1、,I Mesh Friction in Gearing American Gear TECHNICAL PAPER COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesMesh Friction in Gearing . Clifford M. Denny, Consultant The statementsand opinionscontained herein are those of the author and should not be cons
2、trued as an official action or opinion of the American Gear Manufacturers Association. Abstract Gear tooth sliding friction appreciably increases the magnitude of tooth root bending stresses in both the driving and driven gears. Compared to situations devoid of friction, these stresses are decreased
3、 in the approach portion of the mesh but are increased in the recess portion. Even coefficients of friction as low as 0.06 can change the point of tooth-load application for maximum bending stress in the driven pear from the hiehest-point-of-sinple-tooth-contact to the they are not intended to repla
4、ce it. They may be applied to other rating systems and to Finite Element Analysis as well and render similar adjustments to the calculated bending strength. Background In a gear mesh, there is a combination of sliding and rolling between contacting tooth pairs. The locus of points of contact is a li
5、ne that starts at the beginning of contact and ends at the secession of contact. This line passes through the pitch point 0 where it (the line) intersects the line of centers connecting the two gears centers of rotation. Assuming contact begins before the pitch point and ends after the pitch point,
6、the line is divided into two parts: approach and recess. Approach action occurs in the mesh in advance of the pitch point. Recess action occurs after the point of contact passes through the pitch point. During approach, the sliding of one tooth upon the other is toward the others base or root. Conve
7、rsely, in recess, the sliding is reversed, being toward the tips of the meshing pair. At the pitch point, there is no sliding; only pure rolling is present. It is during this sliding of one tooth upon the other where friction forces are generated. The direction of this force acting on one tooth will
8、 be that of the opposing tooth sliding upon it. Therefore, in approach, friction forces are directed toward a tooths root; in recess, toward its tip. In both cases, the direction will be normal to the line of contact and located where contact occurs. In the case of gear teeth with involute profiles,
9、 the line of contact, called the line of action, is straight, and crosses the line of gear centers at an angle, called the operating pressure angle. Only spur gears with involute tooth profiles are considered in this analysis. In plastic gears especially, sliding friction is known often to be higher
10、 than in steel, and material properties are not as well understood. Calculation serves only to estimate performance roughly. Application-specific testing remains an essential step in the gear-design process regardless of the materials employed. Model Studied Table 1 provides the critical rolling ang
11、les for the mesh. Tables 2 and 3 provide the specifications for the particular gears studied in this paper. The AGMA PT basic rackf2 is the origin of the tooth proportions used. Tooth Loading with Friction In the presence of tooth sliding friction, the direction of the tooth-loading force and its ma
12、gnitude change significantly at the operating pitch point. Figure 1 shows a driven gear tooth. In the approach section from tooth tip to operating pitch point, the magnitude of loading force, WL , is high, as is its vertical (compressive) component. At the pitch point, the force changes abruptly in
13、magnitude and I COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicesdirection, its magnitude significantly decreases, and its compressive component all but disappears. In approach, the compressive component decreases tensile bending stresses in the tooths
14、root. In recess, there is little such decrease. Loading just to the recess side of the pitch point produces the greatest tensile tooth bending stress in roots of driven teeth - not at the highest point of single tooth contact. This is true despite the greater magnitude of tooth loading forces in app
15、roach. l9.5- 19 t8.s ie l7.S 17 US- - - - - - I AW VEN GEAR I - 3.4 32 3- 28 2.6 2.4 22 1s 1.6- 1.4 12 I I I I t I -3 -2 -1 e 1 2 3 - - - - - - 2- - - - 1- Loading effects on Critical Section Figure 1 Tooth Bending Stress Maximum tensile tooth bending stresses increase as the point of tooth loading
16、goes from the LPSTC (lowest point of single tooth contact) to the HPSTC (highest point of single tooth contact). In “frictionless” applications, this increase is uninterrupted. But where friction is present, even to a small degree, there is an abrupt change at the pitch point. The direction of this
17、change depends on which tooth is involved, driving or driven. If driving, the change is an abrupt increase in bending stress. If the tooth is the driven one, the change is an abrupt decrease. From there, both resume their smooth upward climb. Figures 2 and 3 show the progression through the single-t
18、ooth contact part of the mesh cycle. In the case studied, any frictional coefficient above 0.06 will maximize the bending stress for loading at the operating pitch point on the driven tooth. The maximum bending stress still occurs on the driving tooth when loading is at the HPSTC. :I 15 - APPROACH -
19、 ! -RCLZSS - af e.e II Bending Stresses on Driving Tooth Figure 2 e.e e .2 8.1 8.3 0.4 8.5 j - AECEJJ - -.APPROACH - B.6 I 1 I I I I II I ie 19 2 21 P P 24 a - Bending Stresses on Driven Tooth Figure 3 Coefficients of Friction Values of the coefficient of sliding friction depend on the materials in
