1、95FTM9 Bending and Compressive Stress Analysis of External Helical Gearsets American Gear Manufacturers TECHNICAL PAPER COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services - STD-AGHA 75FTfl7-ENGL 1775 b87575 00047413 577 D Bending and Compressive Stress A
2、nalysis of External Helical Gearsets of Varying Contact Ratios David Wenthen, New Venture Gear statementsandopinionscontainedhereinarethoseofdieanthorandshouldnotbeconstniedasanofficialactionor opinion of the American Gear Manufacturers Asso&ion.J Abstract in an attempt to better understand how the
3、dipabiiity of a helical gearset is affected by changes in profle (mp) and face contactratio(nfF),ananalyticalinvestigatianwasdoneinwhichnq, wasvariedinlevelsof 1.1,1.6,2.1,and2.6and was varied in leveis of O, 0.6,1.1,1.6,2.1,2.6,3.1, and 3.6. Thirty-two combinations were studied in totai. “he gears
4、were modeied using the hybnd finite element computafiorui rnm Contact Analysis Programmin g Package (CAPP), of Advanced Numericai Solutions. CAPP was chosen for this anaiysis since it has the ability to calculate both bending and compressive stresses in gear teeth without the use of a supcomputer. T
5、he results of this analysis suggest that for a given level of face contact ratio, an advantage in bending and compressive sms exists at the mp=2.1 level over ail of the others considemi. No such optimum level was found fur face contact ratio in this study. As the helix angle and m were inma&, the ma
6、ximum bending stresses increased due to the in- in normaltoothloadandcompressivestressesdecreasedduetotheininncnmalfacewidth. inmasingbothmp and ny. had the effect of smoothing out the bending and compressive stresses when Viewed as a function of roll angle. Copyright O 1995 Amencan Gear Manufacmers
7、 Association 1500 King Street, Suite 201 Aiexandria, Vi 22314 October, 1995 ISBN 1-55589-658-8 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSTD-AGMA 95FTM9-ENGL 1995 H Ob87575 0004749 405 Bending and Contact Stress Analysis of External BelicdL Gears
8、ets with Varying Contact Ratios David W. Wenthen, Gear Design Engineer New Venture Gear, Troy MI 48083 Introduction The influence of contact ratio on the generation of noise from a gear mesh is fairly well known and documented throughout the industry. 1,2 In most instances, increasing the profile co
9、ntact ratio (m, 1 of a spur gearset moves the high point of single tooth contact (HPSTC) closer to the pitch point, which reduces the bending moment arm about the base of the tooth. The effect is a smoother action at the extremes of the meshs arc of action. Helical gears operate more quietly than sp
10、ur gears because their line of contact is not parallel to the axis of rotation, but skewed at an oblique angle, extending the length of the contact line onto adjacent tooth pairs. Increasing the length of the line of contact spreads the transmitted load over a greater area, reducing the load intensi
11、ty of the gear mesh. In addition to the reduced load intensity, each infinitesimally thin slice in the plane of rotation is advanced in roll angle by an amount proportional to the helix angle. Thus, contrary to spur gears, contact across the gear face does not take place at the same roll angle, but
12、rather it is broken up. Neighboring slices contact their mating surfaces at an advanced or retarded roll angle. What limits the gear designer in producing a tooth form with increased contact ratio is the bending and contact stresses of the tooth. In general, higher profile contact ratio gears have l
13、onger and thinner teeth, lower pressure angles and finer pitches. The usual result is a compromise between strength and quietness. Recently the writer redesigned the complete geartrain of a rear wheel transmission for a torque requirement which was lower than that for which the center distance was s
14、elected. Here was an opportunity to design the gear teeth with contact ratios in excess of the usual range (m, = 1.6, m, = 1.6). Pitches were increased, pressure angles were lowered and tooth depths were increased to design for m, 2.0. Helix angles were also increased to design for & 2.0. The streng
15、ths of the gear meshes were still adequate when calculated by conventional methods. In-vehicle testing of initial prototypes proved that the gears would operate quietly, but this was expected with the elevated contact ratios. Some increased durability of the gearsets was also expected, since work ha
16、s been done in this area on spur gears 4-81, reporting that high profile contact ratio gearing could allow products to be designed to 1 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services- STD-AGHA 95FTM9-ENGL 1995 Ob87575 OOOLi75U 127 greater power densi
17、ties. However, no work could be found which studied the strength of helical gears designed with high profile and face contact ratios. With the benefits of reduced noise generation now d-nstrated, it became necessary to re-evaluate the way the gear teeth are stressed under load and how m, and m, effe
