Engineering Tribology 2E Episode 4 pptx

HYDRODYNAMIC LUBRICATION 111 y p max z x h 0 h 1 U FIGURE 4.5 Pressure distribution in the long bearing approximation. Now a boundary condition is needed to solve this equation and it is assumed that at some point along the film, pressure is at a maximum. At this point the pressure gradient is zero, i.e. dp/dx = 0 and the corresponding film thickness is denoted as ‘ h’. h ¯ p max p z x h 0 U dp dx = 0 h 1 Thus the boundary condition is: dp dx = 0 at h = h ¯ Substituting to (4.26) gives: C = −6Uη h ¯ and the final form of the one-dimensional Reynolds equation for the ‘long bearing approximation’ is: dp dx h − h 3 = 6Uη h ¯ (4.27) which is particularly useful in the analysis of linear pad bearings. Note that the velocity ‘U’ in the convention assumed is negative, as shown in Figure 4.1. · Narrow Bearing Approximation Finally it is assumed that the pressure gradient acting along the ‘x’ axis is very much smaller than along the ‘y’ axis, i.e.: ∂p/∂x « ∂p/∂y as shown in Figure 4.6. This is known in the literature as a ‘narrow bearing approximation’ or ‘Ocvirk's approximation’ [3]. Actually this particular approach was introduced for the first time by an Australian, A.G.M. Michell, in 1905. It was applied to the approximate analysis of load capacity in a journal bearing [10]. A similar method was also presented by Cardullo [62]. Michell observed that the flow in a bearing of finite length was influenced more by pressure gradients perpendicular to the sides 112 ENGINEERING TRIBOLOGY of the bearing than pressure gradients parallel to the direction of sliding. A formula for the hydrodynamic pressure field was derived based on the assumption that ∂p/∂x « ∂p/∂y. This work was severely criticized by other workers for neglecting the effect of pressure variation in the ‘x‘ direction when equating for flow in the axial or ‘y’ direction and the work was ignored for about 25 years as a result of this initial unenthusiastic reception. Ocvirk and Dubois later developed the idea extensively in a series of excellent papers and Michell's approximation has since gained general acceptance. The utility of this approximation became apparent as journal bearings with progressively shorter axial lengths were introduced into internal combustion engines. Advances in bearing materials allowed the reduction in bearing and engine size, and furthermore the reduction in bearing dimensions contributed to an increase in engine ratings. The axial length of the bearing eventually shrank to about half the diameter of the journal and during the 1950's this caused a reconsideration of the relative importance of the various terms in the Reynolds equation. During this period, Ocvirk realized the validity of considering the pressure gradient in the circumferential direction to be negligible compared to the pressure gradient in the axial direction. This approach later became known as the Ocvirk or narrow journal bearing approximation. An infinitely narrow bearing is schematically illustrated in Figure 4.6. The bearing resembles a well deformed narrow pad. Also the film geometry is similar to that of an ‘unwrapped’ film from a journal bearing, which will be discussed later. x U L − L 2 L 2 0 y h 0 h max p max B FIGURE 4.6 Pressure distribution in the narrow bearing approximation. In this approximation since L « B and ∂p/∂x « ∂p/∂y, the first term of the Reynolds equation (4.24) may be neglected and the equation becomes: ∂ ∂y h 3 ∂p ∂y () dh dx = 6Uη (4.28) Also, since h ≠ f(y) then (4.28) can be further simplified, dx d 2 p dy 2 h 3 dh = 6Uη (4.29) Integrating once, dx dp dy h 