1、05FTM09Hypoid Gear Lapping Wear Coefficient andSimulationby: C. Gosselin and Q. Jiang, Laval University, K. Jenski andJ. Masseth, American Axle and ManufacturingTECHNICAL PAPERAmerican Gear Manufacturers AssociationHypoid Gear Lapping Wear Coefficient andSimulationClaude Gosselin and Qimi Jiang, Lav
2、al University, Kevin Jenski and JackMasseth, American Axle and ManufacturingThe statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractBecause of the large volume of manufactu
3、red sets, hypoid gears are usually hard finished after heat treatmentusing the lapping process. In the lapping process, a gear set is run at varying operating positions and under alight load in order to lap the tooth surface. An abrasive lapping compound is used as a metal removal media.Because of t
4、he rolling and sliding motion inherent to hypoid gears, the lapping compound abrades and refinesthe tooth surface to achieve smoothness in rolling action and produce high quality gear sets. The pinions andgears are lapped in pairs and must therefore remain as coordinated pairs for the rest of their
5、lives.However, heat treatment distortion can vary significantly from batch to batch, and even within one batch if thetemperature is not consistent throughout the heat treatment furnace. Thus, developing a lapping sequence formanufacturing requires both time and experienced technicians who can establ
6、ish lapping operating positionsand sequence times to produce quality gear sets both in terms of performance and cost. This development isgenerally trial and error as past operator experiences factor heavily into the process.In this paper, the lapping process is simulated using advanced modelling too
7、ls such as gear vectorialsimulation for the tooth surfaces and path of contact and reverse engineering to analyze the tooth contactpattern of existing gear sets under load (static LTCA). Test gear sets are measured using a CMM prior to aspecial lapping cycle where the position of the gear sets on th
8、e lapper does not change, and thenre-measured after lapping in order to establish how much, and where, material was removed. A wearconstant named “wear coefficient” specific to the lapping compound composition is then calculated.Based on the obtained wear coefficient value, an algorithm for simulati
9、ng the lapping process is presented.Gear sets lapped on the production line at American Axle and Manufacturing are used for simulation casestudies. Results show that it is possible to predict how much and where material will be removed on the toothsurface, thereby opening the door to better understa
10、nding of the lapping process.Copyright 2005American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2005ISBN: 1-55589-857-21 HYPOID GEAR LAPPING WEAR COEFFICIENT AND SIMULATION Claude Gosselin1, Qimi Jiang2, Kevin Jenski3, Jack Masseth4 1 Professor,
11、Department of Mechanical Engineering Laval University, Quebec, Canada G1K-7P4 2 Post Doctoral Fellow, Department of Mechanical Engineering Laval University, Quebec, Canada G1K-7P4 3 Gear Development Engineer American Axle and Manufacturing, Detroit, USA 4 Manager, Gear Design and Development America
12、n Axle and Manufacturing, Detroit, USA 1. Introduction Hypoid gears are widely used in vehicles. Two basic cutting processes can be used, either face milling or face hobbing 1-3, the latter now often being preferred for its cost effectiveness. Whichever cutting process is used, tooth surface errors
13、appear, in part because of tooling wear and machine distortion, in part because the gears are heat treated after cutting which releases internal strain and causes tooth flank distortion. Therefore, a finishing operation is necessary. Finishing can be done by grinding the gear teeth, but it is expens
14、ive in terms of tooling and time, and is therefore usually limited to small productions such as aerospace gears. Finishing can also be done by lapping, whereby the gear set is operated for a short time, under limited speed and torque, at varying positions such that the abrasive lapping compound impo
15、rves the contact surfaces. Lapping is normally applied to hypoid gear finishing because it is economical for large production volumes. As a result, excellent smoothness and quietness of operation can be obtained. To evaluate tooth surface errors, gears are measured with a Coordinate Measurement Mach
16、ine (CMM) to obtain the actual tooth flank topography. The CMM coordinates are compared to reference coordinates obtained from a 3 dimensional modelling software where the actual machine settings and cutter dimensions allow precise calculation of the theoretical tooth flank. The CMM coordinates may
17、be used to reverse engineer the actual tooth flank as developed by Gosselin 4. Reverse engineering provides the machine settings of a theoretical reference matching the actual tooth such that the kinematics of an actual gear set may be calculated the same way the theoretical gear set is calculated.
