1、 STD-AGHA 94FTML-ENGL 1994 ab87575 00lib24 154 r m 94FTMlO Computerized Design and Generation of Low-Noise Gears I With Localized Bearing Contact I by: E Litvin, N. Chen, J. Chen and J. Lu University of Illinois at Chicago and R.F. Handschuh, NASA Lewis Research Center I American Gear TECHNICAL PAPE
2、R COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesComputerized Design and Generation of Low-Noise Gears With Localized Bearing Contact E Litvin, N. Chen, J. Chen and J. Lu University of Illinois at Chicago statements and opinions contained herein are t
3、hose of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. ABSTRACT: Thepaperreptesentstheresultsofaccomplishedresearchprojectsduectedatreductionof noisecausedby misalignment of the following gear drives: doublecircular arc helical
4、 gears, modified involute helical gears, face-mied spiral bevel gears and face-miiled formate cut hypoid gears. It is proven by the authors that due to misalignment, periodic almost linear discontinuous functions of transmission errors occur. The period of such functions is the cycle of meshing when
5、 one pair of teeth is changed for another me. Due to the discontinuity of such functions of rransmission errors high vibration and noise are inevitable. The authors propose to predesign a parabolic function of transmission errors that is able to: (i) absorb linear discontinuous functions of transmis
6、sion errors, and (U) turn out the resulting function of transmission enws into a continuous one. The proposed idea was successfully teste by the Gleason Works and NASA Lewis Research Center for a set of spiral bevel gears manufactured by the Bell Helicopter Co. It was found out that the noise was re
7、duced at 1218 decibels in comparison with the existing design. The idea of the predesign of a parabolic function is applied for the reduction of noise of helid and hypoid gears. The effectiveness of the proposed approach has been investigated by developed KA (Tbotb Contact Analysis) programs. The be
8、aring contact for the mentioned above gears is localized. Conditions that aow to avoid edge contact for tlie gear drives have been determined. Manufacturing of helid gears with new topology by hobs and grinding wornis has been investigated. Copyright Q 1994 American Gear Manufacuuer Association 1500
9、 King Street, Suite 201 Alexandria, Virginia, 22314 October, 1994 ISBN 1-55589-645-6 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services STD-AGUA 74FTHLO-ENGL 1774 9 Ob87575 0004b2b T27 . COMPUTERIZED DESIGN AND GENERATION OF LOW-NOISE GEARS WITH LOCALIZE
10、D BEARING CONTACT Faydor. L. Litvin, Ningxin Chen, Jui-Sheng Chen, and Jian Lu The University of Illinois at Chicago Robert F. Handschuh NASA Lewis Research Center 1. Introduction : Vibration and noise of gears are caused due to errors of gear aiign- ment, and deflection of teeth and shafts under th
11、e load. The main attention in this paper is paid to the reduction of transmission errors caused by misalignment, and the stabilization of the bearing contact. Gear misalignment and errors of manufacturing cause transmission errors that act as an oscillator of vibrations. l(a) shows the transmission
12、function (i) Due to elastic deformation, the real contact of gear tooth surfaces is spread over an elliptical area, and the bearing contact is formed as the set of instantaneous contact el- lipses; (ii) The path of contact is chosen in the longitudinal direction, and the center of symmetry of the co
13、ntact ellipse moves as well in the longitudinal direction; this makes the bearing contact more stable and, probably, favors the conditions of lubrication. The discussed above ideas are applied for involute helical gears, double circular-arc helical gears, spiral bevel gears, and hypoid gears (see be
14、low). 2. Involute Helical Gears with Modified Topology : Preliminary Considerations. Ideal involute helical gears are in line contact at every instant. The line of contact is the line of tangency to the helix on the base cylinder (fig. 3). The ideal gears perform rotation with constant gear ratio re
15、presented as where u() (i = 1,2) is the pinion (gear) angular velocity; ri is the radius of the operating pitch circle for nonstandard gears; ri = rpi, where rpi is the pitch circle, for standard gears; Pbi is the radius of the base circle; Ni is the tooth number. The gear axodes are cylinders of ra
16、dii ri, and the line of tangency of these cylinders is the instantaneous axis of rotation. The cross-section of the gear tooth surface is an involute profile (fig. 4(a). The intersection of the tooth surface by a cylinder of radius p is a helix (fig. 4(b). The helix is turned out into a straight lin
17、e when the cylinder is developed onto a plane (fig. 4(c). The lead H represents the axial displacement for one revolution when a point moves along the helix. The lead H of a helical gear is the same for a cylinder of any radius p, and can be expressed as follows (fig. 4(c) H = 2npi tan X,i (2.2) The
