Mechanical Engineering-Tribology In Machine Design Episode 11 pot

Friction, lubrication and wear in higher kinematic pairs 237 Energy dissipated due to plastic defbrmations In the majority of cases, resistance to rolling is dominated by plastic deformation of one or both contacting bodies. In this case the energy is dissipated within the solids, at a depth corresponding to the maximum shear component of the contact stresses, rather than at the interface. With materials having poor thermal conductivity the release of energy beneath the surface can lead to high internal temperatures and failure by thermal stress. Generally metals behave differently than non-metals. The inelastic properties of metals, and to some extent hard crystalline non-metallic solids, are governed by the movement of dislocations which, at normal temperatures, is not significantly influenced either by temperature or by the rate of deformation. The rolling friction characteristics of a material which has an elastic range of stress, followed by rate-independent plastic flow above a sharply defined yield stress, follow a typical pattern. At low loads the deformation is predominantly elastic and the rolling resistance is given by the elastic hysteresis equation (6.8). The hysteresis loss factor as found by experiment is generally of the order of a few per cent. At high loads, when the plastic zone is no longer contained, i.e., the condition of full plasticity is reached, the rolling resistance may be estimated by the rigid-plastic theory. The onset of full plasticity cannot be precisely defined but, from the knowledge of the static indentation behaviour, where full plasticity is reached when W12az2.6 and Ea/YR z 100, it follows that G W/kR z 300, where k is the yield stress in shear of the solid. Energy dissipated due to surface roughness It is quite obvious that resistance to the rolling of a wheel is greqter on a rough surface than on a smooth one, but this aspect of the subject has received little analytical attention. The surface irregularities influence the rolling friction in two ways. First, they intensify the real contact pressure so that some local plastic deformation will occur even if the bulk stress level is within the elastic limit. If the mating surface is hard and smooth the asperities will be deformed plastically on the first traversal but their deformation will become progressively more elastic with repeated traver- sals. A decreasing rolling resistance with repeated rolling contact has been observed experimentally. The second way in which roughness influences resistance is through the energy expended in climbing up the irregularities. It is significant with hard rough surfaces at light loads. The centre-of-mass of the roller moves up and down in its forward motion which is therefore unsteady. Measurements of the resistance force show very large, high- frequency fluctuations. Energy is dissipated in the rapid succession of small impacts between the surface irregularities. Because the dissipation is by impact, the resistance due to this cause increases with the rolling speed. 238 Tribology in machine design 6.6. Lubrication of It is generally necessary to use a lubricant to ensure satisfactory operation cylinders of engineering surfaces in sliding contact. Even surfaces in nominal rolling contact, such as ball-bearings, normally experience some micro-slip, which necessitates lubrication if surface damage and wear are to be avoided. A lubricating fluid acts in two ways. First, it provides a thin adsorbed film to the solid surfaces, preventing the adhesion which would otherwise take place and reducing friction through an interfacial layer of low shear strength. This is the action known as boundary lubrication. The film is generally very thin and its behaviour is very dependent upon the physical and chemical properties of both the lubricant and the solid surfaces. The lubricant may act in a quite different way. A relatively thick coherent film is drawn in between the surfaces and sufficient pressure is developed in the film to support the normal load without solid contact. This action is known as hydrodynamic lubrication. It depends only upon the geometry of the contact and the viscous flow properties of the fluid. The way in which a load-carrying film is generated between two cylinders in rolling and sliding contact is described in this section. The theory can be applied to the lubrication of gear teeth, for example, which experience a relative motion which, as shown in Section 6.2, is instantaneously equivalent to the t~ combined rolling and sliding contact of two cylinders. A thin film of an incompressible lubricating fluid, viscosity p, between two solid surfaces moving with velocities V1 and V2 is shown in Fig. 6.3. With thin, nearly parallel films, velocity components perpendicular to the film are negligible so that the pressure is uniform across the thickness. At a x low Reynolds number, for the case of a thin film and a viscous fluid, the inertia forces are negligible. Then, for two-dimensional steady flow, Figure 6.3 equilibrium of the fluid element gives where v is the stream velocity. Since dpldx is independent of z, eqn (6.14) can be integrated with respect to z. Putting v = V2 and V, at z =O and h, gives a parabolic velocity profile, as shown in Fig. 6.3, expressed by The volume flow rate Q across any section of the film is For continuity of flow, Q is the same for all cross-sections, i.e. where h, is the film thickness at which the pressure gradient dpldx is zero. Friction, lubrication and wear in higher kinematic pairs 239 Eliminating Q gives This is Reynolds equation for a steady two-dimensional flow in a thin lubricating