values of stresses are expected or previously calculated [13,14]. Such a change of the number and sizes of elements also gives the opportunity for the investigation of the convergence of the solution. The main task in this method is to ®nd the appropriate position of the nodal points and to calculate their coordinates. The theory of the hypoid pinion and gear tooth surfaces, presented in paper [15], has been applied and the appropriate equations have been used to determine the coordinates of the nodal points. The corresponding computer program has been developed. By using this program the in¯uence of design parameters and load position on de¯ections and stresses has been investigated. In these investigations, the in¯uence of design parameters on pinion and gear blank geometry is calculated by the method suggested by the Gleason works [16], and the corresponding optimal machine settings are determined by the method presented in papers [17]
Fig. 1. Curved, twenty-node isoparametric element.
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and [18]. On the basis of the obtained FEM results, obtained by performing a big number of computer runs, by using the regression analysis and interpolation functions, equations for the calculation of de¯ections and ®llet stresses have been derived. 2. The ®nite element method A displacement type ®nite element method is applied for the determination of de¯ections and stresses in hypoid gears. Curved, twenty-node, sixty degree of freedom isoparametric solid elements are used to discretize the pinion and the gear teeth (Fig. 1). In the displacement type ®nite element method the only permissible form of loading, other than initial stressing, is by the prescription of concentrated loads at the nodal points. By neglecting the body forces and excluding the thermal loading, the equivalent representation of the tooth contact pressure or a concentrated load is performed. The fully conjugated hypoid gears are theoretically with line contact. In order to decrease the sensitivity of the gear pair to errors in tooth surface manufacture and to mutual position of the mating members, carefully chosen modi®cations are usually introduced into the teeth of one or both members [15]. As a result of these modi®cations point contact of the meshed teeth surfaces appears instead of the linear one, but as the tooth surface modi®cations are relatively small and the conjugation of the meshing surfaces is relatively good, the theoretical point contact changes into line contact even in the case of light loading. To avoid the integration, the tooth contact pressure is approximated by a ®nite number of concentrated loads positioned along these instantaneous contact lines of mating surfaces. The equivalent nodal forces are calculated by the following equation o jN n XF ÿ e Ni xFj , ZFj , zFj Fj 1 Fj j1
The points of the instantaneous contact lines, thus the acting points of the concentrated loads approximating the tooth pressure, are de®ned in the global, orthogonal coordinate system K x, y, z: The corresponding normalized coordinates are calculated by iterations, using the following equation frFj g ÿ
i20 X i1
ÿ Ni xFj , ZFj , zFj fri g 0
2
The vectorial equation (2) is equivalent to two scalar equations. One of the normalized coordinates, namely ZFj is known, its value is equal to 21, and the sign depends on which side of the pinion and the gear tooth ¯ank the load acts. The calculation of the coordinates of the nodal points is based on the real tooth geometry of the mismatched hypoid gears. The points of the instantaneous contact lines under load are determined by using the method developed in paper [15]. 3. Computed stresses and displacements A computer program has been developed for the calculation of stresses and displacements in the pinion and the gear of mismatched hypoid gears. By using this program the stress analysis
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in the hypoid gear pair of the design data given in Table 1 has been carried out. The optimal machine-tool setting parameters for the generation of the pinion and gear tooth blanks are calculated by the method used in the Gleason works and by the method presented by Litvin et al. in Refs. [17] and [18], and are given in Tables 2 and 3. The generated FEM mesh in the pinion and the gear is shown in Figs. 2 and 3. In the stress and de¯ection calculations a much ®ner mesh, with a much bigger number of elements and nodal points, was used than it is presented in these ®gures: the number of elements was 672 and of nodal points 3647. In this mesh the number of rows in the tooth height direction is seven for the tooth ¯ank and two for the root region. With such a FEM mesh a very good convergence of the calculated results has been obtained. Because of the perspicuity a coarser mesh is shown in Figs. 2 and 3., Concentrated load or tooth pressure along the instantaneous contact lines can be applied in this computer program. The aim of the research presented in this paper is the investigation of the in¯uence of the position of a concentrated load acting on the tooth ¯ank on tooth de¯ections and stresses. That is the reason why only a concentrated load, acting in a selected point of the pinion/gear tooth ¯ank is applied. The in¯uence of the number of adjacent teeth included in the calculations has been investigated. The obtained results have shown that the stresses were for 4.4% lower and the de¯ections for 9% higher when the calculation is performed in three teeth instead of one. In the case of ®ve teeth included in the calculations, the further reduction of stresses was 2.6% and the further increase in de¯ections 1%. Therefore, a very good precision can be obtained by making the FEM calculations for three adjacent teeth. Table 1 Pinion and gear design data Pinion Number of teeth Module (mm) Running oset (mm) Outside diameter (mm) Face width (mm) Crown to crossing point (mm) Front crown to crossing point (mm) Mean radius (mm) Mean spiral angle (deg) Pitch angle (deg) Face angle of blank (deg) Root angle (deg) Pitch apex beyond crossing point (mm) Face apex beyond crossing point (mm) Root apex beyond crossing point (mm) Mean addendum (mm) Mean dedendum (mm) Mean working depth (mm) Minimal normal topland width (mm)
11 4.41402 25.4 77.585 31.911 86.998 57.684 27.495 50.2597 18.5400 23.2733 17.5722 3.193 5.735
Gear 41 181.859 27.762 30.671 77.274 32.3007 70.5799 71.5859 65.6684 ÿ0.023 ÿ0.398 0.767 1.083 6.295 6.372 1.930
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Table 2 Gear machine-tool settings Cutter radius (mm) Cutter blade angle (deg) Point width (mm) Basic machine root angle (deg) Basic machine center to back increment (mm) Basic horizontal (mm) Basic vertical (mm)
75.184 19.0 2.032 64.5986 2.385 37.778 66.732
The in¯uence of the following design data on tooth de¯ections and stresses in the pinion and gear teeth is investigated: numbers of pinion and gear teeth (N1, N2), face width bf ), factor of gear mean working depth fhgm ), pinion mean spiral angle cp0 ), hypoid oset (a ), nominal (mean) cutter radius rc ), minimal normal pinion topland width st ), cutter blade angles for pinion ®nishing a11 , a12 ), cutter point width for gear ®nishing wg ), and cutter blade angle for gear ®nishing a2 ). Some of the obtained results are shown in Figs. 4±8. In these ®gures factors kw w=w0 and ks s=s0 represent the ratio of the displacement in the loaded point (w ) and of the maximum ®llet stress s obtained by applying arbitrarily chosen values of design parameters, and the displacement in the same point (w0 ) and the maximum ®llet stress s0 obtained by applying the basic values of design data presented in Table 1. The superscript (p) of factors kw w=w0 and ka s=s0 indicates the pinion, and (g) the gear. It can be noted that with the strongest eect on de¯ections of both members are the numbers of teeth (N1 and N2, Fig. 4) and the cutter blade angle for gear ®nishing a2 , Fig. 8). The de¯ections of the pinion teeth are strongly in¯uenced by the face width bf , Fig. 5), pinion oset (a, Fig. 7), nominal (mean) cutter radius rc , minimal normal pinion topland width st ), and cutter blade angles for pinion ®nishing a11 , a12 ), too. A moderate eect on pinion tooth de¯ections has the mean spiral angle cp0 , Fig. 6). It is interesting, that by the exemption of Table 3 Pinion machine-tool settings
Point radius of the cutter (mm) Cutter blade angle (deg) Machine root angle (deg) Basic tilt angle (deg) Basic swivel angle (deg) Basic cradle angle (deg) Sliding base setting (deg) Basic machine setting to back increment (mm) Basic radial (mm) Blank oset setting (mm) Ratio of roll
Concave
Convex
75.5 10.0 ÿ3.4404 21.2246 ÿ34.1845 79.8130 16.498 ÿ1.117 72.426 23.411 3.5510
75.5 31.0 ÿ3.0231 18.8172 ÿ47.0239 73.2048 22.237 ÿ0.190 69.924 22.336 3.4808
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Fig. 2. Generated mesh in the pinion.
the number of teeth, the cutter blade angle, and the cutter point width for gear ®nishing wg ), all the other design data have a moderate, or almost a negligible in¯uence on the displacement of the loaded point. Also, it can be seen that almost all the design parameters have a much stronger in¯uence on the maximum ®llet stresses than on the de¯ections. By using the same computer program, another investigation has been carried out in order to determine the in¯uence of the position of the loaded point on the de¯ections in the same (F ) and in an arbitrarily chosen point (D ) on the tooth ¯ank (Figs. 9 and 10), and on the maximum ®llet stresses belonging to the normal section passing through tooth surface point D. Because of the limited length of this paper only part of the obtained results is shown in Figs. 11±17. On the basis of results of a large number of computer runs, by using regression analysis and interpolation functions, the following equations are derived for the calculation of the normal de¯ection in tooth surface point D under normal load, Fn N , acting on point F: De¯ection in pinion tooth
Fig. 3. Generated mesh in the gear.
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Fig. 4. The in¯uence of numbers of teeth on de¯ections and ®llet stresses.
Fig. 5. The in¯uence of face width on de¯ections and ®llet stresses.
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Fig. 6. The in¯uence of mean spiral angle on de¯ections and ®llet stresses.
Fig. 7. The in¯uence of pinion oset on de¯ections and ®llet stresses.
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Fig. 8. The in¯uence of cutter blade angle on de¯ections and ®llet stresses.
Fig. 9. Position of the loaded point on the pinion tooth.
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Fig. 10. Position of the loaded point on the gear tooth.
Fig. 11. The in¯uence of the radial position of the loaded point on de¯ection in the same point.
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Fig. 12. The in¯uence of the axial position of the loaded point on de¯ection in the same point.
Fig. 13. De¯ection distributions in the gear tooth height direction for dierent radial load positions.
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Fig. 14. Axial de¯ection distributions in gear tooth for dierent radial load positions.
Fig. 15. The in¯uence of the radial position of the loaded point on maximum ®llet stresses.
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Fig. 16. The in¯uence of the axial position of the loaded point on maximum ®llet stresses.
