Engineering Tribology 2E Episode 6 pdf

COMPUTATIONAL 5 HYDRODYNAMICS 5.1 INTRODUCTION The differential equations which arose from the theories of Reynolds and later workers rapidly exceeded the capacity of analytical solution. For many years some heroic attempts were made to solve these equations using specialized and obscure mathematical functions but this process was tedious and the range of solutions was limited. A gap or discrepancy always existed between what was required in the engineering solutions to hydrodynamic problems and the solutions available. Before numerical methods were developed, analogue methods, such as electrically conductive paper, were experimented with as a means of determining hydrodynamic pressure fields. These methods became largely obsolete with the advancement of numerical methods to solve differential equations. This change radically affected the general understanding and approach to hydrodynamic lubrication and other subjects, e.g. heat transfer. It is now possible to incorporate in the numerical analysis of the bearing common features such as heat transfer from a bearing to its housing. The application of traditional, analytical methods would require to assume that the bearing is either isothermal or adiabatic. Numerical solutions to hydrodynamic lubrication problems can now satisfy most engineering requirements for prediction of bearing characteristics and improvements in the quality of prediction continue to be found. In engineering practice problems like: what is the maximum size of the groove to reduce friction before lubricant leakage becomes excessive, or how does bending of the pad affect the load capacity of a bearing?, need to be solved. In this chapter the application of numerical analysis to problems encountered in hydrodynamic lubrication is described. A popular numerical technique, the ‘finite difference method’ is introduced and its application to the analysis of hydrodynamic lubrication is demonstrated. The steps necessary to obtain solutions for different bearing geometries and operating conditions are discussed. Based on the example of the finite journal bearing it is shown how fundamental characteristics of the bearing, e.g. the rigidity of the bearing, the intensity of frictional heat dissipation and its lubrication regime, control its load capacity. 5.2 NON-DIMENSIONALIZATION OF THE REYNOLDS EQUATION Non-dimensionalization is the substitution of all real variables in an equation, e.g. pressure, film thickness, etc., by dimensionless fractions of two or more real parameters. This process 202 ENGINEERING TRIBOLOGY extends the generality of a numerical solution. A basic disadvantage of a numerical solution is that data is only provided for specific values of controlling variables, e.g. one value of friction force for a particular combination of sliding speed, lubricant viscosity, film thickness and bearing dimensions. Analytical expressions, on the other hand, are not limited to any specific values and are suited for providing data for general use, for example, they can be incorporated in an optimization process to determine the optimum lubricant viscosity. A computer program would have to be executed for literally thousands of cases to provide a comprehensive coverage of all the controlling parameters. The benefit of non- dimensionalization is that the number of controlling parameters is reduced and a relatively limited data set provides the required information on any bearing. The Reynolds equation (4.24) is expressed in terms of film thickness ‘h’, pressure ‘p’, entraining velocity ‘U’ and dynamic viscosity ‘η’. Non-dimensional forms of the equation's variables are following: h* = h c x* = x R y* = y L p* = pc 2 6UηR (5.1) where: h is the hydrodynamic film thickness [m]; c is the bearing radial clearance [m]; R is the bearing radius [m]; L is the bearing axial length [m]; p is the pressure [Pa]; U is the bearing entraining velocity [m/s], i.e. U = (U 1 + U 2 )/2; η is the dynamic viscosity of the bearing [Pas]; x, y are hydrodynamic film co-ordinates [m]. The Reynolds equation in its non-dimensional form is: = ∂ ∂x* ( ( ∂p* ∂x* h* 3 + ( ( R L 2 ( ( ∂p* ∂y* h* 3 ∂h* ∂x* ∂ ∂y* (5.2) All terms in equation (5.2) are non-dimensional apart from ‘R’ and ‘L’ which are only present as a non-dimensional ratio. Although any other scheme of non-dimensionalization can be used this particular scheme is the most popular and convenient. For planar pads, ‘R’ is substituted by the pad width ‘B’ in the direction of sliding. 