18、ct these stresses. Since an empi rica 1 investigation would be too expensive for a study academic in nature, an analytic study using the hybrid finite element program Contact Analysis Programming Package (CAPP) from Advanced Numerical Solutions was used instead. The gear meshes were modeled and the
19、bending and contact stresses were calculated and the results are presented here. Dskiptioa of the Experbat CAPP is a recently developed analysis tool for the study of all kinds of gears 131. The analysis of two gears in contact is a problem which is too complicated for general purpose finite element
20、 packages. When gears are brought into contact, the width of the contact zone is typically an order of magnitude smaller than the other dimensions of the gears. This gives rise to the need for a very highly refined finite element mesh near the contact zone. But given the fact that the contact zone m
21、oves over the surface of the gear as it rolls through its zone of action, one would need a very highly refined mesh all over the contacting surf ace. Finite element models refined to this extent cannot be accommodated on even the largest of todays computers. Compounding this difficulty is the fact t
22、hat the contact conditions are very sensitive to the geometry of the contacting surfaces. General purpose finite element models cannot provide the required level of geometric accuracy. Finally, the difficulties of generating an optimal three- dimensional mesh that can accurately model the stress gra
23、dients in the critical regions while minimizing the number of degrees of freedom of the model have kept the finite element method from being widely used to solve the complete gear contact problem. The approach that CAPP uses is to combine the finite element model with an elastic halfspace model. The
24、 finite element model is used to obtain gross deformations of the gear away from the contact zone, while the elastic halfspace model is used to obtain relative deflections within the contact zone. The finite element deformations are then *matched“ with local deformations in a least squares sense. Wh
25、at results is an accurate representation of the three- dimensional gear contact problem which is able to be solved on an analysis workstation. Transverse and axial plane tooth geometry were kept independent in this experiment. Changes in face contact ratio were achieved entirely through a change in
26、the helix angle, with a corresponding change in normal pressure angle such that the transverse pressure angle remained unchanged. Changes in profile contact ratio were achieved entirely through changes in the tooth height. Thus, not only were the profile and face contact ratios kept as independent v
27、ariables, but also the tooth geometry in the transverse and axial planes as well. Eight levels of face contact ratio were analyzed (spur, 0.6, 1.1, 1.6, ., 3.6) and four levels of profile contact ratio (1.1, 1.6, 2.1, 2.6). Since all the gears were loaded to their design torque (290#), the tocth for
28、ms were relieved to account for tooth deflections. The amount of tip relief was arbitrarily selected to be .0012“ and parabolic in nature. The starting roll angle of modification was determined by trial and error to minimize contact stresses at entry and exit of mesh. Lea& were crowned to .0002“ cir
29、cular modification centered at the midface, without end relief. Discussion of the Results Figure 1 is a surface plot of maximum bending stress plotted 2 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSTD.AGMA 75FTM-ENGL 2775 H Ob87575 0004752 Ob3 m ag
30、ainst profile and face contact ratios for all combinations tested. When viewed along lines of constant face contact ratio one observes that as profile contact ratio is increased from 1.1 to 1.6, bending stress steadily increases also. However, there is a drop in bending stress as the profile contact
31、 ratio is increased further to 2.1. When profile contact ratio is increased further, the bending stress again climbs as a consequence. When viewed along lines of constant profile contact ratio, one observes that the maximum bending stress increases more rapidly between integer values of face contact
32、 ratio. When the integer value are exceeded, the rate of increase of the bending stress decreases but the bending stress continues to climb with increasing face contact ratio. Figures 2a and 2b show bending stress plotted against roll angle for m, = 1.1. The curve for the spur mesh shows that as the
33、 tooth is loaded the bending stress ramps up gradually the contact from the mating tooth moves from the SAP diameter through the pitch point and drops off dramatically as the tooth exits the mesh. The slope of the curve also changes at the point where the second pair of teeth come into mesh. The cur