Fig. 17. Fillet stress distribution in pinion's tooth face direction.
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0:3820 ÿ0:9428 cp0 Fn 0:9337 0:3982 bf1 0:2185 19:1951 N N2 f hgm mE p 1 m 508 0:3685 ÿ1:2476 ÿ0:3937 a rc st a11 ÿ0:2387 a12 0:65893 m m m 108 308
p w n
a2 208
ÿ1:1651
f
p wr0 hFr ,
bFr f
p p wa0 bFr f wr hDr ,
hFr f
p wa bDr ,
De¯ection in gear tooth g w n
ÿ0:3081 Fn ÿ0:0240 ÿ0:4381 bt2 73:1095 N N2 f mE g 1 m
wg m
0:4918
a2 208
0:4743
f
g wr0 hFr ,
bFr f
0:7356 hgm
cp0 508
g g wa0 bFr f wr hDr ,
3
bFr cm
0:4787 0:00966 0:1467 a rc m m
hFr f
g wa bDr ,
hFr
cm 4
where E p and E g (N/cm2) are moduli of elasticity of the pinion and gear materials, respectively. The calculations are made for the Poisson's ratio m 0:3: The module, m, has to be substituted in cm s. In Eqs. (3) and (4) factors fwr0 and fwa0 represent the in¯uence of the radial and axial position of the loaded point F on tooth de¯ection in the same point; factors fwr and fwa describe the de¯ection distributions in the radial and tooth length directions, and represent the in¯uence of the mutual position of points F and D on the de¯ection in point D. The cubic-splines can be used quite eectively for interpolation of curves. This kind of interpolation is especially a superior approximation of the behavior of curves that have local, abrupt changes [19]. In this particular case, the obtained curves are smooth, and they can be quite eectively segment-wise interpolated by second order polynomials (quadratic polynomial, parabola). In addition, the interpolation between these curves, because of the second variable, would have to be made anyway by second order polynomials. On the other hand, the calculated coecient matrices, providing the cubic-spline interpolation, would not be able to be presented in the paper because of their sizes. This is why the interpolation by quadratic polynomials is provided. Table 4 Values of coecients aa , ba , ca , . . . ,cc for the calculation of function f
p wr0
hFr
i
ai
bi
ci
hFr R0:7
a b c a b c
0.25696 0.01551 0.31962 10.173 ÿ25.018 15.845
0.05419 ÿ1.0380 0.74000 12.421 ÿ31.640 19.219
ÿ0.29781 1.5788 ÿ0.86250 ÿ22.439 56.326 ÿ33.887
hFr > 0:7
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Table 5 Values of coecients aa , ba , ca , . . . ,cc for the calculation of function f
g wr0
hFr
i
ai
bi
ci
hFr R0:7
a b c a b c
0.14571 0.27579 0.14494 12.708 ÿ31.244 19.536
0.01700 ÿ0.39862 0.23375 9.1602 ÿ22.949 13.788
0.00208 0.20687 ÿ0.16458 ÿ6.2483 15.490 ÿ9.2417
hFr s > 0:7
The in¯uence of the radial position of the loaded point on the de¯ection in the same point, in the case of the gear, is shown in Fig. 11. For the pinion similar curves have been obtained. The relative radial position of the loaded point, hFr , is de®ned as the ratio of the distance of the loaded point from the tooth ®llet, hF , to the tooth height, htF , in the normal tooth section passing through point F (see Figs. 9 and 10), i.e. hFr hF =htF : The corresponding in¯uence factor, for the pinion/gear tooth, can be approximated by the following function f
p wr0 hFr ,
bFr f
g wr0 hFr ,
bFr a bhFr ch2Fr
5
As this in¯uence depends on the axial position of the loaded point, bF , it follows a aa ba bFr ca b2Fr ;
b ab bb bFr cb b2Fr ; c ac bc bFr cc b2Fr
6
where bFr bF =bf1 for the pinion, bFr bF =bf2 for the gear (Figs. 9 and 10), and represent the relative axial position of the loaded point. For the better ®tting of curves in Fig. 11, they are divided into two parts and the approximations by Eqs. (5) and (6) are separately applied for both parts. The corresponding values of coecients aa , ba , ca , . . . ,cc are given in Table 4 for the pinion, and in Table 5 for the gear. The in¯uence of the axial position of the loaded point on the de¯ection in the same pinion/ gear tooth ¯ank point is shown in Fig. 12. These curves can be approximated by parabolas f
p wa0 bFr
f
g wa0 bFr
a bbFr cb2Fr
7
The corresponding values of coecients a, b, and c are given in Table 6.
Table 6 Values of coecients a, b, c for the calculation of functions f
Pinion, f Gear, f
p wa0
g wa0
p wa0
and f
g wa0
bFr
a
b
c
bFr R0:5 bFr > 0:5 bFr R0:5 bFr > 0:5
1.2894 4.0777 1.2369 2.5483
ÿ2.0985 ÿ10.785 ÿ1.7486 ÿ4.9573
3.0393 9.2590 2.5495 3.7215
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The in¯uence of the relative position of the loaded point F to point D, in which the de¯ection is calculated, is shown in Fig. 13 for the gear tooth. Quite similar curves are obtained for the pinion. These curves represent the de¯ection distributions along the height of the pinion/gear tooth for ®ve dierent radial positions of the loaded point hFr 0:2, 0.4, 0.6, 0.8, 1.0). The corresponding in¯uence factor can be calculated by the following equation f
p wr hDr ,
hFr f
g wr hDr ,
hFr a bhDr ch2Dr
8
where hDr hD =htD (Figs. 9 and 10). The coecients a, b, c can be calculated by Eq. (6). The corresponding values of coecients aa , ba , ca , . . . ,cc are given in Table 7, for the pinion, and in Table 8, for the gear. For better ®tting, the curves shown in Fig. 13 are divided into two parts g at points corresponding to f p wr, lim and f wr, lim , whose values can be calculated by the equation f
p wr, lim
f
g wr, lim
af bf hFr cf h 2Fr
9
The values of coecients af , bf , cf are given in Table 9. In Fig. 14 the axial de¯ection distributions in gear tooth for dierent radial load positions are presented. Respective parts of these curves can be approximated by the function f
p wa bDr ,
hFr f
g wa bDr ,
hDr a bbDr cb2Dr
10
Table 7 Values of coecients aa , ba , ca , . . . ,cc for the calculation of function f p wr, lim
hF , hD
f
p wr ,
hF rhD
f
p p wr Rf wr, lim
f
hFr
i
ai
bi
ci
hFr < 0:6
a b c a b c a b c a b c a b c a b c a b c a b c
4.3506 ÿ40.339 110.61 1.1192 ÿ7.5210 17.991 12.833 ÿ44.750 49.115 85.532 ÿ160.76 67.636 3.9243 ÿ5.7522 ÿ17.482 ÿ33.048 99.455 ÿ81.261 0.42467 ÿ1.1833 0.85006 7.3027 ÿ14.912 7.4717
ÿ11.688 113.09 ÿ306.75 ÿ2.2627 17.374 ÿ36.616 ÿ47.705 180.24 ÿ181.90 ÿ259.74 518.59 ÿ235.92 ÿ22.053 60.655 25.886 132.00 ÿ377.71 291.63 2.6502 ÿ3.0631 1.2250 ÿ26.008 54.140 ÿ26.365
7.9683 ÿ78.604 212.53 1.2361 ÿ10.234 19.580 35.194 ÿ145.60 147.70 186.65 ÿ387.28 186.29 39.095 ÿ111.79 20.820 ÿ114.96 326.57 ÿ244.93 ÿ2.1941 4.2547 ÿ1.8056 26.464 ÿ52.949 25.784
hFr r0:6 f
p wr
>f
p wr, lim
hFr < 0:6 hFr r0:6
hF < hD
f
p p wr rf wr, lim
hFr R0:6 hFr > 0:6
f
p wr
p wa, lim
p wr
hFr R0:6 hFr > 0:6
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Table 8 Values of coecients aa , ba , ca , . . . ,cc for the calculation of function f g wa ,
f
g wr, lim
hF , hD
f
hF rhD
fwr , g Rf
g wr, lim
hFr
i
ai
bi
ci
hFr < 0:6
a b c a b c a b c a b c a b c a b c a b c a b c
7.6854 ÿ71.520 181.57 0.93185 ÿ7.2779 18.191 18.749 ÿ53.253 36.761 109.50 ÿ192.37 64.422 3.8345 ÿ10.934 0.86242 ÿ32.763 94.780 ÿ73.596 0.23649 ÿ0.65741 0.48192 7.0615 ÿ13.551 6.3919
ÿ21.556 203.04 ÿ512.38 ÿ1.8581 15.667 ÿ35.846 ÿ74.436 242.98 ÿ181.15 ÿ339.13 648.74 ÿ261.82 ÿ18.458 64.339 ÿ32.041 134.03 ÿ376.14 278.20 2.9481 ÿ5.6173 2.9907 ÿ25.489 48.105 ÿ21.634
15.072 ÿ142.64 358.87 1.0023 ÿ8.7981 18.496 57.551 ÿ207.09 164.33 246.62 ÿ496.92 221.95 36.917 ÿ112.25 71.033 ÿ115.57 328.22 ÿ239.21 ÿ1.4226 4.9814 ÿ3.0038 27.015 ÿ48.741 21.621
hFr r0:6 f
g wr
>f
g wr, lim
hFr < 0:6 hFr r0:6
hF < hD
f
g g wr rf wr, lim
hFr R0:6 hFr > 0:6
f
g wr
g wr, lim
g wr
hFr R0:6 hFr > 0:6
where bDr bD =bf1 for the pinion, and bDr bD =bf2 for the gear (see Figs. 9 and 10). The coecients of function (10) can be calculated by expressions (6). The corresponding values of coecients aa , ba , ca , . . ., cc are given in Table 10, for the pinion, and in Table 11, for the gear. In the case of the pinion, for bDr R0:33 or bDr r0:67 it is f p wa 0, and in the case of the 0: The values of b gear, for bDr RbDr1 or bDr rbDr2 it is f g Dr1 and bDr2 can be calculated by wa Table 9 Values of coecients af , bf , cf for the calculation of f
Pinion
and f
g wr, lim
hF , hD
hFr
af
bf
cf
hF rhD
hFr < 0:6 hFr r0:6 hFr R0:6 hFr > 0:6 hFr < 0:6 hFr r0:6 hFr R0:6 hFr > 0:6
0.35128 0.35673 0.51202 2.0228 0.36005 0.15320 0.25248 2.6269
0.50652 0.48667 ÿ0.48392 ÿ5.0383 0.14284 0.89631 0.10085 ÿ7.0571
ÿ0.35250 ÿ0.33498 0.79190 4.1243 ÿ0.04000 ÿ0.70479 0.21443 5.4518
hF < hD Gear
p wr, lim
hF rhD hF < hD
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Table 10 Values of coecients aa , ba , ca , . . . ,cc for the calculation of function f p wa bDr
