Wear, 57 (1979) 323  329 @ Elsevier Sequoia S.A., Lausanne  Printed in the Netherlands
TRIBOLOGY
PARAMETER
323
PREDICTION*
D. J. WHITEHOUSE Department of Mechanical Engineering, (Ct. Britain)
University of Warwick, Coventry CV4 7AL
M. S. HAMED Shaubra Faculty of Engineering,
Cairo (Egypt)
(Received June 11,1979)
Summary Recent theoretical advances in the evaluation of surfaces have led to the possibility of predicting many parameters of tribological importance from just three measurements taken from the surface. Two current proposals for three measured parameters are discussed, one derived from the autocorrelation function and one from the power spectral density function. A comparison is made between the two methods and how they apply to practical results obtained from machined surfaces.
1. Introduction Many investigators [ 1  31 have expended their efforts recently in of surface parameters more suited to tribological studying the characteristics applications. Typical of these parameters are peak density, peak height distribution, peak and valley curvature etc. In addition to tribological functions these parameters are also often important in other applications such as electrical and thermal conductivities, painting and appearance. Digital techniques are usually used to evaluate these parameters because of their intrinsic versatility. The accuracy of such techniques is largely affected by the choice of the sampling space (interval) of the data, quantization effects (number of discrete amplitude levels of the profile signal) and the effect of the numerical model. The threepoint model is the most widely accepted numerical model for it is easy to perform and is generally suitable [ 21. The sampling interval seems to be the most important of the three factors affecting the accuracy of evaluation.
*Presented at the International Conference Engineering Surfaces, Leicester, April 18  20,1979.
on Metrology and Properties of
324
This paper relates to an attempt by Whitehouse and Phillips [4] to bridge the gap between the purely random process analysis of surfaces and their measurement. In their paper equations were developed which enable tribologists to measure complicated parameters such as those mentioned earlier without becoming too involved in either digital analysis techniques or random process theory. In particular this paper checks the applicability of the derived equations. To do this surfaces were manufactured by a number of machining processes and examined in two ways: first the tribology parameters were measured directly using digital computation and secondly the prediction equations were applied and a comparison made between the results obtained from the two methods.
2. Theory In order to develop the equations the following assumptions were made: (1) the surfaces are nominally Gaussian; (2) the sampling is not too large to reduce the accuracy of the threepoint method significantly; (3) to achieve a satisfactory result the inequality
has to be obeyed where p 1 is the correlation coefficient between adjacent ordinates spaced a distance h apart and pz is the correlation coefficient between ordinates spaced a distance 2h apart. The prediction equations selected from ref. 4 for examination in this paper are as follows. The mean slope m of the profile can be predicted from m=_
1 1 p1 h i
u2

The correlation be estimated from corr
(C,
Yo)
(2)
1
IT
Corr(C,
=
20

66~1
Ys) between
curvature
and profile height can
Pl)
(3)
+~PZ
Note that since the correlation coefficient predicted from this equation is always positive this points to a feature of tribological importance, i.e. the higher parts of the profile have greater curvature. The density N of peaks can be expressed in the form
I( The peak distribution mean peak height =
(4) parameters
((1 
are
Pl)hY2
2N
(5)
325
variance
mean curvature
variance

of peaks up2 = 1 + 
of curvature
=34Pl
4h%(l
+P2
of peaks 2{(3 841

4~1
Pl) +
+ ~2)(1
correlation
between
of peak height
curvature 1P;
_
+P2
N2 coefficient
P#‘~
N 34Pl

