Engineering Tribology 2E Episode 9 potx

336 ENGINEERING TRIBOLOGY If the contacting solids are of the same material then their thermal constants are also the same and the above equation can be written as: T f maxc = 1.11µW U A 0.5 − U B 0.5  (2l)(2 b) 0.5 (Kρσ) 0.5 (7.40) or in terms of the reduced radius of curvature ‘R'’ and Young's modulus ‘E'’ as: 2l () W 0.75 T f maxc = 0.62µU A 0.5 − U B 0.5  (Kρσ) 0.5 R' () E' 0.25 (7.41) where: E' is the reduced Young's modulus [Pa]. For two solids of the same material: E' = E A /(1 - υ A 2 ) = E B /(1 - υ B 2 ) R' is the reduced radius of curvature of the undeformed surfaces [m]. or in terms of maximum contact pressure as: T f maxc = 2.45µp max 1.5 U A 0.5 − U B 0.5  (Kρσ) 0.5 E' () R' 0.5 (7.42) where: p max is the maximum contact pressure [Pa]. Although the procedure outlined does not always provide precise values of the temperature distribution over the entire interface between contacting solids it greatly facilitates the physical interpretation of frictional temperatures. For example, consider a pin-on-disc machine with the pin and the disc manufactured from the same material. Since the pin is stationary, low speed conditions of heat transfer apply, whereas high speed conditions apply to the disc. The interfacial temperature of the disc will be very much lower than that of the pin since the disc is constantly presenting fresh cool material to the interface. Hence the temperature distribution at the interface will be mostly determined by the heat flow equations in the disc [44,49]. Another example of this effect which is more closely related to EHL is the difference in frictional temperatures between a large and a small gear-wheel when meshed together. The maximum flash temperature rise is located towards the trailing region of the contact and its location depends on the Peclet number as shown in Figure 7.31 [47]. The maximum flash temperature distribution for high speed conditions in circular contacts is shown in Figure 7.32 [44]. It can be seen that the maximum temperature is about T fmax = 1.64T fa and occurs at the centre of the trailing edge of the contact. It should be noted that the heat source considered in the analysis was treated as uniform, i.e. the frictional energy generated is uniformly distributed over the contact area. It has been found that for the non-uniform heat sources arising from the Hertzian pressure distribution, the value of ‘q max ’ is almost unaffected by the non-uniform distribution of ‘q’ [44,49]. However, it was found that the maximum temperature for a circular contact is increased by 16% compared to the uniform heat source, and its location is moved inward from the trailing edge [44]. ELASTOHYDRODYNAMIC LUBRICATION 337 -4 -3 -2 -1 0 1 2 x/b Flash temperature rise Trailing edge Leading edge L =10 5 2 1 0.5 0.2 FIGURE 7.31 Flash temperature rise variations with Peclet number [47]. U T f max = 1.64T a T f a FIGURE 7.32 Flash temperature distribution in circular contacts for the high speed condition (adapted from [44]). As already mentioned the calculated flash temperature rise must be added to the bulk temperature to obtain the temperature of the conjunction, i.e.: T c = T b + T fmax The above equation is valid if the bulk temperatures of the two solids entering the contact are the same. If the bulk temperatures of the contacting solids differ then a new adjusted value of bulk temperature must be calculated and substituted into the above equation. The adjusted value of bulk temperature is evaluated from the formula [51]: T b new = 2 1 ( T b A + T b B ) + 2 1 ()( T b A − T b B ) n + 1 n − 1 (7.43) where: n is a constant calculated from: U A ρ A σ A K A U B ρ B σ B K B ( ) 0.5 338 ENGINEERING TRIBOLOGY T bnew is the new adjusted bulk temperature [°C]; T bA is the bulk temperature of body A [°C]; T bB is the bulk temperature of body B [°C]. The other variables are as already defined. For 0.2 ≤ n ≤ 5 the average bulk temperature can be calculated with sufficient accuracy from: T bnew = 0.5 × ( T bA + T bB ) (7.44) Flash temperature under the surface drops rapidly with depth. For example, for L = 2 at a depth of 15 [µm] below the surface, the temperature drops to about 0.1 of its initial value at the rear end of the source. Thus any thermal stresses or thermally induced microstructure transformations caused by flash temperature fluctuations are very superficial [47]. EXAMPLE Find the maximum flash temperature rise for two