Rolling/ Sliding Contacts 255 which suggests that the elasticity of the rollers causes the minimum film thickness to increase by approximately 100 times. Dowson and coworkers [4, 61 approached the problem from first prin- ciples and simultaneously solved the elasticity and the Reynolds equations. Their formula for the minimum film thickness is given in a dimensionless form as: where tlo U U = speed parameters = - Ee Re - G = material parameter = aEe W = load parameter = - PY EeR Using the same dimensionless groups suggested by Dowson and Higginson [4], the Grubin solution can be given as: H = 1.95 7 ( Gu)0.73 (7.6) What is particularly significant in the EHD theory is the very low depen- dency of the minimum film thickness on load. The important parameters influencing the generation of the fdm are the rolling speed, the effective radius of curvature and the oil viscosity. Consequently, Dowson and Higginson suggested the following simplified formula for practical use: where ho = minimum film thickness (in.) qo = inlet oil viscosity (poise) Re = effective radius (in.) U = rolling speed (in./sec) 256 Chapter 7 7.4 FRICTION IN THE ELASTOHYDRODYNAMIC REGIME The EHD lubrication theory developed over the last 50 years has been remarkably successful in explaining the many features of the behavior of heavily loaded lubricated contacts. However, the prediction of the coeffi- cient of friction is still one of the most difficult problems in this field. Much experimental work has been done [7-211, and many empirical formulas have also been proposed based on the conducted experimental results. Plint investigated the traction in EHD contacts by using three two-roller machines and a hydrocarbon-based lubricant [14]. He found that roller sur- face temperature has a considerable effect on the coefficient of friction in the high-slip region (thermal regime). As the roller temperature increases the coefficient of friction falls linearly until a knee is reached. With further increase in temperature the coefficient of friction rises abruptly and errati- cally and scuffing of the roller surface occurs. He also gave the following equation to correlate all the experimental results, which was obtained from 28 distinct series of tests: 21 300 f = 0.0335 log - (0, + 40) - 44sb3 (7.8) where 0,. is the temperature on the central plane of the contact zone ("C) and h is the radius of the contact zone (inches). Dyson [15] considered a Newtonian liquid and derived the expression for maximum coefficient of friction as: where a = pressureviscosity coefficient K = heat conductivity P = pressure qo = dynamic viscosity ho = minimum oil film thickness y = temperature-viscosity coefficient (7.9) If aP >> I, the coefficient of friction increases rapidly with pressure. Rolling/ Sliding Contacts 257 Sasaki et al. [16] conducted an experimental study with a roller test apparatus. The empirical formula of the friction coefficientf in the region of semifluid lubrication as derived from the tests is given as: (7.10) where rj = lubricant dynamic viscosity U = rolling velocity U' = load per unit width k = function of the slide/roll ratio When slidelroll ratio = 0.3 1, k = 0.037; when slidelroll ratio = 1.22, k = 0.026. Drozdov and Gavrikov [ 171 investigated friction and scoring under conditions of simultaneous rolling and sliding with a roller test machine. The formula for determination off at heavy contact loads from more than 10,000 experiments is found to be: (7.1 I) 1 0.8~:'~ + V,v(Pmax, uO) + 13.4 f= where dl'maxy vg) = 0.47 - 0.12 x 10-4Pmax - 0.4 x 10-3~g uo = kinematic viscosity of the lubricant (cst) at the mean surface V, = sum rolling velocity (sum of the two contact surface velocities, temperature (To) and atmospheric pressure m/sec) P,,, = maximum contact pressure (kg/cm2) O'Donoghue and Cameron [ 181 studied the friction in rolling sliding con- tacts with an Amsler machine and found that the empirical relation relating friction coefficient with speed, load, viscosity, and surface roughness could be expressed as: (7.12) 258 Chapter 7 where S = total initial disk surface roughness (pin. CLA) V, = sliding velocity (difference of the two contact surface velocities) (in./sec) Vr = sum rolling velocity (in./sec) q = dynamic viscosity (centipoises) R = effective radius (in.) Benedict and Kelley [ 191 conducted experiments to investigate the friction in rolling/sliding contacts. The coefficient of friction has been found to increase with increasing load and to decrease with increasing sum velocity, sliding velocity, and oil viscosity when these quantities are varied individu- ally. The viscosity was determined at the temperature of the oil entering the contact zone. The results are combined in a formula, which closely repre- sents the data as below: where (7.13) R = effective radius (in.) S = surface roughness (pin. rms) V, = sliding velocity (in./sec) V, = sum rolling velocity (in./sec) W = load per unit width (lb/in.) qo = dynamic viscosity (cP) The limiting value of S is 30pin. formula: Misharin [20] also studied the friction coefficient and derived the where (7.14) V, = sliding velocity (m/sec) Vr = sum rolling velocity (m/sec) uo = kinematic viscosity (cSt) Rolling/ Sliding Contacts 259 The limiting values are: R: nonsignificant deviation from 1.8 cm slide/roll ratior 0.4-1.3 contact stress 2 2500 kg/cm2 0.08 sf 2 0.02 The accuracy of this empirical formula is reported to be within 15%. Ku et al. [21] conducted sliding-rolling disk scuffing tests over a wide range of sliding and sum velocities, using a straight mineral oil and three aviation gas turbine synthetic oils in combination with two carburized steels and a nitrided steel. It is shown that the disk friction coefficient is dependent not only on the oil-metal combination, but also on the disk surface treat- ment and topography as well as the operating conditions. The quasisteady disk surface temperature and the mean conjection-inlet oil temperature are shown to be strongly influenced by the friction power loss at the contact, but not by the specific make-up of the frictional power loss. They are also influenced by the heat transfer from the disk, mainly by convection to the oil and conduction through the shafts, which are dependent on system design and oil flow rate. For AISI 93 10 steel: I3O + 0.0009 + 0.0003S 0.0666 f=- c5 + W V:.6 + 1965 For AMS 6475 steel: - 0.0041 + 0.0003S 0.0666 130 + c.5 i- W c6 + 1965 where (7.15) (7.16) V, = sum rolling velocity (m/sec) V, = sliding velocity (m/sec) W = load (kN) S = surface roughness (pm CLA) 260 Chapter 7 7.5 DOMAINS OF FRICTION IN EHD ROLLING/SLIDING CONTACTS The coefficient of friction for different slide-to-roll ratio z has three regions of interest as interpreted by Dyson [15]. As illustrated in Fig. 7.2, the first region is the isothermal region in which the shear rate is small and the amount of heat generated is so small as to be negligible. In this region, the lubricant behavior is similar to a Newtonian fluid. The second region is called the nonlinear region where the lubricant is subjected to larger strain rates. The coefficient of friction curve starts to deviate significantly from the Newtonian curve and a maximum coefficient of friction is obtained, after which the coefficient of friction decreases with sliding speed. Thermal effects do not provide an adequate explanation in this region because the observed frictional traction may be several orders of magnitude lower than the cal- culated values even when temperature effects are considered. The third region is the thermal region. The coefficient of friction decreases with increasing sliding speed and significant increase occurs in the temperature of the lubricant and the surfaces at the exit of the contact. Almost all the empirical formulas discussed in the previous section are for the thermal regime. Each formula shows good correlation with the test data from which it was derived, as illustrated in Fig. 7.3, but generally none of these formulas correlates well with the others, as shown in Fig. 7.4. This suggests that these formulas are limited in their range of application and that a unified empirical formula remains to be developed. Slide / Roll Ratio - 2 Figure 7.2 Friction in rolling/sliding contacts. RoNinglSliding Contacts 26 I 0.08 I 0.05 0. M * 0.03 0.02 0.01 nnn /I t /I 0.00 0.01 002 003 004 005 006 0.07 0.06 - 0.08 0.01 - 008 - 005 - 004 - 003 - .c 000 001 002 003 004 005 006 007 C 18 Figure 7.3 (a) Comparison of Drozdov's formula with Drozdov's experiments. (b) Comparison of Cameron's formula with Cameron's experiments. (c) Comparison of Kelley's formula with Kelley's experi- ments. (d) Comparison of Misharin's formula with Misharin's experiments. 