1、 STD-AGMA SbFTML2-ENGL L9Sb BB Ob87575 00047b5 41T 96FTM12 An Investigation of Globoidal Wormgear Drives by: Dr. Ningxin Chen, Peerless-Winsmith, Inc. 1 I TECHNICAL PAPER STD-ALMA SbFTM12-ENGL 277b Db87575 00049bb 35b W An Investigation of Globoidal Wormgear Drives Dr. Ningxin Chen, Peerless-Winsmit
2、h, Inc. The statements and opinions contained herein are those of the author and should not be construed as an officiai action or opinion of the American Gear Manufacturers Association. Abstract This paper investigates the following present globoidal wormgear drives: (i) original and modified Hindle
3、y wormgear drives; (2) Wildhaber worngear drive with inclined plane teeth of wormgear; (3) inclined plane and cone enveloping globoidal wormgear drives; (4) piane, cone and inverted cone enveloping globoidal wormgear drives based on Sakais theory. Meanwhile, a new approach for generation of Hindley
4、wormgearing, and plane and cone enveloping globoidal woxmgear drives is developed in this paper. Contact lines, dual and single contact ratios, relative curvature radii, meshing angles between tangents of contact lines and relative velocities, and sliding ratios of wormgear tooth surfaces of the abo
5、ve globoidal wormgear drives are studied by computerized simulation for numerical examples. Copyright O 1996 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 October, 1996 ISBN: 1-55589-679-0 STD-AGMA SbFTML2-ENGL 1776 W Ob87575 00047b7 272 An Investiga
6、tion of Globoidal Wormgear Drives Dr. Ningxin Chen Manager of Research Peerless-Winsmith, Inc. Springville, NY 14141 1. INTRODUCTION Hindley wormgearing as a main type of globoidal worm- gear drives has been widely used in power transmissions since S. I. Cone modified it in the beginning of 1930s(Co
7、ne 1930, 1931 and 1932). In comparison with cylindrical wormgear drives, as well-known by theoretical and experimental research, the modified Hindley wormgear drives have the following ad- vantages: dual contact lines, large contact ratio, large relative curvature radius and large meshing angle. bet
8、ween tangent of contact line and relative velocity. The above characteris- tic causes better load distribution and lubrication condition, which means high load capability. Since 1960 however, ground(enve1oping) globoidal worms have commanded much attention in research and then in in- dustry because
9、Hindley worms can not be ground which im- plies a constraint to use them for higher load and higher accuracy transmissions. Today, plane and cone enveloping globoidal wormgear drives are playing a more and more im- portant role in globoidal wormgear drives. For plane envelop- ing wormgearing, its wo
10、rmgear tooth surfaces are generated by such hobs that are made in the same or modified forms as Wildhaber worms(Wi1dhaber 1924) that are envelopes of families of planes, so the worm and hob thread surfaces can be ground by plane wheels(Sakai et al 1978, Sakai et al 1980, Ye et al 1990, Zhang and Qin
11、 1988). The cone enveloping globoidal wormgear drives whose worm or hob thread sur- faces are enveloped by conical surfaces of milling or grind- ing wheels are developed by German patent 1969, Pavlov et al 1975, Sakai and Maki 1980, Sakai et al 1980). Toroidal enveloping globoidal wormgear drives wh
12、ose worm and hob thread surfaces are enveloped by surfaces of revolution with circular arc profiles of torus milling and grinding wheels are researched by Hu and Wang 1988, Hu et al 1988, Kobayashi et al 1993. A simpler approach to make the wormgear by two-tooth cutter is reported by Simon 1973 and
13、1988. The purpose of this paper is: (1) to analyze the geome- try of the present industrial Hindley and enveloping globoidal wormgear drives by computerized simulation; (2) to develop new globoidal wormgear drives with better geometry. The dual contact property, dual, single and total contact ratios
14、, relative curvature radius, meshing angle between tangent of contact line and relative velocity and sliding ratio are investi- gated in the same numerical examples with 40 : 1 gear ratio and 100 mm center distance for different globoidal wormgear drives. 2. HINDLEY WORMGEAR DRIVES Hindley wormgeari
15、ng was invented in 1765 and improved by S. I. Cone in 1930s (Cone 1930, 1931 and 1932). Worm and hob thread surfaces of this type of globoidal wormgear drive are generated by straight lines with orthogonal rotation 1 STD-AGMA SbFTML2-ENGL LSSb M Ob87575 00049b8 229 axes between the cutter and worm/h
16、ob, and wormgear tooth surfaces are generated by hobbing, see Fig. 1. The tooth surface of the wormgear can be represented in the following equations: contacts with the worm from 22“ to O“. Theoretically, the tooth surfaces of the worm and the wormgear mesh in dual contact lines from 22“ to 2“, one
17、of which is the ridge line. However, since the ridge lime d-d keeps contacting with the worm thread surface and dwelling at the same position at the wormgear tooth surface for the whole meshing process, it will be rapidly worn away. This means, that only a single contact line is in mesh practically
18、for the original Hindley wormgear b (1) T()(U, 8, 4) = Mgh ($)T(“(., 8) dh9)-n = Asin$+ Bcos$+ C = O in which b), dh) are tooth surfaces of the wormgear and the hob; n and (9) are surface normal and relative velocity be- tween the hob and the wormgear; (u, 8) and 1c, are surface parameters and rotat
19、ion angle of the hob; Mgh is transfor- mation matrix from the hob to the wormgear systems, and A, B and C are coefficients related with surface parameters of the hob and machine settings, respectively. Substitution of the hdf-tangent equation into Eqn. (1) gets two solutions $1 and $2 for unknown $.