steel rollers of radii R A = 10 × 10 -3 [m] and R B = 15 × 10 -3 [m], working under a load of 5 [kN] and rotating at different speeds of U A = 2 [m/s] and U B = 1 [m/s]. Assume that the thermal and material properties of the rollers are the same and are: Young's modulus E = 2.1 × 10 11 [Pa], Poisson's ratio υ = 0.3, thermal conductivity K = 46.7 [W/mK], density ρ = 7800 [kg/m 3 ] and specific heat σ = 460 [J/kgK]. The length of both rollers is 2l = 10 × 10 -3 [m]. The bulk temperature of the rollers is T bA = T bB = 80°C and the coefficient of friction is µ = 0.2. · Reduced Radius of Curvature Since the dimensions of the rollers are the same as in the example already considered the reduced radius of curvature is: R' = 6 × 10 −3 [m] · Reduced Young's Modulus E' = 2.308 × 10 11 [Pa] · Half Width of the Contact Rectangle (Table 7.2) b = πlE' () 4WR' 0.5 = 4 × (5 × 10 3 ) × (6 × 10 −3 ) π × (5 × 10 −3 ) × (2.308 × 10 11 ) () = 1.82 × 10 −4 [m] 0.5 · Thermal Diffusivity χ= ρσ K = 46.7 7800 × 460 = 13.02 × 10 −6 [m 2 /s] · Peclet Number L A = 2χ U A b = 2 × (1.82 × 10 −4 ) 2 × (13.02 × 10 −6 ) = 13.98 ELASTOHYDRODYNAMIC LUBRICATION 339 L B = 2χ U B b = 1 × (1.82 × 10 −4 ) 2 × (13.02 × 10 −6 ) = 6.99 Note that L A and L B > 5. · True Maximum Flash Temperature of the Conjunction T f maxc = 1.11µW U A 0.5 − U B 0.5  (2l)(2 b) 0.5 (Kρσ) 0.5 = 1.11 × 0.2 × (5 × 10 3 ) (10 × 10 −3 ) × (2 × 1.82 × 10 −4 ) 0.5 (46.7 × 7800 × 460) 0.5  2 − 1  = 186.2 [°C] This value is added to the bulk temperature of the rollers in order to obtain the true maximum temperature of the conjunction, i.e.: T c = T b + T fmaxc = 80° + 186.2° = 266.2 [°C] The same result is obtained when applying the two other flash temperature formulae 7.41 and 7.42. For example, calculating flash temperature in terms of the maximum contact pressure equation (7.42) yields: p max = πbl W = = 1748.96 [MPa] π(1.82 × 10 −4 ) × (5 × 10 −3 ) 5 × 10 3 hence: 2.45 × 0.2 × (1748.96 × 10 6 ) 1.5 (46.7 × 7800 × 460) 0.5  2 − 1  = 184.91 [°C] T f maxc = 6 × 10 −3 2.308 × 10 11 () 0.5 Frictional Temperature Rise of Lubricated Contacts All surface lubricating films can affect the surface temperatures through changes in the coefficient of friction. The temperature of the substrate is usually not affected by these films as long as the presence of a film does not affect friction. However, when a lubricating film is relatively thick and the lubricant has sufficiently low thermal conductivity the conjunction temperature can be significantly altered by the presence of a lubricating film. In elastohydrodynamic contacts the surfaces are separated by thin, low thermal conductivity films. The oil viscosity in these contacts varies with pressure from a low value at the entry side to a maximum at the centre to a low value at the exit. Consequently the force needed to shear the lubricating film will also vary along the contact. Thus in the middle of the lamellar elastohydrodynamic film heat is generated at a greater rate since the viscosity attains its highest value. In this region the rates of heat generation are proportional to viscosity [48] and are highest at the centre of the film. Variations in rates of heat generation will obviously affect the temperature distribution on the surfaces but due to a slow temperature response to these variations this effect will be small. The main effect, however, is in the increase of the maximum temperature in the oil film and this maximum temperature has the tendency to 340 ENGINEERING TRIBOLOGY move towards the centre of the parallel film [48]. Experimental measurements of temperature within an EHL film by infra-red spectroscopy have confirmed these theoretical predictions of a temperature maximum at the centre of the EHL film [52,53]. Since the middle of the parallel elastohydrodynamic film is where heat is generated and dissipated at the greatest rate, then the heat distribution between two solids depends on their thermal properties and the EHL film thickness. In effect two surfaces can have different temperatures as long as they are separated by a film. Their temperatures will be the same if the film disappears and the separation ceases to exist. The temperature profiles of a contact lubricated by an EHL film and a dry sliding contact are shown schematically in Figure 7.33. BODY A BODY B T a T b Surface temperatures EHL film Temperature profile at centre of film Temperatures in an EHL contact BODY A BODY B Temperatures in a