Friction and Lubrication in Mechanical Design Episode 2 Part 3 docx

280 Chapter 7.7.4 Effective Viscosity Using the notation: + Th = absolute bulk disk temperature (e.g., Tb = 273.16 "C) A Ts = temperature rise for steel-steel contact ATc = temperature rise from Eq (7.27) using the material properties of the contacting surfaces for steel-coating contact A T = A Tc - A Ts = temperature rise difference between the steel-coating contact and the steel-steel contact AT, = effective temperature rise difference between the steel-coating contact and the steel-steel contact Then: AT'>= A T P (7.31) where B is the coating thickness factor from the previous section Then T', = Tb AT, is used to calculate the viscosity for that coating conditions, and the viscosity is then substituted into Eq (7.24) to calculate the corresponding coefficient of friction The viscosity of 10W30 oil is calculated by the ASTM equation [27]: + lOg(cS + 0.6) = a - b log T , (7.32a) therefore (7.32b) (7.32~) where T, is the absolute temperature ( K or R), cS is the kinematic viscosity (centistokes) a = 7.827 b = 3.045 for 10W30 oil For some commonly used oil, a and values are given in Table 7.3 7.7.5 Coating Thickness Effects on Modulus of Elasticity For the reasons mentioned before, the effective modulus of elasticity, Et,, for coated surface is desirable Using the well-known Hertz equation, one calculates the Hertz contact width for two cylinder contact as [27]: (7.33) 281 RollinglSliding Contacts Table 7.3 Values of a and b for Some Commonly Used Lubricant Oils Oil SAE 10 SAE 20 SAE 30 SAE 40 SAE 50 SAE 60 SAE 70 b a ~~ ~ 11.768 11.583 11.355 I 1.398 10.431 10.303 10.293 4.6418 4.5495 4.4367 4.4385 4.0319 3.9705 3.9567 E' and U are the modulus of elasticity and Poisson's ratio Coating material properties are used for E2 and u2 because coating thickness is an order greater than the deformation depth (this can be seen later) Therefore, the deformation depth is calculated by (Fig 7.18): hd = RsinOtanO is very small, therefore: (7.34) The variation of the deformation depth with load is shown in Fig 7.19 Then the effective modulus of elasticity of the coated surface is proposed as: (7.35a) where Eh = modulus of elasticity of base material E, = modulus of elasticity of coating material E , = modulus of elasticity of coated surface h, = coating film thickness hd = elastic deformation depth r = constant (it is found that r = 13 best fits the test data) 282 Chapter The contact of the shaft and the coated disk Figure 7.18 0.006 , 0.005 ' +Steel-Tin - Steel-Chromium - A Steel-Copper -Steel-steel 0.004 E 0.003 A Y U r 0.002 0.001 a I 1.o * if 0.000 a I 1.5 a ' a ' 2.0 ' a ' ' ' 2.5 ' ' ' 3.0 I ' a ' * I 3.5 a ' a I 4.0 W (Nlm ~10') Figure 7.19 Variation of deformation depth on coated surface with load 283 RollinglSliding Contacts E C # * _ - _ - *-e0- I Eb I I I I I I I I I I I I I I f I I # # f # I 0 II I 10 I 20 I 30 I 40 I 50 I 60 Figure 7.20 shows the variation of effective modulus of elasticity of coated surface for different values of h,/hd The effective combined modulus of elasticity is therefore calculated by (7.35 b) For most metals used in engineering, the variation of U is small, and consequently, the variation in - v2 is smaller Therefore, no significant error is expected from using the Poisson’s ratio of the base material or the coating material Figures 7.21 and 7.22 show the comparison of the calculated coefficient of friction in pure rolling conditions with the test results for chromium and steel Figure 7.23 shows the calculated coefficient of friction