5.3 THE VOGELPOHL PARAMETER The Vogelpohl parameter was developed to improve the accuracy of numerical solutions of the Reynolds equation and was introduced by Vogelpohl [1] in the 1930's. The Vogelpohl parameter ‘M v ’ is defined as follows: COMPUTATIONAL HYDRODYNAMICS 203 M v = p*h* 1.5 (5.3) Substitution into the non-dimensional form of Reynolds equation (5.2) yields the ‘Vogelpohl equation’: = FM v + G ∂ 2 M v ∂x* 2 + ( ( R L 2 ∂ 2 M v ∂y* 2 (5.4) where parameters ‘F’ and ‘G’ for journal bearings are as follows: ∂h* ∂x* + ( ( R L 2 [( ( 2 ∂h* ∂y* ( ( 2 [ 0.75 h* 2 F = ∂ 2 h* ∂x* 2 + ( ( R L 2 [ ∂ 2 h* ∂y* 2 [ 1.5 h* + (5.5) ∂h* ( ( h* 1.5 G = ∂x* (5.6) The Vogelpohl parameter facilitates computing by simplifying the differential operators of the Reynolds equation, and furthermore it does not show high values of higher derivatives in the final solution, i.e. d n M v /dx * n where n > 2, unlike the dimensionless pressure ‘p*’. This is because, where there is a sharp increase in ‘p*’ close to the minimum of hydrodynamic film thickness ‘h*’, ‘M v ’ remains at moderate values. Large values of higher derivatives cause significant truncation error in numerical analysis. The characteristics of ‘M v ’ and ‘p*’ for a journal bearing at an eccentricity of 0.95 are shown in Figure 5.1. 0 100 200 Degrees around bearing Vogelpohl parameter M v Dimensionless pressure p* p*, M v L/D = 1 ε = 0.95 360° bearing FIGURE 5.1 Variation of dimensionless pressure and the Vogelpohl parameter along the centre plane of a journal bearing [4]. 204 ENGINEERING TRIBOLOGY It can be seen from Figure 5.1 that the introduction of the Vogelpohl parameter does not complicate the boundary conditions in the Reynolds equation, since wherever p* = 0, also M v = 0 (zero values of ‘h’, i.e. solid to solid contact, are not included in the analysis). As discussed later in this chapter, wherever cavitation occurs, the gradient of ‘M v ’ adjacent and normal to the cavitation front is zero like that of ‘p*’. Numerical solutions of the Reynolds equation are obtained in terms of ‘M v ’ and values of ‘p*’ found from the definition M v /h * 1.5 = p*. 5.4 FINITE DIFFERENCE EQUIVALENT OF THE REYNOLDS EQUATION Journal and pad bearing problems are usually solved by ‘finite difference’ methods although ‘finite element’ methods have also been employed [2]. The finite difference method is based on approximating a differential quantity by the difference between function values at two or more adjacent nodes. For example, the finite difference approximation to ∂M v /∂x* is given by: ∂M v ∂x* ≈ M v,i+1 − M v,i−1 2δx* ( ( i M v,i−1 M v,i+1 M v x* x* i x* i+1 x* i−1 M v,i 2δx* ∂M v ∂x* ( ( i Approximation to (5.7) where the subscripts i- 1 and i+1 denote positions immediately behind and in front of the central position ‘i’ and ‘δx*’ is the step length between nodes. A similar expression results for the second differential ∂ 2 M v /∂x * 2 . This expression can be found according to the principle illustrated in Figure 5.2. The second differential ∂ 2 M v /∂x * 2 is found by subtracting the expression for ∂M v /∂x* at the i- 0.5 nodal position from the i+0.5 nodal position and dividing by δx*, i.e.: δx* ≈ ∂ 2 M v ∂x* 2 ∂M v ∂x* ( ( i−0.5 ∂M v ∂x* ( ( i+0.5 − ( ( i (5.8) where: ∂M v ∂x* ≈ M v,i+1 − M v,i δx* ( ( i+0.5 ∂M v ∂x* ≈ M v,i − M v,i−1 δx* ( ( i−0.5 substituting