1、99FTMS1 Modeling and U Friction Measurement of Sliding for Gear Analysis by: M. Vaishya and D.R. Houser, The Ohio State University American Gear Manufacturers Association TECHNICAL PAPER Modeling and Measurement of Sliding Friction for Gear Analysis Manish Vaishya and Donald R. Houser, The Ohio Stat
2、e University The statements and opinions contained herein are those of the author and should not be construed as an officiai action or opinion of the American Gear Manufacturers Association. Abstract This paper discusses background studies needed for the prediction of sliding resistance on gear teet
3、h. Various elastohydrodynamic and mixed lubrication theories for coefficient of traction are examined. These theories are evaluated with respect to experimental results from two-disk tests for several parameters that simulate the continuously varying properties during gear engagement. For one mesh c
4、ycle, coefficient of friction for each tooth in contact is predicted as a function of roll angle. Normal load is estimated by the Load Distribution Program and is combined with friction coefficient to compute the total friction force. Dynamics tests are carried out on a pair of gears to measure shaf
5、t displacement in the off-line of action direction. Using the bearing stiffnessand shaft motions, measured friction force is calculated and compared with analytical results predicted from lubrication principles. Frequency response of the test rig is used to eliminate the influence of cross-coupling
6、terms with line-of-action forces and filters are applied to the measured data to curtail the dynamic effects of the test rig. Based on the comparison of theoretical predictions and test results on a spur gear pair, the lubrication models are evaluated, with special emphasis being placed on the dynam
7、ic modeling of friction. Copyright O 1999 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 October, 1999 ISBN: 1-55589-757-6 MODELING AND MEASUREMENT OF SLIDING FRICTION FOR GEAR ANALYSIS Manish Vaishya (Research Associate) Donald R. Houser (Professor)
8、Department of Mechanical Engineering The Ohio State University 206 W, 18 Ave Columbus, OH 432 1 O 1. Introduction Sliding resistance in gear meshing has a considerable effect on off-line-of-action dynamics. Most existing studies i have assumed simplified Coulomb friction to model this phenomenon. In
9、 reality, the lubrication regime includes elastohydrodynamic and boundary lubrication. The coefficient of friction used in lubrication calculations depends upon parameters like radii of curvature, sliding speed and Hertzian pressure and varies significantly during a mesh cycle 2. Furthermore, a unif
10、ied theory for fluid rheological properties is not available and they all depend upon the particular application. This article presents a critical analysis of various tribological principles 3, 4, 5, 6, 7, 8, 93 in the context of pear dynamic behavior. Experiments are specially designed to tackle th
11、e issue of sensitivity, arising due to low magnitude of friction force as compared to the torque transmitting loads. With this information, a comparison could be made between the disk tests and friction measured in a running gear. The validity and limitations of this comparison are discussed. A prel
12、iminary evaluation of lubrication models is done using two-disk tests. Results of these tribology experiments are applied for modeling sliding forces in a pear mesh. Friction forces are computed from load distribution on the gear teeth and instantaneous coefficients of friction. To validate this mod
13、el, dynamics tests are carried out on a gear pair. Shaft displacement is measured in the off-line of action direction and combined with bearing stiffness to get the friction force. Analytical methods are applied to minimize the dynamic and cross-coupling effects of the test rig. O of traction is stu
14、died for four different parameters, namely load, rolling speed, siide-to-roll ratio and temperature. The parameters are selected such that they simulate the mechanism of gear meshing under typical conditions. From the comparison of experimental and predicted coefficients of friction, certain formula
15、tions are selected on the basis of their applicability over this range of parameters. Subsequently, a spur gear pair is modeled as a .quasi-static system. A variable center distance test rig is used to measure shaft deflection under different torque conditions. For quasi-static assumption to hold, t
16、he dynamic effects need to be minimized. This is achieved by restricting the tests to low rotational speeds. Frequency response of the test rip is used to attenuate the signal near resonant conditions. From transmissibility studies, it is shown that the coupling between line of action (LOA) forces,
17、resulting from transmission error or mesh stiffness variation, and off- line of action (OLOA) displacement is negligible. Thus the shaft dispbacement in OLOA direction shall directly provide the information on siiding forces. Finally, different lubrication models are applied to the gear mesh, using
18、the time varying parameters like Hertzian pressure and sliding velocity in the mesh cycle. The meshing load, as predicted by the Load Distribution Program lo, is multiplied with the traction coefficient to get the combined friction force on all teeth in contact. This force is compared with the exper
19、imental values to evaluate different lubrication models. This modeling technique is expected to provide the basis of a full multi-degree of freedom dynamic model of gear meshing, including off-line of action forces. The overall methodology that is applied in this paper is shown as a flowchart (Fig.
