(Ệ i R ll Psum = an ~~ heights (equation [6.44]) N CÌ Probability densities for summit * The surface as a random process 0.4 0.30.2-0.1- e The mean summit curvature, «,,, at any summit on a surface is defined as the mean of the principal curvatures at that point The curvatures at a summit in the x and y directions are —d*x/ax? and —0°z/dy* respectively, and so Km m 82z a*z + os] = —3l|-— 1% =] ~ 1(& ++ &) = 6.47 [6.47] Using the probability distribution for summits with height ¢, and equivalent mean curvature «,, (derived from p(é,, &, , &)) it is possible to derive the expected value of the mean curvature for summits of height €, In dimensionless form this is plotted in Fig 6.5 (for further details see Nayak 1971) The magnitude of the surface gradient, , is defined by £= (6+ &)'” [6.48] and by a standard method the probability density for £ is r= Hạ ep-~=, 2m; ` ero [6.49] The expected value of the gradient is thus é= | £p(0d¿ = (=) | Now the expected value of the absolute slope of a profile is él = :) T 30 - 1/2 [6.50] Surface statistics and so ¿ = lỗ 4T : [6.51] Thus the mean surface gradient is greater than the mean profile slope Discussion The probability density, poun(€*), is plotted in Fig 6.3 for a range of a values (Since a > 1.5 (Longuet-Higgins 1957a) no values of a less than 1.5 appear in that figure.) As a — 1.5 the probability of a high peak increases Similarly, Fig 6.1 gives plots of Ppeax(€*) for a range of a values (for one-dimensional random processes a = 1) The significance of the parameter a is in general related to the breadth of PSDF; a broad spectrum (large a values) has waves with a large number of wavelengths; a narrow spectrum (small a values) has waves of approx- imately equal wavelength (though see sect 6.4 below) Figures 6.4 show a comparison Of Ppeax(é*) and p;um(£*) Clearly the profile distorts the surface in such a way as to show far fewer high peaks and far more low peaks than appear on the surface The distortion is greatest when a = 1.5 and tends to zero as a — ~ In fact a profilemeasuring instrument will travel over the shoulder of an asperity on the surface, rather than the summit A peak will still appear on the profile, but of a smaller height than the summit being sampled The dimensionless mean curvature «,,/./m, is shown in Fig 6.5 for a range of a@ values It is seen that high peaks have a larger expected curvature (small summit radius) then lower peaks For a = 1.5, Km varies linearly with é*, but for larger values of a the mean curvature is constant for summits of all heights The dimensionless peak curvature Kpeak/ m4 is shown in Fig 6.2 for various a values The graphs Fig 6.6 show a comparison between Koeak/^ Tạ and km//mạ distortion of the surface by the profile is again evident, but is small a > For a < 2.5, the profile shows a larger peak curvature than true summit curvature For œ > 2.5, the profile peak curvature is than the summit curvature in The for the less Anisotropic surfaces The study of anisotropic surfaces is considerably complicated by the number of parameters required to describe the surface The simple relations between the moments, m,,, outlined in sect 6.2 for isotropic surfaces no longer apply and closed-form solutions for the summit heights, gradients and curvatures are extremely difficult to obtain Many surfaces do, however, possess a pronounced grain so that the summit curvatures in the grain direction are considerably less than the curvatures across the grain We use this fact as the basis of an approximate treatment of strongly anisotropic rough surfaces (see Bush et al 1978) Initially consider the general theory of anisotropic surfaces as developed by Longuet-Higgins (1957a) The joint probability density of 131 CO a= (a) 1.5; (b) 5; (e) 10 ~ tC) — Probability density Fig 6.4 Comparison of probability densities of peaks and summits for 0.7m 0.6FƑ 0.5 0.4Ƒ 0.3 0.2 Probability density 0.7 =~ ® — Probability density O ~ > — 0.1 0.77 0.6 0.5Ƒ Surface statistics C) _/ Expected summits mean curvature of E + E I~ or Ì L —2.0 | —1.0 | ! | 1.0 ! ! 