Engineering Tribology Episode 2 Part 3 pps

HYDROSTATIC LUBRICATION 275 Gas supply from pump Gas flow to bearing Flexible diaphragm Preload spring FIGURE 6.8 Diaphragm pressure sensor valve developed by Mohsin [5]. There are two basic limitations to diaphragm bearings; it is necessary to adjust the spring force on the diaphragm to match the bearing stiffness and high stiffness is only maintained up to a certain load where bearing deflection rises sharply with any further increase in load. More detailed information about pressure sensing valves can be found in [2]. 6.5 AEROSTATIC BEARINGS In some applications a gas is used to lubricate the bearings and these are known as aerostatic bearings. The mechanism of film generation is the same as in liquid bearings. Gas lubricated bearings offer some advantages such as: · gas viscosity increases with temperature thus reducing heating effects during over- load or abnormal operating conditions, · some gases are chemically stable over a wider temperature range than hydrocarbon lubricants, · a non-combustible gas eliminates the fire hazard associated with hydrocarbons, · if air is selected as the hydrostatic lubricant, then it is not necessary to purchase or recycle the lubricant, · gases can offer greater cleanliness and non-toxicity than fluid lubricants. The viscosities of some gases used in aerostatic bearings are shown in Table 6.1. T ABLE 6.1 Viscosities of various gases at 20°C and 0.1 [MPa] pressure. Gas Viscosity [Pas] Hydrogen 8.80 × 10 -6 Helium 1.96 × 10 -5 Nitrogen 1.76 × 10 -5 Oxygen 2.03 × 10 -5 Carbon Dioxide 1.47 × 10 -5 Air 1.82 × 10 -5 TEAM LRN 276 ENGINEERING TRIBOLOGY Aerostatic bearings are used in the precision machine tool industry, metrology, computer peripheral devices and in dental drills, where the air used in the bearing also drives the drill. They are particularly useful for high speed applications and where precision is required since very thin film thicknesses are possible. The main disadvantage, however, is that the load capacity of gas lubricated bearings or aerostatic bearings is much lower than the load capacity of fluid lubricated bearings of the same size. The most commonly used gas is air, but other gases such as carbon dioxide and helium have been used in specialized systems, e.g. nuclear technology. The analysis of aerostatic bearings is very similar to liquid hydrostatic bearings. The main difference, however, is that the gas compressibility is now a distinctive feature and has to be incorporated into the analysis. Since the pressures generated in these bearings are much lower than in liquid lubricated bearings, ambient pressure cannot be neglected and is also included in the analysis. For example, assuming isothermal behaviour of the gas, the main performance parameters of the aerostatic flat circular pad bearings can be calculated from the following formulae [6]. Pressure Distribution The pressure distribution is found from a simplified form of the compressible Reynolds equation for radial coordinates with angular symmetry and negligible sliding velocity. The derivation of the pressure distribution can be found in [5] and the final form of the equation for pressure at any given radial position between the recess and bearing edge is: p = p s ln(R/R 0 ) [ [ 0.5 ln(r/R 0 ) 1 − [ [ p r p a 1 − ( ( 2 (6.27) where: p s is the supply pressure [Pa]; p r is the recess pressure [Pa]; p a is the ambient pressure [Pa]; R is the outer radius of the bearing [m]; R 0 is the radius of the recess [m]; r is the radius from the centre of the circular thrust bearing [m]. In spite of the apparent complexity of the above expression, the pressure distribution approximates to a linear decline of pressure with distance from the recess. Gas Flow The gas flow in flat circular aerostatic bearings is given by the expression: Q = πh 3 12ηln(R /R 0 ) p r 2 − p a 2 p r (6.28) where: h is the film thickness [m]; η is the gas dynamic viscosity [Pas]; TEAM LRN HYDROSTATIC LUBRICATION 277 Q is