18、 screw parameter p represents the axial displacement along the cylinder for the angle of one radian, and is represented as H 277 p = - =pi tan$ Considering two mating helical gears and using equations (2.1) to (2.3), we obtain that the lead angles of the gears are equal only for helices on the gear
19、cylinders ri and il, and t)l and rb2. However, the mating gears for the case of external meshing have opposite directions of the helices. Equations (2.1) to (2.3) yield that Existing Methods of Crowning. The ideal helical gears are very sensitive to the angular errors of alignment such as crossing o
20、r inter- secting of gear axes (instead of to be parallel), errors of lead angles (caused as error of leads), and the profile angie of the tool. Such errors may cause an edge contact, and therefore the manufacturers used var- ious methods of crowning of gear tooth surfaces to obtain a favorable local
21、ized bearing contact. However, it was not observed that the exist- ing methods of crowning provide the undesirable shape of the function of transmission errors shown in fig. l(b), and vibration and noise of the gears are inevitable. We will illustrate this statement with two particular cases of crow
22、ning. Case 1: Crowning by modification of involute profiles Let us assume that the leads Hl and Ha are observed as the same as for ideal involute helical gears, and they are related with equation (2.4). The crowning is based on the modification of the involute profiles as shown in fig. 5. The point
23、of tangency M of modified profiles is considered in the cross-section of the gear tooth surfaces. Point M coincides with a current point of tangency of the ideal involute profiles. The normal at point M to the theoretical and modified profiles is the same and passes through the instantaneous center
24、of rotation I (fig. 5). However, there is a big difference between the conditions of meshing of ideal involute helical gears and the gears provided with modified tooth surfaces, that can be described as follows: (i) The ideal involute helical gears are in line tangency at every instant. The modified
25、 involute helical gears are in point tangency at every instant. (i;) The instantaneous point of tangency of modified gear tooth surfaces is point M in the respective cross-section. In the process of meshing point M moves along a straight line that passes through M and is parallel to gear axes. (iii)
26、 The bearing contact is localized, and the path of contact is a helix on the cylinder of radius OiM for the pinion, and radius OzM for the gear. (iv) The described method for crowning allows to avoid the edge contact of the gears. However, the function of transmission errors for the crowned gears ha
27、s the unfavorable shape (fig. I(b), and vibra- tion and noise are inevitable. This is the reason why this method of crowning cannot be recommended for application. Case 2: Crowning by modification of leads We will consider the following sub-cases: (i) The profile modification, as described in case 1
28、, is provided and in addition the leads are modified. The modified leads H, and H; satisfy equation (2.4). In this sub-case the main conditions of meshing are the same as in case 1, but there was not a need in modification of the leads. (U) The profile modification, as described in case 1, is provid
29、ed but the modified leads H; and Hi do not satisfy equation (2.4). In this sub-case the bearing contact is localized, the path of contact on the gear tooth surface is a helix, the line of action is a straight line that is parallel to the gear axis, but the gear ratio, although it is constant, differ
30、s from the theoretical one. The gear ratio is determined as (2.5) Since the real gear ratio mi2 differs from the theoretical one deter- N2 mined as m12 = -, the function of transmission errors has the shape Ni shown in fig. l(b), and the transformation of motion is accompanied by a stroke at each cy
31、cle of meshing. (iii) The theoretical involute profiles are not modified, but the leads are changed, and - # -. The contact of gear tooth surfaces is localized, and the path of contact on the gear tooth surface has the shape shown in fig. 6. The disadvantage of such method of crowning is that the co
32、ntact of gear tooth surface (due to small modification of leads) is close to the line contact, the gears are very sensitive to angular errors of misalignment, there is a pwibility of an edge contact, and the function transmission errors has the unfavorable shape shown in fig. l(b). New Approach for
33、Modification of Gear Tooth Surfaces. In the authors opinion the existing methods of crowning do not satisfy the requirements of the design and manufacture of low-noise helical gears Hi NI H Ni 2 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSTD-AGHA