into (5.8) yields: ∂ 2 M v ∂x* 2 ≈ M v,i+1 + M v,i−1 − 2M v,i (δx*) 2 ( ( i COMPUTATIONAL HYDRODYNAMICS 205 M v,i−1 M v,i M v,i+1 Closest approximations to dM v dx* at (x i * − 1 2 δx*) and (x i * + 1 2 δx*) M v dM v dx* x* x* Closest approximation to d 2 M v dx* 2 at x i * x i * − 1 2 δx* x i * + 1 2 δx*x i *x i * −δx* x i * +δx* x i * − 1 2 δx* x i * + 1 2 δx*x i * FIGURE 5.2 Illustration of the principle for the derivation of the finite difference approximation of the second derivative of a function. The finite difference equivalent of (∂ 2 M v /∂x * 2 + ∂ 2 M v /∂y * 2 ) is found by considering the nodal variation of ‘M v ’ in two axes, i.e. the ‘x’ and ‘y’ axes. A second nodal position variable is introduced along the ‘y’ axis, the ‘j’ parameter. The expressions for ∂M v /∂y * and ∂ 2 M v /∂y * 2 are exactly the same as the expressions for the ‘x’ axis but with ‘i’ substituted by ‘j’. The coefficients of ‘M v ’ at the ‘i’-th node and adjacent nodes required by the Reynolds equation which form a ‘finite difference operator’ are usually conveniently illustrated as a ‘computing molecule’ as shown in Figure 5.3. The finite difference operator is convenient for computation and does not create any difficulties with boundary conditions. When the finite difference operator is located at the boundary of a solution domain, special arrangements may be required with imaginary nodes outside of the boundary. The solution domain is the range over which a solution is applicable, i.e. the dimensions of a bearing. There are more complex finite difference operators available based on longer strings of nodes but these are difficult to apply because of the requirement for nodes outside of the solution domain and are rarely used despite their greater accuracy. The terms ‘F’ and ‘G’ can be included with the finite difference operator to form a complete equivalent of the Reynolds equation. The equation can then be rearranged to provide an expression for ‘M v,i,j ’ i.e.: + ( ( R L 2 2C 1 + 2C 2 + F i,j M v,i,j = C 1 ( ( M v,i+1,j + M v,i−1,j C 2 ( ( M v,i,j+1 + M v,i,j−1 − G i,j (5.9) 206 ENGINEERING TRIBOLOGY where: C 1 = 1 δx* 2 C 2 = 1 δy* 2 This expression forms the basis of the finite difference method for the solution of the Reynolds equation. Its solution gives the required nodal values of ‘M v ’. −2 δx* 2 −2 δy* 2 1 δx* 2 1 δx* 2 1 δy* 2 1 δy* 2 δy* δx* To boundary of solution domain j-1 j j+1 i-1 i i+1 FIGURE 5.3 Finite difference operator and nodal scheme for numerical analysis of the Reynolds equation. Definition of Solution Domain and Boundary Conditions After establishing the controlling equation, the next step in numerical analysis is to define the boundary conditions and range of values to be computed. For the journal or pad bearing, the boundary conditions require that ‘p*’ or ‘M v ’ are zero at the edges of the bearing and also that cavitation can occur to prevent negative pressures occurring within the bearing. The range of ‘x*’ is between 0 - 2π (360° angle) for a complete bearing or some smaller angle for a partial arc bearing. The range of ‘y*’ is from -0.5 to +0.5 if the mid-line of the bearing is selected as a datum. A domain of the journal bearing where symmetry can be exploited to cover either half of the bearing area, i.e. from y* = 0 to y* = 0.5, or the whole bearing area is shown in Figure 5.4. Nodes on the edges of the bearing remain at a pre-determined zero value while all other nodes require solution by the finite difference method. When symmetry is exploited to solve for only a half domain, it should be noted that nodes on the mid-line of the bearing are also variable and the finite difference operator requires an extra column of nodes outside the solution domain, as zero values along the edge of the solution domain cannot be assumed. This extra column is generated by adopting node values from the column one step from the mid-line on the opposite side. In analytical