the gas flow (i.e. amount of gas required to maintain the film thickness ‘h’ in a bearing) [m 3 /s]. Load Capacity The expression for load capacity is found by integrating the pressure distribution over the bearing area. The resulting expression is very complex and is available in [5]. To estimate the load capacity it is useful to introduce a proportionality constant between the load capacity and nominal load based on recess pressure and bearing area, i.e.: K* = W p r πR 2 (6.29) where: W is the bearing load capacity [N]; p r is the recess pressure [Pa]; R is the outer radius of the bearing [m]. Values of ‘K*’ as a function of the ratio of bearing radius to recess radius are shown in Figure 6.9. Radius ratio 1 2 5 10 20 50 100 R/R 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Load ratio K* FIGURE 6.9 Values of load factor as a function of bearing geometry for a circular pad aerostatic thrust bearing. Friction Torque The friction torque or friction coefficient of a gas bearing is not usually considered because its value is extremely small unless the bearing is operating at a very high sliding speed. If it is necessary to estimate a friction torque, a Petroff approximation can be used. In other words, the shear rate should be multiplied by the gas viscosity and bearing area. The viscosity of most gases is not greatly affected by temperature so that the ambient temperature value of viscosity can safely be used even if the gas leaves the bearing at a high temperature. At high sliding speeds, turbulent flow of gas in the bearing may occur and this should be checked by computing the local Reynolds number based on film thickness. The friction torque for flat TEAM LRN 278 ENGINEERING TRIBOLOGY circular pad bearings can be calculated from equation (6.11) provided that laminar flow prevails. EXAMPLE Calculate the load capacity of a 0.1 [m] radius flat circular aerostatic bearing with radius ratio of 2, and recess pressure of 1 [MPa]. From Figure 6.9, K* = 0.63. The load capacity is then: W = π × 10 6 × 0.1 2 × 1 × 0.63 = 19 792 [N]. Power Loss The power loss for most slow sliding speed gas bearings is determined by the pumping power required since the friction power loss is extremely small. As mentioned already the pumping power is the product of gas flow rate and the total pressurization required (including losses in capillaries and supply lines). If the friction power loss is significant then this should be added to the pumping power loss to give the total power loss. The friction power loss for flat circular pad bearings can be calculated from equation (6.12). 6.6 HYBRID BEARINGS Hybrid bearings function by the combined action of hydrostatic and hydrodynamic lubrication. A bearing technology resembling a hybrid bearing was described in Chapter 4 where a pressurized lubricant supply is used to prevent metallic contact during starting or stopping of a Michell pad bearing. The principle of augmenting hydrodynamic lubrication with a hydrostatic effect or vice versa has been developed further than this limited application. The distinction between a true ‘hybrid bearing’ and the pressurized lubricant supply to a journal bearing is that in the latter case, the purpose of the extra supply is to supply cool lubricant into the hottest part of the bearing. An important difference between hybrid and hydrostatic bearings is the absence of recesses in the hybrid bearings. Recesses cause reduced hydrodynamic pressures in the loaded parts of the bearing which are where hydrostatic gas or liquid outlets are usually positioned. As discussed previously, lubricant supply outlets are usually located remote from the loaded part of the bearing for efficient hydrodynamic lubrication. An example of a hybrid journal bearing is shown in Figure 6.10. Where several lubricant outlets are used it is important to avoid interconnection of the supply lines. If the supply lines are connected then recirculating flow of lubricant will occur which reduces the hydrodynamic pressure. The basic parameters of these bearings such as pumping power and size are designed as for a hydrostatic bearing and any hydrodynamic effect which improves bearing performance is regarded as a bonus [2]. 