20、 So the wormgear tooth surface is represented in two equations in the following form: -A+ JA2 + B2 - C2 non-orthogonal axes of the hob and the worm- gear(Wi1dhaber 1963): Hindley worms made in accordance with the so-called “basic member gear theory“ in section 5 of this paper(Hose et al 1981) and va
21、rying rolling ratio be- tween the worm and the.cutter(Ye et al 1990). The approach with varying rolling ratio is discussed here since it is the most popular and powerful one. In this approach rotation angle of the cutter is modified with a quadrtic function in the following equation: (3) in which m,
22、 $, and $h are design gear ratio, rotation angles of the cutter and the hobjworm; (a1, a2, a3) are coefficients of the quadratic function, respectively. To compare contact ratios of Merent globoidal wormgear drives, two concepts are defined in this paper: (1) dual contact ratio Q that is contact rat
23、io with dual contact lines in mesh simultaneously at any moment, and (2) single contact ratio c, that is the contact ratio with single contact line in mesh. Total contact ratio Q is equal to the sum of the dual and single contact ratios, that is q = cd + cs (4) The same numerical example with gear r
24、atio m=40:1 and design center distance Ewg = i00 mm is used for the differ- ent globoidal worrngear drives in this paper for comparability. The results of this modified Hindley wormgearing are shown in Figs. 3 and 4. Fig. 3 shows the contact lines and the entrance and exit positions at 24 and -24“ f
25、or one group of contact lines and at 24“ and 3“ for another. The contact ratios are: cd = 2.33, c, = 3.0 and q = 5.33 respectively. The total contact ratio of 5.33 means there are maximal 6 or minimal 5 wormgear teeth contacting with the worm thread surface at 2 STD-AGMA SbFTML2-ENGL LSSb UbA7575 C3
26、00LiSbS Ob5 4 a moment, among them some teeth contact in dual contact lines but some in single contact line. humber of contact lines Nds for a meshing moment varies from 7 to 9 in the meshing process. It is also noted that about one-third of the wormgear tooth surface is free from the mesh. Fig. 4 s
27、hows the relative curvature radii p, the meshing angles qcu between tangents of the contact lines and the relative velocities and sliding ratios of the wormgear tg for the whole wormgear tooth surface, in which the maximum, minimum and average values of them are shown in legends for convenience. The
28、 average values of the above items are pau, = 3.4 in., qn, = 79.9 and taVg = 0.05 respectively. The equations to computerize the above items are shown in another paper of the author(Chen 1996) and are quoted in Appendix for reference. There is a special treatment about the relative curvature radius
29、in this paper: if its value is larger than 20.0 in. it is cut to 20.0 in. without loss of practical meaning. The reasons are that (1) curvature theory of differential geometry is based on local geometry and too large relative curvature radius is not in proportion to the dimension of the meshing area
30、 after tooth deformation; and (2) if the calculating grids are on or near to the intersecting line of two subsurfaces of the wormgear tooth that is a singular line, the relative curvature radii will be infinity or very large, which will increase their average value largely and loses its practical me
31、aning. . The main weaknesses of the Hindley wormgear drives are: (1) the worm thread surfaces can not be ground since they are undevelopable ruled surfaces; and (2) part contact on the wormgear tooth surface. surface at a single contact line, which means they are ;?t suitable for power transmissions
32、. The Wildhabers idea is modified in the following globo:.:d wormgear drives: (1) The wormgear with cone tooth surfi .es milled by a cone milling cutter meshes with the cone enveLD- iag globoidal worm whose thread surfaces are envelope? Df the families of conical surfaces(Popov 1977 and Cmezeno =.id
33、 Maki 1991). (2) The globoidal worm with the thread surfi:,.:?s enveloped by spur or helical involute surfaces meshes :.!I spur or helical gear(Buckingham 1963, Ye et ai 1990). S:r;?:e their geometry property is similar to that of the plane IYX- haber wormgearing, here we only discuss the plane Wild
34、haGer wormgear drives. The spur Wildhaber wormgearing has serious underca:- ting and pointing problems on the worms so it is only +;sed for high gear ratio sets(m 2 40). The example is designed for an inclined Wildhaber wormgear drive with an inclined twth angle 0 = 6“50. The results are shown in Fi