i
ai
bi
ci
0:33 < bDr < 0:42
a b c a b c a b c a b c
0.16415 0.39346 ÿ0.29712 ÿ33.528 136.77 ÿ135.43 15.665 ÿ44.897 31.134 22.011 ÿ69.796 55.169
3.2460 ÿ26.678 50.823 37.293 ÿ150.34 151.51 ÿ45.644 161.59 ÿ140.61 ÿ2.5746 18.862 ÿ22.500
ÿ3.6691 25.184 ÿ42.532 ÿ19.166 76.258 ÿ75.854 36.532 ÿ130.69 115.24 ÿ14.560 40.130 ÿ27.436
0:42RbDr R0:5 0:5 < bDr R0:58 0:58 < bDr < 0:67
the following expressions bDr1 0:49993 ÿ 0:93725hFr 0:52063h2Fr bDr2 0:46355 1:1460hFr ÿ 0:65125h2Fr
11
On the basis of the obtained results, by using regression analysis and interpolation functions, the following equations are derived for the calculation of the maximum ®llet stresses belonging to the normal section passing through tooth surface point D, under normal load, Fn N , acting on point F: Table 11 Values of coecients aa , ba , ca , . . . ,cc for the calculation of function f
g wa
bDr
i
ai
bi
ci
bDr1 < bDr < 0:42
a b c a b c a b c a b c
4.5010 ÿ25.717 36.793 ÿ4.9881 14.442 ÿ4.9313 2.8722 1.5361 ÿ10.561 16.571 ÿ51.054 39.337
ÿ15.337 77.784 ÿ96.453 ÿ7.9141 39.533 ÿ47.409 ÿ16.593 57.930 ÿ49.487 ÿ33.311 113.41 ÿ95.474
10.968 ÿ54.147 65.628 10.036 ÿ47.030 53.914 17.417 ÿ62.448 55.229 20.737 ÿ71.995 61.839
0:42RbDr R0:5 0:5 < bDr R0:58 0:58 < bDr < bDr2
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Table 12 Values of coecients a, b, c for the calculation of functions f
Pinion, f
Gear, f
p sa0
g sa0
p sa0
and f
g sa0
bFr
a
b
c
bFr R0:2 0:2 < bFr R0:6 0:6 < bFr R0:8 bFr > 0:8 bFr R0:2 0:2 < bFr R0:4 0:4 < bFr R0:7 bFr > 0:7
1.6902 1.5354 7.8957 11.006 3.9125 4.3382 1.5200 9.2549
ÿ2.5570 ÿ1.6110 ÿ22.042 ÿ28.182 ÿ10.273 ÿ17.538 ÿ1.4700 ÿ22.366
1.9400 1.0800 17.465 20.280 ÿ2.2667 23.415 0.8600 14.925
Maximum ®llet stresses in pinion teeth ÿ0:3038 Fn 1:8342 0:3170 bf1 p N2 f sRfil 2:3630 2 N 1 m m
ÿ0:7246 hgm
cp0 508
1:1726 0:3736 ÿ1:8549 a rc m m
ÿ0:3042 st a11 ÿ0:1654 a12 0:8402 a2 ÿ1:3042 m 108 308 208 ÿ p p p f sr0 hFr f sa0 bFr f sa bDr , hFr N=cm2 Maximum ®llet stresses in gear teeth p s Rfil
0:1395 Fn ÿ0:0132 ÿ0:3196 bf2 6:6104 2 N 1 N2 f m m
wg m
0:6252
a2 208
0:3233
0:0127 hgm
cp0 508
12
0:4389 ÿ0:3985 0:0058 a rc m m
ÿ g g g f sr0 hFr f sa0 bFr f sa bDr , hFr N=cm2
13
In these equations factors fsr0 and fsa0 represent the in¯uence of the radial and axial position of the loaded point F on maximum ®llet stresses of the normal section passing through the same point; factor fsa describes the distribution of the maximum ®llet stresses in the tooth length direction. The maximum ®llet stresses are proportional to the distance of the loaded point to the tooth ®llet, both for the pinion and the gear, as can be seen in Fig. 15. Therefore, the in¯uence factor fsr0 can be mathematically de®ned by the following equations f
p sr0
0:0692 0:9308hFr ;
f
g sr0
0:1734 0:8266hFr
14
The in¯uence of the axial position of the loaded point on maximum ®llet stresses of the normal section passing through the same point is shown in Fig. 16. These curves can be approximated by parabolas f
p sa0 bFr
f
g sa0 bFr
a bbFr cb2Fr
The corresponding values of coecients a, b, and c are given in Table 12.
15
V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220 Table 13 Values of coecients a, b, c for the calculation of function f
hFr r0:6
hFr 0:4
hFr 0:2
1217
p sa
bFr
a
b
c
bFr R0:32 0:32 < bFr R0:43 0:43 < bFr R0:68 0:68 < bFrR0:79 bFr > 0:79 bFr R0:32 0:32 < bFr R0:43 0:43 < bFr R0:57 0:57 < bFr R0:76 0:76 < bFr R0:87 bFr > 0:87 bFr R0:34 0:34 < bFr R0:43 0:43 < bFr R0:5 0:5 < bFr R0:61 0:76 < bFr R0:87 bFr > 0:87
0.0117 ÿ0.9507 ÿ1.3528 ÿ5.4732 13.077 0.0089 3.0654 ÿ8.8970 ÿ1.1312 15.351 16.577 0.0068 4.7951 0.4978 ÿ9.5439 5.0213 2.7939
ÿ0.8464 2.9918 6.8902 21.097 ÿ26.395 ÿ0.5045 ÿ19.765 37.132 8.0839 ÿ33.290 ÿ33.692 ÿ0.2589 ÿ26.386 ÿ10.428 42.277 ÿ11.768 ÿ5.5889
4.4392 1.8262 ÿ5.0443 ÿ17.029 13.363 2.6696 33.007 ÿ34.696 ÿ7.6261 18.294 17.158 1.2622 36.777 22.865 ÿ42.378 7.0582 2.8057
The load acts in the middle of the pinion tooth bDr 0:5), but the maximum ®llet stresses are out of this plane (see Fig. 17), and the axial stress distribution curves are strongly asymmetric. The segment-wise interpolation of these curves is performed by parabolas: f
p sa bDr
f
g sa bDr
a bbDr cb2Dr
16
The corresponding values of coecients a, b, and c are given in Table 13 for the pinion, and in Table 14 for the gear. For radial load position diering from hFr 0:2, 0.4, 0.6, 0.8, 1.0 further interpolations are necessary, based on the values given in Tables 13 and 14. 4. The application of the developed equations for tooth de¯ections in load distribution calculation The presented equations (3) and (4) for the calculation of pinion and gear teeth de¯ections can be successfully applied in the load distribution calculation of hypoid gears, as it follows. A method and the corresponding computer program for the determination of load distribution in mismatched hypoid gears is developed. The method is very similar to that for double enveloping worm gears presented in [20], includes the bending and shearing de¯ections of teeth, the local contact deformations of the mating teeth surfaces, the gear body bending and torsion, de¯ections of the supporting shafts, and the manufacturing and alignment errors. The governing equations of load distribution represent a system of nonlinear equations and an approximative technique is used to get the solution. The contact lines are discretized in a suitable number of small segments, and the tooth contact pressure, acting along a segment, is approximated by a concentrated load, acting in the mid-point of the segment. The local contact deformations along the segments are treated as Hertzian contacts of cylinders. The
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Table 14 Values of coecients a, b, c for the calculation of function f
g sa
bDr
hFr
bFr
a
b
c
bDr R0:39
hFr r0:6
bDr > 0:39
hFr 1:0
bFr R0:12 0:12 < bFr R0:23 0:23 < bFr R0:39 0:39 < bFr R0:65 0:65 < bFr R0:79 bFr > 0:79 0:39 < bFr R0:65 0:65 < bFr R0:79 bFr > 0:79 0:39 < bFr R0:65 0:65 < bFr R0:79 bFr > 0:79 bFr R0:21 0:21 < bFr R0:32 0:32 < bFr R0:46 0:46 < bFr R0:61 0:61 < bFr R0:77 bFr > 0:77 bFr R0:28 0:28 < bFr R0:43 0:43 < bFr R0:5 0:5 < bFr R0:57 0:57 < bFr R0:65 bFr > 0:65
0.1490 0.1705 ÿ1.1477 ÿ0.3952 5.0977 5.3418 0.0707 4.9637 5.0774 ÿ0.1081 5.8892 4.7611 0.0828 ÿ0.3800 ÿ5.4284 ÿ0.6790 7.5392 2.8141 0.0129 3.3926 ÿ3.8229 16.077 12.222 0.9475
0.1854 0.4176 11.280 6.5393 ÿ10.074 ÿ10.801 4.7533 ÿ10.053 ÿ10.329 5.5970 ÿ12.869 ÿ9.8088 0.2862 4.2694 29.933 8.9473 ÿ18.932 ÿ5.8754 0.0509 ÿ21.273 12.182 ÿ47.124 ÿ37.494 ÿ2.0710
10.906 7.6168 ÿ14.846 ÿ7.5900 4.9702 5.4993 ÿ6.0783 5.1212 5.2884 ÿ7.0654 7.1481 5.0824 4.6960 ÿ3.7757 ÿ34.719 ÿ11.539 12.075 3.0827 0.7400 33.706 ÿ5.0720 33.939 28.935 1.1298
hFr 0:8 hFr 0:6 hFr 0:4
hFr 0:2
actual load distribution, de®ned by the set of these concentrated loads, is obtained by using the successive-over-relaxation method. The tooth bending and shearing deformations are calculated by the ®nite element method, in two ways: by directly applying the FEM and by using the equations for de¯ections, developed on the basis of a big number of results obtained by FEM and presented in this paper (Eqs. (3) and (4)). The investigations have shown that by applying directly the FEM, the CPU time for load distribution calculation on a PC Pentium 100 MHz was 6 h and 5 min, and the convergence error after 10 iterations 4.34%. By using the presented equations for the calculation of tooth de¯ections the CPU time was 2 min and 10 s, and the convergence error after six iterations 0.53%. By using these two methods the dierence in the maximum tooth contact pressure is 3.6% and the dierence in the maximum load per unit length of the contact line is 6.4%. As the mismatch in hypoid gears is relatively small, thus the theoretical point contact, under load spreads over a surface. Usually, it is assumed that the contact spreads over an elliptical area. In this method for load distribution calculation a new approach to tooth contact is introduced: the point contact under load spreads over a surface along the `potential' contact line, which line is made up of the points of the mating tooth surfaces in which the separations of these surfaces are minimal. A method is developed for the determination of the `potential'
V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
1219
Fig. 18. Load distribution and tooth contact pressures.