film. Given the variation in thickness of the film h(x), it can be integrated to give pressure p(x) developed by hydrodynamic action. For a more complete discussion of the Reynolds equation the reader is referred to the books on lubrication listed at the end of Chapter 5. Now, eqn (6.18) will be used to find the pressure developed in a film between two rotating cylinders. Case (i) - Rigid cylinders The geometry of two rotating rigid cylinders in contact is schematically shown in Fig. 6.4. An ample supply of lubricant is provided on the entry side. Within the region of interest the thickness ofthe film can be expressed by h z ho + x2/2R, (6.19) where 1/R = 1/R, + 1/R2 and h is the thickness at x =O. Substituting eqn (6.19) into (6.18) gives dpldx = '5 (6.20) Figure 6.4 By making the substitution <=tan-'[x/(2~h))] eqn (6.20) can be in- tegrated to give an expression for the pressure distribution where 5, =tan-'[xl/(2Rho)*] and xl is the value of x where h=hl and dpldx =O. The values of 5, and A are found from the end conditions. At the start it is assumed that the pressure is zero at distant points at entry and exit, i.e. p =O at x = f co. The resulting pressure distribution is shown by the dotted line in Fig. 6.4. It is positive in the converging zone at entry and equally negative in the diverging zone at exit. The total force W supported by the film is clearly zero in this case. However this solution is unrealistic since a region of large negative pressure cannot exist in normal ambient conditions. In practice the flow at the exit breaks down into streamers separated by fingers of air penetrating from the rear. The pressure is approximately ambient in this region. The precise point of film breakdown is determined by consideration of the three-dimensional flow in 240 Tribology in machine design the streamers and is influenced by surface tension forces. However it has been found that it can be located with reasonable accuracy by imposing the condition at that point. When this condition, together with p =O at x = - oo is imposed on eqn (6.21) it is found that 5, =0.443, whence x, =0.475(2Rh0)*. The pressure distribution is shown by the solid line curve in Fig. 6.4. In this case the total load supported by the film is given by In most practical situations it is the load which is specified. Then, eqn (6.23) can be used to calculate the minimum film thickness ho. To secure effective lubrication, ho must be greater than the surface irregularities. It is seen from eqn (6.23) that the load carrying capacity of the film is generated by a rolling action expressed by (V, + V,). If the cylinders rotate at the same peripheral velocity in opposite directions, then (V, + V2) is zero, and no pressure is developed in the film. Case (ii) - Elastic cylinders Under all engineering loads the cylinders deform elastically in the pressure zone so that the expression for the film profile becomes where u,, and u,, are the normal elastic displacements of the two surfaces and are given by the Hertz theory. Thus This equation and the Reynolds eqn (6.18) constitute a pair of simultaneous equations for the film shape h(x) and the pressure p(x). They can be combined into a single integral equation for h(x) which can be solved numerically. The film shape obtained in that way is then substituted into the Reynolds equation to find the pressure distribution p(x). An important parameter from the point of view of the designer is the minimum film thickness hmin. In all cases hminz0.8hl. The lubrication process in which elastic deformation of the solid surface plays a significant role is known as elastohydrodynamic lubrication. Case (iii) - Variable viscosity of the lubricant It is well known that the viscosity of most practical lubricants is very sensitive to changes in pressure and temperature. In contacts characteristic of higher kinematic pairs, the pressures tend to be high so that it is not Friction, lubrication and wear in higher kinematic pairs 24 1 surprising that an increase in the viscosity with pressure is also a significant factor in elastohydrodynamic lubrication. When sliding is a prevailing motion in the contact, frictional heating causes a rise in the temperature in the film which reduces the viscosity of the film. However, for reasons which will be explained later, it is possible to separate the effects of pressure and temperature. Let us consider an isothermal film in which variation in the viscosity with pressure is given by the equation where po is the viscosity at ambient pressure and temperature and a is a constant pressure coefficient of viscosity. This is a reasonable description of the observed variation in the viscosity of most lubricants. Substituting this relationship into the Reynolds eqn (6.18) gives This modified Reynolds equation for the hydrodynamic pressure in the field must be solved simultaneously with eqn (6.24) for the effect of elastic deformation on the film shape. The solution to this problem can be obtained numerically. There are a number of changes in the contact behaviour introduced by the pressure-viscosity effect. Over an appreciable fraction of the contact area the film is approximately parallel. This results from eqn (6.26). When the exponent ap exceeds unity, the left-hand side becomes small, hence h-h, becomes small, i.e. hz h, =constant. The corresponding pressure distribution is basically that of Hertz for dry contact, but a sharp pressure peak occurs on the exit side, followed by a rapid drop in pressure and thinning of the film where the viscosity falls back to its ambient value po. The characteristic features of highly loaded elastohydrodynamic contacts, that is a roughly parallel film with