34、94FTMZ-ENGL 1994 Ob87575 0004628 BTT D . with localized and stable bearing contact. The approach proposed by the authors is based on the following ideas: (1) The generation of gears is based on application of two imag- inary rack-cutters, with surfaces Et and E, that generate the gear tooth surface
35、E2 and the pinion tooth surface El, respectively. The normal sections of rack-cutter tooth surfaces are shown in figs. 7(b) and (c). We may imagine that both rack-cutters are rigidly connected and generate the pinion and the gear simultaneously. (2) While the rack-cutters translate at the displaceme
36、nt sc (fig. 7(a), the pinion is rotated on the angle However, the gear is rotated through the angle where st = s,. The observation of equation (2.7) allows to provide a predesigned parabolic function with parabola parameter a. The predesigned parabolic function absorbs linear functions of transmissi
37、on errors, and this allows to reduce substantially the noise and vibration. (3) Due to deviation of rack-cutter surfaces E, and Et (fig. 7(b), the gear tooth surfaces are in point contact at every instant, and the bearing contact is localized. In reality, the finishing operations of the manufacture
38、of the gears can be accomplished not by rack-cutters but by application: (a) of two grinding (cutting) worms, or (b) by form grinding. In the case of application of two worms, the following requirements must be observed: (i) The surface of the grinding worm must be determined as the envelope to the
39、family of surfaces of the respective rack-cutter. (i) The relations between the motions of the grinding worm and the pinion are represented by linear functionsbased on equation (2.6). However, the motions of the worm and the gear being generated are nonlinear functions whose derivation is based on e
40、quation (2.7) to pro- vide a predesigned parabolic function. In the case of form-grinding the respective requirements to be ob- served are as follows: (i) The to be ground pinion surface is a helicoid. The tool is a disk- shaped wheel. The relative motion of the pinion (the grinding wheel) is the sc
41、rew motion with the screw parameter p of the helical gears. The required shape of the tool can be determined as described in l. (ii) All the above conditions must be observed when the gear is ground by the disk-shaped grinding wheel. In addition it is required that the tool will be plunged in the pr
42、ocess of grinding in the direction of the shortest distance between the tool and the gear. The plunge of the tool must be varied and controlled in the process of grinding: the plunge is zero in the middle of the gear space, and maximal at the ends of the space. The varied plunge is required to provi
43、de the predesigned parabolic function of transmission errors. (iii) Form-grinding requires the indexing of the pinion (gear) when the neighboring space is to be ground. The authors have developed TCA computer programs to simulate the meshing and contact of the helical gears with the modified surface
44、 topology, and investigate the influence of errors of angular misalign- ment. It was proven that the predesigned parabolic function absorbs indeed the linear functions of transmission errors. The path contact on the gear tooth surface is stable, and slightly deviates from a helix for the misaligned
45、gear drive (fig. 8). 3. Double Circular-Arc Helical Gears with Modified Topology : Preliminary Considerations. The circular arc helical gears (N.- W.) have been proposed by Novikov i and Wildhaber 2. However, there is a significant difference between the ideas proposed by the above inventors. Wildha
46、bers idea is based on generation of the gears by the same imaginary rack-cutter that provides conjugate gear tooth surfaces being in line contact at every instant. Novikov proposed the applica- tion of two mismafched imaginary rack-cutters that provide conjugate gear tooth surfaces being in point co
47、ntact at every instant. The great advantage of Novikovs invention is the possibility to obtain a small value of the relative normal curvature and reduce substantially the contact stresses. The weak point of Novikovs idea was the high value of bending stresses since the gear tooth surfaces are iR poi
48、nt contact at every instant. The successful manufacturing of N.-W. gears has been accomplished by application of two mating hobs based on the idea of two mating imaginary rack-cutters. This idea has been proposed by Kudrjavtsev 131 in the former USSR and Winter and Looman 4) in Germany. The circular
49、 arc helical gears are only a particular case of a gen- eral type of helical gears which can transform rotation with constant gear ratio and are in point contact at every instant. Litvin 5,13 and Davidov 6 simultaneously and independently proposed a method of generation for helical gears by “two rigidly connected tool surfaces. According to this idea, the generating surfaces may be rack-cutter sur- faces, particularly. The kinematics of single circular arc helical gears was the subject of the paper by Litvin and C.-B. Tsay 7. A substantial step forward