contact lines in hypoid gears, presented in [15]. The investigations have shown that, especially on tooth pairs with contact on the toe or on the heel of teeth, the shape of the contact area diers from an ellipse, and there is a dierence in tooth contact pressures, calculated due to the part of an assumed ellipse of contact limited by the physical teeth boundaries ( pel), and calculated due to the actual load distribution ( p, Fig. 18). In Fig. 18 pel-w represents the contact pressure spreading over the whole ellipse, not limited by teeth boundaries, which pressure is only hypothetical. The investigations have also shown, that in the case of misaligned gears the assumption of an elliptical contact is totally unrealistic. Because of the limited length of this paper, the detailed description of the method for load distribution calculation in hypoid gears will be presented in a separate paper. 5. Conclusions A computer program based on ®nite elements has been developed for stress analysis in hypoid gears. By using this program the stresses and displacements have been calculated in the pinion and the gear of a mismatched hypoid gear pair. The obtained results have shown that a good convergence of the solution can be achieved by making the calculation for three adjacent teeth and by using a ®nite element mesh containing 672 elements and 3647 nodal points. By using the same computer program the in¯uence of design parameters and of the position of the loaded point on de¯ections and stresses has been investigated. On the basis of the results obtained by performing a big number of computer runs, by using regression analysis and interpolation functions, equations have been derived for the calculation of de¯ections in an
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arbitrarily chosen tooth ¯ank point under a normal load acting in the same or in a dierent tooth surface point, and for the calculation of ®llet stresses. The calculation of load distribution in hypoid gears has been shown that the use of these equations makes this calculation much easier and faster than it is by the direct use of the ®nite element method. References [1] L.E. Wilcox, An exact analytical method for calculating stresses in bevel and hypoid gear teeth, in: Proceedings of International Symposium on Gearing and Power Transmissions. Tokyo, vol. II, 1981, pp. 115±121. [2] R.F. Handschuh, F.L. Litvin, How to determine spiral bevel gear tooth geometry for ®nite element analysis, in: Proceedings of JSME International Conference on Motion and Power Transmissions, Hiroshima, 1991, pp. 704±710. [3] G.D. Bibel, K. Tiku, A. Kumar, R. Handschuh, Comparison of gap elements and contact algorithm for 3D contact analysis of spiral bevel gears, in: NASA TR ARL-TR-478, 30th Joint Propulsion Conference, Indianapolis, 1994. [4] G.D. Bibel, A. Kumar, S. Reddy, R.F. Handschuh, Contact stress analysis of spiral bevel gears using ®nite element analysis, ASME J. Mech. Des 117 (1995) 235±240. [5] G.D. Bibel, R. Handschuh, Meshing of a spiral bevel gearset with 3D ®nite element analysis, in: Proceedings of 7th International Power Transmission and Gearing Conference, San Diego, 1996, pp. 703±708. [6] R.F. Handschuh, Recent advances in the analysis of spiral bevel gears, in: Proceedings of MTM'97 International Conference on Mechanical Transmissions and Mechanisms, Tianjin, 1997, pp. 635±641. [7] C. Huang, R. Li, C. Zheng, Tooth contact ®nite element analysis for spiral and hypoid gears, in: 6th International Power Transmission and Gearing Conference, Scottsdale, 1992. [8] J.S. Chen, F.L. Litvin, A.A. Shabana, Computerized simulation of meshing and contact of loaded gear drives, in: Proceedings of International Gearing Conference, Newcastle upon Tyne, 1994, pp. 161±166. [9] F.L. Litvin, J.S. Chen, J. Lu, R.F. Handschuh, Application of ®nite element analysis for determination of load share, real contact ratio, precision of motion, and stress analysis, ASME J. Mech. Des 118 (1996) 561±567. [10] C. Gosselin, D. Gingras, J. Brousseau, A. Gakwaya, A review of the current contact stress and deformation formulations compared to ®nite element analysis, in: Proceedings of International Gearing Conference, Newcastle upon Tyne, 1994, pp. 155±160. [11] Ph. Gagnon, C. Gosselin, L. Cloutier, Analysis of spur, helical and straight bevel gear teeth de¯ection by ®nite strip method, in: Proceedings of International Conference on Gears, Dresden, 1996, pp. 909±921. [12] V. Simon, Computer implementation of ®nite element method for stress analysis in machine parts, in: Proceedings of XIIth International Symposium Computer at the University, Cavtat, vol. 78, 1990, pp. 1±6. [13] V. Simon, Automatic ®nite element mesh generation, in: Proceedings of Xth International Symposium Computer at the University, Cavtat, vol. 7.3, 1988, pp. 1±7. [14] V. Simon, Computerized ®nite element mesh generation in hypoid gears, in: Proceedings of 23rd Design Automation Conference in CD-ROM, Sacramento, 1997. [15] V. Simon, Tooth contact analysis of mismatched hypoid gears, in: Proceedings of 7th International Power Transmission and Gearing Conference, San Diego, 1996, pp. 789±798. [16] Gleason Works. Method for designing hypoid gear blanks. Rochester, 1971. [17] F.L. Litvin, Y. Gutman, Methods of synthesis and analysis for hypoid gear-Drives of `Formate' and `Helixform', Part I±III, ASME J. Mech. Des 103 (1981) 83±113. [18] F.L. Litvin, Y. Zhang, M. Lundy, C. Heine, Determination of settings of a tilted head cutter for generation of hypoid and spiral bevel gears, ASME J. Mechsms., Transm., Autom. Des 110 (1988) 495±500; Proceedings of ASME 5th International Power Transmission and Gearing Conference, Chicago, 1989, pp. 719±725. [19] S.C. Chapra, R.P. Canale, Numerical Methods for Engineering, McGraw-Hill, New York, 1989. [20] V. Simon, Load distribution in double enveloping worm gears, ASME J. Mech. Des 115 (1993) 496±501.
