Mechanical Engineering-Tribology In Machine Design Episode 5 pps

Elements of contact mechanics 87 where y =[1+0.87~-~]-' and y ranges from 0.72 at L= 5 to 0.92 at L = 100. 3.8. contact between There are no topographically smooth surfaces in engineering practice. Mica rough surfaces can be cleaved along atomic planes to give an atomically smooth surface and two such surfaces have been used to obtain perfect contact under laboratory conditions. The asperities on the surface of very compliant solids such as soft rubber, if sufficiently small, may be squashed flat elastically by the contact pressure, so that perfect contact is obtained through the nominal contact area. In general, however, contact between solid surfaces is discontinuous and the real area ofcontact is a small fraction of the nominal contact area. It is not easy to flatten initially rough surfaces by plastic deformation of the asperities. The majority of real surfaces, for example those produced by grinding, are not regular, the heights and the wavelengths of the surface asperities vary in a random way. A machined surface as produced by a lathe has a regular structure associated with the depth of cut and feed rate, but the heights of the ridges will still show some statistical variation. Most man- made surfaces such as those produced by grinding or machining have a pronounced lay, which may be modelled, to a first approximation, by one- dimensional roughness. It is not easy to produce wholly isotropic roughness. The usual procedure for experimental purposes is to air-blast a metal surface with a cloud of fine particles, in the manner of shot-peening, which gives rise to a randomly cratered surface. 3.8.1. Characteristics of random rough surfaces The topographical characteristics of random rough surfaces which are relevant to their behaviour when pressed into contact will now be discussed briefly. Surface texture is usually measured by a profilometer which draws a stylus over a sample length of the surface of the component and reproduces a magnified trace of the surface profile. This is shown schematically in Fig. 3.9. It is important to realize that the trace is a much distorted image of the actual profile because of using a larger magnification in the normal than in the tangential direction. Modern profilometers digitize the trace at a suitable sampling interval and send the output to a computer in order to extract statistical information from the data. First, a datum or centre-line is established by finding the straight line (or circular arc in the case of round components) from which the mean square deviation is at a minimum. This implies that the area of the trace above the datum line is equal to that below it. The average roughness is now defined by Figure 3.9 88 Tribology in machine design where z(x) is the height ofthe surface above the datum and L is the sampling length. A less common but statistically more meaningful measure of average roughness is the root mean square (r.m.s.) or standard deviation a of the height of the surface from the centre-line, i.e. The relationship between a and R, depends, to some extent, on the nature of the surface; for a regular sinusoidal profile a = (n/2 JZ)R, and for a Gaussian random profile a = (n/2)*~,. The R, value by itself gives no information about the shape of the surface profile, i.e. about the distribution of the deviations from the mean. The first attempt to do this was by devising the so-called bearing area curve. This curve expresses, as a function of the height z, the fraction of the nominal area lying within the surface contour at an elevation z. It can be obtained from a profile trace by drawing lines parallel to the datum at varying heights, z, and measuring the fraction of the length of the line at each height which lies within the profile (Fig. 3.10). The bearing area curve, however, does not give the true bearing area when a rough surface is in contact with a smooth flat one. It implies that the material in the area of interpenetration vanishes and no account is taken of contact deformation. An alternative approach to the bearing area curve is through elementary statistics. If we denote by $(z) the probability that the height of a particular point in the surface will lie between z and z +dz, then the probability that the height of a point on the surface is greater than z is given by the cumulative probability function: @(z)=j: $(zl)dz'. This yields an S- shaped curve identical to the bearing area curve. It has been found that many real surfaces, notably freshly ground '" @('' surfaces, exhibit a height distribution which is close to the normal or Figure 3.10 Gaussian probability function: where a is that standard (r.m.s.) deviation from the mean height. The cumulative probability, given by the expression can be found in any statistical tables. When plotted on normal probability graph paper, data