in the thermal regime compared with test results for tin, steel, chromium, and copper Figure 7.24 shows the calculated coefficient of friction in the thermal regime compared with the results from Drozdov’s [ 171, Cameron’s [ 181, Kelley’s [ 191, and Misharin’s [20] experiments Figures 7.25-7.28 show sample comparisons of the experimental results with the curves, which are constructed by using the calculated f,,f,,andf,, and appropriate curves against slide/roll ratios Figure 7.29 shows the comparison of Plint’s test data with prediction It can be seen that the correlations are excellent 284 Chapter 0.08 A U=0.303ml8 u=1.44mls U=2.76m/s 0.06 y 0.04 0.02 n 0 U 0.00 0.5 I A A 1.o I I I 1.5 2.0 2.5 A I I 3.0 3.5 4.0 W (N/mx106) 0.06 cc - A U=0.303mlr U=l.44m/s U=2.76mls 0.04 0.02 0.00 0.5 (b) 1.0 1.5 2.0 2.5 3.0 3.5 4.0 W (Nlm x l 06) Figure 7.21 Comparison of experimental coefficient of rolling friction vs load with prediction for (a) tin; (b) chromium; (c) copper; (d) steel Rolling}Sliding Contacts 285 0.08 0.06 - A O * O [ 0.00 0.5 (c) 1.o , 1.5 - ,; 2.0 I 2.5 ,; 3.0 U=0.303m/r U=1.44mlfs U=2.76mls I 3.5 W (NImxl0') 0.08 0.06 cc - 0.04 - U=0.303m/s U=l.44m/s U = 2.76 m/s - 0.02 rn A A 4.0 Chapter 286 A v 0.00 I 0.0 I 0.5 I 1.o (a) 0.08 0.06 I c 0.04 W-94703Wm W=l69408)(lm W=284109N/m W = 378812 N/m I I I 1.5 2.0 2.5 3.0 W (Nlmxl0') i m A v W = 94703 Nlm W=189406Nlm W=284109Wm W = 310092 N/m - O*02! 0.0 (b) 0.5 1.o 1.5 2.0 2.5 3.0 w (N/m J C 04 ~ Figure 7.22 Comparison of experimental coefficient of rolling friction vs rolling speed with prediction for (a) tin; (b) chromium; (c) copper; (d) steel 287 RollinglSliding Contacts 0.08 I ' - m W=94703N/m W=l89406Nlm W=284409Nlm v W = 376812 Wm A 0.06 0.04 - 0.02 cc - 0.00 0.0 (c) 0.08 I 0.5 1.o I I I 1.5 2.0 2.5 U (mw 0.06 - cc 0.04 - 0.00 - A v W=94703N/m W = 189406N/m W = 284109 N/m W = 378812 N/m 288 Chapter 0.08 0.06 - I I r - A 0.04 0.02 ' 0.00 I 0.0 I I I I I I I 0.5 1.0 1.5 2.0 2.5 3.0 3.5 (a) 0.06 4.0 W (N/mx106) I - 0.04 I I - - A I A A - 0.02 cc 0 U=0.303mlr U3:1.44ml8 A 0.00 I 0.0 (b) U+2.76ds I I I I I I I 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 W (Nlm XI 07 Comparison of experimental coefficient of thermal friction vs load with prediction for (a) tin; (b) chromium; (c) copper; (d) steel Figure 289 RollinglSliding Contacts 0.08 L 0.06 - A 0.04 - 0.02 cc - U m U=O.303mls U=1.44mls U=2.?6mls I A A 0.00 I I I I I 1.5 2.0 2.5 3.0 3.5 I 1.o 0.5 A (c) 4.0 W (Nlmxl06) U = 1.44 m/s 0.06 Ic - 0.04 r 0.001 0.5 (d) I I I I I I 1.0 1.5 2.0 2.5 3.0 3.5 W (Nlmxl OS) 4.0 290 Chapter (a) f (predicted) 0.08 0.07 0.06 A c 0.05 ‘t 0.04 am - E 0.03 I 0.00 (b) 0.01 0.02 I I I I 0.03 0.04 0.05 0.06 0.07 0.08 f (predicted) Figure 7.24 Comparison of test data with prediction: (a) Cameron; (b) Misharin; (c) Kelley; (d) Drozdov 29 I Rolling/ Sliding Contact s 0.08 0.07 0.06 0.05 n Q) 0.04 Y 0.03 0.02 0.01 0.00 0.00 (a (d) O.O? 0.02 0.03 0.04 0.05 f (predicted) f (predicted) 0.06 0.07 0.08 292 Chapter 0.08 W = 94703Nhn A A 0.04 0.03 0.02 A V 0.0 0.08 0.07 Tin Chnnnium 0.1 0.2 COpQer Steel 0.4 0.3 - U == W 94703N h 0.303 m h Y 0.06 0.05 rc 0.04 0.03 o 0.01 - 0.00- 0.0 (b) Tin Chromium 0.02 A A Copper copper Steel I ' v ' a ' ' I 0.1 ' ' * ' I 0.2 ' ' ' ' 0.3 ' 0.4 z Figure 7.25 Coefficient of friction vs slide/roll ratio ( W = 94,703 N/m, U = 0.303 mjsec): (a) experimental; (b) calculated 293 RollinglSliding Confacts 0.0 " " " " l " ' " " " 0.1 (a) 0.2 Steel v 0.4 0.3 Z 0.06 0.05 o Tin Chromium A A 0.0 (b) 0.1 0.2 0.3 0.4 z Figure 7.26 Coefficient of friction vs slide/roll ratio ( W = 94,703 N/m, U = 0.303 m/sec): (a) experimental; (b) calculated Chapter 294 c 0.08 I 0.07 W=87M12Nhn U=2.7Omh o f l n = chromium Ir coPP@r steel 0.0 0.2 0.1 (a) 0.06 - 0.4 0.3 Z - U = 2.76 mh Ur2.7Omh I 0.05 0.04 - c 0.03 0.02 Tin Chromium COPPM 0.01 A steel 0.00 0.0 (b) l 0.1 i l 0.2 l l 0.3 0.4 Z Coefficient of friction vs slide/roll ratio ( W = 378,8 12 N/m, U = 2.76 mlsec): (a) experimental; (b) calculated Figure 7.27 * " t " 0.0 " t 0.1 * " " " 0.2 (a) 0.08 0.07 Steel 0.4 0.3 Z W =378812 Nhn U=l.Umh 0.06 0.05 cc 0.04 0.03 0.02 rn 0.01 0.00 0.0 (b) l 0.1 , 0.2 Tin Chromium * Copper v Steel , 0.3 0.4 Z Coefficient of friction vs slidelroll ratio ( W = 378,812 N/m, U = 1.44m/sec): (a) experimental; (b) calculated Figure 7.28 296 Chapter 0.10 I 0.09 0.08 0.07 0.06 rc 0.05 0.04 0.03 0.02 0.01 0.00 0.00 0.05 0.10 0.15 0.20 0.25 z Figure 7.29 7.7.6 Comparison of Plint's test data with prediction Surface Chemical layer Effects All the test data used so far are obtained from ground or rougher surface contacts The chemical layer on the contact surfaces resulting from the manufacturing process and during operation is ignored because it wears off relatively quickly on a rough surface, especially at the real areas of contact where high shear stress is expected On the other hand, the surface chemical layer can be expected to play an important role in smooth surface contacts The properties of the contact surfaces are affected by this chemical layer because it can remain on the surfaces more easily than on a rough surface In this case, Eq (7.27) can be used to account for this effect Cheng [7], Hirst [8], and Johnson [9] conducted experimental investigations on very smooth surface contacts All the contact surfaces were superfinished The A values are roughly between 80 and 100 for Cheng's test, 50 and 60 for Hirst's test, and 40 and 60 for Johnson's test (A = ho/Sc, and Sc = where S1 and S2 are surface roughness of contact surfaces, CLA.) Because the surface chemical layers usually are very thin, they are assumed to have little effect on the elastic properties of the surfaces ,/:fs:, Rolling/ Sliding Contacts 297 However, they affect the temperature rise in the contact zone significantly In order to use Eq (7.27) to account for this effect, the thermal-physical properties and thickness of the chemical layer are needed Because these values are not known, the following thermal-physical properties are used as an approximation: p = 3792 kg/m3 c = 840 J/(kg - "C) E = 3.45 x 10" Pa K = 0.15W/(m - "C) TheS, values from the above tests are used to find the thickness of the corresponding chemical layers inversely The result is shown in Fig 7.30 It is found that the thickness decreases as the load increases as expected because the higher the load, the higher the shear stress in the lubricant, which results in a thinner chemical layer In Figs 7.31-7.33, the test data are compared with prediction It can be seen that the chemical layer makes a -6-5 - A t l Johnron*rTest Cheng'r Test Hirrt'rTert 4- -3 - -2 - -1 - -0 I 0.o I 2.0~10~ I I I I 4.Ox1O6 6.0~1 OS I I 8.0x105 I 1.ox1o6 W (N/m) Figure 7.30 Inversely calculated surface chemical layer thickness vs load for superfinished surface contacts (roughness = pin CLA) Chapter 298 0.05 - 0.04 - 0.03 - 0.02 - 0.01 Figure 7.91 2SOp.l n E % E I= i i i ll5pd -H h r s Data A C.ku).t.df,withoutm#.r.thof.mlyrr - 12 1Mp.l A Comparison of Cheng's experimental data with prediction 16 14 md p - e CJoul.tedf,wittlGono#.rclbknof.