www.elsevier.com/locate/mechmt
FEM stress analysis in hypoid gears Vilmos Simon Szent IstvaÂn Egyetem, GeÂpeÂszmeÂrnoÈki Kar, Mechanika eÂs MuÈszaki AÂbraÂzolaÂs TanszeÂk, 2103 GoÈdoÈlloÈ, PaÂter KaÂroly u.1, Hungary Received 9 July 1997; received in revised form 14 September 1998; accepted 15 September 1999
Abstract Stress analysis in hypoid gears by using ®nite element method is performed in order to develop simple equations for the calculation of tooth de¯ections and stresses. A displacement type ®nite element method is applied with curved, twenty-node isoparametric elements. A method is developed for the automatic ®nite element discretization of the pinion and the gear. The full theory of mismatched hypoid gears is applied for the determination of the nodal point coordinates on the teeth surfaces. The corresponding computer program is developed. By using this program the in¯uence of design parameters and load position on tooth de¯ections and ®llet stresses is investigated. On the basis of the results, obtained by performing a big number of computer runs, by using regression analysis and interpolation functions, equations for the calculation of tooth de¯ections and ®llet stresses are derived. The advantages of the use of these equations in load distribution calculation in hypoid gears are shown. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Hypoid gear; Finite element method; Displacements; Stresses; Approximation
1. Introduction The ®nite element method (FEM) is particularly well suited to study tooth de¯ections, contact stresses and tooth ®llet stresses in gears. Most of the research by FEM has been directed towards the spur and helical gears, mostly with two-dimensional geometry. Because of the diculty of correctly identifying the three-dimensional geometry, only a few researchers have investigated hypoid and spiral bevel gears, especially hypoid gears. For hypoid gears the ®rst attempt has been made by Wilcox [1], using the ¯exibility matrix in combination with the ®nite element method. Handschuh and Litvin [2] developed an analytical method to solve the gear tooth surface coordinates and provide input to a three-dimensional geometric modeling program that enables stress analysis in spiral bevel gears by FEM. A similar method was 0094-114X/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 9 9 ) 0 0 0 7 1 - 3
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Nomenclature a bf1 , bf 2 E p , E g fhgm fF e i g fFj g Fn fw , fs kw , ks m NF Ni N1 , N2 rc frFj g fri g st wg g w p n , wn a11 , a12 a2 xFj , ZFj , zFj cp0 g s p Rfil , sRfil
hypoid oset pinion and gear face widths, moduli of elasticity of the pinion and gear materials, respectively factor of gear mean working depth equivalent nodal force concentrated load normal load displacement and stress in¯uence factors displacement and stress factors module number of concentrated loads acting on the element interpolation function numbers of pinion and gear teeth nominal (mean) cutter radius position vector of the loaded point expressed by normalized coordinates position vector of the element nodal points in the global coordinate system minimal normal pinion topland width cutter point width for gear ®nishing de¯ections of pinion and gear teeth, respectively cutter blade angles for pinion ®nishing cutter blade angle for gear ®nishing normalized coordinates of the loaded point pinion mean spiral angle maximum ®llet stresses in pinion and gear teeth
applied by Bibel et al. [3] to make a multi-tooth model (four gear and three pinion teeth) for FEM stress analysis in spiral bevel gears. Gap elements are used in [4] for the evaluation of stresses by FEM in spiral bevel gears. In the paper published by Bibel and Handschuh [5] the data for a commercially available 3D solid modeler was generated by numerically evaluating the kinematic motion of the manufacturing process for spiral bevel gears and the contact between the deformable gear teeth was modeled with the automatic generation of nonpenetration constraints. Handschuh [6] uses the FEM for the thermal and for the structural/ contact analysis in spiral bevel gears. Huang et al. [7] presented a new method of tooth contact ®nite element analysis for spiral bevel and hypoid gears, which combines 3D ®nite element contact stress analysis, in which the friction was considered, with loaded tooth contact analysis. In the paper written by Chen et al. [8] a computerized simulation of meshing and contact of loaded gear drives was presented, determining the instantaneous contact ellipse and the contact force distributed over the contact ellipse, the de¯ection of contacting teeth under the load, the load share, the real contact ratio and the real function of transmission errors caused by gear misalignment and tooth de¯ection, and the maximum bending stresses by FEM. The method is
V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
1199
applied to a hypoid gear pair. This method is further improved by Litvin et al. [9]. In the paper published by Gosselin et al. [10], the contact deformations are calculated by FEM for line contact (spur) and point contact (spiral bevel) gear pairs and compared with recognized analytical formulations. The use of the ®nite strip method for the determination of de¯ections in spur, straight bevel and helical gear teeth is introduced by Gagnon et al. [11]. By the author of this paper research has been ongoing in an attempt to predict stresses and to calculate the load distribution in hypoid gears by using ®nite element method. A displacement type ®nite element method is applied [12] for the stress analysis in the pinion and the gear. Curved, twenty-node isoparametric elements are used (see Fig. 1). A great deal of work has gone into ®nite element modeling of teeth and bodies of the pinion and the gear. A method has been developed for the automatic ®nite element discretization of the pinion and the gear by arbitrarily chosen number and sizes of elements in the main directions and in the dierent regions of the teeth in order to get a ®ner mesh where extreme values of stresses are expected or previously calculated [13,14]. Such a change of the number and sizes of elements also gives the opportunity for the investigation of the convergence of the solution. The main task in this method is to ®nd the appropriate position of the nodal points and to calculate their coordinates. The theory of the hypoid pinion and gear tooth surfaces, presented in paper [15], has been applied and the appropriate equations have been used to determine the coordinates of the nodal points. The corresponding computer program has been developed. By using this program the in¯uence of design parameters and load position on de¯ections and stresses has been investigated. In these investigations, the in¯uence of design parameters on pinion and gear blank geometry is calculated by the method suggested by the Gleason works [16], and the corresponding optimal machine settings are determined by the method presented in papers [17]
Fig. 1. Curved, twenty-node isoparametric element.
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V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
and [18]. On the basis of the obtained FEM results, obtained by performing a big number of computer runs, by using the regression analysis and interpolation functions, equations for the calculation of de¯ections and ®llet stresses have been derived. 2. The ®nite element method A displacement type ®nite element method is applied for the determination of de¯ections and stresses in hypoid gears. Curved, twenty-node, sixty degree of freedom isoparametric solid elements are used to discretize the pinion and the gear teeth (Fig. 1). In the displacement type ®nite element method the only permissible form of loading, other than initial stressing, is by the prescription of concentrated loads at the nodal points. By neglecting the body forces and excluding the thermal loading, the equivalent representation of the tooth contact pressure or a concentrated load is performed. The fully conjugated hypoid gears are theoretically with line contact. In order to decrease the sensitivity of the gear pair to errors in tooth surface manufacture and to mutual position of the mating members, carefully chosen modi®cations are usually introduced into the teeth of one or both members [15]. As a result of these modi®cations point contact of the meshed teeth surfaces appears instead of the linear one, but as the tooth surface modi®cations are relatively small and the conjugation of the meshing surfaces is relatively good, the theoretical point contact changes into line contact even in the case of light loading. To avoid the integration, the tooth contact pressure is approximated by a ®nite number of concentrated loads positioned along these instantaneous contact lines of mating surfaces. The equivalent nodal forces are calculated by the following equation o jN n XF ÿ e Ni xFj , ZFj , zFj Fj 1 Fj j1
The points of the instantaneous contact lines, thus the acting points of the concentrated loads approximating the tooth pressure, are de®ned in the global, orthogonal coordinate system K x, y, z: The corresponding normalized coordinates are calculated by iterations, using the following equation frFj g ÿ
i20 X i1
ÿ Ni xFj , ZFj , zFj fri g 0
2
The vectorial equation (2) is equivalent to two scalar equations. One of the normalized coordinates, namely ZFj is known, its value is equal to 21, and the sign depends on which side of the pinion and the gear tooth ¯ank the load acts. The calculation of the coordinates of the nodal points is based on the real tooth geometry of the mismatched hypoid gears. The points of the instantaneous contact lines under load are determined by using the method developed in paper [15]. 3. Computed stresses and displacements A computer program has been developed for the calculation of stresses and displacements in the pinion and the gear of mismatched hypoid gears. By using this program the stress analysis
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in the hypoid gear pair of the design data given in Table 1 has been carried out. The optimal machine-tool setting parameters for the generation of the pinion and gear tooth blanks are calculated by the method used in the Gleason works and by the method presented by Litvin et al. in Refs. [17] and [18], and are given in Tables 2 and 3. The generated FEM mesh in the pinion and the gear is shown in Figs. 2 and 3. In the stress and de¯ection calculations a much ®ner mesh, with a much bigger number of elements and nodal points, was used than it is presented in these ®gures: the number of elements was 672 and of nodal points 3647. In this mesh the number of rows in the tooth height direction is seven for the tooth ¯ank and two for the root region. With such a FEM mesh a very good convergence of the calculated results has been obtained. Because of the perspicuity a coarser mesh is shown in Figs. 2 and 3., Concentrated load or tooth pressure along the instantaneous contact lines can be applied in this computer program. The aim of the research presented in this paper is the investigation of the in¯uence of the position of a concentrated load acting on the tooth ¯ank on tooth de¯ections and stresses. That is the reason why only a concentrated load, acting in a selected point of the pinion/gear tooth ¯ank is applied. The in¯uence of the number of adjacent teeth included in the calculations has been investigated. The obtained results have shown that the stresses were for 4.4% lower and the de¯ections for 9% higher when the calculation is performed in three teeth instead of one. In the case of ®ve teeth included in the calculations, the further reduction of stresses was 2.6% and the further increase in de¯ections 1%. Therefore, a very good precision can be obtained by making the FEM calculations for three adjacent teeth. Table 1 Pinion and gear design data Pinion Number of teeth Module (mm) Running oset (mm) Outside diameter (mm) Face width (mm) Crown to crossing point (mm) Front crown to crossing point (mm) Mean radius (mm) Mean spiral angle (deg) Pitch angle (deg) Face angle of blank (deg) Root angle (deg) Pitch apex beyond crossing point (mm) Face apex beyond crossing point (mm) Root apex beyond crossing point (mm) Mean addendum (mm) Mean dedendum (mm) Mean working depth (mm) Minimal normal topland width (mm)
11 4.41402 25.4 77.585 31.911 86.998 57.684 27.495 50.2597 18.5400 23.2733 17.5722 3.193 5.735
Gear 41 181.859 27.762 30.671 77.274 32.3007 70.5799 71.5859 65.6684 ÿ0.023 ÿ0.398 0.767 1.083 6.295 6.372 1.930
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Table 2 Gear machine-tool settings Cutter radius (mm) Cutter blade angle (deg) Point width (mm) Basic machine root angle (deg) Basic machine center to back increment (mm) Basic horizontal (mm) Basic vertical (mm)
75.184 19.0 2.032 64.5986 2.385 37.778 66.732
The in¯uence of the following design data on tooth de¯ections and stresses in the pinion and gear teeth is investigated: numbers of pinion and gear teeth (N1, N2), face width bf ), factor of gear mean working depth fhgm ), pinion mean spiral angle cp0 ), hypoid oset (a ), nominal (mean) cutter radius rc ), minimal normal pinion topland width st ), cutter blade angles for pinion ®nishing a11 , a12 ), cutter point width for gear ®nishing wg ), and cutter blade angle for gear ®nishing a2 ). Some of the obtained results are shown in Figs. 4±8. In these ®gures factors kw w=w0 and ks s=s0 represent the ratio of the displacement in the loaded point (w ) and of the maximum ®llet stress s obtained by applying arbitrarily chosen values of design parameters, and the displacement in the same point (w0 ) and the maximum ®llet stress s0 obtained by applying the basic values of design data presented in Table 1. The superscript (p) of factors kw w=w0 and ka s=s0 indicates the pinion, and (g) the gear. It can be noted that with the strongest eect on de¯ections of both members are the numbers of teeth (N1 and N2, Fig. 4) and the cutter blade angle for gear ®nishing a2 , Fig. 8). The de¯ections of the pinion teeth are strongly in¯uenced by the face width bf , Fig. 5), pinion oset (a, Fig. 7), nominal (mean) cutter radius rc , minimal normal pinion topland width st ), and cutter blade angles for pinion ®nishing a11 , a12 ), too. A moderate eect on pinion tooth de¯ections has the mean spiral angle cp0 , Fig. 6). It is interesting, that by the exemption of Table 3 Pinion machine-tool settings
Point radius of the cutter (mm) Cutter blade angle (deg) Machine root angle (deg) Basic tilt angle (deg) Basic swivel angle (deg) Basic cradle angle (deg) Sliding base setting (deg) Basic machine setting to back increment (mm) Basic radial (mm) Blank oset setting (mm) Ratio of roll
Concave
Convex
75.5 10.0 ÿ3.4404 21.2246 ÿ34.1845 79.8130 16.498 ÿ1.117 72.426 23.411 3.5510
75.5 31.0 ÿ3.0231 18.8172 ÿ47.0239 73.2048 22.237 ÿ0.190 69.924 22.336 3.4808
V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
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Fig. 2. Generated mesh in the pinion.
the number of teeth, the cutter blade angle, and the cutter point width for gear ®nishing wg ), all the other design data have a moderate, or almost a negligible in¯uence on the displacement of the loaded point. Also, it can be seen that almost all the design parameters have a much stronger in¯uence on the maximum ®llet stresses than on the de¯ections. By using the same computer program, another investigation has been carried out in order to determine the in¯uence of the position of the loaded point on the de¯ections in the same (F ) and in an arbitrarily chosen point (D ) on the tooth ¯ank (Figs. 9 and 10), and on the maximum ®llet stresses belonging to the normal section passing through tooth surface point D. Because of the limited length of this paper only part of the obtained results is shown in Figs. 11±17. On the basis of results of a large number of computer runs, by using regression analysis and interpolation functions, the following equations are derived for the calculation of the normal de¯ection in tooth surface point D under normal load, Fn N , acting on point F: De¯ection in pinion tooth
Fig. 3. Generated mesh in the gear.