terms this is achieved by setting: COMPUTATIONAL HYDRODYNAMICS 207 M v,i,jnode+1 = M v,i,jnode−1 (5.10) where ‘jnode’ is the number of nodes in the ‘j’ or ‘y*’ direction. A split domain reduces the number of nodes but when analyzing a non-symmetric or misaligned bearing then a domain covering the complete bearing area is necessary. The domain in this case is the complete bearing with limits of ‘y*’ from -0.5 to +0.5 and the mid- line boundary condition vanishes. Sliding direction Load line i=1  x* = 0  x* = π i=INODE x* = 2π j=1 j=JNODE  half bearing j=JNODE full bearing Extra row for half bearing Extra row of nodes for overlap with x* = 0 FIGURE 5.4 Nodal pressure or Vogelpohl parameter domains for finite difference analysis of hydrodynamic bearings. Calculation of Pressure Field It is possible to apply the direct solution method to calculate the pressure field but this is quite complex. In practice the pressure field in a bearing is calculated by iteration procedure and this will be discussed in the next section. Calculation of Dimensionless Friction Force and Friction Coefficient Once the pressure field has been found, it is possible to calculate the friction force and friction coefficient from the film thickness and pressure gradient data. As discussed already in Chapter 4 the frictional force operating across the hydrodynamic film is calculated by integrating the shear stress ‘τ’ over the bearing area, i.e.: F = ⌠ ⌡ 0 L ⌠ ⌡ 0 2πR τdxdy (5.11) where the shear stress ‘τ’ is given by: τ = ηU h + h 2 dp dx (5.12) where: τ is the shear stress [Pa]; η is the dynamic viscosity of the lubricant [Pas]; U is the entraining velocity [m/s]; 208 ENGINEERING TRIBOLOGY h is the hydrodynamic film thickness [m]; p is the hydrodynamic pressure [Pa]; F is the friction force [N]; x is the distance in the direction of sliding [m]; y is the distance normal to the direction of sliding [m]. In a manner similar to the computation of pressure, the equation for friction force can be expressed in terms of non-dimensional quantities. From (5.1) h = h*c, x = x*R and p = p*(6UηR)/c 2 and substituting into (5.12) yields: τ = ηU c 1 h* ) + ch* 2 6UηR c 2 1 R dp* dx* = Uη c 1 h* + 3h* ()( dp* dx* (5.13) Substituting for ‘x’ and ‘y’ from (5.1) the ‘τdxdy’ in terms of non-dimensional quantities is: τdxdy = τdx*dy*RL (5.14) Substituting (5.14) and (5.13) into (5.11) results in an expression for frictional force in terms of non-dimensional quantities: F = ⌠ ⌡ 0 L ⌠ ⌡ 0 2πR τdxdy = RL ⌠ ⌡ 0 1 ⌠ ⌡ 0 2π τdx*dy* = ⌠ ⌡ 0 1 ⌠ ⌡ 0 2π dx*dy* RLηU c ) 1 h* + 3h* dp* dx* ( (5.15) It can be seen from equation (5.15) that the non-dimensional shear stress ‘τ*’ is expressed as: τ* = 1 h* + 3h* dp* dx* (5.16) and equation (5.15) can be re-written in the following form: F = ⌠ ⌡ 0 1 ⌠ ⌡ 0 2π τ*dx*dy* = F* RLηU c RLηU c )( (5.17) The coefficient of friction is calculated by dividing the friction force by the load. A similar quantity is also found when the dimensionless friction is divided by the dimensionless load. Thus in journal bearings the friction coefficient is the ratio of circumferential friction force divided by the load: µ = F W = ⌠ ⌡ 0 L ⌠ ⌡ 0 2πR τdxdy ⌠ ⌡ 0 L ⌠ ⌡ 0 2πR pdxdy (5.18) COMPUTATIONAL HYDRODYNAMICS 209 where: µ is the coefficient of friction; W is the bearing load [N]. Load on a journal bearing is often expressed as: ⌠ ⌡ 0 L ⌠ ⌡ 0 2πR −cos(x*)pdx dyW = (5.19) where the term -cos(x*) arises from the fact that load supporting pressure is located close to x* = π or cos(x*) = -1. Any pressure close to x* = 0 merely imposes an extra load on the bearing since it acts in the direction of the load. The negative sign refers to the fact that the load vector does not coincide with the position of maximum film thickness. Expressing equation (5.19) in terms of non-dimensional quantities yields: 6UηR c 2 () RL ⌠ ⌡ 0 1 ⌠ ⌡ 0 2π −cos(x*)p*dx*dy* = W* 6R 2 LUη c 2 () W = (5.20) Substituting (5.17) and (5.20) into (5.18) gives