3 dh = 6Uη y + C 1 (4.30) HYDRODYNAMIC LUBRICATION 113 and again gives: dxh 3 dh p = 6Uη + C 1 y + C 2 2 y 2 (4.31) From Figure 4.6 the boundary conditions are: p = 0 at y = ± L/2 i.e. at the edges of the bearing and = 0 at y = 0 dy dp i.e. the pressure gradient is always zero along the central plane of the bearing. Substituting these into (4.30) and (4.31) gives the constants ‘C 1 ’ and ‘C 2 ’: C 1 = 0 dxh 3 dh C 2 = −3Uη 4 L 2 (4.32) and the pressure distribution in a narrow bearing approximation is expressed by the formula: dxh 3 dh p = 3Uη 4 L 2 () y 2 − (4.33) The infinitely long bearing approximation is acceptable when L/B > 3 while the narrow bearing approximation can be used when L/B < 1/3. For the intermediate ratios of 1/3 < L/B < 3, computed solutions for finite bearings are applied. Bearing Parameters Predicted from Reynolds Equation From the Reynolds equation most of the critical bearing design parameters such as pressure distribution, load capacity, friction force, coefficient of friction and oil flow are obtained by simple integration. · Pressure Distribution By integrating the Reynolds equation over a specific film shape described by some function h = f(x,y) the pressure distribution in the hydrodynamic lubricating film is found in terms of bearing geometry, lubricant viscosity and speed. · Load Capacity When the pressure distribution is integrated over the bearing area the corresponding load capacity of the lubricating film is found. If the load is varied then the film geometry will change to re-equilibrate the load and pressure field. The load that the bearing will support at a particular film geometry is: W = ⌠ ⌡ 0 L pdxdy ⌠ ⌡ 0 B (4.34) 114 ENGINEERING TRIBOLOGY The obtained load formula is expressed in terms of bearing geometry, lubricant viscosity and speed, hence the bearing operating and design parameters can be optimized to give the best performance. · Friction Force Assuming that the friction force results only from shearing of the fluid and integrating the shear stress ’τ’ over the whole bearing area yields the total friction force operating across the hydrodynamic film, i.e.: F = 0 L τdxdy 0 B ⌠ ⌡ ⌠ ⌡ (4.35) The shear stress ‘τ’ is expressed in terms of dynamic viscosity and shear rates: dz du τ = η where du/dz is obtained by differentiating the velocity equation (4.11). After substituting for ‘τ’ and integrating, the formula, expressed in terms of bearing geometry, lubricant viscosity and speed for friction force per unit length is obtained. The derivation details of this formula are described later in this chapter. = ⌠ ⌡ 0 B L F dp dx Uη h − dx h 2 dx ⌠ ⌡ 0 B ± ‘+’ and ‘−’ refer to the upper and lower surface respectively. The ‘±’ sign before the first term may cause some confusion as it appears that the friction force acting on the upper and the lower surface is different which apparently conflicts with the law of equal action and reaction forces. The balance of forces becomes much clearer when a closer inspection is made of the force distribution on the upper surface as schematically illustrated in Figure 4.7. z U α α W Wtanα reaction from pressure field W x FIGURE 4.7 Load components acting on a hydrodynamic bearing. It can be seen from Figure 4.7 that the reaction force from the pressure field acts in the direction normal to the inclined surface while a load is applied vertically. Since the load is at an angle to the normal there is a resulting component ‘Wtanα’ acting in the opposite direction from the velocity. This is in fact the exact amount by which the frictional force acting on the upper surface is smaller than the force acting on