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V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
Fig. 4. The in¯uence of numbers of teeth on de¯ections and ®llet stresses.
Fig. 5. The in¯uence of face width on de¯ections and ®llet stresses.
V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
Fig. 6. The in¯uence of mean spiral angle on de¯ections and ®llet stresses.
Fig. 7. The in¯uence of pinion oset on de¯ections and ®llet stresses.
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V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
Fig. 8. The in¯uence of cutter blade angle on de¯ections and ®llet stresses.
Fig. 9. Position of the loaded point on the pinion tooth.
V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
Fig. 10. Position of the loaded point on the gear tooth.
Fig. 11. The in¯uence of the radial position of the loaded point on de¯ection in the same point.
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V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
Fig. 12. The in¯uence of the axial position of the loaded point on de¯ection in the same point.
Fig. 13. De¯ection distributions in the gear tooth height direction for dierent radial load positions.
V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
Fig. 14. Axial de¯ection distributions in gear tooth for dierent radial load positions.
Fig. 15. The in¯uence of the radial position of the loaded point on maximum ®llet stresses.
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V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
Fig. 16. The in¯uence of the axial position of the loaded point on maximum ®llet stresses.
Fig. 17. Fillet stress distribution in pinion's tooth face direction.
V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
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0:3820 ÿ0:9428 cp0 Fn 0:9337 0:3982 bf1 0:2185 19:1951 N N2 f hgm mE p 1 m 508 0:3685 ÿ1:2476 ÿ0:3937 a rc st a11 ÿ0:2387 a12 0:65893 m m m 108 308
p w n
a2 208
ÿ1:1651
f
p wr0 hFr ,
bFr f
p p wa0 bFr f wr hDr ,
hFr f
p wa bDr ,
De¯ection in gear tooth g w n
ÿ0:3081 Fn ÿ0:0240 ÿ0:4381 bt2 73:1095 N N2 f mE g 1 m
wg m
0:4918
a2 208
0:4743
f
g wr0 hFr ,
bFr f
0:7356 hgm
cp0 508
g g wa0 bFr f wr hDr ,
3
bFr cm
0:4787 0:00966 0:1467 a rc m m
hFr f
g wa bDr ,
hFr
cm 4
where E p and E g (N/cm2) are moduli of elasticity of the pinion and gear materials, respectively. The calculations are made for the Poisson's ratio m 0:3: The module, m, has to be substituted in cm s. In Eqs. (3) and (4) factors fwr0 and fwa0 represent the in¯uence of the radial and axial position of the loaded point F on tooth de¯ection in the same point; factors fwr and fwa describe the de¯ection distributions in the radial and tooth length directions, and represent the in¯uence of the mutual position of points F and D on the de¯ection in point D. The cubic-splines can be used quite eectively for interpolation of curves. This kind of interpolation is especially a superior approximation of the behavior of curves that have local, abrupt changes [19]. In this particular case, the obtained curves are smooth, and they can be quite eectively segment-wise interpolated by second order polynomials (quadratic polynomial, parabola). In addition, the interpolation between these curves, because of the second variable, would have to be made anyway by second order polynomials. On the other hand, the calculated coecient matrices, providing the cubic-spline interpolation, would not be able to be presented in the paper because of their sizes. This is why the interpolation by quadratic polynomials is provided. Table 4 Values of coecients aa , ba , ca , . . . ,cc for the calculation of function f
p wr0
hFr
i
ai
bi
ci
hFr R0:7
a b c a b c
0.25696 0.01551 0.31962 10.173 ÿ25.018 15.845
0.05419 ÿ1.0380 0.74000 12.421 ÿ31.640 19.219
ÿ0.29781 1.5788 ÿ0.86250 ÿ22.439 56.326 ÿ33.887
hFr > 0:7
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V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
Table 5 Values of coecients aa , ba , ca , . . . ,cc for the calculation of function f
g wr0
hFr
i
ai
bi
ci
hFr R0:7
a b c a b c
0.14571 0.27579 0.14494 12.708 ÿ31.244 19.536
0.01700 ÿ0.39862 0.23375 9.1602 ÿ22.949 13.788
0.00208 0.20687 ÿ0.16458 ÿ6.2483 15.490 ÿ9.2417
hFr s > 0:7
The in¯uence of the radial position of the loaded point on the de¯ection in the same point, in the case of the gear, is shown in Fig. 11. For the pinion similar curves have been obtained. The relative radial position of the loaded point, hFr , is de®ned as the ratio of the distance of the loaded point from the tooth ®llet, hF , to the tooth height, htF , in the normal tooth section passing through point F (see Figs. 9 and 10), i.e. hFr hF =htF : The corresponding in¯uence factor, for the pinion/gear tooth, can be approximated by the following function f
p wr0 hFr ,
bFr f
g wr0 hFr ,
bFr a bhFr ch2Fr
5
As this in¯uence depends on the axial position of the loaded point, bF , it follows a aa ba bFr ca b2Fr ;
b ab bb bFr cb b2Fr ; c ac bc bFr cc b2Fr
6
where bFr bF =bf1 for the pinion, bFr bF =bf2 for the gear (Figs. 9 and 10), and represent the relative axial position of the loaded point. For the better ®tting of curves in Fig. 11, they are divided into two parts and the approximations by Eqs. (5) and (6) are separately applied for both parts. The corresponding values of coecients aa , ba , ca , . . . ,cc are given in Table 4 for the pinion, and in Table 5 for the gear. The in¯uence of the axial position of the loaded point on the de¯ection in the same pinion/ gear tooth ¯ank point is shown in Fig. 12. These curves can be approximated by parabolas f
p wa0 bFr
f
g wa0 bFr
a bbFr cb2Fr
7
The corresponding values of coecients a, b, and c are given in Table 6.