34、ves for the low axial contact ratio helical (m, 600,000 psi) contact stresses at the entry and exit points of the mesh when unrelieved tooth forms are used at their design load. The relieved tooth forms used in this experiment did not display this characteristic and, therefore, are more representati
35、ve of a gear mesh with a properly developed tooth contact 3 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSTD-AGHA 95FTM9-ENGL 1775 = Ub87575 0004752 TTT pattern. Conclusions With regard to bending stress, one can draw the following conclusions from
36、this analysis: 1. Increasing the helix angle causes the normal tooth load and, therefore, the root bending stress to increase. The benefit of additional load sharing from the increase in face contact ratio is not sufficient to overcome the detrimental effects of the higher normal tooth loads. 2. Inc
37、reasing the tooth height has no effect on the normal tooth load, but increases the moment arm about which the base of the tooth is loaded as a cantilevered beam. In this case, the benefit of increased load sharing when a second pair of teeth are present to carry the load (i.e. m, 2.0) is enough to c
38、ause a drop in bending stress at the root of the gear tooth. This is seen as the profile contact ratio was increased from 1.6 to 2.1. Increasing profile contact ratio further only serves to increase the bending stress at the root of the tooth. 3. In cases where the total contact ratio exceeds 2.0, t
39、he maximum root bending stress occurs when the normal tooth load passes through the pitch diameter. This was seen for spur gears as well as for helical gears in this investigation. Thus, the high contact ratio spur gears analyzed in this study can be seen to display a characteristic of helical gears
40、 of conventional profile contact ratio. The last item above is a statement having powerf u1 ramifications. Products which use spur gears may be designed at a loner cost than those which use helical gears, since the thrust force associated with helical gears necessitates the use of larger and more ex
41、pensive bearings. If a spur gear can be designed to operate with the characteristics of a helical gear these larger bearings would not be required. A further benefit would be seen in reduced deflections of the mounting system. Even the finest gear will operate poorly in a product with excessive defl
42、ections in its shafts and mounts. However, it must be stated that this analysis assumes that the gears have been manufactured to very precise tolerances. Index errors would cause the assumption of increased load sharing to be invalid. If a long and thin tooth which has been designed to share bending
43、 load with an adjacent tooth is forced to carry the entire load itself, a catastrophic failure could result. With regard to contact stresses, the following conclusions can be drawn: 1. Increasing the active length of the line of action will spread the contact pattern over a larger area and reduce th
44、e contact stresses. This was seen as the profile contact ratio was increased for constant levels of face contact ratio, where the normal tooth load remains constant. As the face contact ratio increases with profile contact ratio held constant, the normal tooth load increases due to the increased hel
45、ix angle. However, even though the tooth is more heavily loaded, the load is spread out over a larger length of action and the contact stress decreases. 2. Conventionally designed helical gears with profile contact ratios less than 2.0 must be designed such that the maximum contact stress is determi
46、ned at the HPSTC or the LPSTC. High contact ratio helical gears may be designed such that the maximum contact stresses are determined at the operating pitch diameter, provided that index errors are sufficiently small to ensure proper load sharing throughout the entire arc of action. 3. As either pro
47、file or face contact ratio is increased, there is a general trend toward higher bending 4 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services STD.AGMA 95FTMS-ENGL 1775 Ob87575 0004753 93b stresses and lower contact stresses, indicating an expected change
48、in the mode of failure from surface spalling to tooth breakage. In closing, one must address some of the problems associated with high profile contact ratio gears, since it can not be assumed that the benefits outlined here come without cost. High profile contact ratio gears are prone to scoring fai
49、lures since their deep tooth forms result in high sliding velocities. Deep tooth forms with low pressure angles may also result in a start of active profile very near the base circle diameter and, therefore, very difficult to manufacture. Also, the narrow tips of the gear teeth will be through-hardened and brittle making them very sensitive to handling damage, even in the hardened state. Index errors are also of greater concern with this type of gear tooth form. Finally, if shaving is to be the finishing process of these gears, all of the problems just mentioned a
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