dry contact T T T a = T b Identical temperatures at conjunction FIGURE 7.33 Temperature profiles in an EHL contact and in a dry contact. The difference in temperature between the centre of the lubricant film and the surfaces can be as large as 60°C or even higher in spite of the extreme thinness of an EHL film which is usually only 0.1 to 1 [µm] thick. The high temperatures occurring in the EHL contacts can explain why the EHL film, once formed, can fail, resulting, for example, in scuffing in gears. The temperatures within the EHL conjunction may be high enough for the lubricant to decompose and cause lubricating film failure. It is possible to discriminate between the temperature of the lubricating oil and the surface temperatures of a contacting solid in an EHL contact by the difference in emissivity between the oil and the metal or sapphire surfaces. An example of the temperature fields of the contacting surface and the peak oil temperature [52] are shown in Figure 7.34. The measurements were taken at the maximum contact stress of 1.05 [GPa], sliding speed 1.4 [m/s] and a naphthenic mineral oil was used as the lubricant. The most important difference between the temperature field of the surface of the contacting solid (a steel ball) and the lubricant oil temperature field is the much greater variability of the latter. The surface temperature varies at a near uniform gradient with position from a temperature maximum close to the exit restriction of the EHL contact. In contrast, the lubricant temperature field reveals a number of peaks or ‘hot spots’ distributed along the exit constriction of the EHL contact. The high lubricant temperature may be the result of intense viscous heat generation in the oil which has a much lower heat capacity/unit volume than steel. Under these experimental conditions the EHL film is on the point of sustaining thermally induced collapse or ‘burn out’ and this mode of failure may be the fundamental limit to all EHL films. Even when the EHL film is not subjected to high levels of sliding and load, thermal effects can still be significant. At high rolling speeds, viscous shear heating at the inlet will cause the lubricant to enter the EHL contact with reduced viscosity and a corresponding lowered film thickness [54]. The relation between film thickness and rolling speed as predicted by the Dowson-Higginson formulae (7.26 and 7.27) is an overestimate of the film thickness at high rolling speeds. Errors in the Dowson-Higginson formulae become significant at film ELASTOHYDRODYNAMIC LUBRICATION 341 thicknesses higher than 1 [µm] and rolling speeds greater than 10 [m/s] [54]. However, a maximum or limiting film thickness was not observed even for the highest of rolling speeds. Temperature distribution in the EHL film Temperature distribution on the contacting ball Hertzian contact radius (180 µm) 90°C Minimum film thickness Inlet Outlet 100° 120° 110° 130° 140° 150° 160° 170° 173° max 60° 70° 80° 90° 100° 110° FIGURE 7.34 Experimental measurements of surface temperature and oil temperature in an EHL contact (adapted from [52]). Mechanism of Heat Transfer Within the EHL Film Frictional heat is transmitted through lubricating oil film mainly by convection and conduction [55]. Other forms of heat transfer such as by bubble nucleation during boiling may occur in EHL contacts operating at a high sliding speed and load but these mechanisms of heat transfer have not yet been observed. As mentioned in Chapter 4, the balance between convection and conduction depends largely on the lubricating oil film thickness. An example of an approximate calculation of the relative importance of convection and conduction is shown below. EXAMPLE Find the ratio of convected to conducted heat in a rolling contact bearing operating with a surface velocity of U = 15.71 [m/s]. The contact width between the inner race and a roller is B = 0.0001 [m] and the film thickness is h = 0.5 [µm]. The bearing is lubricated by mineral oil of thermal diffusivity χ = 8.4 × 10 -6 [m 2 /s]. Substituting these values into equation (4.126) yields: = 0.084 × 10 −6 2 × 0.0001 15.71(0.5 × 10 −6 ) 2 H cond H conv = 4.278 This result implies that conduction is a far more significant mechanism than convection for thin EHL films as opposed to the much thicker lubricant films found in journal or pad bearings. Some reductions in EHL film temperatures may therefore be achieved by supplying lubricants with conductivities much