Table 6 Values of coecients a, b, c for the calculation of functions f
Pinion, f Gear, f
p wa0
g wa0
p wa0
and f
g wa0
bFr
a
b
c
bFr R0:5 bFr > 0:5 bFr R0:5 bFr > 0:5
1.2894 4.0777 1.2369 2.5483
ÿ2.0985 ÿ10.785 ÿ1.7486 ÿ4.9573
3.0393 9.2590 2.5495 3.7215
V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
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The in¯uence of the relative position of the loaded point F to point D, in which the de¯ection is calculated, is shown in Fig. 13 for the gear tooth. Quite similar curves are obtained for the pinion. These curves represent the de¯ection distributions along the height of the pinion/gear tooth for ®ve dierent radial positions of the loaded point hFr 0:2, 0.4, 0.6, 0.8, 1.0). The corresponding in¯uence factor can be calculated by the following equation f
p wr hDr ,
hFr f
g wr hDr ,
hFr a bhDr ch2Dr
8
where hDr hD =htD (Figs. 9 and 10). The coecients a, b, c can be calculated by Eq. (6). The corresponding values of coecients aa , ba , ca , . . . ,cc are given in Table 7, for the pinion, and in Table 8, for the gear. For better ®tting, the curves shown in Fig. 13 are divided into two parts g at points corresponding to f p wr, lim and f wr, lim , whose values can be calculated by the equation f
p wr, lim
f
g wr, lim
af bf hFr cf h 2Fr
9
The values of coecients af , bf , cf are given in Table 9. In Fig. 14 the axial de¯ection distributions in gear tooth for dierent radial load positions are presented. Respective parts of these curves can be approximated by the function f
p wa bDr ,
hFr f
g wa bDr ,
hDr a bbDr cb2Dr
10
Table 7 Values of coecients aa , ba , ca , . . . ,cc for the calculation of function f p wr, lim
hF , hD
f
p wr ,
hF rhD
f
p p wr Rf wr, lim
f
hFr
i
ai
bi
ci
hFr < 0:6
a b c a b c a b c a b c a b c a b c a b c a b c
4.3506 ÿ40.339 110.61 1.1192 ÿ7.5210 17.991 12.833 ÿ44.750 49.115 85.532 ÿ160.76 67.636 3.9243 ÿ5.7522 ÿ17.482 ÿ33.048 99.455 ÿ81.261 0.42467 ÿ1.1833 0.85006 7.3027 ÿ14.912 7.4717
ÿ11.688 113.09 ÿ306.75 ÿ2.2627 17.374 ÿ36.616 ÿ47.705 180.24 ÿ181.90 ÿ259.74 518.59 ÿ235.92 ÿ22.053 60.655 25.886 132.00 ÿ377.71 291.63 2.6502 ÿ3.0631 1.2250 ÿ26.008 54.140 ÿ26.365
7.9683 ÿ78.604 212.53 1.2361 ÿ10.234 19.580 35.194 ÿ145.60 147.70 186.65 ÿ387.28 186.29 39.095 ÿ111.79 20.820 ÿ114.96 326.57 ÿ244.93 ÿ2.1941 4.2547 ÿ1.8056 26.464 ÿ52.949 25.784
hFr r0:6 f
p wr
>f
p wr, lim
hFr < 0:6 hFr r0:6
hF < hD
f
p p wr rf wr, lim
hFr R0:6 hFr > 0:6
f
p wr
p wa, lim
p wr
hFr R0:6 hFr > 0:6
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V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
Table 8 Values of coecients aa , ba , ca , . . . ,cc for the calculation of function f g wa ,
f
g wr, lim
hF , hD
f
hF rhD
fwr , g Rf
g wr, lim
hFr
i
ai
bi
ci
hFr < 0:6
a b c a b c a b c a b c a b c a b c a b c a b c
7.6854 ÿ71.520 181.57 0.93185 ÿ7.2779 18.191 18.749 ÿ53.253 36.761 109.50 ÿ192.37 64.422 3.8345 ÿ10.934 0.86242 ÿ32.763 94.780 ÿ73.596 0.23649 ÿ0.65741 0.48192 7.0615 ÿ13.551 6.3919
ÿ21.556 203.04 ÿ512.38 ÿ1.8581 15.667 ÿ35.846 ÿ74.436 242.98 ÿ181.15 ÿ339.13 648.74 ÿ261.82 ÿ18.458 64.339 ÿ32.041 134.03 ÿ376.14 278.20 2.9481 ÿ5.6173 2.9907 ÿ25.489 48.105 ÿ21.634
15.072 ÿ142.64 358.87 1.0023 ÿ8.7981 18.496 57.551 ÿ207.09 164.33 246.62 ÿ496.92 221.95 36.917 ÿ112.25 71.033 ÿ115.57 328.22 ÿ239.21 ÿ1.4226 4.9814 ÿ3.0038 27.015 ÿ48.741 21.621
hFr r0:6 f
g wr
>f
g wr, lim
hFr < 0:6 hFr r0:6
hF < hD
f
g g wr rf wr, lim
hFr R0:6 hFr > 0:6
f
g wr
g wr, lim
g wr
hFr R0:6 hFr > 0:6
where bDr bD =bf1 for the pinion, and bDr bD =bf2 for the gear (see Figs. 9 and 10). The coecients of function (10) can be calculated by expressions (6). The corresponding values of coecients aa , ba , ca , . . ., cc are given in Table 10, for the pinion, and in Table 11, for the gear. In the case of the pinion, for bDr R0:33 or bDr r0:67 it is f p wa 0, and in the case of the 0: The values of b gear, for bDr RbDr1 or bDr rbDr2 it is f g Dr1 and bDr2 can be calculated by wa Table 9 Values of coecients af , bf , cf for the calculation of f
Pinion
and f
g wr, lim
hF , hD
hFr
af
bf
cf
hF rhD
hFr < 0:6 hFr r0:6 hFr R0:6 hFr > 0:6 hFr < 0:6 hFr r0:6 hFr R0:6 hFr > 0:6
0.35128 0.35673 0.51202 2.0228 0.36005 0.15320 0.25248 2.6269
0.50652 0.48667 ÿ0.48392 ÿ5.0383 0.14284 0.89631 0.10085 ÿ7.0571
ÿ0.35250 ÿ0.33498 0.79190 4.1243 ÿ0.04000 ÿ0.70479 0.21443 5.4518
hF < hD Gear
p wr, lim
hF rhD hF < hD
V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
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Table 10 Values of coecients aa , ba , ca , . . . ,cc for the calculation of function f p wa bDr
i
ai
bi
ci
0:33 < bDr < 0:42
a b c a b c a b c a b c
0.16415 0.39346 ÿ0.29712 ÿ33.528 136.77 ÿ135.43 15.665 ÿ44.897 31.134 22.011 ÿ69.796 55.169
3.2460 ÿ26.678 50.823 37.293 ÿ150.34 151.51 ÿ45.644 161.59 ÿ140.61 ÿ2.5746 18.862 ÿ22.500
ÿ3.6691 25.184 ÿ42.532 ÿ19.166 76.258 ÿ75.854 36.532 ÿ130.69 115.24 ÿ14.560 40.130 ÿ27.436
0:42RbDr R0:5 0:5 < bDr R0:58 0:58 < bDr < 0:67
the following expressions bDr1 0:49993 ÿ 0:93725hFr 0:52063h2Fr bDr2 0:46355 1:1460hFr ÿ 0:65125h2Fr
11
On the basis of the obtained results, by using regression analysis and interpolation functions, the following equations are derived for the calculation of the maximum ®llet stresses belonging to the normal section passing through tooth surface point D, under normal load, Fn N , acting on point F: Table 11 Values of coecients aa , ba , ca , . . . ,cc for the calculation of function f
g wa
bDr
i
ai
bi
ci
bDr1 < bDr < 0:42
a b c a b c a b c a b c
4.5010 ÿ25.717 36.793 ÿ4.9881 14.442 ÿ4.9313 2.8722 1.5361 ÿ10.561 16.571 ÿ51.054 39.337
ÿ15.337 77.784 ÿ96.453 ÿ7.9141 39.533 ÿ47.409 ÿ16.593 57.930 ÿ49.487 ÿ33.311 113.41 ÿ95.474
10.968 ÿ54.147 65.628 10.036 ÿ47.030 53.914 17.417 ÿ62.448 55.229 20.737 ÿ71.995 61.839
0:42RbDr R0:5 0:5 < bDr R0:58 0:58 < bDr < bDr2
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V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
Table 12 Values of coecients a, b, c for the calculation of functions f
Pinion, f
Gear, f
p sa0
g sa0
p sa0
and f
g sa0
bFr
a
b
c
bFr R0:2 0:2 < bFr R0:6 0:6 < bFr R0:8 bFr > 0:8 bFr R0:2 0:2 < bFr R0:4 0:4 < bFr R0:7 bFr > 0:7
1.6902 1.5354 7.8957 11.006 3.9125 4.3382 1.5200 9.2549
ÿ2.5570 ÿ1.6110 ÿ22.042 ÿ28.182 ÿ10.273 ÿ17.538 ÿ1.4700 ÿ22.366
1.9400 1.0800 17.465 20.280 ÿ2.2667 23.415 0.8600 14.925
Maximum ®llet stresses in pinion teeth ÿ0:3038 Fn 1:8342 0:3170 bf1 p N2 f sRfil 2:3630 2 N 1 m m
ÿ0:7246 hgm
cp0 508
1:1726 0:3736 ÿ1:8549 a rc m m
ÿ0:3042 st a11 ÿ0:1654 a12 0:8402 a2 ÿ1:3042 m 108 308 208 ÿ p p p f sr0 hFr f sa0 bFr f sa bDr , hFr N=cm2 Maximum ®llet stresses in gear teeth p s Rfil
0:1395 Fn ÿ0:0132 ÿ0:3196 bf2 6:6104 2 N 1 N2 f m m
wg m
0:6252
a2 208
0:3233
0:0127 hgm
cp0 508
12
0:4389 ÿ0:3985 0:0058 a rc m m
ÿ g g g f sr0 hFr f sa0 bFr f sa bDr , hFr N=cm2
13
In these equations factors fsr0 and fsa0 represent the in¯uence of the radial and axial position of the loaded point F on maximum ®llet stresses of the normal section passing through the same point; factor fsa describes the distribution of the maximum ®llet stresses in the tooth length direction. The maximum ®llet stresses are proportional to the distance of the loaded point to the tooth ®llet, both for the pinion and the gear, as can be seen in Fig. 15. Therefore, the in¯uence factor fsr0 can be mathematically de®ned by the following equations f
p sr0
0:0692 0:9308hFr ;
f
g sr0
0:1734 0:8266hFr
14
The in¯uence of the axial position of the loaded point on maximum ®llet stresses of the normal section passing through the same point is shown in Fig. 16. These curves can be approximated by parabolas f
p sa0 bFr
f
g sa0 bFr
a bbFr cb2Fr
The corresponding values of coecients a, b, and c are given in Table 12.