262 Chapter 7 00 n n= I oo I 00 I 0.04 1 ’c / 0.00 0.01 0.02 0.03 0.04 0.05 0.08 0.07 0.08 (a) f 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 r t 0: .”/ .OO 0.01 0.02 0.03 0.04 0.05 0.08 0.07 0.08 0.08 0.07 - 0.08 - 0.05 - L 0.03 - 0.00 0.01 0.02 0.03 0.04 0.05 0.08 0.07 0 @) f w) Figure 7.4 (a) Comparison of Drozdov’s formula with Cameron’s experiments. (b) Comparison of Drozdov’s formula with Misharin’s experiments. (c) Comparison of Kelley’s formula with Cameron’s experi- ments. (d) Comparison of Kelley’s formula with Misharin’s experiments. R ollingl Sliding Contacts 263 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 (9) f 0.08 I 0.01 0.08 0.05 0.04 0.03 0.02 0.01 0.00 * / 0 0 1 1 1 1 1 1 .OO 0.01 0.02 0.03 0.04 0.05 0.08 0.07 (h) f Figure 7.4 (Cont ’d.) (e) Comparison of Misharin’s formula with Kelley’s experiments. (f) Comparison of Misharin’s formula with Drozdov‘s experiments. (8) Comparison of Cameron’s formula with Misharin’s experiments. (h) Comparison of Misharin’s formula with Cameron’s experiments. 264 Chapter 7 No formulas are available in the literature for determination of pure rolling friction in the EHD regime. 7.6 EXPERIMENTAL EVALUATION OF THE FRICTIONAL COEFFICIENT An experimental study was undertaken by Li [22] to simulate typical engi- neering conditions, and explore and evaluate the effects of different para- meters such as loads, speeds, slide/roll ratios, materials, oil viscosi ties, and machining processes on the coefficient of friction. The results were then used to derive general empirical formulas for the coefficient of friction, which cover the different lubrication regimes. These formulas will also be com- pared with other published experimental data to further evaluate their gen- eral applicability. The formulas developed by Rashid and Seireg [23] are used to calculate the temperature rise in the film. The experimental setup used in this study is schematically shown in Fig. 7.5. It is a modified version of that used by Hsue [24]. The shaft remained unchanged during the tests, whereas the disks were changed to provide different coated surfaces. The shaft was ground 4350 steel, diameter 61 mm, and the disks were ground 1020 steel, diameter 203.2mm. The coat- ing materials used for the disks were tin, chromium, and copper. Uncoated steel disks were also used. The coating was accomplished by electroplating with a layer of approximately 0.0127 mm for all the three coated disks, and the contact width was 3.175mm for all the disks. The disk coated with tin and the one coated with chromium were machined before plating. The measured surface roughness is shown in Table 7.1 and the material proper- ties are shown in Table 7.2. A total of 240 series of tests were run. The disk assembly was mounted on two 1 in. ground steel shafts which could easily slide in four linear ball bearing pillow blocks. The load was applied to the disk assembly by an air bag. This limited the fluctuation of load caused by the vibration which may result from any unbalance in the disk. The frictional signal obtained from the torquemeter was relatively constant in the performed tests. A variable speed transmission was used to adjust the rolling speed to any desired value. A toothed belt system guaranteed the accuracy of sliding- rolling ratios. This was particularly important for the rolling friction tests. The lubricant used was 10W30 engine oil with a dynamic viscosity of 0.09Pa-s at 26°C; the loads were 94,703, 189,406,284,109, and 378,8 12 N/m; the slide/roll ratios were 0,0.08,0.154,0.222,0.345; the rolling speeds varied from 0.3 to 2.76 m/s, and the sliding speeds were in the range 0 to 0.95m/s. [...]... (7 .20 ) Sec is calculated according to Eq (7 .25 ), and p = 0.865 (which is an approximate value for most lubricating oils used in test conditions) All the other variables are defined in the following notation: U = rolling speed = UI + U2 L U , , U2 = rolling speeds of rollers 1, 2 RI R2 R = effective radius = RI +R2 R I ,R2 = radii of rollers 1, 2 E’ = effective modulus of elasticity = 1 1 1 -~ - U : 2. .. Tin 10W30 0. 