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Comprehensive study of parameters for characterizing three-dimensional surface topography I: Some inherent properties of parameter variation
Wear, 159 (1992) 161-171
161
Comprehensive study of parameters for characterizing dimensional surface topography I: Some inherent properties of parameter variation
three-
W. P. Dong, P. J. Sullivan and K. J. Stout School of Manufacturing
and Mechanical
Engineering
The University of Birmingham,
Edgbaston,
Birmingham,
BI5
2TT
(UK)
(Received August 28, 1991; revised and accepted January 8, 1992)
Abstract The characterization of surface topography is critically important in many disciplines and industrial applications including triboiogy. Surfaces are often characterized through the use of parameters (usually using statistical techniques) which attempt to provide an indication of some attribute of the surface such as amplitude variation. However, many questions regarding the use of such parameter characterization remain unanswered such as (i) the degree to which the parameters are effective at representing the geometry of the surface and (ii) the dependence of the parameters on measurement conditions and characterization procedures. These and many other questions still need to be considered in some detail before an effective characterization of surface topography is possible. Through the analysis of surfaces, which have been measured using three-dimensional surface topographic techniques, we present the first of a series of reports on parameters for characterizing surface topography. This paper addresses inherent properties of two-dimensional parameter variation for the case of two orthogonal directions for some typical machined surfaces. Some experimental results are presented to illustrate that parameter variation is determinate rather than random in some cases, and that the variation is unable to be affected by measurement and signal-processing conditions. The factors which produce determinate and significant parameter variation are proposed through the analysis of surface topography features. The sensitivity of parameter variation to changes in surface topography is also discussed in the paper.
1. Introduction It is well known [l] that surface roughness greatly influences the mechanical and physical properties of contacting parts. The understanding of this behaviour is important in many applications such as wear, friction, lubrication, sealing tightness of joint, contact rigidity, contact stress, loaded area and thermal conductivity. Therefore engineers are very concerned about the surface roughness of their products. As early as the 193Os,researchers started to use profilometer techniques to characterize surface roughness. These characterization techniques included a visual display of the profile; however, in order to quantio and standardize measurements, the use of statistical parameters developed. Although the subject of surface topography has developed over the last 60 years, some important questions about the significance of parameter characteristics still arise. For example (i) which of the many parameters available may be used to fully characterize surface topography and (ii) how effective are the parameters calculated from a profile at representing the surface
0043-1648192/$5.00
character? For the first question, many researchers [14] have considered the proliferation of parameters (the “parameter rash”) for characterizing a surface. For an answer to the second question, engineers and scholars all agreed that some parameters vary significantly even with a very small area [5-131. For example Thomas and Charlton [5] pointed out that, on machined surfaces, variations of 50% are not uncommon. In recent years, coinciding with the development of microcomputers, digital instrumentation and other measurement techniques, three-dimensional (3D) surface topography measurement has become realistic. Many 3D systems have been proposed and developed [14-221, with the view that the topography of a surface could be represented by area data, as long as the sampling interval is small enough to satisfy the sampling theorem. However, although the surface data can be obtained relatively easily using these systems, the problem still remains of how to characterize such data from surfaces manufactured in different ways, both mathematically and functionally. Although some analysis of 3D surface topography has been carried out, very few parameters
0
1992 - Elsevier Sequoia. All rights reserved
162
U! P. Dong Ed al. / Three-dimensional