6.7 STABILITY OF HYDROSTATIC AND AEROSTATIC BEARINGS Hydrostatic bearings are subject to vibrational instability particularly under variable loads or where a gas is used as the lubricant. The mechanism causing vibrational instability is the same as discussed already in hydrodynamic lubrication, i.e. a resonance dependent on the stiffness and damping coefficients of the load-carrying film and the coupled mass. ‘Oil whirl’ can also occur in hydrostatic and hybrid journal bearings at high speeds in a similar manner TEAM LRN HYDROSTATIC LUBRICATION 279 to hydrodynamic bearings. Most high-speed externally pressurized bearings are aerostatic and the analysis of the corresponding vibrational stability is highly specialized. For the practical prevention of vibrational instability it is important to minimize the depth of the recess in a hydrostatic bearing since this increases the storage capacity of energy in the bearing, particularly so if gas is used. The depth of the recess should only be a small multiple of the design film thickness, e.g. only 10 to 20 times larger but not 100 times larger. A high bearing stiffness can raise the critical vibration speed in these bearings. The bearing stiffness may be increased by supplying a gas under higher pressure. The method of gas or liquid supply is also important, restrictors and capillary compensation are associated with vibration problems. More information on bearing stability can be found in [2] and [5]. a) b) FIGURE 6.10 Hybrid journal bearing; a) with slots as gas outlets, b) with holes as gas outlets. 6.8 SUMMARY Hydrostatic lubrication provides complete separation of sliding surfaces to ensure zero or negligible wear and very low friction. Hydrostatic lubrication is based on the same physical principles as hydrodynamic lubrication but has certain fundamental differences. There is no friction force at infinitesimal sliding speeds unlike hydrodynamic lubrication which is a uniquely useful characteristic in the design and operation of precision control systems. The disadvantage of hydrostatic lubrication is a complete reliance on an external pressurized supply of lubricant which means that the pump must be reliable and the supply lines free of dirt that might block the flow of lubricant. Hydrostatic lubrication with a gas, which is known as aerostatic lubrication, can provide very low friction even at extremely high sliding speeds because of the low viscosity of gases. Quasi-ideal characteristics of zero wear and friction are obtained with hydrostatic or aerostatic lubrication at low to medium contact stresses but a more complicated technology, e.g. the application of an external high pressure pump, is required in comparison to other forms of lubrication. Bearing stiffness in these bearings can also be manipulated more easily than with other types of bearings to suit specific design requirements. REFERENCES 1 L.G. Girard, Nouveaux Systeme de Locomotion Sur les Chemins de Fer, Bachelier, Paris, 1852, pp. 40. 2 W. B. Rowe, Hydrostatic and Hybrid Bearing Design, Butterworths, 1983. 3 A. Cameron, Basic Lubrication Theory, Ellis Horwood Ltd, 1981. 4 J.P. O'Donoghue and W.B. Rowe, Hydrostatic Bearing Design, Tribology, Vol. 2, 1969, pp. 25-71. TEAM LRN 280 ENGINEERING TRIBOLOGY 5 M.E. Mohsin, The Use of Controlled Restrictors for Compensating Hydrostatic Bearings, Advances in Machine Tool Design and Research, Pergamon Press, Oxford, 1962. 6 W.A. Gross, Fluid Film Lubrication, John Wiley and Sons, 1980. 7 J.N. Shinkle and K.G. Hornung, Frictional Characteristics of Liquid Hydrostatic Journal Bearings, Transactions ASME, Journal of Basic Engineering, Vol. 87, 1965, pp. 163-169. 8 H. Opitz, Pressure Pad Bearings, Conf. Lubrication and Wear, Fundamentals and Application to Design, London, 1967, Proc. Inst. Mech. Engrs., Vol. 182, Pt. 3A, 1967-1968, pp. 100-115. 