Engineering Tribology Episode 2 Part 4 docx

300 ENGINEERING TRIBOLOGY derive simplified expressions for the elliptic integrals required for the stress and deflection calculations in Hertzian contacts. The derived formulae apply to any contact and eliminate the need to use numerical methods or charts such as those shown in Figures 7.12 and 7.13. The formulae are summarized in Table 7.4. Although they are only approximations, the differences between the calculated values and the exact predictions from the Hertzian analysis are very small. This can easily be demonstrated by applying these formulae to the previously considered examples, with the exception of the two parallel cylinders. In this case the contact is described by an elongated rectangle and these formulae cannot be used. In general, these equations can be used in most of the practical engineering applications. 1.5 2.0 0.5 1.0 1 2 5 10 0 0 0.2 0.4 0.6 0.8 1.00.1 0.3 0.5 0.7 0.9 k 0 k 3 k 2 k 1 FIGURE 7.12 Chart for the determination of the contact coefficients ‘k 1 ’, ‘k 2 ’ and ‘k 3 ’ [13]. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1.00.1 0.3 0.5 0.7 0.9 k 2 /k 1 k 4 & k 5 k 4 k 5 (line contact) (point contact) FIGURE 7.13 Chart for the determination of contact coefficients ‘k 4 ’ and ‘k 5 ’ [13]. TEAM LRN ELASTOHYDRODYNAMIC LUBRICATION 301 T ABLE 7.4 Approximate formulae for contact parameters between two elastic bodies [7]. a = 6k 2 εWR' πE' () 1/3 ellipse p max = 3W 2πab p average = W πab δ = ξ Contact area dimensions Average contact pressure Simplified elliptical integrals Maximum contact pressure Maximum deflection a b 4.5 εR' [( )( 1/3 b = 6εWR' πkE' () 1/3 W πkE' ) 2 ] Ellipticity parameter ε = 1.0003 + 0.5968R x R y ξ = 1.5277 + 0.6023ln R y R x () k = 1.0339 R y R x () 0.636 where: ε and ξ are the simplified elliptic integrals; k is the simplified ellipticity parameter. The exact value of the ellipticity parameter is defined as the ratio of the semiaxis of the contact ellipse in the transverse direction to the semiaxis in the direction of motion, i.e. k = a/b. The differences between the ellipticity parameter ‘ k ’ calculated from the approximate formula, Table 7.4, and the ellipticity parameter calculated from the exact formula, k = a/b, are very small [7]. The other parameters are as defined already. EXAMPLE Find the contact parameters for a steel ball in contact with a groove on the inside of a steel ring (as shown in Figure 7.7). The normal force is W = 50 [N], radius of the ball is R ax = R ay = R A = 15 × 10 -3 [m], the radius of the groove is R bx = 30 × 10 -3 [m] and the radius of the ring is R by = 60 × 10 -3 [m]. The Young's modulus for both ball and ring is E = 2.1 × 10 11 [Pa] and the Poisson's ratio is υ = 0.3. · Reduced Radius of Curvature Since the radii of the ball and the grooved ring are R ax = 15 × 10 -3 [m], R ay = 15 × 10 -3 [m] and R bx = -30 × 10 -3 [m] (concave surface), R by = -60 × 10 -3 [m] (concave surface) respectively, the reduced radii of curvature in the ‘x’ and ‘y’ directions are: = 1 R x + 1 R ax 1 R bx =⇒R x = 0.03 [m]+=33.33 1 15 × 10 −3 1 −30 × 10 −3 = 1 R y + 1 R ay 1 R by =+ =50.0 1 15 × 10 −3 1 −60 × 10 −3 ⇒ R y = 0.02 [m] Since 1/R x < 1/R y condition (7.3) is not satisfied. According to the convention it is necessary to transpose the directions of the coordinates, so ‘R x ’ and ‘R y ’ become: R x = 0.02 [m] and R y = 0.03 [m] TEAM LRN 302 ENGINEERING TRIBOLOGY and the reduced radius of curvature is: 1 R' =+ 1 R x 1 R y = 50.0 + 33.33 = 83.33 ⇒ R' = 0.012 [m] · Reduced Young's Modulus E' = 2.308 × 10 11 [Pa] · Contact Coefficients The angle between the plane containing the minimum principal radius of curvature of the ball and the plane containing the minimum