20、i). 2. Methodology Two lubricants with distinct properties are tested on a two-disk friction testing machine. The coefficient Mesh Kinematics V, Vs, R Lube 2 Fi (experimental) Figure 1. Flowchart showing the analysis method 3. Two-Disk Friction Tests For testing rheological properties of fluids, 2-d
21、isk experiments are usually applied to emulate the varying gear mesh conditions. Here, the two-disk friction test rig (Fig. 2) located in the Tribology Lab at the Ohio State University has been used to test two different lubricants, Automatic Transmission Fluid (fluid A) and high viscosity gear oil
22、(fluid B). The selection of lubricants was based on their widely distinct properties (Table i), so the Friction effects measured during gear tests should have pronounced differences. The rig consists of two disks of radii inch and 1.5 inches, each crowned at a radius of 3 inches across the face. The
23、 disks have a hardness of 50 HRC and 20 p-in Ra surface roughness value. The friction disks are propelled by two motors whose individual speeds can be varied using a controller. Normal load between the disks is applied pneumatically and is measured using a load cell. The other parameters that are re
24、corded are the inlet temperature and output torque. The coefficient of friction is calculated from the output torque and the normal load. Table 1. Properties of test lubricants Fluid A Fluid B Viscosity index Pressure-viscosit coeff 2.2e-8 2.2e-8 AGMA number 7 EP Since most lubrication models are re
25、levant only over a limited range of parameter values, the required range for a gear application is established. In this study, a unity ratio spur gear pair with 28 teeth, quarter inch face width and diameteral pitch of 8, was analyzed. These gears have a profile contact ratio of 1.638. For any given
26、 running condition, say 1200 rpm and 900 in- Ib, the various meshing parameters can be estimated (Table 2). Figure 2. Two-disk lubricant test rig 2 Table 2. Gear mesh parameters (a) At beginning of contact Mean rolling speed ids 75.2 Sliding speed ids 76.0 , Hertz contact stress psi Mean rollinp spe
27、ed (b) At the pitch point Range Baseline value ids 12 - 150 97.5 I Pinion 1 Gear Radius of curvature I in I 0.599 I 0.599 (Hertz stress) Inlet temperature Lubricani (ksi) (257) OC 33 -73 33 A, B Surface speed T ids 1 75.2 1 75.2 Mean rolling speed I ids 1 75.2 Sliding speed 1 ids I 0.0 Hertz contact
28、 stress 1 p si I 200,000 From these representative values, parameters for any running condition can be estimated, since all velocities are proportional to the rpm and the contact stresses are proportional to florque)“. TO simulate a wide range of speeds and torque, within the constraints of the test
29、 machine, the following conditions were selected (Table 3). Only one controlled parameter was varied at a time and others were approximately held at baseline values. Table 3. Parameters for 2-disk tests Slide-to-roll ratio 1 1 o- 1.2 1 1 .O8 Load Ib I 11-135 1 67 Figure 3 depicts the behavior of the
30、 coefficieni of friction (1) for the above conditions for both the lubricants The four parameters that were independently varied are the rolling velocity. slide-to- roll ratio, normal load and temperature. In a gear mesh, these would correspond to the rpm, position within a mesh cycle. torque and ge
31、arbox design respectively. p has been calculated from the average of 100 points, sampled at 4 Hz. - 0.05 - -_ 0.05 O 50 100 150 0.6 0.8 1 1.2 roll speed, ids Slide to Roll ratio 0.1 jl 0.1 - lb O 200 400 600 30 40 50 60 70 Figure 3. Variation in coefficient of friction for Load, N temp, deg C lubric
32、ants A and B It can be seen from Figure 3 that for all the conditions, ATF has a value of p nearly one-half of that of fluid B. Hence, it can be deduced that viscosity of the lubiicant is one of the primary determinants of its traction properties. These graphs also show a strong dependence on rollin
33、g velocity and contact stress. On the other hand, 1 increases very gradually as the inlet temperarure is raised, and drops with increasing specific sliding velocity All four graphs show a asymptotic behavior at extreme conditions. In the following section, these results will be evaluated against the
34、 available theoretical predictions. 4. Lubricant Formulations Gear tribology depends upon various parameters, including the surface roughness, lubricant rheology and meshing conditions. During one tooth roll cycle, sliding friction can vary significantly due to changes in speed. load distribution an
35、d surface profile. As a result, the lubrication regime may oscillate between elastohydrodynamic to thin film boundary lubrication. Miscellaneous theories exist under such varying conditions, which may depend upon totally different sets of parameters. In particular, equations for gear friction have l