9 R. Bassani and B. Piccigallo, Hydrostatic Lubrication: Theory and Practice, Elsevier, Amsterdam, 1992. 10 J.K. Scharrer and R.I. Hibs Jr., Flow Coefficients for the Orifice of a Hydrostatic Bearing, Tribology Transactions, Vol. 33, 1990, pp. 543-550. TEAM LRN ELASTOHYDRODYNAMIC 7 LUBRICATION 7.1 INTRODUCTION Elastohydrodynamic lubrication can be defined as a form of hydrodynamic lubrication where the elastic deformation of the contacting bodies and the changes of viscosity with pressure play fundamental roles. The influence of elasticity is not limited to second-order changes in load capacity or friction as described for pivoted pad and journal bearings. Instead, the deformation of the bodies has to be included in the basic model of elastohydrodynamic lubrication. The same refers to the changes in viscosity due to pressure. The existence of elastohydrodynamic lubrication was suspected long before it could be proved or described using specific scientific concepts. The lubrication mechanisms in conformal contacts such as those encountered in hydrodynamic and hydrostatic bearings were well described and defined and the reasons for their effectiveness were well understood. However, the mechanism of lubrication operating in highly loaded non-conformal contacts, such as those which are found in gears, rolling contact bearings, cams and tappets, although effective was poorly understood. The wear rates of these devices were very low which implied the existence of films sufficiently thick to separate the opposing surfaces, yet this conclusion was in direct contradiction to the calculated values of hydrodynamic film thicknesses. The predicted values of film thickness were so low that it was inconceivable for the contacting surfaces to be separated by a viscous liquid film. In fact, the calculated film thicknesses suggested that the surfaces were lubricated by films only one molecule in thickness. In experiments specifically designed to permit only lubrication by monomolecular films, much higher wear rates and friction coefficients were obtained. This apparent contradiction between the empirical observation of effective lubrication and the limits of known lubrication mechanisms could not be explained for a considerable period of time. The entire problem acquired an aura of mystery and many elaborate experiments and theories were developed as a result. From the view point of an engineer, the answers to the questions of what controls the lubrication mechanism and how it can be optimized are very important, since heavily loaded point contacts are often found, and provision for effective lubrication of these contacts is critical. In the 1940's a substantial amount of work was devoted to resolving elastohydrodynamics and the first realistic model which provided an albeit approximate solution for elastohydrodynamic film thickness was proposed by Ertel and Grubin. The work was published by Grubin in 1949 [1]. It was found that the combination of three effects: TEAM LRN 282 ENGINEERING TRIBOLOGY hydrodynamics, elastic deformation of the metal surfaces and the increase in the viscosity of oil under extreme pressures are instrumental to this mechanism. This lubrication regime is referred to in the literature as elastohydrodynamic lubrication which is commonly abbreviated to EHL or EHD. At this stage, it should be realized that elastohydrodynamic lubrication is effectively limited to oils as opposed to other viscous liquids because of the pressure-viscosity dependence. It was shown theoretically that under conditions of intense contact stress a lubricating oil film can be formed. The lubricated contacts in which these three effects take place are said to be operating elastohydrodynamically, which effectively means that the contacting surfaces deform elastically under the hydrodynamic pressure generated in the layer of lubricating film. The lubricating films are very thin, in the range of 0.1 to 1 [µm], but