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1、05FTM06 A Model to Predict Friction Losses of Hypoid Gears by: H. Xu, A. Kahraman, D.R. Houser, The Ohio State University TECHNICAL PAPER American Gear Manufacturers Association Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/111111100
2、1, User=Wing, Bernie Not for Resale, 04/18/2007 11:19:22 MDTNo reproduction or networking permitted without license from IHS -,-,- A Model to Predict Friction Losses of Hypoid Gears Hai Xu, Ahmet Kahraman, Donald R. Houser, The Ohio State University The 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 A model to predict friction-related mechanical efficiency losses of hypoid gear pairs is proposed, which combines a commercial available finite element based gear
4、contact analysis model and a friction coefficient model with a mechanical efficiency formulation. The contact analysis model is used to provide contact pressuresandothercontactparametersrequiredbythefrictioncoefficientmodel. Theinstantaneousfriction coefficient is computed by using a validated new f
5、ormula that is developed based on a thermal elastohydrodynamiclubrication(EHL)modelconsideringnon-Newtonianfluid. Computedfrictioncoefficient distributions are then used to calculate the friction forces and the resultant instantaneous mechanical efficiencylossesofthehypoidgearpairatagivenmeshangle.
6、Themodelis appliedtostudytheinfluenceof speed, load, surface roughness, and lubricant temperature as well as assembly errors on the mechanical efficiency of an example face-hobbed hypoid gear pair. Copyright 2005 American Gear Manufacturers Association 500 Montgomery Street, Suite 350 Alexandria, Vi
7、rginia, 22314 October, 2005 ISBN: 1-55589-854-8 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:19:22 MDTNo reproduction or networking permitted without license from IHS -,-,-
8、 1 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 The Ohio State University 650 Ackerman Road, OH 43202 1 Introduction Gear mesh friction has attracted a nu
9、mber 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 dynamical behavior including vibration and noise 2,3. Both sliding and rolling actions at the gear mesh contac
10、t 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 rolling motion 4. Coefficient of friction that is used widely in the literature usually refers to the coef
11、ficient 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. The first group of studies focused on measuring power losses of gear pair directly 8- 17. Several others me
12、asured 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 studies 18-25 resulted in well-known and widely used empirical formulae for . These empirical formulae indic
13、ate 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 viscosity. A group of efficiency models 31-33 investigated the efficiency of a spur gear pair by assumi
14、ng 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 parameters of the spur gears. As a result, the amount of reduction of torque transmitted to the driven
15、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 short in terms of the definition of , as a user- defined constant value must be used for every contacting
16、 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 gears and many complicating effects of the tooth bending and contact deformations, tooth profile modificati
17、ons 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 considered spur 35-37, 40 and helical 34 gear pairs and calculated the parameters required to define acco
18、rding 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 empirical formula typically represents a certain type of lubricant, operating temperature, speed and loa
19、d 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 EHL model to predict instead of relying on the user or the empirical formulae 42-54. Among them, Dowson
20、 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 the contact. Adkins and Radzimovsky 49 developed a model for lightly loaded spur gears under hydrodynamic
21、 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 spur gears with smooth surfaces considering the elastic displacement of the surface due to fluid pressure
22、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 mixed-EHL contacts and applied it to calculate the frictional power losses of Copyright American Gear M
23、anufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:19:22 MDTNo reproduction or networking permitted without license from IHS -,-,- 2 spur gears. The roughness was modeled such that all the asperities hav
24、e the same radius of curvature whose heights have a Gaussian distribution. Mihalidis et al 54 included the influence of the asperity contacts as well 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 ex
25、pense of significantly more computational effort. While they were relatively enhanced in EHL aspects of the problem, the applications were limited to simple spur gears with ideal load distributions and no tooth bending deformations. A small number of efficiency studies on helical gears were found 34
26、,40,55-58. Literature on hypoid gear efficiency is even sparser. Buckingham 59 proposed an approximated formula for the power loss of hypoid gears, which 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 hyp