18、In practice, lapping is an abrasive wear process. The pinions and gears are lapped in pairs and must therefore remain as coordinated pairs in operation. Because of the variability in tooth surface errors caused by different machine production lines, varying tool wear, and different positions in the
19、heat treatment furnace, tooth surface errors vary significantly from batch to batch, piece to piece, and even tooth to tooth. As a result, the contact pattern is different from gear set to gear set, while the lapping cycle must be same. There are many factors affecting the lapping wear rate (weight
20、removal per unit of time), including grain size, shape, material and hardness of the lapping grit, applied torque, pinion RPM, hardness of the gear set, etc. Thus, developing a lapping cycle for production requires both time and experienced technicians to establish lapping operating positions and cy
21、cle time to produce quality gear sets. So far, theoretical knowledge on the relationship between material removal, lapping time and applied torque is limited. To help address this problem, one key parameter dubbed “wear coefficient” can be determined. Since the wear coefficient can be affected by fa
22、ctors such as grain size, shape, material and hardness of the lapping grit, this work is based on lapping tests made using the same lapping compound and conditions such that the above factors can be neglected. In other words, this work focuses on presenting an algorithm working with data measured fr
23、om hypoid gear sets lapped 2 under the same conditions (same production lines). The lapping process is reproduced using advanced modelling tools such as gear tooth simulation for the tooth surfaces and reverse engineering to analyze the tooth contact of the test gear sets. The test gear sets are mea
24、sured using a CMM prior to a special lapping cycle in which the position of the pinion and gear does not change; the gear sets are measured after lapping in order to establish the distribution of material removal. A wear coefficient is thus calculated and used to simulate the lapping process of prod
25、uction gear sets. Results show significant scattering of tooth heat treatment distortion, from tooth to tooth and from gear set to gear set, which makes the simulation process difficult. However, results also provide a rather constant wear coefficient, which is used to predict how much and where mat
26、erial will be removed on the teeth. 2. Main Nomenclature A contact area on the surface (m2) H Knoop hardness of the soft material ( 2mN ). L sliding distance, (m) mV volume of material removed by wear, (m3) W load normal to the tooth surface, (N) h thickness of removed material. (m) k abrasive wear
27、coefficient, jr ratio of the radius at point jP to the radius at the mean point. jdS difference in residual surface errors after and before lapping at corner point jP ( HBorHTTTTBj ,= ) (m); i average contact pressure at point i , (Mpa) iv average relative rolling and sliding speed between the tooth
28、 surfaces of pinion and gear at point i (m/s) rotation speed of the pinion or gear (RPM). 3. Wear Coefficient Calculation 3.1 Abrasive Wear Basically, there are two abrasive wear mechanisms in hypoid gear lapping, generally referred to as two-body and three-body abrasive wear respectively 8. In two-
29、body wear, a hard rough surface is in contact with another softer surface; in three-body wear, large hard abrasive grains are in contact with two softer surfaces. Literature results show that the wear equations of these two basic types of abrasive wear have the same form 9-10: HLWkVm = (1) Eqn.(1) s
30、hows that the observed wear rate is proportional to the load and the sliding distance, but inversely proportional to hardness. Kruschov 11, Rabinowicz et al. 12, Aleinikov 13, Mulhearn and Samuels 14, Avient et al. 15 verified experimentally the validity of Eqn. (1). So, the details of the two-body
31、and three-body abrasive wear mechanisms will be neglected. Avient et al. 15, Spurr et al. 16, Lopa 17, Kruschov and Babichev 18, Samuels 19, Toporov 20, Rabinowicz et al. 12, 21 experimented with different materials, and obtained typical values for k . Wear rate, and therefore the wear coefficient,
32、can be affected by factors such as hardness 22-25, the size of abrasive materials 12, 14, 15, the atmospheric moisture content 26, etc. Kallas investigated the wear energy in the abrasive wear process 27. Kato discussed the abrasive wear mode and abrasive wear rate from the viewpoints of effective w
33、ork for plastic deformation and fracture 28. Bradley et al. analyzed the relative abrasive wear rate using wall friction instead of hardness 29. Mason and Rooij investigated the abrasive wear between rough surfaces in deep drawing 30. From the above, one concludes that it is impossible to obtain a c