a constriction at the exit and a pressure distribution which approximates to Hertz but has a sharp peak near the exit, are now well established and supported by experiments. It is sufficiently accurate to assume that the minimum film thickness is about 75 per cent of the thickness in the parallel section. The important practical problem is to decide under what conditions it is permissible to neglect elastic deformation and/or variable viscosity. Some guidance in this matter can be obtained by examining the values of the two non-dimensional parameters, the viscosity parameter g, and the elasticity parameter g, which are presented and discussed in Chapter 2, Section 2.12.1. The mechanism of elastohydrodynamic lubri- cation with a pressure dependent lubricant is now clear. The pressure develops by hydrodynamic action in the entry region with a simultaneous very large increase in the viscosity. The film thickness at the end of the converging zone is limited by the necessity of maintaining a finite pressure. This requirement virtually determines the film thickness in terms of the speed, roller radii and the viscous properties of the lubricant. Increasing the load increases the elastic deformation of the rollers with only a minor 242 Tribology in machine design influence on the film thickness. The highly viscous fluid passes through the parallel zone until the pressure and the viscosity get back to normal at the exit. This means a decrease in thickness of the film. The inlet and exit regions are effectively independent. They meet at the end of the parallel zone with a discontinuityin the slope ofthe surface which is associated with a sharp peak in the pressure. 6.7. Analysis of line In this section line contact lubrication is presented in a way which can be contact lubrication directly utilized by the designer. The geometry of a typical line contact is shown in Fig. 6.5. The minimum film thickness occurs at the exit of the region and can be predicted by the formula proposed by Dowson and Higginson for isothermal conditions where G =aE is the dimensionless material parameter, V=[po (V, + V2)]/2ER is the dimensionless speed parameter, W=w/ERL is the dimensionless load parameter, u is the pressure-viscosity coefficient based I b) pressure d~str~but~on I I I \ inlet I ~ertzlab region Figure 65 Ic) film th~ckness d~stribut~on Friction, lubrication and wear in higher kinematic pairs 243 on the piezo-viscous relation p =poeZP and reflects the change of viscosity with pressure po is the lubricant viscosity at inlet surface temperature, V1, V2 are the surface velocities relative to contact region where the plus sign assumes external contact (both surfaces convex) and the minus sign denotes internal contact (the surface with the larger radius of curvature is concave), w is the total load on the contact and L is the length of the contact. The viscosity of the lubricant at the temperature of the surface of the solid in the contact inlet region is the effective viscosity for determining the film thickness. This temperature may be considerably higher than the lubricant supply temperature and therefore the inlet viscosity may be substantially lower than anticipated, when based on the supply temperature. Usually the inlet surface temperature is an unknown quantity in design analyses. The solution to this problem is to use the lubricant system outlet temperature or an average of the inlet and the outlet temperatures to obtain an estimate of the film thickness. It should be pointed out that the predicted film thickness may be too large when the system supply temperature is used and the flow of lubricant is not sufficient to keep the parts close to the lubricant inlet temperature. Input data characterizing the lubricant, that is its viscosity, pressure-viscosity coefficient and temperature-viscosity coefficient are usually available from catalogues of lubricant manufacturers. The best way to illustrate the practical application ofeqn (6.27) is to solve a numerical problem. Two steel rollers of equal radius R1 = R2 = 100 mm and length L = 100 mm form an external contact. Using the following input data estimate the thickness of the lubricating film Rl =100mm, R2 = 100 mm, w = 30.0 kN, El = E2 = 207 GPa, vl =v2 =0.33, a = 14.5 (GPa)- ', V1 =V2=15.0ms-', po = 27.6 mPa s. Equivalent radius of the contact R lR2 (100)(100) R= - = 50 mm. Rl+R2 100+100 244 Tribology in machine design Equivalent Young modulus 6*8. Heating at the inlet to the contact = 233 GPa. El E2 Material parameter Speed parameter Load parameter Therefore The assumption in the analysis presented in the previous section, was that the lubricant properties are those at the inlet zone temperature and the system is isothermal. The inlet zone lubricant temperature can be, and frequently is, higher than the bulk lubricant temperature in the system. There are basically two mechanisms responsible for the increase in the lubricant temperature at the inlet to the contact. The first is viscous heating of the lubricant in the inlet zone and the second is the conduction of thermal energy accumulated in the bulk of the contacting solids to the inlet zone. This second mechanism is probably only important in pure sliding where the conduction can occur through the stationary solid. The heating at the inlet zone is significant only at high surface velocities and can be subjected to certain simplified analysis. Under conditions of high surface velocity or high lubricant viscosity the effect of inlet heating due to shear on effective viscosity ought to be considered. The engineering approach to this problem is to use a thermal reduction factor, TI, which can be multiplied by the isothermal film