20、contact, the contact pressure, the velocity of sliding, and the degree of wear among other things including the actual test conducted and the controls placed upon it. Understandably, there is generally wide variance in test results even when well controlled. The value can be estimated somewhere with
21、in the range of test results. Typical values of the friction coefficient of materials against steel follow 6lI1l2lI13lIW 2 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicesc Gear Mesh: P18-UWM G36-UWM CD 27.2 mm Pinion Start of Active Profile e, degrees
22、 7.792 Lowest Point Single-Tooth Contact ob degrees 18.621 Operating Pitch Point e, degrees 22.1 16 Highest Point Single-Tooth Contact ed degrees 27.788 End of Active Profile ee degrees 38.617 L Gear 13.866 19.309 22.1 16 23.892 29.335 AI alloys Bronze 80-1 0-1 O Zinc Polyketone Acetal Acetal+PTFE A
23、ceta I+ Lu be Nylon 66 Acetal vs. Acetal Nylon 46 vs. Acetal+PTFE and, 0.33 0.15 0.15 0.505 . 0.32 - 0.41 0.13-0.24 0.17 - 0.33 0.17 - 0.33 0.42 - 0.50 0.10- 0.15 Conclusions The presence of sliding friction in a mesh has a sizable affect on the bending stresses in the teeth, increasing it substanti
24、ally above its magnitude as calculated when friction is ignored. The effect is more pronounced on the driving tooth (1) than on the driven tooth (2). For a friction coefficient of 0.3, the bending stresses increase by about 36% in the driving tooth, and about 10% in the driven tooth. A 0.5 coefficie
25、nt produced relative increases of about 48% and 20%. Whereas the point of load application for maximum bending stress remains at the HPSTC on the driving tooth (l), the point of application on the driven tooth has shifted. The maximum bending stress in the driven tooth (2) occurs for loading at the
26、operating pitch point at the beginning of recess. In both cases, driving tooth and driven tooth, the maximum bending stresses occur for contact in recess even though the magnitudes of contact forces are less than their values during approach. With material-to-material contact, the coefficient of sli
27、ding friction was held constant throughout the mesh cycle as was the driven load. Reduced normal forces in recess result in reduced frictional losses in that part of the mesh cycle. This merely masquerades as a reduction in the coefficient of friction there. Frictional forces in recess serve to help
28、 drive the load rather than reinforce it. Remarks The manner in which friction affects the normal tooth contact load was noted as it arose in the stress calculation process. It is increased in the approach region of the mesh cycle, and reduced in the recess region. It is therefore conceivable that t
29、ooth surface durability could be increased if recess action were increased to some maximum. Efficiency and wear as they relate to tooth friction in approach and recess could be the subject of further study and experimentation, especially where polymeric materials are used. Acknowledgement I wish to
30、thank Irving Laskin, P.E., for his encouragement and support in pursuing this investigation. 3 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesAbbreviated Pinion Specifications Table 2 Abbreviated Gear Specifications Table 3 e 4 COPYRIGHT American Gear
31、 Manufacturers Association, Inc.Licensed by Information Handling Servicesh APPENDIX The Normal Contact Force a The friction effect takes into consideration the direction of the friction force that results from tooth sliding and which acts upon the tooth under consideration. Its magnitude varies with
32、 the location of contact along the line of action. Consider a load .that remains constant at all times, and driven by a gear pair. The torque driving the pair then fluctuates depending upon the magnitude of frictional forces in the mesh. These frictional forces combine with the normal contact force
33、with which the driving tooth propels the tooth it contacts. The normal force lies along the line of action; the frictional force is perpendicular to it at the point of contact. Figure 4 shows a diagram of the meshing gear pair. RB1 and RB2 are the base circle radii of the driving pinion (1) and driv
34、en gear (2) respectively. Likewise, Rol and Ro2 are the respective operating pitch radii. Terms rxi and rx2 are radii to the point of contact between the driving and driven teeth. This point of contact (x) here happens to lie in the region of approach on the line of action. Here it is a distance (e)
35、 from the base end of the line of action at the driving gear (I), and a distance (9) from the base end at the driven gear (2). An input torque (T,) drives the constant load torque (L) through the gear pair as shown in the diagram. The input torque varies as necessary to overcome mesh friction and ma
36、intain the torque the load demands. Figures 5 are free-body diagrams of the gear (2) and pinion (1) for single-tooth contact. Figures 5a and 5c are for gear and pinion contact in approach; figures 5b and 5d are for the respective contact in recess. W, is the normal contact force lying on the line of
37、 action in the presence of friction, and pW, is the friction force acting on the teeth and lying perpendicular to the line of action. p is the coefficient of sliding friction. Figure 5c shows, pWnf directed “toward” the pinions center in approach, while in recess, figure 5d shows it directed “away”.