35、gs. 7 and 8. The wormgear enters the mesh with the worm at 20 ?.rid leaves at -10“ after contacting only half of the worvxgear tooth surface in single contact line(Fig. 7), The conta ra- tios are: Q = c, = 3.33 and cd = 0.0: and the numb-;. of contact lines Ncis for a meshing moment varies from 3 eo
36、 4 The average relative curvature radius pa, and average rl.d. ing angle qcv are 2.7 in. and 69“ that are good enough, biit average sliding ratio is tOug = 1.5 and sliding ratios cb:,oge very sharply that results in non-uniform wear(Fig. 8). 4. INCLINED PLANE AND CONE ENVELOF i.iVG GLOBOIDAL WORMGEA
37、R DRIVES 3. WILDHABER WORMGEAR DRIVES The Wildhaber wormgear drive was invented in 1924 by E. Wildhaber(Wi1dhaber 1924), in which hob and worm thread surfaces are generated by spur or inclined planes in ortho- gonal rotation =-(Figs. 5(a) and 6(a), and wormgear tooth surfaces are spur or inclined pl
38、anes nilled by plane milling cutters(Figs. 5(b) and 6(b). Modified Wildhaber wormgear drives are reported by Ishida et al 1978 and Ye et al 1990. The tooth surfaces of Wildhaber wormgear can not be represented by Eqn. (2) so they do not have the property of dual con- tact that is an exception for gl
39、oboidal wormgear drives. How- ever, the advantage of Wildhaber wormgear drives is that both tooth surfaces of the worm and the wormgear ot al 1988, Ye et al 1990). The wormgear tooth surface of this wormgearing car: 5e represented by Eqns. (1) and (2) so it possesses the dual :un- tact property. To
40、get better tooth geometry, the worn :*4 hob are modified usually in cutter tooth number N, + b.J: center distances E, # E, and inclined tooth angle : !z:z E, and E, are center distances between the cutter and ;*e worm and between the worm and the wormgear, see Fie 5. The results of the inclined plan
41、e enveloping globoidal wc J: gearing are shown in Figs. 10 and 11. Two subsurfaces defi, .-? 3 by Eqn. (2) intersect about the middle of the wormgear tooth surface and enter and leave the mesh with the worm from 22“ to -10“ and from 21“ and -10“ respectively(Fg. 10). The contact ratios are cd = 3.44
42、, c, = 0.11 and Q = 3.55. The number of contact lines Nd, for a meshing mkment varies from 6 to 9 in the meshing process. The average relative curvature radius paUg, average meshing angle qm and average sliding ratio are 6.1 in., 58“ and 0.019 respectively(Fig. 11). 4.2 Inclined Cone Enveloping Glob
43、oidal Wormgear Drive To reduce the grinding cost of the worm thread surfaces of the plane enveloping globoidal worms(sing1e flank grinding), cone enveloping globoidal wormgear drives(dup1ex grinding) are developed in German patent 1969, and modified by Pavlov et al 1975). The same machine settings(F
44、ig. 9) are used and the generating surfaces for the worm or hob thread surfaces are replaced by conical surfaces of cone ding cutter or cone grinding wheel, see Fig. 15(b). The results of the inclined cone enveloping globoidal worm- gearing are shown in Figs. 12 and 13. The wormgear tooth surface en
45、ters and leaves the mesh with the worm thread SUI- face at 23“ and -8“ for one group of contact lines and 22“ and -8“ for another(Fig. 12). The contact ratios are cd = 3.33, c, = 0.11 and q = 3.44. The number of contact lines Nci5 for a meshing moment varies from 6 to 9 in the meshing process. The a
46、verage relative curvature radius povo, average meshing angle qcu avg and average sliding ratio Eavg are 5.3 in., 75.7“ and 0.027 respectively(Fig. 13). 5. ENVELOPING GLOBOIDAL WORMGEAR DRIVES BASED ON SAKAIS THEORY 5.1 Sakais Basic Member Gear Theory Sakais (basic member gear theory“ was developed f
47、or hy- poid gearing in Sakai 1955 and was firstly utilized in globoidal wormgearing in Sakai et al 1978. The basic idea is that the contact lines between the tooth surfaces of the cutter and the worm/hob will coincide with the Antact lines between the tooth surfaces of the worm and the wormgear at a
48、ny moment for one group of contact lines if the following three equations are satisfied: N, = N, cos 6 - N, sin 6 E, = Ewg COS 6 1, = E,+, sin 6 cos 6 (5) here N, Ng and N, are tooth numbers of cutter, wormgear and worm; 6, Ewg and E, are tilt angle of cutter axis z, from wormgear axis zg, center di
49、stances between the worm and the wormgear and between the worm and the cutter, $, and 1, are rotation angle and translation of the cutter about and along its axis z,: respectively(Fig. 14). There are two differences of Sakais approach from others: (1) The cutter performs screw motion about and along its axis z,; (2) Among the four design parameters (Ne, E, 5, Z,), only one, say tilt angle 5, is independent but other three are determined by Eqn. (5). Sakais “basic member gear theory“ are used for Hindley wormgearing(H0se 1981), pla