the lower surface. HYDRODYNAMIC LUBRICATION 115 · Coefficient of Friction The coefficient of friction is calculated from the load and friction forces: µ = ⌠ ⌡ 0 L pdxdy ⌠ ⌡ 0 B ⌠ ⌡ 0 L τdxdy ⌠ ⌡ 0 B W F = (4.36) Bearing parameters can then be optimized to give, for example, a minimum value of the coefficient of friction. This means, in approximate terms, minimizing the size of the bearing by allowing the highest possible hydrodynamic pressure. This is discussed in greater detail later as bearing optimization involves many other factors. · Lubricant Flow By integrating the flow expressions ‘q x ’ and ‘q y ’ (4.18) and (4.19) over the edges of the bearing, lubricant leakage out of the sides and ends of the bearing is found. Q x = q x dy ⌠ ⌡ 0 L Q y = q y dx ⌠ ⌡ 0 B (4.37) Lubricant flow is extremely important to the operation of a bearing since enough oil must be supplied to the hydrodynamic contact to prevent starvation and consequent failure. The flow formulae are expressed in terms of bearing parameters and the lubricant flow can also be optimized. Summary In summary it can be stated that the same basic analytical method is applied to the analysis of all hydrodynamic bearings regardless of their geometry. Initially the bearing geometry, h = f(x,y), must be defined and then substituted into the Reynolds equation. The Reynolds equation is then integrated to find the pressure distribution, load capacity, friction force and oil flow. The virtue of hydrodynamic analysis is that it is concise, simple, and the same procedure applies to all kinds of bearing geometries, i.e. linear bearings, step bearings, journal bearings, etc. The solution to the Reynolds equation becomes more complicated if other effects such as heating, locally varying viscosity, elastic deformation, cavitation, etc., are introduced to the analysis. The basic method of analysis, however, remains unchanged. It is always necessary to start with a definition of the bearing geometry and to perform the integration procedure, taking into account extra terms and equations describing the additional effects that we wish to consider. In the next section, some typical bearing geometries are considered and analysed. The one- dimensional Reynolds equation is used to study the linear pad and journal bearings since it provides qualitative indications of the effect of varying the controlling parameters such as load and speed. This approach is very useful as a method of rapid estimation and is widely applied in engineering analysis. For more exact treatments, the two-dimensional (2-D) Reynolds equation has to be employed. The solution of the 2-D Reynolds equation requires the application of numerical methods, and this will be discussed in the next chapter devoted to ‘Computational Hydrodynamics’. 116 ENGINEERING TRIBOLOGY 4.3 PAD BEARINGS Pad bearings, which consist of a pad sliding over a smooth surface, are widely used in machinery to sustain thrust loads from shafts, e.g. from the propeller shaft in a ship. An example of this application is shown in Figure 4.8. The simple film geometry of these bearings, as compared to journal bearings, renders them a suitable example for introducing hydrodynamic bearing analysis. Infinite Linear Pad Bearing The infinite linear pad bearing, as already mentioned, is a pad bearing of infinite length normal to the direction of sliding. This particular bearing geometry is the easiest to analyse. It has been described in many books on lubrication theories [e.g. 3,4]. The basic procedures involved in the analysis are summarized in this section. Ship’s hull Propeller thrust Engine drive Reaction