principal radius of the ring is: φ = 0° The contact coefficients are: k 0 = − 1 R ax 1 R ay [( ) − 1 R bx 1 R by () + 22 + 2 − 1 R ax 1 R ay () − 1 R bx 1 R by () cos2φ ] 1/2 + 1 R ax ( + 1 R ay + 1 R bx 1 R by ) = − 1 15 × 10 −3 1 15 × 10 −3 [( ) − 1 −60 × 10 −3 1 −30 × 10 −3 () + 22 + 2 − 15 × 10 −3 1 15 × 10 −3 () − 1 −60 × 10 −3 1 −30 × 10 −3 () cos0° ] 1/2 + 1 15 × 10 −3 ( + 1 15 × 10 −3 + 1 −60 × 10 −3 1 −30 × 10 −3 ) 1 = 0.2 = 16.67 83.33 From Figure 7.12, for k 0 = 0.2: k 1 = 1.17, k 2 = 0.88 and k 3 = 1.98 and from Figure 7.13 where k 2 /k 1 = 0.88/1.17 = 0.75, the other constants have the following values: k 4 = 0.33 and k 5 = 0.54 · Contact Area Dimensions a = k 1 3WR' E' () 1/3 = 1.17 3 × 50 × 0.012 2.308 × 10 11 () 1/3 = 2.32 × 10 −4 [m] b = k 2 3WR' E' () 1/3 = 0.88 3 × 50 × 0.012 2.308 × 10 11 () 1/3 = 1.75 × 10 −4 [m] TEAM LRN ELASTOHYDRODYNAMIC LUBRICATION 303 · Maximum and Average Contact Pressures p max = 3W 2πab = 3 × 50 2π(2.32 × 10 −4 ) × (1.75 × 10 −4 ) = 588.0 [MPa] p average = W πab = 50 π(2.32 × 10 −4 ) × (1.75 × 10 −4 ) = 392.0 [MPa] · Maximum Deflection δ= 0.52k 3 W 2 E' 2 R' () 1/3 = 0.52 × 1.98 50 2 (2.308 × 10 11 ) 2 0.012 () 1/3 = 1.6 × 10 −6 [m] · Maximum Shear Stress τ max = k 4 p max = 0.33 × 588.0 = 194.0 [MPa] · Depth at which Maximum Shear Stress Occurs z = k 5 b = 0.54 × (1.75 × 10 −4 ) = 9.5 × 10 −5 [m] It can easily be found that the Hamrock-Dowson approximate formulae (Table 7.4) give very similar results, e.g.: · Ellipticity Parameter = 1.3380k = 1.0339 R y R x () 0.03 0.02 () 0.636 = 1.0339 0.636 · Simplified Elliptical Integrals = 1.3982ε = 1.0003 + 0.5968R x R y = 1.0003 + 0.5968 × 0.02 0.03 = 1.7719ξ = 1.5277 + 0.6023ln R y R x () = 1.5277 + 0.6023ln 0.03 0.02 () · Contact Area Dimensions = 6 × 1.3380 2 × 1.3982 × 50 × 0.012 π(2.308 × 10 11 ) () 1/3 = 2.32 × 10 −4 [m]a = 6k 2 εWR' πE' () 1/3 = 1.73 × 10 −4 [m] b = 6εWR' πkE' () 1/3 = 6 × 1.3982 × 50 × 0.012 π × 1.3380 × (2.308 × 10 11 ) () 1/3 TEAM LRN 304 ENGINEERING TRIBOLOGY · Maximum and Average Contact Pressures p max = 3W 2πab = 3 × 50 2π(2.32 × 10 −4 ) × (1.73 × 10 −4 ) = 594.8 [MPa] p average = W πab = 50 π(2.32 × 10 −4 ) × (1.73 × 10 −4 ) = 396.5 [MPa] · Maximum Deflection = 1.7719 = 1.6 × 10 −6 [m] δ = ξ 4.5 εR' [( )( 1/3 W πkE' ) 2 ] 4.5 1.3982 × 0.012 [( )( 1/3 50 π1.3380 × (2.308 × 10 11 ) ) 2 ] When comparing the results obtained by the Hertz theory and the Hamrock-Dowson approximation it is apparent that the differences between the results obtained by both methods are very small. Errors due to the approximation on reading values of contact coefficients from Figures 7.12 and 7.13 may contribute significantly to the difference. The benefits of applying the Hamrock-Dowson formulae to the evaluation of contact parameters are demonstrated by the simplification of the calculations without any compromise in accuracy. Hence the Hamrock-Dowson formulae can be used with confidence in most practical engineering applications. Total Deflection In some practical engineering applications, such as rolling bearings, the rolling element is squeezed between the inner and outer ring and the total deflection is the sum of the deflections between the element and both rings, i.e.: δ T = δ o + δ i (7.11) where: δ T is the total combined deflection between the rolling element and the inner and outer rings [m]; δ o is the deflection between the rolling element and the outer ring [m]; δ i is the deflection between the rolling element and the inner ring [m]. According to the formula from Table 7.4, the maximum deflections for the inner and outer conjunctions can be written as: δ i = ξ i 4.5 ε i R i ' [( )( 1/3 W πk i E' ) 2 ] (7.12) δ o = ξ o 4.5 ε o R o ' [( )( 1/3 W πk o E' ) 2 ] TEAM LRN