34、onstant value for k which can be applied to all materials and situations. Thus, it is not possible to select from the literature 3 t )(ti)(,( 11 tt i )(,( 22 tt i )(,( 33 tt i )(,( 44 tt i )(,( 55 tt i Figure 2: Contact pressure in one mesh a suitable value for the abrasive wear coefficient k that w
35、ould be directly applicable to lapping as lapping is a different abrasive wear process from the above literature. 3.2 Formulation and Procedure To address wear in hypoid gear lapping, one may rewrite Equation (1) as: AHLWkh = (2) At any contact point, under load a contact region is created between t
36、he hypoid pinion and gear tooth surfaces where contact pressure is distributed in an elliptical fashion. If one divides the contact area into many small elements, as shown in Figure1, the worn thickness of element ie during a period of time t can be expressed as: HtvkHAtvWkh iiiiii= (3) So the wear
37、rate (thickness of removed material) of element ie will be: Hvkth iiii = (4) When the size of element ie becomes very small, i will be the wear rate of point iP . During one mesh, the sliding speed of point iP can be regarded as unchangeable with time if the operating position of the gear set remain
38、s constant: ii vtv =)( . However, the contact pressure of point iP is a function of time, )(tii = , as shown in Figure 2. Thus, the wear rate of point iP is a function of time over one mesh. Suppose the contacting time of point iP , that is the time over which point iP is submitted to contact pressu
39、re, is it . The worn thickness at iP for one mesh will be: =iiitiit iitiidttHkvdtH vtkdtth0001)()()(5) Contact pressure is obtained through a Loaded Tooth Contact Analysis software module 4, 5. Five values for the contact pressure, at every contact point, are used for calculation. As shown in Figure
40、 2, the five contact pressure values occur at: 1) time 1t when contact point iP comes into mesh where the contact pressure can be regarded as 0, i.e. 0)( 1 =ti ; 2) time 2t between 1t (above) and the moment when the contact pressure becomes maximum; 3) time 3t when the contact pressure becomes maxim
41、um; ieiVFigure1: Discretized contact area iP4 4) time 4t between 3t and when meshing at point iP ends; 5) time 5t when meshing at point iP ends, i.e. 0)( 5 =ti . In Figure 2, as a general rule, Hertzian pressure is not symmetric about point 3t . Building up a coordinate frame as in Figure 2, the con
42、tact pressure )(ti can be interpolated with two parabolas as in Equation (6) such that the integral in Equation (5) can be calculated using numerical methods. + + +=)()()()()()()()()()()()()()()()()()()()()()()()()(5453434535335454545343325244252325453434533542332322223223322332ttttttttttttttttttttt
43、tt ttttttttttttt ttttttttttt ttttttttt tttttiiiiiiiiiii g24g22g22g87g87g87g73g82g85g87g87g19g73g82g85(6) To calculate the wear coefficient, two test gear sets underwent special lapping where the values of the gear set positionning axes do not change, i.e. the values of the three lapping parameters (
44、i) hypoid offset E , (ii) pinion mounting distance P and (iii) gear mounting distance G are kept constant. Suppose wear thickness of point iP after the gear set has been lapped for T seconds is ih . Then the wear coefficient value calculated at this point will be: =itiiii dttvTHhk0)(60(7) However, t
45、he shape of the contact pattern changes as wear progresses during special lapping. Figures 3 and 4 respectively show the calculated contact patterns on the tooth surface of the gear member before and after special lapping (both have been reverse-engineered). Figure 3 : Calculated contact pattern bef
46、ore lapping gear IB tooth flank (using the error surface) Figure 4 : Calculated contact pattern after lapping gear IB tooth flank (using the error surface) The origin is set at toe-bottom. The grid shows the location of the CMM measured points (CMM grid). Because of the variation in contact pressure
47、 caused by material removal as lapping proceeds, it is reasonable to average the calculated contact 5 pressure obtained before and after lapping to establish the wear coefficient. For convenience, several equidistant contact lines discretize the contact pattern. For example, in Figures 3 and 4, 11 c
48、ontact lines discretize the contact patterns before and after lapping. Nine equidistant contact points discretize a contact line such that 99 contact points are available. Since the contact pressure at each end of a contact line is close to zero, the two end points of each line are omitted from the
49、calculation process; thus, the number of valid contact points in Figures 3 and 4 change from 99 to 55. In addition, to avoid extrapolation outside the CMM grid, the contact points that do not lie within the CMM grid are not considered either (in Figures 3 and 4, all contact points lie within the CMM grid). Wear thickness is calculated at each contact point; then, interpol