manage to separate the interacting surfaces, resulting in a significant reduction of wear and friction. Although this regime generally operates between non- conforming surfaces, it can also occur under certain circumstances in the contacts classified as conformal such as highly loaded journal and pad bearings which have a significant component of contact and bending deformation. However, enormous loads are required for this to occur and very few journal or pad bearings operate under these conditions. Significant progress has been made towards a complete understanding of the mechanism of elastohydrodynamic lubrication. The pioneering work in this field was conducted by Martin (1916) [2], followed by Grubin (1949) [1] and was continued by Dowson and Higginson [3], Crook [4], Cameron and Gohar [5] and others. In this chapter the fundamental mechanisms of film generation in elastohydrodynamic contacts, together with the methods for calculating the minimum film thickness in rolling bearings and gears will be outlined. Some particular characteristics of elastohydrodynamic contacts such as traction and flash temperature will be discussed, along with the methods of their evaluation. 7. 2 CONTACT STRESSES From elementary mechanics it is known that two contacting surfaces under load will deform. The deformation may be either plastic or elastic depending on the magnitude of the applied load and the material's hardness. In many engineering applications, for example, rolling contact bearings, gears, cams, seals, etc., the contacting surfaces are non-conformal hence the resulting contact areas are very small and the resulting pressures are very high. From the view point of machine design it is essential to know the values of stresses acting in such contacts. These stresses can be determined from the analytical formulae, based on the theory of elasticity, developed by Hertz in 1881 [6-8]. Hertz developed these formulae during his Christmas vacation in 1880 when he was 23 years old [7]. Simplifying Assumptions to Hertz's Theory Hertz's model of contact stress is based on the following simplifying assumptions [6]: · the materials in contact are homogeneous and the yield stress is not exceeded, · contact stress is caused by the load which is normal to the contact tangent plane which effectively means that there are no tangential forces acting between the solids, · the contact area is very small compared with the dimensions of the contacting solids, · the contacting solids are at rest and in equilibrium, · the effect of surface roughness is negligible. TEAM LRN ELASTOHYDRODYNAMIC LUBRICATION 283 Subsequent refinements of Hertz's model by later workers have removed most of these assumptions, and Hertz's theory forms the basis of the model of elastohydrodynamic lubrication. Stress Status in Static Contact Consider two bodies in contact under a static load and with no movement relative to each other. Since there is no movement between the bodies, shearing does not occur at the interface and therefore the shear stress acting is equal to zero. According to the principles of solid mechanics, the planes on which the shear stress is zero are called the principal planes. Thus the interface between two bodies in a static contact is a principal plane on which a principal stress ‘σ 1 ’ is the only stress acting. It is also known from solid mechanics that the maximum shear stress occurs at 45° to the principal plane, as shown in Figure 7.1. If the contact load is sufficiently high then the maximum shear stress will exceed the yield stress of the material, i.e. τ max > k, and plastic deformation takes place. Material will then deform (slip) along the line of action of maximum shear stress. The maximum shear stress ‘τ max ’, also referred to in the literature as ‘τ 45° ’ since it acts on planes inclined at 45° to the interface (in static contacts), is given by: τ max = τ 45° =±k = ± σ 1 − σ 3 2 ( ( (7.1) The stresses ‘σ x ’, ‘σ z ’ and ‘τ’ vary with depth below the interface. An