1、05FTM06A Model to Predict Friction Lossesof Hypoid Gearsby: H. Xu, A. Kahraman, D.R. Houser, The Ohio State UniversityTECHNICAL PAPERAmerican Gear Manufacturers AssociationA Model to Predict Friction Losses of Hypoid GearsHai Xu, Ahmet Kahraman, Donald R. Houser, The Ohio State UniversityThe stateme
2、nts 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.AbstractA model to predict friction-related mechanical efficiency losses of hypoid gear pairs is proposed, whichcombines a commercial a
3、vailable finite element based gear contact analysis model and a friction coefficientmodel with a mechanical efficiency formulation. The contact analysis model is used to provide contactpressures and other contact parameters required by the friction coefficient model. The instantaneous frictioncoeffi
4、cient is computed by using a validated new formula that is developed based on a thermalelastohydrodynamic lubrication (EHL) model considering non-Newtonian fluid. Computed friction coefficientdistributions are then used to calculate the friction forces and the resultant instantaneous mechanicaleffic
5、iency losses of the hypoid gear pair at a given mesh angle. The model is applied to study the influence ofspeed, load, surface roughness, and lubricant temperature as well as assembly errors on the mechanicalefficiency of an example face-hobbed hypoid gear pair.Copyright 2005American Gear Manufactur
6、ers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2005ISBN: 1-55589-854-81 A Model to Predict Friction Losses of Hypoid Gears Hai Xu Ahmet Kahraman Donald R. Houser Graduate Research Associate Assoc. Professor Professor Emeritus Department of Mechanical Engineering T
7、he Ohio State University 650 Ackerman Road, OH 43202 1 Introduction Gear mesh friction has attracted a number of researchers for more than a century 1. The friction between gear teeth plays an important role in defining the efficiency of the system as well as influencing scoring limits and the dynam
8、ical behavior including vibration and noise 2,3. Both sliding and rolling actions at the gear mesh contact contribute to gear mesh friction. Sliding friction is a direct product of the relative sliding between the two contacting surfaces while rolling friction originates from the resistance to the r
9、olling motion 4. Coefficient of friction that is used widely in the literature usually refers to the coefficient of sliding friction. A significant number of studies have been published especially within the last forty years on friction and efficiency of gear trains as reviewed by references 5-7. Th
10、e first group of studies focused on measuring power losses of gear pair directly 8-17. Several others measured using twin-disk test machines under conditions simulating a gear pair so that this friction coefficient can be used to predict the efficiency of a gear pair 18-30, 38-39. Some of these stud
11、ies 18-25 resulted in well-known and widely used empirical formulae for . These empirical formulae indicate that is a function of a list of parameters such as sliding and rolling velocities, radii of curvature of the surfaces in contact, load or contact pressure, surface roughness, and the lubricant
12、 viscosity. A group of efficiency models 31-33 investigated the efficiency of a spur gear pair by assuming a uniform along the entire contact surface. A tangential friction force along the sliding direction was computed by using a given constant friction coefficient , and the geometric and kinematic
13、 parameters of the spur gears. As a result, the amount of reduction of torque transmitted to the driven gear was used to calculate the mechanical efficiency of the gear pair. These models were useful in bringing a qualitative understanding to the role of spur gear geometry on efficiency. They fell s
14、hort in terms of the definition of , as a user-defined constant value must be used for every contacting point on the tooth surface. However, the published experiments on sliding/rolling contacts indicate that many parameters might influence 18-25. In addition, these studies were limited to spur gear
15、s and many complicating effects of the tooth bending and contact deformations, tooth profile modifications and manufacturing errors were not included. Another group of efficiency models 34-37, 40 relied on published experimental formulae such as those in references 18-21. The models in this group co
16、nsidered spur 35-37, 40 and helical 34 gear pairs and calculated the parameters required to define according to the particular empirical formula adapted. While they are potentially more accurate than the constant models, their accuracy is limited to the accuracy of the empirical formula used. Each e