greater than conventional oils. 342 ENGINEERING TRIBOLOGY Effect of Surface Films on Conjunction Temperatures Surface films of solids such as metal oxides can also affect the surface temperature to a varying degree, depending on their thermal properties. If a solid layer is a good heat conductor then the surface temperature will be lowered, whereas if its thermal conductivity is low relative to the bulk material then the surface temperature will be increased. Significant modification of frictional temperature rises are, however, only found when the thickness of solid film material is much greater than molecular dimensions [47,48]. Films of solid material, particularly oxides, are also instrumental in controlling the friction coefficient, but this aspect of frictional temperatures in dry or lubricated contacts is discussed in later chapters. Measurements of Surface Temperature in the EHL Contacts The temperature in EHL contacts has recently been measured by infra-red spectroscopy. One of the contacting surfaces was made of material transparent to infra-red radiation and hard enough to sustain high Hertzian contact stresses. Sapphire or diamond has been used for this purpose [56,57] in an arrangement shown schematically in Figure 7.35. The high shear stresses and shear rates prevailing in an EHL contact and the relatively small volume of liquid available to dissipate the frictional heating ensure that high contact temperatures are reached even when very little power is dissipated by friction. An example of a surface temperature rise above bulk inlet temperature, measured by infra-red spectroscopy through the centre of a sphere-on-plane contact, is shown in Figure 7.36 [56]. The lubricant used in the experiments was a synthetic perfluoroether tested at sliding speeds ranging from 0.5 to 2 [m/s] and a maximum contact pressure of 1.05 [GPa] [56]. Sliding velocity EHL film Infra-red radiation IR radiometric microscope Sapphire disc Hertzian width FIGURE 7.35 Schematic diagram of the apparatus for the determination of surface temperature profile by infra-red spectroscopy [56]. A temperature rise approaching 80°C at the centre of the contact was found even under the relatively mild conditions of the test. However, the exact mechanism by which contact temperatures can prevent effective EHL is still poorly understood [56-58]. A fundamental limitation of the infra-red spectroscopic measurement of surface temperature is the need for a special transparent window to function as one of the contacting surfaces. This requirement precludes common engineering materials such as steel from being used for both contacting surfaces. A conventional thermocouple embedded in the contacting material is unsuitable for the measurements because the temperature rise is confined to the surface. The only way that a thermocouple can be used with accuracy is if a lamellar thermocouple is attached to the surface. A lamellar thermocouple is made by depositing on the surface successive thin films approximately 0.1 [µm] thick of insulants and two metals. The specialized form of thermocouple required to measure flash temperatures is illustrated schematically in Figure 7.37. ELASTOHYDRODYNAMIC LUBRICATION 343 x b x = -2 -1 0 1 2 Surface temperature rise [°C] 0 10 20 30 40 50 60 70 80 0.5 1.0 2.0 Inlet Outlet Hertzian width Sliding speed [m/s] Bulk oil temperature = 60°C FIGURE 7.36 Surface temperature profiles within an EHL contact determined by infra-red spectroscopic measurements [56]. Bimetallic couple Insulating layer of e.g. alumina 0.1µm 0.1µm Contact width V FIGURE 7.37 Lamellar thermocouple suitable for the measurement of flash temperature. The lamellar thermocouple requires elaborate coating equipment for its manufacture and is not very durable against wear. The measurement of surface temperature can be a very difficult experimental task and for most studies it is more appropriate to estimate it by calculating the range of surface temperatures that may be found in the particular EHL contact. 