~by.r C.lo~t.df".ndi c.kul.t.df, x 10d 8- 6% GPa 8 0.0 0.5 A A I 1.o I 1.5 Sliding Speed (rnrs) Figure 7.92 Comparison of Hirst's test data with prediction I 2.0 A Rolling} Sliding Contacts 299 0.08 I I 0.07 0.06 - ‘6 0.05 - 0.04 e t Q) # C B, 0*03 ; Q B, : B, 0.01 0.02 B, - 0 I 0.00 I 10 Figure 3 A 1.01OP 0.76OPa 1.38GP8 0.60 V 0.43GP I I I I I 20 30 40 50 60 Sliding Speed, (U,-U,) (ink) iIi IK 70 Comparison of Johnson’s experimental data with prediction great difference in the coefficient of friction for conditions with large A ratios (>40) in the thermal regime Load can have a significant effect on the chemical layer thickness and Fig 7.30 can be used for evaluating the thickness as a function of normal load 7.7.7 General Observations on the Results The empirical formulas were checked for different regimes of lubrication, surface roughness, load, speed, and surface coating The formulas were used for evaluating rolling friction, and traction forces in the isothermal, nonlinear, and thermal regimes of elastohydrodynamic lubrication Because of the current interest in surface coating, the formulas were also applied for determining the coefficient of friction for cylinders with surface layers of any arbitrary thickness and physical and thermal properties It can be seen from the empirical formulas that: It appears that in general, the slide/roll ratio has little direct effect on the coefficient of friction in the thermal region (slidelroll > 0.27) Chapter 300 The surface roughness effect is treated in this study as a function of the surface generating process rather than the traditional surface roughness measurements The oil film thickness is found to be better represented for friction calculation in a nondimensional form by normalizing it to the effective radius rather than the commonly used film thickness to roughness ratio A Coating has a significant effect on the temperature rise in the contact zone This is represented by a factor B, as shown in Eq (7.30) Coating has an effect on the modulus of elasticity as shown in Eq (7.35) This is represented by using an effective modulus of elasticity for the coated surface For tin (whose modulus of elasticity differs from that of the base material most significantly among the three coating materials used), this correction produces a 50% increase in the effective modulus of elasticity PROCEDURES FOR CALCULATION OF THE COEFFICIENT OF FRICTION 7.8.1 Unlayered Steel-Steel Contact Surfaces Given: contact surface radii r l , 1-2 (m, in.) Surface velocities, U1, U , (mlsec, in./sec) Dynamic viscosity of lubricant oil at entry condition qo (Pa-sec, reyn) Load F (n, lbf) Surface roughness S1, S2 (m CLA, in CLA) or manufacturing processes Density of lubricant p (kg/m3, (lb/in.3) x 0.0026) Modulus of steel E (Pa, psi) Contact width of surfaces y (m, in.) Poisson’s ratio for steel v (dimensionless) Pressure-viscosity coefficient of lubricant a! ( 1/Pa, 1/psi) Calculate: E Effective modulus of elasticity E’ = 1-3 Mean rolling velocity U = U ~ + U2 30 I Rolling/Sliding Conlac ts rl r2 Effective radius R = YI +r2 F Load per unit width W = Y 7.8.2 Find Sr.2,from Fig 7.13a by using S1 and S2 (or by manufacturing processes) For S < 0.05 pm, take S,, = 0.05 pm Then s,, =4 Calculate dimensionless U , q, W , S from Eqs (7.17bt7.20) Calculate coefficient of rolling friction fi from Eq (7.21) Calculate coefficient of friction in the nonlinear region and its location.f, is calculated from Eq (7.22), z* is calculated from Eq (7.23) Calculate minimum oil film thickness ho from Eq (7.26), where G = @E' Find.fo from Fig 7.12 Find (S,,(./R)(, from Fig 7.13b Calculate coefficient of thermal