which follow the normal or Gaussian distribution will fall on a straight line whose gradient gives a measure of the standard deviation. It is convenient from a mathematical point of view to use the normal probability function in the analysis of randomly rough surfaces, but it must be remembered that few real surfaces are Gaussian. For example, a ground surface which is subsequently polished so that the tips of the higher asperities are removed, departs markedly from the straight line in the upper height range. A lathe turned surface is far from random; its peaks are nearly all the same height and its valleys nearly all the same depth. Elements of contact mechanics 89 So far only variations in the height of the surface have been discussed. However, spatial variations must also be taken into account. There are several ways in which the spatial variation can be represented. One of them uses the r.m.s. slope a, and r.m.s. curvature ak. For example, if the sample length L of the surface is traversed by a stylus profilometer and the height z is sampled at discrete intervals of length h, and if zi- and zi+ are three consecutive heights, the slope is then defined as and the curvature by The r.m.s. slope and r.m.s. curvature are then found from where n = L/h is the total number of heights sampled. It would be convenient to think of the parameters a, a, and ak as properties of the surface which they describe. Unfortunately their values in practice depend upon both the sample length L and the sampling interval h used in their measurements. If a random surface is thought of as having a continuous spectrum of wavelengths, neither wavelengths which are longer than the sample length nor wavelengths which are shorter than the sampling interval will be recorded faithfully by a profilometer. A practical upper limit for the sample length is imposed by the size of the specimen and a lower limit to the meaningful sampling interval by the radius of the profilometer stylus. The mean square roughness, a, is virtually independent of the sampling interval h, provided that h is small compared with the sample length L. The parameters a, and a,, however, are very sensitive to sampling interval; their values tend to increase without limit as h is made smaller and shorter, and shorter wavelengths are included. This fact has led to the concept of function filtering. When rough surfaces are pressed into contact they touch at the high spots of the two surfaces, which deform to bring more spots into contact. To quantify this behaviour it is necessary to know the standard deviation of the asperity heights, a,, the mean curvature of their peaks, &, and the asperity density, q,, i.e. the number of asperities per unit area of the surface. These quantities have to be deduced from the information contained in a profilometer trace. It must be kept in mind that a maximum in the profilometer trace, referred to as a peak does not necessarily correspond to a true maximum in the surface, referred to as a summit since the trace is only a one-dimensional section of a two- dimensional surface. The discussion presented above can be summarized briefly as follows: (i) for an isotropic surface having a Gaussian height distribution with 90 Tribology in machine design standard deviation, a, the distribution of summit heights is very nearly Gaussian with a standard deviation The mean height of the summits lies between 0.50 and 1.5~ above the mean level of the surface. The same result is true for peak heights in a profilometer trace. A peak in the profilometer trace is identified when, of three adjacent sample heights, zi-, and zi+ ,, the middle one zi is greater than both the outer two. (ii) the mean summit curvature is of the same order as the r.m.s. curvature of the surface, i.e. (iii) by identifying peaks in the profiIe trace as explained above, the number of peaks per unit length of trace q, can be counted. If the wavy surface were regular, the number ofsummits per unit area q, would be qi. Over a wide range of finite sampling intervals Although the sampling interval has only a second-order effect on the relationship between summit and profile properties it must be emphasized that the profile properties themselves, i.e. ak and a, are both very sensitive to the size of the sampling interval. 