thickness, ho, to give a better estimate of the actual film thickness. The thermal correction factor is a weak function of load and material parameters. As a first approximation, the following expression may be used to determine the thermal correction factor for line contacts In eqn (6.28) the thermal loading factor, Tl, is defined as Friction, lubrication and wear in higher kinematic pairs 245 6.9. Analysis of point contact lubrication la) geometry of entry lb) geometry of contact Figure 6.6 where po is the l'ubricant viscosity at the atmospheric pressure and the inlet surface temperature, V = (V, + V2)/2 is the average surface velocity, [m/s], 6 is the lubricant viscosity-temperature coefficient, [=(l/p)/Ap/AT); "C- '1 and p is the lubricant thermal conductivity. The viscosity-temperature coefficient, 6, can be adequately estimated from a temperature viscosity chart of the lubricant. It is necessary to select two temperatures about 20" apart near the assumed inlet temperature and divide the viscosity difference by the average viscosity and temperature difference corresponding to the viscosity difference. Thermal conductivity, p, is relatively constant for classes of lubricants based on chemical composition. For mineral oils, suitable values for these calculations are 0.124.15 W/mK. The lower range applies for lower viscosity either resulting from lower molecular weight or higher tempera- ture. To illustrate the procedure outlined above the numerical example solved earlier is used. Thus, assuming that the lubricant is SAElO mineral oil, inlet temperature is 55 "C, corresponding 6 value is 0.045 "C-' and lubricant thermal conductivity is 0.12 W/m K, eqn (6.29) gives Then, using eqn (6.28) T, =0.857 -0.0234(2.32) +0.000168(2.32)2 =0.8. Finally, the thermally corrected film thickness is his' means a 20 per cent reduction of the film thickness as a result of the heating at the inlet zone. In contrast with the heavily loaded line contacts which have been investigated very fully, the understanding of point contact lubrication is less advanced. Any analysis of the problem naturally relies to a considerable extent on a knowledge of the local shape ofthe contact, which usually is not known in detail. The foundation of the theoretical solution to the problem was laid down by Grubin. He proposed that: (i) the pressure distribution under lubricated conditions was almost Hertzian; (ii) the shape of the entry gap was determined by the Hertz pressure alone, i.e. the fluid pressure at the entry to the contact zone had negligible effect. The application of the Grubin approximation is quite simple in the case of line contacts. The point contact case is far more complex due to side leakage effects. Figure 6.6 shows the geometry of the point contact. The various formulae for load, peak pressure, contact radius and surface deformation can be easily found in any standard textbook on elasticity (see Chapter 3 for more details). The relationships needed here are as follows: 246 Tribology in machine design (i) the applied load PI/ producing a circular contact of radius a: where p,,, is the peak pressure at the centre; (ii) the contact radius a is found from the relationship where R is the equivalent radius defined by 1/R = l/Rl + 1/R2 and E is the equivalent materials modulus where vdenotes the Poisson ratio and the subscripts 1 and 2 refer to the two materials in contact; (iii) the gap outside the main contact regon at any radial distance r is (see Fig. 6.6) The expression in square brackets can be approximated over the region a<r<2a by By denoting that E = (1 1.43 W)/4aEho, eqn (6.32) becomes By writing a non-dimensional version of the Reynolds equation in polar coordinates, and considering the quadrant of the annulus ADCBA, only the final differential equation can be derived. Obviously, the equation does not have an exact analytical solution and it is usually solved numerically. This has been done by many workers and the solution in a simplified form is as follows where po is the oil viscosity at atmospheric pressure, cc is the pressure- viscosity coefficient and V is the surface velocity. Equation (6.34) has been verified experimentally and it is now clear that despite an unfavourable geometric configuration, an elastohydrodynamic lubrication film exists at the nominal point contacts over a very wide range of conditions. 6.10. Cam-follower The schematic geometry of a cam-flat-follower nose is shown in Fig. 6.7. system The film parameter 3, for a cam-follower system can be calculated by the [...]... geometric errors The main cause ofchanges in the friction torque are errors in the geometric shape of the bearing elements The process leading to the formation of the friction torque is shown in Fig 7.6 The line of action is formed at every given instant In an ideal bearing this line passes through the centre of the loaded ball In a real bearing the line of action does not coincide with the direction... for thrust ball-bearings 0.0013, for deep groove ball-bearings 0.0015, for tapered roller-bearings 0.0018, and needle roller-bearings 0.0045 7.3 Deformations in rolling . bodies at the point of contact and K is a coefficient depending on the function F(p) defined as 252 Tribology in machine design and is usually determined from tables contained in the textbooks. to find the pressure developed in a film between two rotating cylinders. Case (i) - Rigid cylinders The geometry of two rotating rigid cylinders in contact is schematically shown in Fig the lubricant in bearing operation. Finally, vibration and acoustic emission in rolling-element bearings are discussed as they are inherently associated with the running of the bearing. Some methods