7.7 TRACTION AND EHL Traction is the application of frictional forces to allow the transmission of mechanical energy rather than its dissipation. The most common example of the distinction between traction and friction is the contact between a wheel and a road. When the wheel rolls without skidding, traction is obtained and the frictional forces available enable propulsion of a vehicle. When skidding occurs, the same frictional forces will now dissipate any mechanical energy applied to the wheel. Thus the difference between traction and friction is in the way that the mechanical energy is processed, e.g. in the case of traction this energy is transmitted between the contacting bodies (i.e. one body is driving another) whereas with friction it is dissipated. Traction can also be applied to lubricated contacts in spite of the relatively low coefficients of friction involved. EHL contacts can provide sufficiently high traction to be used as interfaces for variable speed transmissions. Unique features of variable speed transmissions such as infinitely variable output speed, almost a constant torque over the speed range and low noise make them particularly attractive for applications in computers, 344 ENGINEERING TRIBOLOGY machine tools and the textile industry, or even in motor vehicles. A lubricated contact is selected to suppress wear which would otherwise shorten the lifetime of the transmission. The operating principles of these transmissions are shown schematically in Figure 7.38. Input shaft Output shaft Input shaft Output shaft S p eed reduction S p eed increase Tractional contact FIGURE 7.38 Schematic diagram of the operating principles of a variable speed toroidal transmission. The level of traction in an EHL contact also accelerates wear of the corresponding rolling/sliding elements and traction should therefore be controlled wherever possible. This topic is discussed further in later chapters. It is possible to analyze the traction force in an EHL contact and obtain a good agreement between theory and experiment although the models involved are fairly complex [55,59]. A basic simplification used in most analyses of traction is to assume a uniform film thickness inside the EHL contact and ignore the end constriction. This is a film geometry similar to Grubin's original model of EHL. When traction is applied, there is a small but non-zero sliding speed between the contacting surfaces. This non-zero sliding speed is inevitable since all the tractional force in an EHL contact is the result of viscous shear. The envisaged simplified film geometry and velocity profiles of the sheared lubricant are shown in Figure 7.39. Hertzian contact pressure U 2 U 1 Inlet h Traction force 2b Traction force Viscous shearing Outlet FIGURE 7.39 Simplified film geometry and generation of traction in an EHL contact. ELASTOHYDRODYNAMIC LUBRICATION 345 From an elementary analysis of the relationship between shear rate and shear stress in a Newtonian fluid, discussed in Chapter 2, it can be seen that the traction force is a product of contact area, local viscosity and velocity difference between the surfaces divided by film thickness, i.e.: F = η × A × ∆U/h (7.45) For the purposes of argument it is assumed that either the local viscosity remains constant in the EHL contact or that its average value can be found. In more refined analyses, however, the local variation of viscosity is included. Under conditions of constant load, geometry and lubricant characteristics, the contact area, local viscosity and film thickness remain almost invariant. A ‘coefficient of traction’ is obtained by dividing the traction force by load, i.e.: µ T = F/W (7.46) where: µ T is the traction coefficient; F is the traction force [N]; W is the contact load [N]. Substituting for traction force (7.45) yields: µ T = ηA∆U / hW = κ × ∆U (7.47) where: η is the dynamic viscosity of the lubricant [Pas]; A is the contact area [m 2 ]; ∆U is the surface velocity (i.e. velocity difference between the contacting surfaces) [m/s]; h is the film thickness [m]; κ is constant defined as: κ = ηA / hW [s/m]. The velocity difference is often normalized as a coefficient which is obtained by division with the larger velocity. This coefficient is known in the literature as the ‘slide to roll ratio’ and is defined as: ∆U/U = (U A - U B )/U A where: ∆U is the velocity difference [m]; U A , U B are the surface velocities of body ‘A’ and ‘B’ respectively [m]. The relationship between the traction coefficient and the slide to roll ratio then is: µ T = κ' × ∆U/U A where: κ' is the coefficient defined as: κ' = κU A . The velocity difference between the contacting surfaces is usually extremely small and for EHL traction systems a single ‘velocity’ value is often given in the literature instead of accurate values of ‘U A ’ and ‘U B ’. [...]... 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