42 0.38 0.17 0 .20 Material Properties 43 40 1 94 67 0.145 7800 8930 7135 728 0 888 473 386 450 22 2 1880 20 3.4 103 25 0 46 - Chapter 7 26 6 The experimental results cover rolling friction, the isothermal regime, the nonlinear regime, and the thermal regime The variables in the tests include load, speed, slide/roll ratio, surface roughness, and the properties of the coated layer The following... higher coefficient friction as in the isothermal and nonlinear regimes However, the RolIinglSliding Contacts 0.05 c 26 7 8 0.04 c 0.03 0.5 oq0.03 1.0 1.5 2. 0 2. 5 3.0 3.5 ~ LL+ +- 0.00 O.O1 0.5 40 w (WmXlO', (8) 1 1.0 2. 0 1.5 2. 5 3.0 3.5 4 3 w (Nlm x1oq @) 0.03 - -W 0 cW +W - 0 0. 02 = 94703 Nlm = 198408 Nln = 28 4109 Nln I W = 3768 12 Nlm c 0.01 0.00 ' 0.0 (c) 1 I I 1 I 0.5 1.0 1.5 2. 0 2. 5 U (mlr) 1 3.0... 7.6.1 Friction Regimes Although many investigators have conducted experimental investigations on the coefficient of friction, no experimental results have been reported in the literature for the rolling friction with EHD lubrication This is probably due to the difficulties of measuring the very small rolling friction force to be expected in pure rolling It is found in the performed tests that rolling friction. .. coefficient of frictionf, can be calculated from: (7 .22 ) where a = 0.0191 - 1.15 x 10-4Jij - B = 0 .26 5 + 6.573 x 10-3 T,I 27 2 Chapter 7 and its location z* is calculated from: where a' = 0 .21 9(1 - e - 1 f i 6 3 6 8 ) 4 + 0.0 122 In the thermal regime, where slide/roll > 0 .27 : s=jb - [ a ( ~ eh)] - (7 .24 ) where j b = coefficient of friction at ho = 0, from Fig 7.11 U = 0.0864 - 1.3 72 x 103(%) 6 ' h0 - (7 .25 ) b=-... = EN 32 steel cast hardened to 750 VPN to a depth of 0. 025 in ~ 7 1 SAE 8 622 carburized and hardened to Rockwell hardness 60 [18] Steel 38XMI-OA [19] 27 5 RollinglSliding Contacts (El.otroplat8d Machining) 0.1 - (Soperfinldng) 1 0.1 I I: 20 0.1 (b) 1 10 mm i 100 10 1000x106 Figure 7.1 3 (a) Proposed effective surface roughness for various manufacturing processes (b) Effective (Sec/R)eagainst nominal... temperature rise in the contact zone is calculated by the empirical formulas developed by Rashid and Seireg [23 ]: where 9, = I& - U2IWf' where all the variables are defined in the notation except the film thickness h: h = Eh0 E is a factor proposed by Wilson and Sheu [25 ]: & = 1 I + 0 .24 1[(1 + 14.8 z0.83)80.64] (7 .28 ) RollinglSliding Contacts 27 7 where z = sliding/rolling ratio 6 = - rloY u2 K qo = lubricant... manufacturing process The test data used in developing the proposed formula cover the following range: contact surfaces: steel-steel effective radius R = 0.0109 - 0. 027 4m lubricant viscosity q = 2. 65 - 20 00cP surface processing operation = grinding (0.1 - 1.6 pm AA) film/surface roughness h = 0 .21 - 14.31 slide/roll ratio z = 0 .26 8 - 0.455 sliding speed V , = 1.35 - 5 m/sec rolling speed U, = 3 .2 - 15m/sec... of friction The surface roughness also increases friction The coefficient of friction is also found to increase with load and decrease with rolling speed Figures 7.7-7.10 show the variation of coefficient of friction with slide/roll ratio It can be seen from these figures that the coefficient of friction for steel and copper coating reaches its maximum in the nonlinear regime For chromium and tin coatings,... of the rolling coefficient of friction The second point isf,, which gives the coefficient of friction in the nonlinear region, and z*, its location This point is assumed to approximately define the end of the isothermal region or the maximum value in the nonlinear regime The third one is the thermal coefficient of friction, f,, and the corresponding slidelroll ratio location is chosen as 0 .27 , after . 1 .22 , k = 0. 026 . Drozdov and Gavrikov [ 171 investigated friction and scoring under conditions of simultaneous rolling and sliding with a roller test machine. The formula for determination. Benedict and Kelley [ 191 conducted experiments to investigate the friction in rolling/sliding contacts. The coefficient of friction has been found to increase with increasing load and to. 94,703, 189,406 ,28 4,109, and 378,8 12 N/m; the slide/roll ratios were 0,0.08,0.154,0 .22 2,0.345; the rolling speeds varied from 0.3 to 2. 76 m/s, and the sliding speeds were in the range 0