the expression for coefficient of friction: RLηU c () = F W 6R 2 LUη c 2 () F* W* = ( c 6R )( F* W* ) µ = (5.21) hence: F* W* = ( ) 6R c µ (5.22) The presence of cavitation in the bearing adds some complication to the calculation of the coefficient of friction. Within the cavitated region, the proportion of clearance space between shaft and bush that is filled by lubricant is: h* cav h* where: h cav * is the dimensionless film thickness at the cavitation front; h* is the dimensionless film thickness at a specified position down-stream of the cavitation front. The average or ‘effective’ coefficient of friction is proportional to the lubricant filled fraction of the clearance space and within the cavitated region ‘p*’ and ‘dp*/dx*’ are equal to zero. The symbol for ‘effective’ dimensionless shear stress is ‘τ e * ’. Assuming a simple proportionality between fluid filled volume and total shear force, an average value of ‘τ e * ’ that allows for zero shear stress between streamers of lubricant, is given by: 210 ENGINEERING TRIBOLOGY τ* e = h* cav h* 2 (5.23) This value of dimensionless shear stress is included in the integral for dimensionless friction force (eq. 5.17) with no further modification. Values of h*, ∂h*/∂x*, ∂h*/∂y* and ∂ 2 h*/∂x * 2 are also required in computation and the expressions for these are: h* = y*tcos (x*) + εcos(x* − β) + 1 (5.24) where: ε is the eccentricity ratio; t is the misalignment factor; β is the attitude angle. Note that the variation in ‘h*’ due to misalignment is dependent on ‘x*’ whereas the variation in ‘h*’ due to eccentricity is also controlled by the attitude angle. The derivatives of ‘h*’ are found by direct differentiation of (5.24), i.e.: =−y*tsin (x*) −εsin(x* −β) dh* dx* (5.25) = tcos(x*) dh* dy* (5.26) =−y*tcos (x*) −εcos(x* −β) d 2 h* dx* 2 (5.27) Numerical Solution Technique for Vogelpohl Equation The nodal values of ‘M v ’ are conveniently arranged in a matrix with ‘i’ and ‘j’ as the column and row ordinates. The coefficients in equation (5.9) can also be organized into a ‘sparse’ matrix with all coefficients lying close to the main diagonal. It is therefore possible to solve equation (5.9) by matrix inversion but this requires elaborate computation. Programming is greatly simplified when iterative solution methods are applied. The Gauss-Seidel iterative method is used in this chapter. All node values are assigned an initial zero value and the finite difference equation (5.9) is repeatedly applied until convergence is obtained. 5.5 NUMERICAL ANALYSIS OF HYDRODYNAMIC LUBRICATION IN IDEALIZED JOURNAL AND PARTIAL ARC BEARINGS A numerical solution to the Reynolds equation for the full and partial arc journal bearings is necessary for the calculation of pressure distribution, load capacity, lubricant flow rate and friction coefficient when the bearings are neither ‘infinitely long’ nor ‘infinitely narrow’. This condition is valid for bearings with L/D ratio in the range 1/3 < L/D < 3, where ‘L’ is the bearing length and ‘D’ is the bearing diameter. Equation (5.9) is solved numerically in order to find the dimensionless pressure field corresponding to equation (5.2) and the other [...]... (5 .65 ) where: ηp is the predicted dynamic viscosity of the lubricant [Pas]; η0 is the dynamic viscosity of the lubricant at some reference temperature ‘T0’ [Pas]; γ is an exponent of viscosity-temperature dependence (typically γ = 0.05) [K-1] or rearranged as: dηp = −γηp dTp (5 .66 ) Substituting (5 .66 ) into (5 .63 ) and (5 .64 ) yields: Sp = −γS ( Sc = S 1 − (5 .67 ) ) ( ) Tp dη = S 1 + γTp ηp dTp,old (5 .68 )... and 9 rows in the ‘y*’ direction yields: dimensionless load W* = 0 .60 29 and maximum dimensionless pressure p* = 1. 268 4 max Since the load ‘W’ and pressure ‘p’ expressed in non-dimensional terms are: W = W* p = p* 6R 2LUη c2 6RUη c2 substituting the bearing data yields: 2 10 W = 0 .60 29 6 × 0.1 × 0.2 × 2 × 0.05 0.0002 × p max = 1. 