The correlation obtained using
(7)
2Nh2 {7r(l  ~~)}l’~
+P2 _PI)
(6)
4nN2 34P1
of peaks =
lP,
(8)
1 and curvature
can be
and peak height 112
+P2
(3  4p i + p 2) variance (peak height)
The value of the correlation coefficient obtained from this equation always positive which means that higher peaks have larger curvature. The density of crossing is estimated from crossing density
1 = z cos1 p 1
(9)
I
is
(10)
In these formulae the variance of the surface (the root mean square (r.m.s.) value I?:) is taken as unity. To apply the equations all the quantities have to be adjusted accordingly, e.g. in eqn. (2)
becomes
3. Experimental
conditions
Components usually critical in tribological situations are normally produced by finishing processes such as grinding, lapping and honing and as such can be reasonably expected to adhere to the Gaussian assumption made in the derivation of the prediction equations (eqns. (2)  (10)). However, because even in turning, shaping and similar processes a substantial component of the waveform is often random because of tool wear or micro
326
fracture of the surface caused through builtup edge, it seemed worthwhile to explore the potential of the technique in nominally nonGaussian processes. For this reason the surfaces examined in this paper fall into two basic categories: (1) random which includes ground surfaces using surface, cylindrical and centreless grinders, linishing and spark erosion; (2) deterministic which includes turning and shaping. 3.1. Technological details of the specimens The specimens were made of ball bearing SKF steel containing 1% carbon and 1.5% chromium; the material hardness was 240 kgf mmU2. Surface treatment details are given in Table 1.
4. Results The results considered are for two of the most important tribological parameters: peak density and peak curvature. Other parameters show similar results. Table 2 gives the measured and predicted parameters for both random and deterministic processes. The results are the average of ten independent readings on each surface. The standard deviations (s.d.) are also quoted so that the scatter of the measured and predicted values can be examined. The ratios of the average measured parameters to the average predicted parameters are plotted as a function of R, and skewness of the asperity heights. Each point represents a surface machined using any of the chosen processes. The coefficients of variation for the ratio of the measured parameter to the predicted parameter are shown. The definition of coefficient of variation used here is coefficient
of variation
=
spread (standard
deviation)
mean value
5. Discussion The equations developed by Whitehouse and Phillips [4] to predict tribological surface parameters have been applied to different machined surfaces. These equations are based on the measurement of the autocorrelation function in two places which is straightforward especially with the availability of a digital correlator. Moreover, it is now possible to obtain the autocorrelation function graphically with considerable accuracy from a profile trace of the surface [ 51. Therefore the parameters can be obtained easily without the use of expensive equipment. The results presented in Table 2 show close agreement (error of 2  5%) between measured and predicted values for surfaces produced by random machining processes except those produced by spark erosion. For surfaces treated by deterministic processes, i.e. turning and shaping, the equations
327 TABLE
1
Surface grinding Cutting speed Table speed Machine depth of cut Type of cut Condition of cut
31.7 m sl 17 m minl 5pm Plunge Dry
Cylindrical grinding Cutting speed Table speed Machine depth of cut Type of cut Condition of cut
27.8 m sl 11.4 m mine1 10 I_lrn Plunge Dry
Centreless grinding Cutting speed Feed rate Machine depth of cut Type of cut Condition of cut
28.8 m sl 0.876 m minl 7.5 /Jrn Traverse With coolant
Linishing Cutting speed Force 60 grit Al,Oa
420 m minl 5N Coated abrasive
Spark erosion Erosion time Servo adjust Feed rate
5 min 30 Automatic
Shaping No. of strokes Depth of cut Feed rate
34 min’ 0.2 mm 0.125 mm per stroke 0.25 mm per stroke
(fine cut) (rough cut)
Fine cut 800 rev mine1 0.1 mm 0.063 mm revl
Rough cut 480 rev minl 0.3 mm 0.18 mm revl
belt