force from ship Bearing view Gear Stern bearing seal Thrust bearing Tilted pads Bearing plate Pad FIGURE 4.8 Example of a pad bearing application to sustain the thrust loads from the ship propeller shaft. Consider an infinitely long linear wedge with L/B > 3 as shown in Figure 4.9, where ‘L’ and ‘B’ are the pad dimensions normal to and parallel to the sliding direction, i.e. pad length and width, respectively. Assume that the bottom surface is moving in the direction shown, dragging the lubricant into the wedge which results in pressurization of the lubricant within the wedge. The inlet and the outlet conditions of the wedge are controlled by the maximum and minimum film thicknesses, ‘h 1 ’ and ‘h 0 ’ respectively. · Bearing Geometry As a first step in any bearing analysis the bearing geometry, i.e. h = f(x), must be defined. The film thickness ‘h’ in Figure 4.9 is expressed as a function: B h 1 − h 0 h = h 0 + xtanα = h 0 + x or simply: HYDRODYNAMIC LUBRICATION 117 h 0 h 1 − h 0 h = h 0 B x () 1 + The term (h 1 - h 0 )/h 0 is often known in the literature as the convergence ratio ‘K’ [3,4]. The film geometry can then be expressed as: h = h 0 B Kx () 1 + (4.38) p max p z x h 0 h ¯ h 1 h U x B ∂p ∂x = 0 x dx α FIGURE 4.9 Geometry of a linear pad bearing. · Pressure Distribution As mentioned already, the pressure distribution can be calculated by integrating the Reynolds equation over the specific film geometry. Since the pressure gradient in the ‘x’ direction is dominant, the one-dimensional Reynolds equation for the long bearing approximation (4.27) can be used for the analysis of this bearing. dp dx h − h 3 = 6Uη h ¯ (4.27) There are two variables ‘x’ and ‘h’ and the equation can be integrated with respect to ‘x’ or ‘h’. Since it does not really matter with respect to which variable the integration is performed we choose ‘h’. Firstly one variable is replaced by the other. This can be achieved by differentiating (4.38) which gives ‘dx’ in terms of ‘dh’: dx = dh B Kh 0 (4.39) Substituting into (4.27) yields: 118 ENGINEERING TRIBOLOGY h − h 3 = 6Uη h ¯ dp dh B Kh 0 and after simplifying and separating variables: dhdp = Kh 0 h − h 3 6UηB h ¯ (4.40) which is the differential formula for pressure distribution in this bearing. Equation (4.40) can be integrated to give: + Cp = − Kh 0 1 h6UηB h ¯ 2h 2 + (4.41) The boundary conditions, taken from the bearing's inlet and outlet, are (Figure 4.9): p = 0 at h = h 0 p = 0 at h = h 1 (4.42) Substituting into (4.41) the constants ‘ h ’ and ‘C’ are: h = 2h 0 h 1 h 1 + h 0 C = 1 h 1 + h 0 (4.43) The maximum film thickness ‘h 1 ’, can also be expressed in terms of the convergence ratio ‘K’: K = h 1 - h 0 h 0 Thus: h 1 = h 0 (K + 1) (4.44) Substituting into (4.43) the constants ‘ h ’ and ‘C’ in terms of ‘K’ are: = 2h 0 (K + 1) h ¯ (K + 2) C = 1 h 0 (K + 2) (4.45) Substituting into (4.41) gives: p = − Kh 0 1 h6UηB h 0 h 2 + (K + 1) (K + 2) + 1 h 0 (K + 2) or: HYDRODYNAMIC LUBRICATION 119 p = Kh 0 1 h h 0 h 2 + (K + 1) (K + 2) + 1 h 0 (K + 2) − () 6UηB (4.46) Note that the velocity ‘U’, in the convention assumed, is negative, as shown in Figure 4.1. It is useful to find the pressure distribution in the bearing expressed in terms of bearing geometry and operating parameters such as the velocity ‘U’ and lubricant viscosity ‘η’. A convenient method of finding the controlling influence of these parameters is to introduce non-dimensional parameters. In bearing analysis non-dimensional parameters such as pressure and load are used. Equation (4.46) can be expressed in terms of a non-dimensional pressure, i.e.: p* = K 1 h h 0 h 2 + (K + 1) (K + 2) + 