36、ong been debated because of the lack of reliable friction measurement. This problem arose due to the small magnitude of friction forces and the dynamic effects during a gear mesh 1 i. The following table (see Table 4) gives a summary of some common formulations to calculate the coefficient of fricti
37、on, along with the factors that contribute to its calculation. First, the minimum fluid film thickness is calculated, which determines the applicable lubrication regime. Two alternative formulations are presented here (list of symbols is provided at the end of the paper): 3 Table 4. Traction formula
38、tions and influencing parameters e e ho, k etc Evans 06 0.04 I 10 12 14 16 10 20 22 roll angle. deg (a) I“ tooth Figure 6. Coefficient of friction through I mesh cycle on individual teeth, at 200 rpm. 1200 in-lb, - calculated for different friction theories 7 0.16 I 2 0.081 LI 1 I f 10 12 14 16 18 2
39、0 22 24 ml1 angle, deg (b)2“d tooth Figure 6. (contd.) In Fig 6 (b), zero value of p implies that the 2“ tooth pair is not in contact. The subsequent portion of the graph shows the third tooth pair in mesh due to edge contact. From the value of p and the normal load, the instantaneous hction force c
40、an be found from k (6.41 where k = tooth number in contact The variation in friction force with roll angle on each individual pinion tooth is shown in Figure 7. Finally, Figure 8 shows the total friction force on the pinion computed from the four lubrication theories. 300. ;= -100. -200 . -300 -400.
41、 - -500 1 I 10 12 14 16 18 20 22 24 roll angel. deg (a) i“ tooth Figure 7. Friction force on individual pinion teeth in contact. at 200 rpm, 1200 in-lb, calculated from different friction theories -350 10 12 14 16 18 20 22 24 roll angel, deg (b) 2“d tooth Figure 7. (contd.) sum of fnction forces 500
42、 400 - 300. 200. -300 . !: ,. 10 12 14 16 18 20 22 24 roil angle, deg O Figure 8. Total friction force at 200 rpm and 1200 in- Ib, predicted by different friction theories 7. Dynamic Gear Tests Dynamic experiments were performed for measurement of friction force by means of input shaft deflection. T
43、he unity ratio gear pair is mounted on the Variable Center Distance Test rig in the Gear Dynamics and Noise Research lab at the Ohio State University. A 150 HP DC eddy current brake dynamometer provides the transmitted torque and gcars are run wiih a 200 HP DC motor. Test lubricant is injected direc
44、tly into the exit side of tooth meshing area. Shafr displacement is measured ai one of the inpui shaft bearings. both in LOA and OLOA directions, using non-contact eddy current displacement probes. Gears are tested at a speed of 200 rpm to simulate quasi-static conditions. The torque value is set at
45、 1200 in-lb, which is close to rated load of this gear. Data is O 8 shows the unfiltered measured shaft displacement in the OLOA direction over one mesh cycle. 16- 14- 2 1.2 Figure 9. Measured input shaft displacement at 1200 in-lb torque, unfiltered data 6.5 6- 1 35 Even though low speeds were used
46、 during the tests. the dynamic effects of the test rig may not necessarily be negligible. A further distortion in data mav occur due to contribution from LOA forces. which I ,- e. i3 1- 1 -0.2 O 200 400 600 800 1000 mesh load. Ibf Figure 10. Static loading results in OLOA and LOA direction for singl
47、e tooth contact are also much higher in magnitude. These issues are discussed in the following section. e 8. Dynamic and Coupling Effects The measured data for shaft displacement needs to be rectified for cross-coupling, if any, between OLOA displacernent of input shaft and LOA forces. Coupling term
48、 was estimated by static calibration of the tesi gears with linearly increasing torque, thereby increahing the LOA load. Under these conditions. dellcction in LOA and OLOA directions is measured (Fig. IO). OLOA motion was recorded by simulating thc input shaft as both driving and driven. thus revers
49、ing the direction of friction. Taking the mean of the two cancels out hysteresis, effects o-f tooth friction and other components of the test rig. The magnitude of OLOA motion is less than 2% of LOA values. It follows that bearing cross coupling does not affect the OLOA system response due to line of aciion forces. This was also verified with the aid of transmissibiliiy tests. An impulse hammer is applied on pinion ouler diameier in lhe OLOA and LOA directions. The frequency response of OLOA shaft displacement to these forces is measured and recorded
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