3.8.2. Contact of nominally flat rough surfaces Although in general all surfaces have roughness, some simplification can be achieved if the contact of a single rough surface with a perfectly smooth surface is considered. The results from such an argument are then reasonably indicative of the effects to be expected from real surfaces. Moreover, the problem will be simplified further by introducing a theoretical model for the rough surface in which the asperities are considered as spherical cups so that their elastic deformation charac- teristics may be defined by the Hertz theory. It is further assumed that there is no interaction between separate asperities, that is, the displacement due to a load on one asperity does not affect the heights of the neighbouring asperities. Figure 3.1 1 shows a surface of unit nominal area consisting of an array of identical spherical asperities all of the same height z with respect to some reference plane XX'. As the smooth surface approaches, due to the smooth Figure 3.1 1 'reference plane on rough surf ace Elements of contact mechanics 9 1 application of a load, it is seen that the normal approach will be given by (z - d), where d is the current separation between the smooth surface and the reference plane. Clearly, each asperity is deformed equally and carries the same load Wi so that for q asperities per unit area the total load W will be equal to q Wi. For each asperity, the load Wi and the area of contact Ai are known from the Hertz theory and where 6 is the normal approach and R is the radius of the sphere in contact with the plane. Thus if p is the asperity radius, then and and the total load will be given by that is the load is related to the total real area of contact, A =qAi, by This result indicates that the real area ofcontact is related to the two-thirds power of the load, when the deformation is elastic. If the load is such that the asperities are deformed plastically under a constant flow pressure H, which is closely related to the hardness, it is assumed that the displaced material moves vertically down and does not spread horizontally so that the area of contact A' will be equal to the geometrical area 2nPS. The individual load, Wi, will be given by Thus that is, the real area of contact is linearly related to the load. It must be pointed out at this stage that the contact of rough surfaces should be expected to give a linear relationship between the real area of contact and the load, a result which is basic to the laws of friction. From the simple model of rough surface contact, presented here, it is seen that while a plastic mode of asperity deformation gives this linear relationship, the elastic mode does not. This is primarily due to an oversimplified and hence 92 Tribology in machine design unrealistic model ofthe rough surface. When a more realistic surface model is considered, the proportionality between load and real contact area can in fact be obtained with an elastic mode of deformation. It is well known that on real surfaces the asperities have different heights indicated by a probability distribution of their peak heights. Therefore, the simple surface model must be modified accordingly and the analysis of its contact must now include a probability statement as to the number of the asperities in contact. If the separation between the smooth surface and that reference plane is d, then there will be a contact at any asperity whose height was originally greater than d (Fig. 3.12). If 4(z) is the probabilitydensity of the asperity peak height distribution, then the probability that a particular asperity has a height between z and z +dz above the reference plane will be 4(z)dz. Thus, the probability of contact for any asperity of height z is 5 prob(z > d) = J d(z) dz. d tZ I ,smmth surface 'distribution Figure 3.12 of peak heights ~(ZI If we consider a unit nominal area of the surface containing asperities, the number of contacts n will be given by Since the normal approach is (z-d) for any asperity and Ni and Ai are known from eqns (3.48) and (3.49), the total area of contact and the expected load will be given by and 5 3 N =$q/3*E1 f (Z - d)'4(z) dz. d It is convenient and usual to express these equations in terms of standardized variables by putting h =d/a and s =z/a, a being the standard deviation of the peak height distribution of the surface. Thus n =qF,(h), A = xqBgFl(h), N = 9q~f ot~l~+ (h), Elements of contact mechanics 93 where +*(s) being the probability density standardized by scaling it to give a unit standard deviation. Using these equations one may evaluate the total real area, load and number of contact spots for any given height distribution. An interesting case arises where such a distribution is exponential, that is, In this case so that These equations give N=CIA and N=C2n, where C1 and C, are constants of the system. Therefore, even though the asperities are deforming elastically, there is exact linearity between the load and the real area ofcontact. For other distributions ofasperity heights, such a simple relationship will not apply, but for distributions approaching an exponential shape it will be substantially true. For many practical surfaces the distribution of asperity peak heights is near to a Gaussian shape. Where the asperities obey a plastic deformation law, eqns (3.53) and (3.54) are modified to become m A' = 2nqP J (Z - d)+(z) dz, d It is immedately seen that the load is linearly related to the