1 h 0 (K + 2) − () h 0 (4.47) where the non-dimensional pressure ‘p*’ is: p* = h 0 2 6UηB p (4.48) It is clear that hydrodynamic pressure is proportional to sliding speed ‘U’ and bearing width ‘B’ for a given value of dimensionless pressure and proportional to the reciprocal of film thickness squared. If a quick estimate of hydrodynamic pressure is required to check, for example, whether the pad material will suffer plastic deformation, a representative value of dimensionless pressure can be multiplied by the selected values of sliding speed, viscosity and bearing dimensions to yield the necessary information. · Load Capacity The total load that a bearing will support at a specific film geometry is obtained by integrating the pressure distribution over the specific bearing area. W = ⌠ ⌡ 0 L pdxdy ⌠ ⌡ 0 B This can be re-written in terms of load per unit length: = pdx ⌠ ⌡ 0 B L W and substituting for ‘p’, equation (4.46), yields: Kh 0 1 h h 0 h 2 + (K + 1) (K + 2) + 1 h 0 (K + 2) − () 6UηB dx ⌠ ⌡ 0 B = L W (4.49) Again there are two variables in (4.49), ‘x’ and ‘h’, and one has to be replaced by the other before the integration can be performed. Substituting (4.39) for ‘dx’, 120 ENGINEERING TRIBOLOGY Kh 0 1 h h 0 h 2 + (K + 1) (K + 2) + 1 h 0 (K + 2) − () 6UηB dh ⌠ ⌡ h 0 h 1 = L W Kh 0 B and integrating yields: Kh 0 h 0 h − lnh − (K + 1) (K + 2) + h h 0 (K + 2) () 6UηB = L W Kh 0 B h 0 h 0 (K + 1) K 2 h 0 2 − ln(K + 1) + 2K K + 2 () 6UηB 2 = L W (4.50) Equation (4.50) is the total load per unit length the bearing will support expressed in terms of the bearing's geometrical and operating parameters. In terms of the non-dimensional load ‘W*’ equation (4.50) can be expressed as: K 2 − ln(K + 1) + 2K K + 2 () 1 W* = (4.51) where: 6UηB 2 L h 0 2 W* = W (4.52) Bearing geometry can now be optimized to give maximum load capacity. By differentiating (4.51) and equating to zero an optimum value for ‘K’ is obtained which is: K= 1.2 for maximum load capacity for the bearing geometry analysed. The inlet ‘h 1 ’ and the outlet ‘h 0 ’ film thickness can then be adjusted to give the maximum load capacity. From (4.38) it can be seen that the maximum load capacity occurs at a ratio of inlet and outlet film thicknesses of: h 1 h 0 = 2.2 · Friction Force The friction force generated in the bearing due to the shearing of the lubricant is obtained by integrating the shear stress ‘τ’ over the bearing area (eq. 4.35): F = 0 L τdxdy 0 B ⌠ ⌡ ⌠ ⌡ The friction force per unit length is: [...]... can be estimated from the following equation: 136 ENGINEERING TRIBOLOGY TABLE 4. 2 Pivot position for various ‘K’ ratios X B K h1 h0 0 0 .48 2 0 0.2 1 1.2 0 .46 6 0 .45 3 0 .44 2 0 .43 1 0 .4 0.6 0.8 1.0 1 .4 1.6 1.8 2.0 0 .42 2 0 .41 4 1.2 1 .4 2.2 2 .4 0 .40 6 0.399 0.393 0.387 1.6 1.8 2.0 2.2 2.6 2.8 3.0 3.2 0.381 0.376 0.371 2 .4 2.6 2.8 3 .4 3.6 3.8 0.366 3.0 4. 0 ;;;;;;;; ;;;;;;;; ; ; ;; ; ; ;; ; ;;;;;;;; ;; ; ; ;; ; ;;;;;;;;... and substituting this into (4. 84) yields: p* = 2 π [ ] 1 π (θ + sinθ cosθ) − (sinθcos2 θ + 2sinθ) 2 8 (4. 87) The pressure distribution for the Full-Sommerfeld condition is shown in Figure 4. 24 p pmax B −pmax z ¯ h h0 x,θ −π 2 0 ¯ θ π 2 U FIGURE 4. 24 Pressure distribution with the Full-Sommerfeld boundary condition HYDRODYNAMIC LUBRICATION 141 It can be seen from Figure 4. 24 that the Full-Sommerfeld... the bearing: L Qx = ⌠ qx dy ⌡ 0 substituting for ‘qx’ (eq 4. 18): (4. 66) HYDRODYNAMIC LUBRICATION ( L Qx = ⌠ − ⌡ 0 ) h3 ∂p Uh + dy 12η ∂x 2 125 (4. 67) The boundary conditions shown in Figure 4. 9 are: dp =0 dx at h = h (point of maximum pressure) ¯ (4. 68) substituting into (4. 