friction fr from Eq (7.24) Use.fr,.fn,z*, and fi to construct the coefficient of friction curve versus sliding/rolling ratio, as in Fig 7.1 1, where sliding speed = I U ] - U , I, and rolling speed = U Layered Surfaces Given: contact surface radii r l , rl (m, in.) surface velocities U U2 (m/sec, in./sec) , load F (N, lbf) surface roughness S , , S2 (m CLA, in CLA) or manufacturing processes density of lubricant p (kg/m3, (lb/in.3) x 0.0026) modulus of steel E (Pa, psi) contact width of surface y (m, in.) Poisson's ratio for steel U (dimensionless) Lubricant oil properties: Thermal conductivity KO (W/m-'C), (BTU/(sec-in.-OF)) x 9338) Specific heat CO (J/(kg-"C), (BTU/lb-OF) x 3,604,437) Pressure-viscosity coefficient a (I/Pa, l/psi) Temperature-viscosity coefficient /3 (1 /"C, Dynamic viscosity at entry condition qo (Pa-sec, reyn) / O F ) Disk base material properties: Modulus of elasticity Ehl(Pa, psi) Poisson's ratio uhl (dimensionless) 302 Chapter Thermal conductivity Kbl (W/(m-OC), (BTU/(sec-in-OF) x 9338) Specific heat cbl (J/(kg-"C), (BTU/(lb-OF) x 3,604,437) Density pbl (kg/m3, (lb/in.3) x 0.0026) Disk surface material properties: Modulus of elasticity ECl (Pa, psi) Poisson's ratio ucl (dimensionless) Thermal conductivity Kcl (W/(m-"C), (BTU/(sec-in.-OF) x 9338) Specific heat ccl (J/(kg-"C), (BTU/(lb-OF) x 3,604,437) Density pcl (kg/m3, (lb/in.3) x 0.0026) Thickness hCl (m, in.) Disk base material properties: Modulus of elasticity Eb2 (Pa, psi) Poisson's ratio ub2 (dimensionless) Thermal conductivity Kb2 (W/(m-OC), (BTU/-sec-in.-"F) x 9338) Specific heat cb2 (J/kg-"C), (BTU/(lb-OF) x 3,604,437) Density pb2 (kg/m3, (Ib/in.3) x 0.0026) Disk surface material properties: Modulus of elasticity Ec2 (Pa, psi) Poisson's ratio uc (dimensionless) Thermal conductivity Kc2 (W/(m-OC), (BTU/(sec-in,-"F) x 9338) Specific heat cc2 (J/(kg-"C), (BTU/(lb-OF) x 3,604,437) Density pc2 (kg/m3, (lb/in.3) x 0.0026) Thickness hc2 (m, in.) Steel properties: Modulus of elasticity Es (Pa, psi) Poisson's ratio vs (dimensionless) Thermal conductivity Ks (W/(m-OC), (BTU/(sec-in.-OF) x 9338) Specific heat cs (J/(kg-"C), (BTU/(lb-OF) x 3,604,437) Density ps (kg/m3, (lb/in.3) x 0.0026) Bulk temperature Tb (K, R) (K = "C 273.16) Use previous section to calculate fr for steel-steel contact surfaces Substitute ATs, K s , cs, ps, and Es for A T , K1, c1, p1, El and K2, p2, E2 in Eqs (7.27) and (7.28), where f is replaced c2, byf,, L is replaced by L = J ' m + U + U2 Mean rolling velocity U = rl r2 Effective radius R = rl + r2 Rolling/Sliding Contacts 303 F Load per unit width W =X Effective modulus of elasticity - - WR E' Half contact width = 1.6,/- Calculate hd by using Eq (7.34) Substitute Ebl , Ecl, hcl for Eb, E,, h, in Eq (7.35) to calculate E e l Substitute Eb2, Ec2, hc2 for E b , Ec, h, in Eq (7.35) to calculate Ee2 Calculate the effective modulus of elasticity of the layered surfaces by: Find Sel, from Fig 7.13a by using Sl and S2 (or by manuSe2 facturing processes) For S < 0.05 pm take Se = 0.05 pm.Then S - ec = , / S x Calculate dimensionless U, q, W , S from Eqs (7.17)-(7.20) except that E' is replaced by E,& 10 Calculate coefficient of rolling friction fr from Eq (7.2 1) 11 Calculate coefficient of friction in the nonlinear region and its location fn is calculated from Eq (7.22), z* is calculated from Eq (7.23) 12 Calculate minimum oil film thickness ho from Eq (7.26), where G = aE,I, 13 Calculate E by using Eq (7.28), where z = 0.27 14 Calculate A T, by Eq (7.27), where K l , c1, p1, El are replaced by & I * cc19 Pcl9 & I * K2 c2 * P2 * E2 are replaced by Kc2 * cc2 * Pc2 , Ec2 * L = 26,f =f!from step for steel-steel contact surfaces 15 Calculate D 1by using Eq (7.29) where K, c, p, U are substituted c1 by K1, , p l , El Use Eq (7.30) to calculate p1 where h, = hCl, D = 01 16 Calculate D2 by using Eq (7.29) where K, c, p, U are substituted by K , c2,p2, E2 Use Eq (7.30) to calculate p2 where h, = hr2, D = 02 17 Calculate AT, by using Eq (7.31) where /? = (PI P2)/2 18 Use Eq (7.32) to calculate a and for theparticular lubricant as follows Suppose that the viscosity is rnl at T1 and m2 at T2 (where T1 and T2 are absolute temperature, say, K = 273.16 "C, ml and m2 are kinematic viscosity in centi9 + + 304 Chapter 19 20 21 22 23 7.9 stokes), substitute ml, T l and m2, T2 into Eq (7.32), respectively, and solve these two linear equations simultaneously to get a and (If lubricant is SAE 10, SAE 20, SAE 30, SAE 40, SAE 50, SAE 60, or SAE 70, use Table 7.3.) Use Eq (7.32) to calculate the viscosity q at temperatufe T,(q = pou where po is the lubricant density, U is the kinematic viscosity) Use Eq (7.26) to find ho where qo = q, G = al& Findfo from Fig 7.12 Find (Sc,r/R)', from Fig 7.13b Calculate coefficient of thermal frictionf, from Eq (7.24) If the difference betweenf, value in step 21 andf, value in step 14 does not satisfy your accuracy requirement, go back to step 14, replace ft by fr value in step 21 and iterate until the accuracy requirement is satisfied Use f;.fn, z*, and J to construct the coefficient of friction curve versus sliding/rolling ratio as in Fig 7.1 1, where sliding speed = lU, - U21, and rolling speed = U SOME NUMERICAL RESULTS The following are some illustrative examples for the application of the developed empirical formulas in sample cases Figure 7.34 shows calculated coefficient of friction versus sliding/rolling ratio for different rolling speeds, T = 26°C (78.8"F), steel-steel contact, ground surfaces, S = 0.03 pm (12 pin.), W = 378,8 12 N/m (2 160 lbf/in.), 10W30 oil, R = 0.0234m (0.92 in.) Figure 7.35 shows calculated coefficient of friction versus sliding/rolling ratio for different normal loads, T = 26°C (78.8"F), steel-steel contact, ground surfaces, S = 0.03 pm (12 pin.), U1 = 3.2 m/sec (126 in./sec), 10W30 oil, R = 0.0234 m (0.92 in.) Figure 7.36 shows calculated coefficient of friction versus slidinglrolling ratio for different effective radii, T = 26°C (78.8"F), steel-steel contact ground surfaces, S = 0.03 pm (12 pin.), W = 378,8 12 N/m (2 160 lbf/in.), 10W30 oil, U1 = 3.2m/sec (126in./sec) Figure 7.37 shows calculated coefficient of friction versus sliding/ rolling ratio for different viscosity, steel-steel contact, ground surfaces, S = 0.03 pm (12pin.), W = 378,812N/m (21601bf/in.), 10W30 oil, U1 = 3.2 m/sec (126 in./sec), R = 0.0234 m (0.92 in.) Figure 7.38 shows calculated coefficient of friction versus sliding/ rolling ratio for different materials, T = 26°C (78.8"F), ground surfaces, ... "C, ml and m2 are kinematic viscosity in centi9 + + 30 4 Chapter 19 20 21 22 23 7.9 stokes), substitute ml, T l and m2, T2 into Eq (7. 32 ) , respectively, and solve these two linear equations simultaneously... 10. 431 10 .30 3 10 .29 3 4.6418 4.5495 4. 436 7 4. 438 5 4. 031 9 3. 9705 3. 9567 E'' and U are the modulus of elasticity and Poisson''s ratio Coating material properties are used for E2 and u2 because coating thickness... surfaces, S = 0. 03 pm ( 12 pin.), W = 37 8,8 12 N/m (2 160 lbf /in. ), 10W30 oil, R = 0. 0 23 4m (0. 92 in. ) Figure 7 .35 shows calculated coefficient of friction versus sliding/rolling ratio for different