real area of contact by N'= HA' and this result is totally independent of the height distribution +(z), see eqn (3.51). The analysis presented has so far been based on a theoretical model of the rough surface. An alternative approach to the problem is to apply the concept of profilometry using the surface bearing-area curve discussed in Section 3.8.1. In the absence of the asperity interaction, the bearing-area curve provides a direct method for determining the area of contact at any given normal approach. Thus, if the bearing-area curve or the all-ordinate distribution curve is denoted by $(z) and the current separation between the smooth surface and the reference plane is d, then for a unit nominal 94 Tribology in machine design surface area the real area of contact will be given by so that for an ideal plastic deformation of the surface, the total load will be given by To summarize the foregoing it can be said that the relationship between the real area of contact and the load will be dependent on both the mode of deformation and the distribution of the surface profile. When the asperities deform plastically, the load is linearly related to the real area of contact for any distribution of asperity heights. When the asperities deform elastically. the linearity between the load and the real area ofcontact occurs only where the distribution approaches an exponential form and this is very often true for many practical engineering surfaces. 3.9. Representation of Many contacts between machine components can be represented by machine element cylinders which provide good geometrical agreement with the profile of the contacts undeformed solids in the immediate vicinity of the contact. The geometrical errors at some distance from the contact are of little importance. For roller-bearings the solids are already cylindrical as shown in Fig. 3.13. On the inner race or track the contact is formed by two convex contact (1) contact (2) equ~valent cylinders equivalent cylinders and planes r R, R=- r(R,+2r) R,+ r Figure 3.13 = - R,+ r Elements of contact mechanics 95 Figure 3.14 Figure 3.1 5 cylinders of radii r and R,, and on the outer race the contact is between the roller of radius r and the concave surface of radius (R, + 2r). For involute gears it can readily be shown that the contact at a distances from the pitch point can be represented by two cylinders of radii, R,,, sin$? s, rotating with the angular velocity of the wheels. In this expression R represents the pitch radius of the wheels and $ is the pressure angle. The geometry of an involute gear contact is shown in Fig. 3.14. This form of representation explains the use of disc machines to simulate gear tooth contacts and facilitate measurements ofthe force components and the film thickness. From the point of view of a mathematical analysis the contact between two cylinders can be adequately described by an equivalent cylinder near a plane as shown in Fig. 3.15. The geometrical requirement is that the separation of the cylinders in the initial and equivalent contact should be the same at equal values of x. This simple equivalence can be adequately satisfied in the important region of small x, but it fails as x approaches the radii of the cylinders. The radius of the equivalent cylinder is determined as follows : Using approximations and For the equivalent cylinder Hence, the separation of the solids at any given value of x will be equal if The radius of the equivalent cylinder is then If the centres of the cylinders lie on the same side of the common tangent at the contact point and R, > Rb, the radius of the equivalent cylinder takes the form From the lubrication point of view the representation of a contact by an 96 Tribology in machine design equivalent cylinder near a plane is adequate when pressure generation is considered, but care must be exercised in relating the force components on the original cylinders to the force components on the equivalent cylinder. The normal force components along the centre-lines as shown in Fig. 3.15 are directly equivalent since, by definition The normal force components in the direction of sliding are defined as Hence and For the friction force components it can also be seen that where To,., represents the tangential surface stresses acting on the solids. References to Chapter 3 1. S. Timoshenko and J. N. Goodier. Theory of Elasticity. New York: McGraw- Hill, 1951. 2. D. Tabor. The Hardness of Metals. Oxford: Oxford University Press, 1951. 3. J. A. Greenwood and J. B. P. Williamson. Contact of nominally flat surfaces. Proc. Roy. Soc., A295 (1966), 300. 4. J. F. Archard. The temperature of rubbing surfaces. Wear, 2 (1958-9), 438. 5. K. L. Johnson. Contact Mechanics. Cambridge: Cambridge University Press, 1985. 6. H. S. Carslaw and J. C. Jaeger. Conduction of Heat in Solids. London: Oxford University Press, 1947. 