67) the flow is: Qx = ⌠ L ⌡ 0 U¯ h dy 2 (4. 69) substituting for ‘h’ (eq 4. 45): Qx = ⌠ ⌡ 0 L ( ) K+1 U 2 h0 dy K+2 2 and simplifying... in Figure 4. 20 The moment of force about the bearing outlet is: L B WX =⌠ ⌠ px dxdy ⌡⌡ 0 0 or per unit length: WX = L B ⌠ pxdx ⌡ 0 (4. 77) HYDRODYNAMIC LUBRICATION 135 substituting for ‘p’ (eq 4. 46), ‘x’ (from eq 4. 38) and ‘dx’ (eq 4. 39) gives: ( ) 1 W X 6 UηB B2 ⌠ h1 1 h (K + 1) = + − + 0 (h − h0 )dh L Kh0 K2 h02 ⌡ h2 (K + 2) h0 (K + 2) h h 0 (4. 78) After substituting for W/L (eq 4. 50) into (4. 78) and... directly from Figure 4. 11 126 ENGINEERING TRIBOLOGY p pmax z h0 h1 x ZONE 2 B2 U ZONE 1 B1 B FIGURE 4. 11 Geometry of the Rayleigh step bearing ZONE 1 ( ) (4. 71) ZONE 2 ( ) (4. 72) dp p = − max dx 1 B1 dp p = max dx 2 B2 dp dp Note that physically, for the configuration shown in Figure 4. 11, dx is positive while dx is 1 2 negative In the entry zone the oil flow per unit length (eq 4. 18) into the bearing... 2 (K + 2)ln(K + 1) 6 K − 3 (K + 2)ln(K + 1) (4. 64) where: µ* = B µ h0 (4. 65) The optimum bearing geometry which gives a minimum value of coefficient of friction can now be calculated Differentiating (4. 64) with respect to ‘K’ and equating to zero gives: K = 1.55 which is the optimum convergence ratio for a minimum coefficient of friction 1 24 ENGINEERING TRIBOLOGY As stated previously, the maximum load... Figure 4. 26 which is a plan view of the hydrodynamic pressure field p pmax B z h ¯ ¯ h h0 x,θ −π 2 Streamers −θ ¯ dp p= =0 dx 0 ¯ θ π 2 ;;;;;;;;;;; ;;;;;;;;;;; ;;;;;;;;;;; dp =0 dx Continuous film FIGURE 4. 26 Pressure distribution with Reynolds boundary condition It can be seen from Figure 4. 26 that the Reynolds boundary condition assumes: U 144 ENGINEERING TRIBOLOGY π 2 p=0 at θ= p=0 at θ θ=−¯ when (4. 91)... into equation (4. 83) yields the constants ‘C’ and ‘secθ’: C = −0.03685 (4. 92) θ sec ¯ = 1.1228 Substituting into (4. 84) gives the non-dimensional pressure for the Reynolds boundary condition: p* = 2 π [ 1 (θ + sinθ cosθ) − 0.3 743 (sin θcos2 θ + 2sinθ) − 0.03685 2 ] (4. 93) It can also be found from (4. 92) that the maximum pressure is reached at θ = 27.05° and is zero at θ = -27.05° (Figure 4. 26) Load Capacity... for ‘dx’ (4. 82): W 2 B⌠ θ2 = pdθ L π ⌡ θ (4. 94) 1 where ‘θ 1’ and ‘θ 2’ mark the extent of the pressure field, which depends on the boundary condition applied Substituting for ‘p’ (4. 83): [ ] 6 Uη 2 B 2 B⌠ θ2 1 sec ¯ θ W (θ + sinθ cosθ) − (sinθcos2 θ + 2sinθ) + C dθ = h02 π π ⌡ 2 3 L θ 1 Integrating yields: ( ) 2 2 sin2 θ sec ¯ cos3 θ θ W 6 UηB 4 θ + + 2cosθ + Cθ = + 2 2 4 3 h0 3 π 4 L θ2 θ1 (4. 95) HYDRODYNAMIC... UηB dx = ln(1 + K) h h0 K (4. 57) Substituting (4. 56) and (4. 57) into (4. 55) the expression for friction force per unit length for a linear pad bearing is obtained: F Kh0 W UηB = − ln(1 + K) h0 K L 2B L (4. 58) Note that calculating the friction force for the upper surface, i.e for z = h, and subtracting from equation (4. 58) yields Kh 0W/BL = Wtanα/L Substituting for ‘W’ (eq 4. 50), W= ( 6UηB2 L 2K − ln(K . h 1 = h 0 (K + 1) (4. 44) Substituting into (4. 43) the constants ‘ h ’ and ‘C’ in terms of ‘K’ are: = 2h 0 (K + 1) h ¯ (K + 2) C = 1 h 0 (K + 2) (4. 45) Substituting into (4. 41) gives: p =. outlet, are (Figure 4. 9): p = 0 at h = h 0 p = 0 at h = h 1 (4. 42) Substituting into (4. 41) the constants ‘ h ’ and ‘C’ are: h = 2h 0 h 1 h 1 + h 0 C = 1 h 1 + h 0 (4. 43) The maximum film. − h 3 6UηB h ¯ (4. 40) which is the differential formula for pressure distribution in this bearing. Equation (4. 40) can be integrated to give: + Cp = − Kh 0 1 h6UηB h ¯ 2h 2 + (4. 41) The boundary