ELASTOHYDRODYNAMIC LUBRICATION 305 where ‘i’ and ‘o’ are the indices referring to the inner and outer conjunction respectively. Note that each of these conjunctions has a different contact geometry resulting in a different reduced radius ‘R'’, ellipticity parameter ‘ k’ and simplified integrals ‘ ξ ’ and ‘ ε ’ . Introducing coefficients which are a function of the contact geometry and material properties, i.e.: K i = πk i E' 4.5ξ i 3 () ε i R i ' 1/2 (7.13) K o = πk o E' 4.5ξ o 3 () ε o R o ' 1/2 The deflections can be written as: δ i = () W 2/3 K i δ o = () W 2/3 K o and δ T = () W 2/3 K T Substituting into equation (7.11) yields: = () W 2/3 () W 2/3 + () W 2/3 K T K o K i (7.14) By rearranging the above expression the coefficient ‘ K T ’ for the total combined deflection, in terms of the ‘ K i’ and ‘ K o ’ coefficients, can be obtained [7], i.e.: = 1 [(() 1 2/3 + () 1 2/3 ] 3/2 K T K o K i (7.15) It should be realized that the deflections and furthermore the pressures resulting from different loads cannot be superimposed. This is because Hertzian deflections are not linear functions of load. 7.4 ELASTOHYDRODYNAMIC LUBRICATING FILMS The term elastohydrodynamic lubricating film refers to the lubricating oil which separates the opposing surfaces of a concentrated contact. The properties of this minute amount of oil, typically 1 [µm] thick and 400 [µm] across for a point contact, and which is subjected to extremes of pressure and shear, determine the efficiency of the lubrication mechanism under rolling contact. TEAM LRN 306 ENGINEERING TRIBOLOGY Effects Contributing to the Generation of Elastohydrodynamic Films The three following effects play a major role in the formation of lubrication films in elastohydrodynamic lubrication: · the hydrodynamic film formation, · the modification of the film geometry by elastic deformation, · the transformation of the lubricant's viscosity and rheology under pressure. All three effects act simultaneously and cause the generation of elastohydrodynamic films. · Hydrodynamic Film Formation The geometry of interacting surfaces in Hertzian contacts contains converging and diverging wedges so that some form of hydrodynamic lubrication occurs. The basic principles of hydrodynamic lubrication outlined in Chapter 4 apply, but with some major differences. Unlike classical hydrodynamics, both the contact geometry and lubricant viscosity are a function of hydrodynamic pressure. It is therefore impossible to specify precisely a film geometry and viscosity before proceeding to solve the Reynolds equation. Early attempts by Martin [2] were made, for example, to estimate the film thickness in elastohydrodynamic contacts using a pre-determined film geometry, and erroneously thin film thicknesses were predicted. · Modification of Film Geometry by Elastic Deformation For all materials whatever their modulus of elasticity, the surfaces in a Hertzian contact deform elastically. The principal effect of elastic deformation on the lubricant film profile is to interpose a central region of quasi-parallel surfaces between the inlet and outlet wedges. This geometric effect is shown in Figure 7.14 where two bodies, i.e. a flat surface and a roller, in elastic contact are illustrated. The contact is shown in one plane and the contact radii are ‘∞’ and ‘R’ for the flat surface and roller respectively. x h f h e B W R U h g = x 2 2R y h e A Body A Body B FIGURE 7.14 Effects of local elastic deformation on the lubricant film profile. The film profile in the ‘x’ direction is given by [15]: TEAM LRN ELASTOHYDRODYNAMIC LUBRICATION 307 h = h f + h e + h g where: h f is constant [m]; h e is the combined elastic deformation of the solids [m], i.e. h e = h e A + h e B ; h g is the