7. H. Blok. Surface Temperature under Extreme Pressure Conditions. Paris: Second World Petroleum Congress, 1937. 8. J. C. Jaeger. Moving sources of heat and the temperature of sliding contacts. Proc. Roy. Soc. NSW, 10, (1942), 000. [...]... any plane figure have a common point of intersection, motion is reduced t o turning about that point For a simple turning pair in which the profile is circular, the common point of intersection is fixed relatively to either element, and continuous turning is possible 98 Tribology in machine design 4.2 The concept of friction angle - Figure 4.1 represents a body A supporting a load W and free to slide... self-sustaining Thus, if x = 4 =6", corresponding to the value of f=0.1, then when the load is being raised Figure 4.10 efficiency tanx 0.1 051 - 49 .5 % = tan(x + 4 ) =0.2126 and On the other hand, for the value x = 42", corresponding t o the maximum efficiency given above and the mechanical advantage is reduced in the ratio 4. 75 :0.9 = 5. 23 : 1 In general, the following is approximately true: a machine will... efficiency is then 81 per cent There are two disadvantages in the use of a large thread angle when the screw is used as a lifting machine, namely low mechanical advantage and Friction, lubrication and wear in lower kinematic pairs 1 05 the fact that when r > + the machine will not sustain the load when the effort is removed Thus, referring to the inclined plane, Fig 4.9, if the motion is reversed the reaction... due to the obstacle and F the downward force in the centre-line of the sluice Taking the moment about A, Resolving vertically ( P + Q ) s i n @ + R= F Resolving horizontally 100 Tribology in machine design and so P=Q To calculate the perpendicular distance z we have tan 4 =f=0. 25; 4 = 14'2' and and so z = A B sin(% 4 )= 3.84 sin 65' 36' + = 3.48 m Substituting, the above equations become from this R... replaced by the inclined plane or wedge and that B moves in parallel guides The angle of friction is assumed to be the same at all rubbing surfaces The system, shown in Fig 4.6, is so proportioned that, 102 Tribology in machine design Figure 4.6 as the wedge moves forward under the action of a force P, the reaction R, at S must pass above 0, the point of intersection of R, and W Hence, tilting will tend... +)] according to eqn (4.14), one full revolution results in lifting the load a distance p, so that tan a =- P 2nr' + If the limiting angle of friction for the contacting surfaces is assumed constant, then it is possible to determine the thread angle a which will give maximum efficiency; thus, differentiating eqn (4. 15) with respect to a and equating to zero that is sin 2(a + 4) =sin 2a and finally so... It follows directly that the elements are then in contact over their surfaces, and that motion will result in sliding, which may be either in curved or rectilinear paths This sliding may be due to either turning or translation of the moving element, so that the lower pairs may be subdivided t o give three kinds of constrained motion: (a) a turning pair in which the profiles are circular, so that the... a - R2 sin 4b - Wx C - Figure 4 .5 The limitingcase occurs when this couple is zero, i.e when the line of action of R2 passes through the intersection 0 of the lines of action of Wand R, The three forces are then in equilibrium and have no moment about any point Hence But and W=R1cos4-Rzsin& from which R1 = W - cos 4 cos 2 4 and R2 = W - sin 4 cos 2 4 ' Substituting these values of R, and R2 in eqn (4.6)... and J(sec2 a + tan2 $, ) sec $, = sec u and eliminating $, efficiency = + tan2 $,) sina +fcos2 a,/(sec2 a + tan2 I, )) sin a -f sin2aJ(sec2 a Friction, lubrication and wear in lower kinematic pairs 4 .5 Plate clutch - mechanism of operation Figure 4.16 11 1 A long line of shafting is usually made up of short lengths connected together by couplings, and in such cases the connections are more or less permanent... about C' and the saddle will move freely The limitingcase occurs when P and Q intersect at C on the line of action of F, in which case and Hence, to ensure immunity from jamming f must not exceed the value given by eqn (4 .5) By increasing the ratio x:y, i.e by making y small, the maximum permissible value off greatly exceeds any value likely to be attained in practice Numerical example Figure 4.4 A rectangular . the downward force in the centre-line of the sluice. Taking the moment about A, Resolving vertically (P+Q)sin@+R =F. Resolving horizontally 100 Tribology in machine design and so P=Q corresponding to the maximum efficiency given above and the mechanical advantage is reduced in the ratio 4. 75 : 0.9 = 5. 23 : 1. In general, the following is approximately true: a machine will. profilometry using the surface bearing-area curve discussed in Section 3.8.1. In the absence of the asperity interaction, the bearing-area curve provides a direct method for determining the area