17、mpirical formula typically represents a certain type of lubricant, operating temperature, speed and load ranges, and surface roughness conditions of roller specimens that might differ from those of the gear pair that is being modeled. The models in the last group are more advanced since they use an
18、EHL model to predict instead of relying on the user or the empirical formulae 42-54. Among them, Dowson and Higginson 47, and Martin 48 used a smooth surface EHL model to determine the surface shear stress distribution caused by the fluid film, and hence, the instantaneous friction coefficient at th
19、e contact. Adkins and Radzimovsky 49 developed a model for lightly loaded spur gears under hydrodynamic lubrication condition and assumed that the gear tooth is rigid without deflections and local deformations. Simon 50 provided an enhancement by using point contact EHL model for heavily crowned spu
20、r gears with smooth surfaces considering the elastic displacement of the surface due to fluid pressure distributions. Larsson 51 and Wang et al 52 analyzed involute spur gear lubrication by using a transient thermal-EHL model with smooth surfaces. Wu and Cheng 53 developed a friction model based on
21、mixed-EHL contacts and applied it to calculate the frictional power losses of 2 spur gears. The roughness was modeled such that all the asperities have the same radius of curvature whose heights have a Gaussian distribution. Mihalidis et al 54 included the influence of the asperity contacts as well
22、in calculating and hence efficiency. These models 47-54 were successful in eliminating to a certain extent the need for prior knowledge of , at the expense of significantly more computational effort. While they were relatively enhanced in EHL aspects of the problem, the applications were limited to
23、simple spur gears with ideal load distributions and no tooth bending deformations. A small number of efficiency studies on helical gears were found 34,40,55-58. Literature on hypoid gear efficiency is even sparser. Buckingham 59 proposed an approximated formula for the power loss of hypoid gears, wh
24、ich is the sum of the losses of a spiral bevel gear and a worm gear. Naruse et al 8,10 conducted several tests on scoring and frictional losses of hypoid gears of Klingelnberg type. Coleman 60 used a simple formula to calculate hypoid gear efficiency with a constant or a formula with a very limited
25、number of parameters included 61. Smooth-surface EHL formulations were found applied to hypoid gears by Simon 62 and Jia et al 63. 1.1 Objectives and Scope Efficiency losses in a gearbox are originated from several sources including gear mesh sliding and rolling friction, windage, oil churning, and
26、bearing friction 34. When gears are loaded, a gear contact under load experiences combined sliding and rolling, both of which result in frictional losses. The amount of sliding frictional loss is directly related to the coefficient of friction, normal tooth load and relative sliding velocity of the
27、surfaces while the rolling friction occurs due to the deformation of the two contacting surfaces. When the contact is lubricated, rolling frictional losses are originated from the formation of the EHL film 35. Efficiency can be improved by reducing the coefficient of friction via precision manufactu
28、ring and smoothening the contact surfaces and enhancement of lubricant properties. Existing approaches of improving efficiency are based mostly on experimental trial-and-error type procedures focusing on such parameters, while the predictive capabilities have been limited. The main objective of this
29、 study is to develop a mechanical efficiency model for hypoid gears. The model allows an analysis of both face-hobbed and face-milled hypoid gears. The efficiency model will allow two methods of calculating , i.e. published empirical formulae and a thermal EHL formulation. The differences amongst th
30、ese approaches will be described. Parametric studies will be performed to investigate the influence of several relevant parameters such as speed, load, surface roughness, lubricant temperature as well as the assembly errors on the mechanical efficiency of hypoid gears. This study is focused primaril
31、y on the mechanical efficiency losses related to tooth friction, including sliding and rolling friction, while it relies on the published studies in terms of losses associated with windage, oil churning and bearings 34,35,64-69 when necessary. Figure 1. Flowchart for the efficiency prediction. 2 Eff
32、iciency Model 2.1 Technical Approach Figure 1 illustrates a flowchart of the efficiency computation methodology used in this study. Three main components are the gear contact analysis model, the friction coefficient computation model, and the gear pair mechanical efficiency computation formulation.