2.0 ! | 3.0 &* the variables &(i = 1,2, , 6) given by equations [6.34] and [6.35], where in the anisotropic case: N= Mo 0 Mo ~™M290 —M1, —Mo2 My 0 ™y, Moo 0 —M2 0 Mao m3) M2 —m, 0 m3) M2 M13 —M2 0 M2 13 Moa [6.52] Choosing the x-axis to be in the direction of the grain, and allowing the autocorrelation function to be even in x and y implies that the PSDF is also even in x and y; that is, G(u, v) = G(-u, v) = G(u, -v) Symmetry then implies that My, = mại = m3 = [6.53] The area of intersection of a plane with a rough surface is the bearing area and for an asperity model the bearing area consists of an aggregate of ellipses which are found by the intersection of the plane with the asperities If the semi-major axis of an ellipse makes an angle B with the x-axis then the probability distribution of has mean zero and variance depending on m2 In particular var(8) > as M22, — OQ, the case of interest for strongly anisotropic surfaces In this case it is sufficient to consider the probability density of the variables é,, &, &, &, and &; that is, P(E» ar By Ease) = tGOmm.=rn exp(~3£T Mộ where E “(6 6,6, 6u 6), NÚ=ÑT, CÁ = deLÑ (6.54 [6.55] 133 Fig 6.6 (a) Comparison of peak and summit * curvatures for a = (a) 1.5; (b) 5; (c) 10 Curvature Curvature ™~ Curvature C -2.0 —1.0 1.0 2.0 3.0 Surface statistics ~ ì } and N=| mạc 0 —Hạo —mqa mạo O 0 0 —mạo —M mạ 0 mạo - 0Ö 0 [6.56] tọa Inverting N we obtain:  = mo mạo mọa mạa mạok, mẽ Moo M _ =| 0 Px —- 0 0 Be — 0 0 1ơ 8; VBi Bo Mao VƠ Mp4" 409 vs 1-8; 129 By Mo2 Ba M20 M20 Br [6.57] Mo2 ] Moz [6.58] Vv Mo4Mo4 Mo4 J where w= 1- ( ~ Bi B2) MoolM 40 Ba mo 0¡:=—= › ' > Moo œ=—=—— : B: o4 mã; [6.59] C and ay, a; are the bandwidth parameters in the x- and y-directions respectively Following the analysis of sect 6.2 the probability distribution for summits per unit area is Psum(€1, &, &) — l&u¿@|p(&:, 0, 0, &4, &) [6.60] Using equation [6.60] it is possible after some analysis (see Bush et al 1978 and Keogh 1977) to derive various statistics for a strongly anisotro- pic surface; examples of these are: The summit Dgam = 2S (27)? The Km mean Jn density, Dyum, | iS moras) (ors summit L curvature, v§ (VJmoa + man) 6.61 k,,, is [6.62] 135 | NRE ONIN NETH r meres = ete The surface as a random process O ~ The summit mean plane, Agun, is } Moot \!/2 hsum = ("2") (V8: + V8›) [6.63] Discussion Theories of isotropic surfaces are not applicable to the important practical case of ground surfaces which are strongly anisotropic Five parameters are required to describe such surfaces: mạo, the variance of the surface height about the mean plane; Moz and myo, the principal mean square slopes; Moq4 and myo, the principal mean square curvatures These moments can be obtained from two profile measurements; one taken in the direction of the grain and the other across the grain Both profiles have variance zmọo The number of zeros, D,.,,, and extrema, Dextrema > per unit length of profile are related to the surface moments by the equations: grain) ale Dyero (across grain) Dextrema (across grain) ( ) /2 | š lề š lš šlš (along ate extrema ale Đ„¿;o (along grain) aị— C) | Cre) (re) ) [6.64] (Restricting the theory to the case of highly eccentric asperities with their semi-axes closely aligned to the x-direction leads to m2) being negligible and, by symmetry, the moments, m,,, m3 and m3, are zero.) The bandwidth parameters, a, along and across the grain respectively can be calculated directly from equation [6.64] 6.4 Measurement of spectral moments (R S Sayles) From a single profile, apart from actually computing the moments from a generated spectrum, three techniques are available: The first and most straightforward method is that suggested by Longuet-Higgins and Nayak of simply counting zero crossings and extrema, and measuring the RMS roughness The principal moments required are specified in terms of the profile zero-crossing density and extrema density by equations [6.13] and [6.15]: m= m4 mo D zero m?mạD 2eroD extrema L6 65 ] [6 66] Measurement of spectral moments Longuet-Higgins demonstrated how these moments can be spe- cified in terms of the derivatives of the autocorrelation function at the origin Rice (1944) in a similar analysis obtained the same results via equations [6.65] and [6.66] By considering the spatial probability of a maximum and a zero crossing within a small but finite length of signal, Papoulis (1965) also arrived at the density of maxima and zero crossings by a much simpler method His final expressions again were effectively those of equations [6.65] and [6.66], with the moments defined as auto- correlation function derivatives Equations [6.65] and [6.66] in some form are therefore