38、 Summing moments in these two cases, we have: Tr = Wn ( RB1 - w) , 1 a) and Tr = W,( RBI + pel . b) Now, e = RB1 x eX1 2) where Ix1 is the roll angle on the pinions (1) involute profile to the point of contact (x). Therefore, after rearrangement, equations 1 ) 3a) n Meshing Pair in Single-Tooth Cont
39、act Figure 4 rxi Roi recess .I (dl Free Body Diagrams Figures 5 - in approach, and COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesIntroduce a term “s“ that shifts value according to the portion of the mesh cycle in which the tooth contact takes place
40、as shown in the following chart. approach I pitch pt. 1 recess f 12- r 0 1.1 1 1“ 88 “rir = Tf 1 RBI (1 + pox1 11 1 b) in recess. Figure 5a shows, pWnf directed “toward“ the gears center in approach, while in recess, figure 5b shows it directed “away“. Summing moments in these two cases, we have: L=
41、Wnf(RBZ-pg) v 44 and L = Wnf ( RBZ + b) Now, = RBZ x 6x2 5) where eX2 is the roll angle on the involute profile of gear (2) to the point of contact (x). Therefore, equations 4) become after rearrangement: Ga) in approach, and b) in recess. Figures 6 show the resultant contact force WL acting in the
42、mesh in both approach 6a and in recess 6b and how sliding friction affects its direction with respect to the normal force W, which is always coincident with the line of action. wnf = L 1 RB2 (1- Pb )I I Wnf = L 1 RBZ (1 + ciex2 )I 8.4, 8.4 8.7 8.2 8.1 $1 - wf 8.0 8.8 8.1 8.2 9 8.3 8.4 o IYl 1 1 +-,
43、RCCCJS - - APPROKH 8.711“l f/NiON (1) Recess (b) Friction Effects in Single-Tooth Contact Figures 6 “s“ may be substituted into equations 3) and 6) to become: 7) for the pinion (1) in single-tooth contact, and 8) for the driven gear (2) in single-tooth contact. Now, in the frictionless case, p = O.
44、Let W, be the normal contact force between teeth in mesh in this frictionless case for single-tooth contact. Equation 8) becomes: W, = TI/ RBI (1 + spe, 11 wnf = L I RB2 (? + speie, W,= L / Raz 9) Substitute equation 9) into 8), eliminate L, W,/Wn= 1 / (1+ Sp,) 10) and rearrange into the ratio: When
45、 p O, the normal contact force W, is greater in approach (s = -1) than it is in recess (s = +i). Figure 7 shows this effect. Normal Contact Load Figure 7 The Tangential Force Let m, be the gear ratio, where: Combine equations 7), 8), and 1 l), and solve m, = Rez / RB1 . 11) for the driving torque Tf
46、. 6 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicesh, where T = driving torque without friction WTf = tangential force with friction WT = tangential force without friction 13) WTf = Tf 1 rol 14) WT= T /roi 15) e Also, T = Um, where rol = pinions (1) o
47、perating pitch radius Combine equations 13), 14), and 15) into equation 12) and solve for the ratio of tangential forces that drive the pinion (1). WTf/wT = (l+spexl) / (l+speu). 16) When p O, as the mesh goes through its cycle, figure 8 shows this ratio descending from some value at the initiation
48、of single-tooth contact to unity at the operating pitch point (s = O) from whence it again ascends to some other value at the secession of single-tooth contact. The initiation of single-tooth contact is in the approach portion of the mesh cycle (s = -1). The point where single-tooth contact ceases i
49、s in the recess region (s = +1). For gear teeth in mesh, the relationships between Oxl and Ou are: eX1 = eo(l + mg)-8Qxmg, 17a) where 8, is the roll angle of either the pinion or the gear at the operating pitch point where its the same in either case. and eu = eo(l + l/mg)-exll/mg, b) 186- t 9 182- F 1- 5 185- 2 1.M- Y = 183- z 1.m- 8 .5 8.4 8.3 8.2 8.1 8.8 Tangential Load Figure 8 The Friction Load Factor The friction load factor Kf is the ratio of the single-tooth contact force Wu acting on the tooth in the presence of friction to the no