separation due to the geometry of the undeformed solids [m], i.e. for the ball on a flat plate shown in Figure 7.14 h g = x 2 /2R; R is the radius of the ball [m]. · Transformation of Lubricant Viscosity and Rheology Under Pressure The non-conformal geometry of the contacting surfaces causes an intense concentration of load over a very small area for almost all Hertzian contacts of practical use. When a liquid separates the two surfaces, extreme pressures many times higher than those encountered in hydrodynamic lubrication are inevitable. Lubricant pressures from 1 to 4 [GPa] are found in typical machine elements such as gears. As previously discussed in Chapter 2, the viscosity of oil and many other lubricants increases dramatically with pressure. This phenomenon is known as piezoviscosity. The viscosity-pressure relationship is usually described by a mathematically convenient but approximate equation known as the Barus law: η p = η 0 e αp where: η p is the lubricant viscosity at pressure ‘p’ and temperature ‘θ’ [Pas]; η 0 is the viscosity at atmospheric pressure and temperature ‘θ’ [Pas]; α is the pressure-viscosity coefficient [m 2 /N]. As an example of the radical effect of pressure on viscosity, it has been reported that at contact pressures of about 1 [GPa], the viscosity of mineral oil may increase by a factor of 1 million (10 6 ) from its original value at atmospheric pressure [15]. With sufficiently hard surfaces in contact, the lubricant pressure may rise to even higher levels and the question of whether there is a limit to the enhancement of viscosity becomes pertinent. The answer is that indeed there are constraints where the lubricant loses its liquid character and becomes semi-solid. This aspect of elastohydrodynamic lubrication is the focus of present research and is discussed later in this chapter. For now, however, it is assumed that the Barus law is exactly applicable. Approximate Solution of Reynolds Equation With Simultaneous Elastic Deformation and Viscosity Rise An approximate solution for elastohydrodynamic film thickness as a function of load, rolling speed and other controlling variables was put forward by Grubin and was later superseded by more exact equations. Grubin's expression for film thickness is, however, relatively accurate and the same basic principles that were originally established have been applied in later work. For these reasons, Grubin's equation is derived in this section to illustrate the principles of how the elastohydrodynamic film thickness is determined. The derivation of the film thickness equation for elastohydrodynamic contacts begins with the 1-dimensional form of the Reynolds equation without squeeze effects (i.e. 4.27): TEAM LRN 308 ENGINEERING TRIBOLOGY dp dx = 6Uη h − h h 3 () where the symbols follow the conventions established in Chapter 4 and are: p is the hydrodynamic pressure [Pa]; U is the surface velocity [m/s]; η is the lubricant viscosity [Pas]; h is the film thickness [m]; h is the film thickness where the pressure gradient is zero [m]; x is the distance in direction of rolling [m]. Substituting into the Reynolds equation the expression for viscosity according to the Barus law yields: dp dx = 6Uη 0 e αp h − h h 3 () (7.16) To solve this equation, Grubin introduced an artificial variable, known as the ‘reduced pressure’, defined as: q = () 1 α 1 − e −αp (7.17) Differentiating gives: dq dx = e −αp dp dx When this term is substituted into the Reynolds equation (7.16), a separation of pressure and film thickness is achieved: dq dx = 6Uη 0 h − h h 3 () (7.18) Two independent controlling variables, i.e. ‘x’ and ‘h ’, however, still remain and replacement of either of these variables by the other (since x = f(h)) is