well- established expressions, and Bendat (1958) gives several examples of good agreement with measured data These equations can be used to define mg, mz and m, in terms of either the autocorrelation, autocovar- iance or structure functions With digital signals where first-difference techniques can be employed, any one of these functions will yield the required moments In terms of the profile principal moments required and the profile autocovariance function R(r) (which is the same as the surface autocovariance function for stationary isotropic data): Moo = Mo = R(0) tọa = Mo = —dˆ2R(0)/dr? [6.67] Mog = May = d*R(0)/dr* [6.68] By first-difference techniques the derivatives can be expressed as follows using the first three terms of Maclaurin’s series: R(ô:) = R(0) + R*(0) âr?/2 if 57 is small, mạ = 2(Rọ — R\)/ðr? where Ry = R(0) and R, = R(6z), the first point of the autocovariance function Similarly, m, = RỲỲ(0) = 2(Rĩ - Ro)/dr? which reduces to ta = 2(3Ro — 4R, + Rạ)/ôr! In terms of the structure function (defined in sect 5.3.2), the values of mz and mz, are simply: mạ = S/®r? = (4S, [6.69] and mM, _~ %;)ðr! [6.70] Thus, by means of the first three autocovariance coefficients, or the first two structure function points and the RMS roughness, the principal spectrum moments can be fully defined 137 The surface as a random process } A third method is simply to measure the variance of the distributions of profile heights, slopes and curvatures as these in fact are My, mạ and my, respectively This can be shown by differentiation of the autocovariance function and showing that equations [6.67] and [6.68] represent the variance of slopes and curvatures respectively Nayak’s bandwidth parameter a is given by equation [6.16] as a> mom,/mé Cartwright and Longuet-Higgins (1956) use what is in some cases a more representative bandwidth parameter: € = (mom, — m3)/mom, which is equivalent to: e=1-1/a ‘Thus as @ varies from 20 to ~, ¢ changes only from 0.975 to 1.0 which, for many of the statistical properties, reflects more closely the order of change occurring The interpretation of a or € as a pure spectrum width parameter can be misleading For example, low-pass white noise with a straight cut-off has a constant value of a = 9/5 This is independent of how high a cut-off is employed Another, perhaps more practical surface spectrum model exhibiting a constant value of a is a triangular spectrum decaying from some finite power at the origin Here a = 12/5 and again is completely independent of the width of spectrum over which the decay occurs The parameter ¢ is defined as a measure of the RMS width of the spectrum, which implies that it is also sensitive to the way in which the spectrum decays Because the surfaces in most cases have no detectable short spatial variation or short-wavelength limit, properties such as m, and m,, and as a result a, become functions of sampling interval This in no way invalidates the theory, it simply shifts the emphasis to the sampling process Figure 6.7 shows variations of mg, m2, m4 and a with sampling interval for an isotropic (spark-eroded) surface The sets of experimental points represent the first and third methods discussed for obtaining the spectrum moments The second method is considered the most accurate, and is represented by the solid curves Two important effects are evident from the surface Firstly the second and fourth moments, and consequently a, increase rapidly as the sampling interval reduces and no indication of an asymptote is evident Secondly, with increasing sampling interval, which effectively filters out more and more of the small-scale features, m2, m4, and @ reduce, but mg, the surface variance, remains almost constant This is an indication that the vast majority of the surface power is contained in the large-scale features and yet it is the small-scale features which have most influence on the moments and parameters of the spectrum moment theory Of the three methods depicted the results gained by counting zeros and peaks to predict the moments differ from the others This is due to long wavelengths reducing the number of zero crossings and hence mz —— ni C` Measurement of spectral moments Fig 6.7 (Variation of spectral moments ~ with sampling interval measured in three different ways for a sparkeroded surface (Sayles & Thomas 1979) ~~ ‘ 10? Ag — m fy Lo N ; ee