15
V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220 Table 13 Values of coecients a, b, c for the calculation of function f
hFr r0:6
hFr 0:4
hFr 0:2
1217
p sa
bFr
a
b
c
bFr R0:32 0:32 < bFr R0:43 0:43 < bFr R0:68 0:68 < bFrR0:79 bFr > 0:79 bFr R0:32 0:32 < bFr R0:43 0:43 < bFr R0:57 0:57 < bFr R0:76 0:76 < bFr R0:87 bFr > 0:87 bFr R0:34 0:34 < bFr R0:43 0:43 < bFr R0:5 0:5 < bFr R0:61 0:76 < bFr R0:87 bFr > 0:87
0.0117 ÿ0.9507 ÿ1.3528 ÿ5.4732 13.077 0.0089 3.0654 ÿ8.8970 ÿ1.1312 15.351 16.577 0.0068 4.7951 0.4978 ÿ9.5439 5.0213 2.7939
ÿ0.8464 2.9918 6.8902 21.097 ÿ26.395 ÿ0.5045 ÿ19.765 37.132 8.0839 ÿ33.290 ÿ33.692 ÿ0.2589 ÿ26.386 ÿ10.428 42.277 ÿ11.768 ÿ5.5889
4.4392 1.8262 ÿ5.0443 ÿ17.029 13.363 2.6696 33.007 ÿ34.696 ÿ7.6261 18.294 17.158 1.2622 36.777 22.865 ÿ42.378 7.0582 2.8057
The load acts in the middle of the pinion tooth bDr 0:5), but the maximum ®llet stresses are out of this plane (see Fig. 17), and the axial stress distribution curves are strongly asymmetric. The segment-wise interpolation of these curves is performed by parabolas: f
p sa bDr
f
g sa bDr
a bbDr cb2Dr
16
The corresponding values of coecients a, b, and c are given in Table 13 for the pinion, and in Table 14 for the gear. For radial load position diering from hFr 0:2, 0.4, 0.6, 0.8, 1.0 further interpolations are necessary, based on the values given in Tables 13 and 14. 4. The application of the developed equations for tooth de¯ections in load distribution calculation The presented equations (3) and (4) for the calculation of pinion and gear teeth de¯ections can be successfully applied in the load distribution calculation of hypoid gears, as it follows. A method and the corresponding computer program for the determination of load distribution in mismatched hypoid gears is developed. The method is very similar to that for double enveloping worm gears presented in [20], includes the bending and shearing de¯ections of teeth, the local contact deformations of the mating teeth surfaces, the gear body bending and torsion, de¯ections of the supporting shafts, and the manufacturing and alignment errors. The governing equations of load distribution represent a system of nonlinear equations and an approximative technique is used to get the solution. The contact lines are discretized in a suitable number of small segments, and the tooth contact pressure, acting along a segment, is approximated by a concentrated load, acting in the mid-point of the segment. The local contact deformations along the segments are treated as Hertzian contacts of cylinders. The
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V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
Table 14 Values of coecients a, b, c for the calculation of function f
g sa
bDr
hFr
bFr
a
b
c
bDr R0:39
hFr r0:6
bDr > 0:39
hFr 1:0
bFr R0:12 0:12 < bFr R0:23 0:23 < bFr R0:39 0:39 < bFr R0:65 0:65 < bFr R0:79 bFr > 0:79 0:39 < bFr R0:65 0:65 < bFr R0:79 bFr > 0:79 0:39 < bFr R0:65 0:65 < bFr R0:79 bFr > 0:79 bFr R0:21 0:21 < bFr R0:32 0:32 < bFr R0:46 0:46 < bFr R0:61 0:61 < bFr R0:77 bFr > 0:77 bFr R0:28 0:28 < bFr R0:43 0:43 < bFr R0:5 0:5 < bFr R0:57 0:57 < bFr R0:65 bFr > 0:65
0.1490 0.1705 ÿ1.1477 ÿ0.3952 5.0977 5.3418 0.0707 4.9637 5.0774 ÿ0.1081 5.8892 4.7611 0.0828 ÿ0.3800 ÿ5.4284 ÿ0.6790 7.5392 2.8141 0.0129 3.3926 ÿ3.8229 16.077 12.222 0.9475
0.1854 0.4176 11.280 6.5393 ÿ10.074 ÿ10.801 4.7533 ÿ10.053 ÿ10.329 5.5970 ÿ12.869 ÿ9.8088 0.2862 4.2694 29.933 8.9473 ÿ18.932 ÿ5.8754 0.0509 ÿ21.273 12.182 ÿ47.124 ÿ37.494 ÿ2.0710
10.906 7.6168 ÿ14.846 ÿ7.5900 4.9702 5.4993 ÿ6.0783 5.1212 5.2884 ÿ7.0654 7.1481 5.0824 4.6960 ÿ3.7757 ÿ34.719 ÿ11.539 12.075 3.0827 0.7400 33.706 ÿ5.0720 33.939 28.935 1.1298
hFr 0:8 hFr 0:6 hFr 0:4
hFr 0:2
actual load distribution, de®ned by the set of these concentrated loads, is obtained by using the successive-over-relaxation method. The tooth bending and shearing deformations are calculated by the ®nite element method, in two ways: by directly applying the FEM and by using the equations for de¯ections, developed on the basis of a big number of results obtained by FEM and presented in this paper (Eqs. (3) and (4)). The investigations have shown that by applying directly the FEM, the CPU time for load distribution calculation on a PC Pentium 100 MHz was 6 h and 5 min, and the convergence error after 10 iterations 4.34%. By using the presented equations for the calculation of tooth de¯ections the CPU time was 2 min and 10 s, and the convergence error after six iterations 0.53%. By using these two methods the dierence in the maximum tooth contact pressure is 3.6% and the dierence in the maximum load per unit length of the contact line is 6.4%. As the mismatch in hypoid gears is relatively small, thus the theoretical point contact, under load spreads over a surface. Usually, it is assumed that the contact spreads over an elliptical area. In this method for load distribution calculation a new approach to tooth contact is introduced: the point contact under load spreads over a surface along the `potential' contact line, which line is made up of the points of the mating tooth surfaces in which the separations of these surfaces are minimal. A method is developed for the determination of the `potential'
V. Simon / Mechanism and Machine Theory 35 (2000) 1197±1220
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Fig. 18. Load distribution and tooth contact pressures.
contact lines in hypoid gears, presented in [15]. The investigations have shown that, especially on tooth pairs with contact on the toe or on the heel of teeth, the shape of the contact area diers from an ellipse, and there is a dierence in tooth contact pressures, calculated due to the part of an assumed ellipse of contact limited by the physical teeth boundaries ( pel), and calculated due to the actual load distribution ( p, Fig. 18). In Fig. 18 pel-w represents the contact pressure spreading over the whole ellipse, not limited by teeth boundaries, which pressure is only hypothetical. The investigations have also shown, that in the case of misaligned gears the assumption of an elliptical contact is totally unrealistic. Because of the limited length of this paper, the detailed description of the method for load distribution calculation in hypoid gears will be presented in a separate paper. 5. Conclusions A computer program based on ®nite elements has been developed for stress analysis in hypoid gears. By using this program the stresses and displacements have been calculated in the pinion and the gear of a mismatched hypoid gear pair. The obtained results have shown that a good convergence of the solution can be achieved by making the calculation for three adjacent teeth and by using a ®nite element mesh containing 672 elements and 3647 nodal points. By using the same computer program the in¯uence of design parameters and of the position of the loaded point on de¯ections and stresses has been investigated. On the basis of the results obtained by performing a big number of computer runs, by using regression analysis and interpolation functions, equations have been derived for the calculation of de¯ections in an
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arbitrarily chosen tooth ¯ank point under a normal load acting in the same or in a dierent tooth surface point, and for the calculation of ®llet stresses. The calculation of load distribution in hypoid gears has been shown that the use of these equations makes this calculation much easier and faster than it is by the direct use of the ®nite element method. References [1] L.E. Wilcox, An exact analytical method for calculating stresses in bevel and hypoid gear teeth, in: Proceedings of International Symposium on Gearing and Power Transmissions. Tokyo, vol. II, 1981, pp. 115±121. [2] R.F. Handschuh, F.L. Litvin, How to determine spiral bevel gear tooth geometry for ®nite element analysis, in: Proceedings of JSME International Conference on Motion and Power Transmissions, Hiroshima, 1991, pp. 704±710. [3] G.D. Bibel, K. Tiku, A. Kumar, R. Handschuh, Comparison of gap elements and contact algorithm for 3D contact analysis of spiral bevel gears, in: NASA TR ARL-TR-478, 30th Joint Propulsion Conference, Indianapolis, 1994. [4] G.D. Bibel, A. Kumar, S. Reddy, R.F. Handschuh, Contact stress analysis of spiral bevel gears using ®nite element analysis, ASME J. Mech. Des 117 (1995) 235±240. [5] G.D. Bibel, R. Handschuh, Meshing of a spiral bevel gearset with 3D ®nite element analysis, in: Proceedings of 7th International Power Transmission and Gearing Conference, San Diego, 1996, pp. 703±708. [6] R.F. Handschuh, Recent advances in the analysis of spiral bevel gears, in: Proceedings of MTM'97 International Conference on Mechanical Transmissions and Mechanisms, Tianjin, 1997, pp. 635±641. [7] C. Huang, R. Li, C. Zheng, Tooth contact ®nite element analysis for spiral and hypoid gears, in: 6th International Power Transmission and Gearing Conference, Scottsdale, 1992. [8] J.S. Chen, F.L. Litvin, A.A. Shabana, Computerized simulation of meshing and contact of loaded gear drives, in: Proceedings of International Gearing Conference, Newcastle upon Tyne, 1994, pp. 161±166. [9] F.L. Litvin, J.S. Chen, J. Lu, R.F. Handschuh, Application of ®nite element analysis for determination of load share, real contact ratio, precision of motion, and stress analysis, ASME J. Mech. Des 118 (1996) 561±567. [10] C. Gosselin, D. Gingras, J. Brousseau, A. Gakwaya, A review of the current contact stress and deformation formulations compared to ®nite element analysis, in: Proceedings of International Gearing Conference, Newcastle upon Tyne, 1994, pp. 155±160. [11] Ph. Gagnon, C. Gosselin, L. Cloutier, Analysis of spur, helical and straight bevel gear teeth de¯ection by ®nite strip method, in: Proceedings of International Conference on Gears, Dresden, 1996, pp. 909±921. [12] V. Simon, Computer implementation of ®nite element method for stress analysis in machine parts, in: Proceedings of XIIth International Symposium Computer at the University, Cavtat, vol. 78, 1990, pp. 1±6. [13] V. Simon, Automatic ®nite element mesh generation, in: Proceedings of Xth International Symposium Computer at the University, Cavtat, vol. 7.3, 1988, pp. 1±7. [14] V. Simon, Computerized ®nite element mesh generation in hypoid gears, in: Proceedings of 23rd Design Automation Conference in CD-ROM, Sacramento, 1997. [15] V. Simon, Tooth contact analysis of mismatched hypoid gears, in: Proceedings of 7th International Power Transmission and Gearing Conference, San Diego, 1996, pp. 789±798. [16] Gleason Works. Method for designing hypoid gear blanks. Rochester, 1971. [17] F.L. Litvin, Y. Gutman, Methods of synthesis and analysis for hypoid gear-Drives of `Formate' and `Helixform', Part I±III, ASME J. Mech. Des 103 (1981) 83±113. [18] F.L. Litvin, Y. Zhang, M. Lundy, C. Heine, Determination of settings of a tilted head cutter for generation of hypoid and spiral bevel gears, ASME J. Mechsms., Transm., Autom. Des 110 (1988) 495±500; Proceedings of ASME 5th International Power Transmission and Gearing Conference, Chicago, 1989, pp. 719±725. [19] S.C. Chapra, R.P. Canale, Numerical Methods for Engineering, McGraw-Hill, New York, 1989. [20] V. Simon, Load distribution in double enveloping worm gears, ASME J. Mech. Des 115 (1993) 496±501.