example of the stress field beneath the surface of two parallel cylinders in static contact is shown in Figure 7.2 [9]. For a circular contact, the subsurface stress field is very similar [9]. It can be seen from Figure 7.2 that the maximum shear stress ‘τ 45° ’ occurs at some depth below the surface. This depth depends on load and is therefore related to the contact area. In a circular contact for example, the maximum shear stress occurs at approximately 0.6a, where ‘a’ is the radius of the contact area. Stress Status in Lubricated Rolling and Sliding Contacts In a lubricated rolling contact, the contact stresses are affected by the lubricating film separating the opposing surfaces and the level of rolling and sliding. The hydrodynamic film generated under these conditions and the relative movement of the surfaces cause significant changes to the original static stress distribution and will be discussed later. Rolling, in general, results in an increase in contact area and a subsequent modification of the Hertzian stress field in both dry and lubricated conditions. The most critical influence on subsurface stress fields, however, is exerted by sliding. To illustrate the effect of sliding on the stress distribution, consider two bodies in contact with some sliding occurring between them. Frictional forces are the inevitable result of sliding and cause a shear stress to act along the interface, as shown in Figure 7.3. The frictional stress acting at the interface is balanced by rotating the planes of principal stresses through an angle ‘φ’ from their original positions when frictional forces are absent. The magnitude of the angle ‘φ’ depends on the frictional stress µq acting at the interface according to the relation: φ = 1/2cos −1 (µq/k) TEAM LRN 284 ENGINEERING TRIBOLOGY p p kk σ 1 σ 1 σ 3 W Body A Body B Interface 45° FIGURE 7.1 Stress status in a static contact; σ 1 , σ 3 are the principal stresses, p is the hydrostatic pressure, k is the shear yield stress of the material. x b σ p max -1 0 1 τ max p max σ x p max σ z p max z b 1 2 0.173 0.236 0.267 0.283 0.290 0.295 0.300 0.251 FIGURE 7.2 Subsurface stress field for two cylinders in static contact; p max is the maximum contact pressure, b is the half width of the contact rectangle [9]. The variation with depth below the interface of the principal shear stress ‘τ max ’ for a cylinder and the plane on which it slides is shown in Figure 7.4. The contours show the principal shear stress due to the combined normal pressure and tangential stress for a coefficient of friction µ = 0.2 [9]. It can clearly be seen that as friction force increases, the maximum shear stress moves towards the interface. Thus there is a gradual increase in shear stress acting at the interface as the friction force increases. This phenomenon is very important in crack formation and the subsequent surface failure and will be discussed later. 7.3 CONTACT BETWEEN TWO ELASTIC SPHERICAL OR SPHEROIDAL BODIES Elastic bodies in contact deform and the contact geometry, load and material properties determine the contact area and stresses. The contact geometry depends on whether the contact occurs between surfaces which are both convex or a combination of flat, convex and TEAM LRN [...]... 11 + E' 2 EA EB 2 2.1 × 10 2. 1 × 1011 · Contact Area Dimensions a= · ( 3 WR' E' 3 × 5 × (3 × 10 3) 2. 30 8 × 1011 ) 1/ 3 = 5.799 × 10−5 [m] = 709.9 [MPa] W 5 = π a2 π (5.799 × 10−5 )2 = 4 73. 3 [MPa] Maximum Deflection δ = 1. 039 7 ( ) ( ) W2 1/ 3 52 = 1. 039 7 2 E' R' (2. 30 8 × 1011 )2 3 × 10 3 1 /3 = 5.6 × 10−7 [m] Maximum Shear Stress τmax = · = 3W 3 5 = 2 π a2 2 π (5.799 × 10−5 )2 paverage = · ) ( 1/ 3 Maximum... 1 1 1 = + = 100 + 100 = 20 0 R' Rx Ry · ⇒ R' = 5 × 10 3 [m] Reduced Young's Modulus E' = 2. 30 8 × 1011 [Pa] TEAM LRN 29 4 ENGINEERING TRIBOLOGY · Contact Area Dimensions a= · ( 3 WR' E' ) ( 1/ 3 = 3 × 5 × (5 × 10 3) 2. 30 8 × 1011 ) 1/ 3 = 6.88 × 10−5 [m] Maximum and Average Contact Pressures pmax = 3W 3 5 = 2 π a2 2 π (6.88 × 10−5 )2 = 504.4 [MPa] W 5 = π a2 π (6.88 × 10−5 )2 = 33 6 .2 [MPa] paverage = · Maximum... 