268 4 6 × 0.1 × 10 2 0.05 0.0002 = 90.435 [kN] = 9.513 [MPa] The difference... lubricant: S=η () ∂u ∂z 2 (5 .61 ) where: S is the intensity of viscous heating [W/m3]; The controlling equation for ‘Sp’ and ‘Sc’ is based on the assumption of a linear dependence of the heat source term on temperature: S = Sc + Sp Tp (5 .62 ) The precise forms of ‘Sp’ and ‘Sc’ are given by: Sp = S dη ηp dTp,old (5 .63 ) 2 26 ENGINEERING TRIBOLOGY ( Sc = S 1 − Tp dη ηp dTp,old ) (5 .64 ) where all terms are as... of the peak pressure P% 100% Pressure along the load line Outlet 120 96 8 84 10 72 L 0.75 L 0.5 L Inlet 0.25 L 12 0 FIGURE 5.7 24 36 60 48 es gre De Computed pressure profile for 120° perfectly aligned partial bearing 100% P% Pressure along the load line Outlet 120 96 8 84 10 72 L 0.75 L 0.5 L Inlet 0.25 L 0 FIGURE 5.8 12 24 36 60 48 es gre De Computed pressure field for 120° misaligned partial bearing... equation (5.45) simplifies to: v=M dp dy (5. 46) An expression for ‘p’ in terms of derivatives with respect to ‘x’ and ‘y’ axis can now be derived This expression is comparable to the isothermal Reynolds equation The continuity of flow condition is given by: ∂⌠ h ∂ h u∂z + ⌠ v ∂z = 0 ∂x⌡ ∂y⌡ 0 0 (5.47) 222 ENGINEERING TRIBOLOGY Substituting (5.40) and (5. 46) into (5.47) yields: ( ) ( ) h ∂ ∂p⌠ h ∂ ∂p⌠... the continuity of flow in the ‘x’ direction is maintained In algebraic terms this means that the integral of lubricant velocity across the film thickness is constant, i.e.: 220 ENGINEERING TRIBOLOGY ∂ ∂x ( ⌠ ⌡0 ) h udz = 0 (5. 36) The velocity ‘u’ can be expressed in terms of parameters ‘M’ and ‘N’ which are composed of the integrals of viscosity ‘η’ [8], i.e.: u=M dp + NU dx (5.37) where the terms ‘M’... pronounced influence on maximum hydrodynamic pressure In Figure 5 .6 this effect is illustrated for a 120° arc bearing, L/D = 1 and an eccentricity ratio of 0.7 Maximum dimensionless pressure p* max 2 1.5 1 0.5 0 0.1 0.2 0.3 0.4 0.5 0 .6 t Effect of misalignment on maximum hydrodynamic pressure in a partial arc bearing Misalignment parameter FIGURE 5 .6 It can be seen that the maximum pressure is more than doubled... to 0.5 The limiting value of misalignment before contact occurs between the shaft and the bush is 0 .6 for a value of eccentricity ratio of 0.7 This is based on equation (5.24) which implies that the sum of eccentricity and half the misalignment must be less than 1 for no solid contact 214 ENGINEERING TRIBOLOGY This effect can also be demonstrated easily by comparing the pressure fields for perfectly... bearing to the other (i.e the distance between the axes of the tilted and non-tilted shaft measured at the edges of the bearing) is: m = tc Substituting gives: m = 0.4 × 0.0002 = 8 × 10 -5 [m] 2 16 ENGINEERING TRIBOLOGY The minimum film thickness has two components: one due to eccentricity (as described in Chapter 4, i.e h min,ecc = c(1 − ε)) and one due to misalignment and can be estimated from the formula:... out load End FIGURE 5.14 Flow chart of the program for the analysis of a thermohydrodynamic pad bearing 230 ENGINEERING TRIBOLOGY A strong effect of pad heat transfer on the temperatures inside the lubricant film is clear The maximum temperature in the isothermal bearing is 71°C, compared to 1 16 C for the adiabatic bearing The location of the maximum temperature is also different for these bearings . dimensionless load W* = 0 .60 29 and maximum dimensionless pressure p max * = 1. 268 4. Since the load ‘W’ and pressure ‘p’ expressed in non-dimensional terms are: W = W* 6R 2 LUη c 2 p = p* 6RUη c 2 substituting. p* 6RUη c 2 substituting the bearing data yields: W = 0 .60 29 6 × 0.1 2 × 0.2 × 10 × 0.05 0.000 2 2 = 90.435 [kN] p max = 1. 268 4 6 × 0.1 × 10 × 0.05 0.000 2 2 = 9.513 [MPa] The. expression for coefficient of friction: RLηU c () = F W 6R 2 LUη c 2 () F* W* = ( c 6R )( F* W* ) µ = (5.21) hence: F* W* = ( ) 6R c µ (5.22) The presence of cavitation in the bearing adds