required for the solution. The argument used to achieve this reduction in unknown variables is perhaps the most original and innovative part of Grubin's analysis. Grubin observed that at the inlet of the EHL contact, the contact pressure rises very sharply as predicted by Hertzian contact theory. If a hydrodynamic film is established, then the hydrodynamic pressure should also rise sharply at the inlet. This sharp rise in pressure can be approximated as a step jump to some value in pressure comparable to the peak Hertzian contact pressure. If this pressure is assumed to be large enough then the term e −αp « 1 and it can be seen from equation (7.17) that q ≈ 1/α. Grubin reasoned that since the stresses and the deformations in the EHL contacts were substantially identical to Hertzian, the opposing surfaces must almost be parallel and thus the film thickness is approximately uniform within the contact. Inside the contact therefore, the film thickness h = constant so that h = h. TEAM LRN ELASTOHYDRODYNAMIC LUBRICATION 309 Since ‘ h ’ occurs where ‘p max ’ takes place Grubin deduced that there must be sharp increase in pressure in the inlet zone to the contact as shown in Figure 7.15. It therefore follows that according to this model q ≈ 1/α = constant, dq/dx = 0 and h = h within the contact. Grubin’s model of contact pressure p max p Hertzian pressure BODY A BODY B Steep pressure jump at inlet h ¯ FIGURE 7.15 Grubin's approximation to film thickness within an EHL contact. A formal expression for ‘q’ is found by integrating (7.18); q = 6Uη 0 ⌠ ⌡ h ∞ h 1 dx h − h h 3 () (7.19) where: h 1 is the inlet film thickness to the EHL contact [m]; h ∞ is the film thickness at a distance ‘infinitely’ far from the contact [m]. Since q ≈ 1/α the above equation (7.19) can be written in the form: ⌠ ⌡ h ∞ h 1 q ==6Uη 0 1 α dx h − h h 3 () (7.20) After replacing one variable with another (i.e. expressing ‘x’ in terms of ‘h’), this integral is solved numerically by assuming that the values of film thickness ‘h’ are equal to the distance separating the contacting dry bodies plus the film thickness within the EHL contact. The constant of integration is zero for the selected limits of this integral since at any position remote from the contact, p = 0 and therefore q = 0. The following approximation was calculated numerically for the integral as applied to a line contact: dx = 0.131 LE'R' () W −0.625 R' 2 () b R' () h −1.375 ⌠ ⌡ h ∞ h 1 h − h h 3 () (7.21) where: R' is the reduced radius of curvature [m]; E' is the reduced Young's modulus [Pa]; L is the full length of the EHL contact, i.e. L = 2l, [m]; TEAM LRN [...]... parameter, k = 1, is shown in Figure 7 .20 [7] 7 10 6 5 × 10 10 000 4 000 -e l as v zo Pi ez ov is co us e Pi Lubrication regime: Isoviscous-elastic Isoviscousrigid boundary Dimensionless elasticity parameter 5 × 105 106 10 20 0 100 o isc 2 500 1 500 1 000 700 500 20 0 127 5 × 1 04 5 10 5 2 × 10 20 00 1000 500 us 500 1000 20 00 2 × 1 04 4 10 5000 id rig 50 100 20 0 2 × 10 5 10 5 × 1 04 6 000 tic 5 Dimensionless minimum^... 1 .27 5 0. 625 = h R' −1.375 (7 .23 ) Expressing equation (7 .23 ) as a unit power of h / R ' yields: () ( ) ( ) −0 .45 45 h R '2 −0. 727 3 W = 1.193 R' bU η0 α L E' R' (7 . 24 ) Substituting for contact width ‘b’ the Hertzian contact formula (Table 7 .2) yields a more convenient expression for routine film thickness calculation The expression for ‘b’ (Table 7 .2) is: b= ( 4 WR' π l E' ) ( 1/ 2 = 8 WR' π L E' ) 1/ 2. .. spike [18] p E' Dimensionless pressure p* = 0.0 020 0.0015 Dimensionless speed parameter U = 5.0500 × 10-11 Maximum Hertzian Stress 0. 841 6 × 10-11 0. 841 6 × 10- 12 0.0010 0.0005 0 h −6 R' 100 × 10 Dimensionless film thickness H = Dimensionless speed parameter U = 5.0500 × 10-11 −6 80 × 10 −6 60 × 10 −6 40 × 10 0. 