27、oid 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 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
28、Scope Efficiency losses in a gearbox are originated from several sources including gear mesh sliding and rolling friction, windage, oil churning, and bearing friction 34. When gears are loaded, a gear contact under load experiences combined sliding and rolling, both of which result in frictional los
29、ses. The amount of sliding frictional loss is directly related to the coefficient of friction, normal tooth load and relative sliding velocity of the surfaces while the rolling friction occurs due to the deformation of the two contacting surfaces. When the contact is lubricated, rolling frictional l
30、osses are originated from the formation of the EHL film 35. Efficiency can be improved by reducing the coefficient of friction via precision manufacturing and smoothening the contact surfaces and enhancement of lubricant properties. Existing approaches of improving efficiency are based mostly on exp
31、erimental trial- and-error type procedures focusing on such parameters, while the predictive capabilities have been limited. The main objective of this 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. T
32、he efficiency model will allow two methods of calculating , i.e. published empirical formulae and a thermal EHL formulation. The differences amongst these approaches will be described. Parametric studies will be performed to investigate the influence of several relevant parameters such as speed, loa
33、d, surface roughness, lubricant temperature as well as the assembly errors on the mechanical efficiency of hypoid gears. This study is focused primarily on the mechanical efficiency losses related to tooth friction, including sliding and rolling friction, while it relies on the published studies in
34、terms of losses associated with windage, oil churning and bearings 34,35,64-69 when necessary. Figure 1. Flowchart for the efficiency prediction. 2 Efficiency Model 2.1 Technical Approach Figure 1 illustrates a flowchart of the efficiency computation methodology used in this study. Three main compon
35、ents are the gear contact analysis model, the friction coefficient computation model, and the gear pair mechanical efficiency computation formulation. The same methodology was applied by these authors earlier to spur and helical gears successfully 70. It was also shown the parallel-axis efficiency m
36、odel compares well with the gear pair efficiency experiments 77. The gear contact analysis model uses the gear design parameters, operating conditions and errors associated with assembly, mounting and manufacturing of the tooth profile to predict load and contact pressure distributions at every cont
37、act point during each mesh position. Predicted load distribution or contact No Overall Efficiency m = m+1 m+1 = m+ Yes Gear Design/Cutter/ Machine Parameters Operating Conditions Assembly/Manufacturing Errors Gear Contact Analysis Mechanical Efficiency () m Friction Coefficient 1. Published formulae
38、 2. EHL analysis 3. EHL-based formula Surface Roughness Lubricant Parameters X,P,R ,V mM ( , ,) m z Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:19:22 MDTNo reproduction or
39、 networking permitted without license from IHS -,-,- 3 pressure together with other geometric and kinematic parameters are input to the friction coefficient model to determine the instantaneous friction coefficient ( , ,) m z of every contact point ( , )z on the gear tooth surface. ( , ,) m z is the
40、n used by the mechanical efficiency computation module to determine the instantaneous efficiency () m of the gear pair at 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
41、 ( m m=) to cover an entire mesh cycle. These instantaneous mechanical efficiency values () m are then averaged over 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
42、in Fig. 1 are described in detail. 2.2 Contact Analysis of Hypoid Gears A commercial available finite element (FE) based 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 awa
43、y from the contact zone with a surface integral formulation applied at and near the contact zone 72. The contact analysis 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
44、 automatically on hypoid gear teeth to capture the entire contact zone. These grid cells are much finer than the regular 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 g
45、rid cells is shown in Fig. 2(b). Along the face width, there are 2n+1 divisions, and at each division, there is a principal 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
46、in contact due to tooth deflections and local surface deformations. (a) (b) (c) (d) Figure 2. (a) Moving grids for contact zones, (b) moving grid setup, (c) grid in tangent plane for calculation, and (d) principal directions and contact ellipse. i = - n j = -m i = 0 i = n j = 0 j = m q 2 4 5 tp26t 1
47、 3 6 7 8 q 1 2 3 7 6 5 8 4 n t26 t48 f e h e s e q e t2 q tp26t 1 q 2 q y x (1) p V (1) t V (2) t V (2) p V Hertzian contact ellipse Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/20
48、07 11:19:22 MDTNo reproduction or networking permitted without license from IHS -,-,- 4 The calculation is carried 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, with points 1 to 8 along the edges, is the grid in the tangent plane