1. 039 7 W2 1/ 3 E '2 R' ) 1 /3 52 −4 (2. 30 8 × 10 ) × (7.5 × 10 ) 11 2 = 8.9 × 10−7 [m] Maximum Shear Stress τmax = · ) 3 × 5 × (7.5 × 10−4) 1 /3 2. 30 8 × 1011 3W 3 5 = 2 π a2 2 π (3. 65 × 10−5 )2 paverage = · = Maximum and Average Contact Pressures pmax = · ) ( 1/ 3 1 1 p = 1791.9 3 max 3 = 597 .3 [MPa] Depth at which Maximum Shear Stress Occurs z = 0. 638 a = 0. 638 × (3. 65 × 10−5) = 2. 3 × 10−5 [m] TEAM LRN ELASTOHYDRODYNAMIC... 10 3 15 × 10 3 ⇒ Rx = 6 × 10 3 [m] 1 1 1 1 1 = + = + = 166.67 Ry Ray Rby 10 × 10 3 15 × 10 3 ⇒ Ry = 6 × 10 3 [m] Note that 1/R x = 1/Ry, i.e condition (7 .3) is satisfied (circular contact), and the reduced radius of curvature is: TEAM LRN 29 2 ENGINEERING TRIBOLOGY 1 1 1 = 166.67 + 166.67 = 33 3 .34 = + R' Rx Ry · ⇒ R' = 3 × 10 3 [m] Reduced Young's Modulus [ ] [ ] 1 1 − υA2 1 − υB2 1 1 − 0. 32 1 − 0. 32 . .. 10 3 ∞ ⇒ Ry = 0.0015 [m] Since 1/Rx = 1/Ry condition (7 .3) is satisfied and the reduced radius of curvature is: 1 1 1 = + = 666.67 + 666.67 = 133 3 .34 R' Rx Ry · ⇒ R' = 7.5 × 10−4 [m] Reduced Young's Modulus E' = 2. 30 8 × 1011 [Pa] · Contact Area Dimensions a= · ( 3 WR' E' = 3. 65 × 10−5 [m] = 1791.9 [MPa] W 5 = π a2 π (3. 65 × 10−5 )2 = 1194.6 [MPa] Maximum Deflection δ = 1. 039 7 ( ) ( = 1. 039 7 W2 1/ 3 E '2. .. 10 3) paverage = · ) ( 1/ 2 = 55.4 [MPa] W 5 = 4bl 4 × (5.75 × 10−6) × (5 × 10 3) = 43. 5 [MPa] Maximum Deflection δ = 0 .31 9 = 0 .31 9 [ ][ [ W E'l ( )] 2 4 RA RB + ln 3 b2 ][ ( )] 5 2 4 × (10 × 10 3) × (15 × 10 3) + ln (2. 30 8 × 1011) × (5 × 10 3) 3 (5.75 × 10−6 )2 = 2. 40 × 10−8 [m] · Maximum Shear Stress τmax = 0 .30 4 pmax = 0 .30 4 × 55.4 · = 16.8 [MPa] Depth at which Maximum Shear Stress Occurs z = 0.786... = 504.4 [MPa] W 5 = π a2 π (6.88 × 10−5 )2 = 33 6 .2 [MPa] paverage = · Maximum Deflection δ = 1. 039 7 · ( ) ( ) W2 1/ 3 52 = 1. 039 7 2 E' R' (2. 30 8 × 1011 )2 5 × 10 3 1 /3 = 4.7 × 10−7 [m] Maximum Shear Stress τmax = · 1 1 pmax = 504.4 3 3 = 168.1 [MPa] Depth at which Maximum Shear Stress Occurs z = 0. 638 a = 0. 638 × (6.88 × 10−5) = 4.4 × 10−5 [m] · Contact Between Two Parallel Cylinders The contact area... parameters between two spheres Contact area dimensions a= ( circle a 3 WR' E' Maximum contact pressure ) 1/ 3 pmax = 3W 2 π a2 Average contact pressure paverage = Maximum deflection W π a2 Hemispherical pressure distribution δ = 1. 039 7 Maximum shear stress ( ) W2 1 /3 E '2 R' τmax = 1 p 3 max at a depth of z = 0. 638 a TEAM LRN 29 0 ENGINEERING TRIBOLOGY where: a is the radius of the contact area [m]; W is the... summarized in Table 7 .3 TABLE 7 .3 Formulae for contact parameters between two elastic bodies; elliptical contacts, general case Contact area dimensions a = k1 Maximum contact pressure ( ( 3W R' E' b = k2 3W R' E' ellipse ) ) 1/ 3 pmax = 3W 2 π ab Average contact pressure paverage = Maximum deflection W π ab δ = 0.52k3 Maximum shear stress ( ) W2 1 /3 E '2 R' 1/ 3 b τmax = k4 pmax ≈ 0.3pmax at a depth of... ) 1/ 3 , δ = 1 .31 ( ) W2 1/ 3 E '2 R' EXAMPLE Find the contact parameters for two steel balls The normal force is W = 5 [N], the radii of the balls are RA = 10 × 10 -3 [m] and RB = 15 × 10 -3 [m] The Young's modulus for both balls is E = 2. 1 × 1011 [Pa] and the Poisson's ratio of steel is υ = 0 .3 · Reduced Radius of Curvature Since Rax = Ray = RA = 10 × 10 -3 [m] and Rbx = Rby = RB = 15 × 10 -3 [m] . −υ B 2 E B =+ 1 − 0 .3 2 2.1 × 10 11 [] 1 2 1 − 0 .3 2 2.1 × 10 11 ⇒ E' = 2. 30 8 × 10 11 [Pa] · Contact Area Dimensions a = 3WR' E' () 1 /3 = 3 × 5 × (3 × 10 3 ) 2. 30 8 × 10 11 () 1 /3 = 5.799. p max = 3W 2 a 2 = 3 × 5 2 (5.799 × 10 −5 ) 2 = 709.9 [MPa] p average = W πa 2 = 5 π(5.799 × 10 −5 ) 2 = 4 73. 3 [MPa] · Maximum Deflection δ= 1. 039 7 W 2 E' 2 R' () 1 /3 = 1. 039 7 5 2 (2. 30 8. LRN 29 2 ENGINEERING TRIBOLOGY 1 R' =+ 1 R x 1 R y = 166.67 + 166.67 = 33 3 .34 ⇒ R' = 3 × 10 3 [m] · Reduced Young's Modulus = 1 E' + 1 −υ A 2 E A [] 1 2 1 −υ B 2 E B =+ 1