33、The same methodology was applied by these authors earlier to spur and helical gears successfully 70. It was also shown the parallel-axis efficiency model compares well with the gear pair efficiency experiments 77. The gear contact analysis model uses the gear design parameters, operating conditions
34、and errors associated with assembly, mounting and manufacturing of the tooth profile to predict load and contact pressure distributions at every contact point during each mesh position. Predicted load distribution or contact NoOverall Efficiency m = m+1m+1 = m+YesGear Design/Cutter/ Machine Paramete
35、rs Operating Conditions Assembly/Manufacturing Errors Gear Contact Analysis Mechanical Efficiency ()m Friction Coefficient 1. Published formulae 2. EHL analysis 3. EHL-based formulaSurface RoughnessLubricant ParametersX,P,R ,V mM (, , )mz 3 pressure together with other geometric and kinematic parame
36、ters are input to the friction coefficient model to determine the instantaneous friction coefficient (, , )mz of every contact point (, )z on the gear tooth surface. (, , )mz is then used by the mechanical efficiency computation module to determine the instantaneous efficiency ()m of the gear pair a
37、t the m-th incremental rotational position defined by angle m . The above sequential procedure is repeated for an M number of discrete positions ( 1, 2, ,mM= ) spaced at an increment of (mm =) to cover an entire mesh cycle. These instantaneous mechanical efficiency values ()m are then averaged over
38、a complete mesh cycle to obtain the average mechanical efficiency loss of the gear pair due to tooth friction. In the following sections, main components of this methodology as shown in Fig. 1 are described in detail. 2.2 Contact Analysis of Hypoid Gears A commercial available finite element (FE) ba
39、sed hypoid gear analysis package CALYX 71 is used as the contact analysis tool. Both face-hobbed and face-milled versions of this model are available. The model combines FE method away from the contact zone with a surface integral formulation applied at and near the contact zone 72. The contact anal
40、ysis model used in this study has a special setup for the finite element grids inside the instantaneous contact zone. As shown in Fig. 2(a), a set of very fine contact grid is defined automatically on hypoid gear teeth to capture the entire contact zone. These grid cells are much finer than the regu
41、lar size of finite element meshes elsewhere on the tooth surfaces and they are attached to the contact zones that result in more accurate contact analysis. A schematic view of these grid cells is shown in Fig. 2(b). Along the face width, there are 2n+1 divisions, and at each division, there is a pri
42、ncipal contact point (shown in dot) if contact occurs. In the profile direction, there are 2m+1 grid cells within each division for capturing potential contact points, which would be in contact due to tooth deflections and local surface deformations. (a) (b) (c) (d) Figure 2. (a) Moving grids for co
43、ntact zones, (b) moving grid setup, (c) grid in tangent plane for calculation, and (d) principal directions and contact ellipse. i = - n j = -mi = 0i = nj = 0j = mq2 45tp26t13678 q 12 376 58 4nt26t48feheseqe t2q tp26t1q 2q yx (1)pV(1)tV(2)tV(2)pVHertzian contact ellipse 4 The calculation is carried
44、out in the grid of principal contact point in the tangent plane as shown in Fig. 2(c), which is a magnified grid cell for the principal contact point q . The surface formed by dotted lines, with points 1 to 8 along the edges, is the grid on the real tooth surface and the plane formed by solid lines,
45、 with points 1 to 8 along the edges, is the grid in the tangent plane for this particular contact point q . Points 2(2), 4(4), 6(6) and 8(8) are the mid point at each side of the grids. Vector t26that connects point 2 and 6 is approximated as the instant line of contact and vector tp26tis normal to
46、t26in the tangent plane. n is the surface normal vector at the contact point q . When is obtained for this principal contact point, same value will be assigned to the potential contact point within the same face width division. While the load distribution and contact pressure at each grid is provide
47、d by the contact analysis model, surface velocities and curvatures are calculated in the following sections. Definition of Principal Contact Points. Assume pinion surface and gear surface are defined by 111(,)s tr and 222(,)s tr respectively, where 1s ,1t and2s ,2t are the surface curvilinear parame
48、ters. The principal contact point is determined and located when 1r and 2r become the closest to each other 72. Surface 111(,)s tr was discretized into a grid of points 1111(,)ij i js t=rr and for each of these grid points, an effort was made to locate 2222(, )ij i js t=rr such that ()( )111 2 22,s
49、tstrr is minimized with respect to the variable 2s and 2t . This extremization is equivalent to solving the following system of nonlinear equations 72 1222 222 21222 222 2(, ) (, ) 0(, ) (, ) 0ij i j i j iij i j i j jst st sst st s= =rr rrr r(1) First, solutions from Eq. (1) for each of the grid point 1ijr were obtained by the Newton-Raphson method. Then a new grid, which is