841 6 × 10-11 20 × 10−6 - 12 0. 841 6 × 10 0 -2 -1 0 1 x = x b FIGURE 7.17 Effects of speed parameter...310 ENGINEERING TRIBOLOGY b is the half width of the EHL contact [m]; h is the film thickness where the pressure gradient is zero, i.e Grubin's EHL film thickness as shown in Figure 7.15 [m]; W is the contact load [N] Rearranging (7 .20 ) gives: ( ) h1 ⌠ ⌡∞ h h−h 1 dx = h3 6U η0 α (7 .22 ) The integral term is then eliminated by substituting equation (7 .22 ) into equation (7 .21 ), i.e.: ( ) () W R '2 bU... shows pressure and film thickness profiles for ‘k’ ranging from 1 .25 to 6 for the following values of the non-dimensional controlling parameters: U = 1.683 × 10 - 12, W = 1.106 × 10 -7 and G = 4. 522 × 103 [7] The profile is shown for a section codirectional with the rolling velocity The pressure ‘spike’ is predicted for k = 1 .25 and 2. 5 but not for k = 6 The film thickness appears to increase in proportion... formula [7]: ^ [ ^ Hmin = Hc = 128 αa λb2 0.131 tan−1 () ] αa + 1.683 2 2 where: H min is the non-dimensional minimum film thickness; Hc is the non-dimensional central film thickness; αa and λb are coefficients which can be calculated from: αa = RB ≈ 0.955 k RA ( λb = 1 + k 0.698 k ) −1 is the ellipticity parameter as previously defined TEAM LRN (7 .28 ) 318 ENGINEERING TRIBOLOGY It can be seen that the... on steel even up to maximum pressures of 3 -4 [GPa] [11] The numerically derived formulae for the central and minimum film thicknesses, as shown in Figure 7.16, are in the following form [7]: ( )( )( ) ( ) 0.53 hc Uη0 0.67 W −0.067 = 2. 69 α E' 1 − 0.61e−0.73k R' E'R' E'R '2 ( )( )( ) ( ) 0 .49 h0 Uη0 0.68 W −0.073 = 3.63 α E' 1 − e−0.68k R' E'R' E'R '2 (7 .26 ) (7 .27 ) where: hc is the central film thickness... corresponding changes in the film thickness In this manner a calibration curve, allocating a specific film thickness to a particular colour, is obtained y [mm] Constriction 0 .2 0.1 Hertzian contact radius Outlet -0 .2 -0.1 0.1 Inlet 0 .2 x [mm] Plateau -0.1 Cavitation area -0 .2 FIGURE 7 .22 Schematic representation of the interferometric image of the contact area under EHL conditions [69] It may be noticed... or partial lubrication prevails The theory describing the mechanism of partial elastohydrodynamic lubrication was developed by Johnson, Greenwood and Poon [30] It was found that during partial lubrication, the average surface separation between two rough surfaces is about the same as predicted for smooth surfaces It has also been found that the average asperity pressure TEAM LRN 3 24 ENGINEERING TRIBOLOGY. .. 20 00 1000 500 us 500 1000 20 00 2 × 1 04 4 10 5000 id rig 50 100 20 0 2 × 10 5 10 5 × 1 04 6 000 tic 5 Dimensionless minimum^ film-thickness parameter Hmin 5000 4 10 4 2 × 10 2 × 106 106 5 × 105 20 Dimensionless viscosity parameter GV GE FIGURE 7 .20 Map of lubrication regimes for an ellipticity parameter k =1 [7] Elastohydrodynamic Film Thickness Measurements Various elastohydrodynamic film thickness measurement . shown in Figure 7 .20 [7]. 100 20 0 500 1000 20 00 5000 10 4 2 × 10 4 5 × 10 4 10 5 2 × 10 5 5 × 10 5 10 6 2 × 10 6 5 × 10 6 10 7 20 0 500 1000 20 00 5000 10 4 2 × 10 4 5 × 10 4 10 5 2 × 10 5 5 × 10 5 10 6 100 50 20 10 Piezoviscous-elastic Piezoviscous-rigid Lubrication. p max = 3W 2 ab = 3 × 50 2 (2. 32 × 10 4 ) × (1.75 × 10 4 ) = 588.0 [MPa] p average = W πab = 50 π (2. 32 × 10 4 ) × (1.75 × 10 4 ) = 3 92. 0 [MPa] · Maximum Deflection δ= 0.52k 3 W 2 E' 2 R' () 1/3 =. 1.39 82 × 50 × 0.0 12 π × 1.3380 × (2. 308 × 10 11 ) () 1/3 TEAM LRN 3 04 ENGINEERING TRIBOLOGY · Maximum and Average Contact Pressures p max = 3W 2 ab = 3 × 50 2 (2. 32 × 10 4 ) × (1.73 × 10 4 ) =