Turning Cutting speed Depth of cut Feed rate
predict higher values. The error for the peak curvature does not exceed 5% while that for peak density is higher, of order 20  30%. These results are confirmed by the values of the coefficient of variation for the data where a marked increase is noticed for surfaces with R, values over 4 pm and skewness values of 0.3 and over. The prediction of the tribological function of two mating components (properties of the gap) requires that the surface parameters must refer to both components. Unfortunately it is not normally valid to add or subtract the surface parameter of one component to that of another to obtain the gap properties, e.g. R, values should not be added. This is even more important
328 TABLE
2
Process
Skew
Surface
grinding
Peak density
Peak curvature
Measured
Predicted
Measured Predicted ____.
Mean s.d.
0.155 0.01
0.8 0.13
0.270 0.007
0.281 0.006
52.46 1.61
46.97 2.2
Mean s.d.
0.724 0.07
0.6 0.2
0.179 0.005
0.210 0.005
91.57 3.4
89.36 3.4
Mean s.d.
0.346 0.032
0.2 0.1
0.211 0.008
0.241 0.006
56.27 2.25
54.48 2.65
Spark erosion
Mean s.d.
2.78 0.21
0.29 0.13
0.051 0.006
0.113 0.009
61.48 7.62
80.67 10.09
Linishing
Mean s.d.
3.61 0.06
0.47 0.19
0.166 0.01
0.162 0.003
108.45 3.2
126.6 5.4
Rough shaping
Mean s.d.
5.68 0.42
0.42 0.11
0.098 0.028
0.164 0.013
98.82 5.24
105.07 14.7
Rough turning
Mean s.d.
1.775 0.17
0.21 0.07
0.136 0.01
0.179 0.006
97.7 6 6
Cylindrical Centreless
grinding grinding
98.76 5.16
for tribological parameters such as the standard deviation of peak curvature. What is needed is something like dimensional tolerances which can be integrated from one component to another in order to test whether the fit, for example, will be satisfied. The gap properties can be found from the properties of the two mating surfaces by working out three composite correlation values for the gap: u,” = u1”+ 0; Pig=
PPg =
GPl
oqp;
(11)
+o;p2
2 QIz
+fJ;p; 2 ug
(12)
(13)
where ug, u1 and u2 are the r.m.s. values of the gap for the first and the second surface respectively, p lg,p 1 and p 2 are the correlation coefficients between adjacent ordinates spaced a distance h apart in the gap and on surfaces (1) and (2) respectively, and p 2g,p ; and p; are the correlation coefficients between ordinates spaced a distance 2h apart in the gap and on surfaces (1) and (2) respectively. These values are inserted into the relevant formula to give any tribological parameter. The use of peak density and zero crossing instead of correlation coefficients to predict tribological parameters [6] is not easy because they are not simply additive and they are also very sensitive to instrumental error: ug2= L7; + Cl;
(14)
329
=
Hg
(15)
where D is the density
H is the density H=1
!?71 ( m0
of peaks given by
of zeros given by r/s 1
and mj is the jth moment of the power spectral density. The coefficients of variation for four of the surface parameters are shown in Table 3 for the different machined surfaces under examination. From the table it can be seen that in general the autocorrelation method is more reliable than the moment method. TABLE
3
Coefficient
of variation for some measured surface parameters
Process
Cylindrical grinding Centreless grinding Surface grinding Rough turning Rough shaping Spark erosion Linishing
(2) 2.23 1.88 4.91 0.41 0.05 0.19 0.07
6.04 3.75 7.23 1.2 0.16 0.68 0.16
Peak density
Zero crossing
(“ro)
(%)
3.49 6.43 2.73 7.47 28.6 12.6 10.7
11.6 3.73 5.77 15.9 18.1 7.61 6.61
References 1 J. A. Greenwood and J. B. P. Williamson, Development in the theory of surface topography, Proc. Leeds/Lyon Conf. on Tribology, Lyon, 1977. 2 D. J. Whitehouse, The digital measurement of peaks on surface profiles, J. Me&. Eng. Sci., 20 (1978). 3 D. J. Whitehouse and J. F. Archard, The properties of random surfaces of significance in their contact, Proc. R. Sot. London, Ser. A, 316 (1970) 97. 4 D. J. Whitehouse and M. J. Phillips, Discrete properties of random surfaces, to be published. 5 D. J. Whitehouse, Approximate methods of assessment of surface topography parameters, CIRP Ann. Znt. Inst. Prod. Eng. Res., 25 (1976) 461. 6 P. R. Nayak, Random process model of rough surfaces, Trans. ASME, (1971) 398. 7 J. Peklenik, New developments in surface characterization, Proc. Inst. Mech. Eng., 182 (3K) (1967  68) 108.