and approaches to characterizing 3D surfaces have been proposed because of the complexity of surfaces and mathematical description difficulties. Some problems such as the following still have to be solved. (1) What set of parameters are able to characterize surface topography? (2) How do the parameters vary with sampling area and sampling interval? (3) How may the parameters correlate with a functional sense? (4) How may we define and identify lay structure or randomness of a surface? (5) How might we perform data pre-processing such as filtering and levelling for logged area data? Owing to the existence of these problems, a comprehensive and thorough study for 3D surface topography seems to be very necessary and urgent. This paper is the first in a series of reports towards carrying out a comprehensive study of parameters for characterizing 3D surface topography. It addresses the serious problem of parameter variation encountered in two-dimensional (2D) analysis through the analysis of 3D surface topography data. In analysing the variation of digital profile parameters it is helpful to classily the causes of parameter variation. The degree of variability due to these causes is different for different parameters. Cause A: variation in the geometry of the real machined surfaces. There are many possible causes for a variation in the geometry of a machined surface. Some causes are (i) manufacturing effect where factors such as tool wear and damage, and the presence of cutting debris will result in a variation in the geometry of the surface being produced and this variation will be superimposed on the rougher machining process which may be more or less random and (ii) surface geometric integrity with some features of surfaces described in terms of homogeneity, isotropy, stationarity and normality. The geometric variation of a machined surface is the main interest of metrologists in investigating profile measurement variation, Cause B: variation due to the measurement and data processing of real machined surfaces. There are many factors which determine a digital measurement system’s ability to obtain information about the real continuous surface. The inability of an instrument to record information about a real surface will result in variation in the parameters derived from such data. These factors may be classified into (i) inherent limitations of the measurement process such as finite stylus geometry and load, temperature and vibration effects, signal drift and quantization effects and (ii) selectable measurement conditions such as the influence of the profile length, sample spacing, filter type and cut-off, reference line or reference surface. The measurement
surface
topography
orientation may also cause profile measurement variation. Previous research work has been mainly concerned about whether parameter variation exists, or whether it is affected by machining and measurement conditions. To the authors’ knowledge no work has been carried out to date on the behaviour of the variation characteristics, and the degree of variation for a given machined surface. Thomas and Charlton [5] examined ten profiles for different surfaces obtained with different manufacturing methods. Attention was paid to the external factors which affect the parameter variation, e.g. how the variation changes with the measurement conditions such as cut-off, magnification range and skid. Spedding [6] developed statistical models for illustrating the variation which might be expected when estimates of skew and kurtosis were made from uncorrelated or correlated observations of gaussian or non-gaussian surfaces. The scatter of skew and kurtosis for measured surface data and simulated data was also addressed in his paper. King [7] also conducted theoretical and experimental examination of the distribution of skew and kurtosis of normal and non-normal populations, and suitable sample sizes for required parameter confidence limits were suggested. Davis and Stout [S] explored several sources which result in parameter variation. These sources are considered to be tracking inaccuracy, filtering of data by the choice of skid or datum, and surface damage caused by stylus and skid, or by particles attaching themselves to the surface under assessment. Recently Lin et al. [9] assessed joint effects of sample length and sample interval on the variation of roughness parameters. Multiple-regression analysis was carried out to infer statistical properties of parameter variation. Previous researchers all recognized that significant parameter variation exists, and they attempted to analyse the problem from different perspectives. Generally these investigations may be categorized into three areas. (i) Some researchers [l, 6, 71 have attempted to describe surface geometry theoretically and thus have developed random data models of surfaces such as normal or non-normal, and correlated or non-correlated. This approach is limited by the accuracy of the modelling process which is necessarily based upon many assumptions and it is impossible to model all engineering surfaces completely. (ii) Other workers [5, 81 examined the effects of measurement conditions on parameter variation such as the cut-off, the range of magnification, and the effects of the use of a skid. (iii) A fu r th er group [7, 9, 121 investigated the effects of (post-measurement) signal-processing conditions on parameter variation such as sampling interval, sampling length and filtering conditions.
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U! P. Dong et al. / Three-dimensional surface topography
Clearly, in (ii) and (iii) the parameter variation is affected by sampling and processing conditions with different parameters being more or less affected by different conditions. However, all these considerations are subject to measurement choices as mentioned above in cause B (ii), and revealed just one aspect of parameter variation. Another aspect to be considered is the inherent variation in the surface geometry of manufactured surfaces, cause A. This variation results in surfaces that cannot be simply modelled by normal or nonnormal distributions and, as long as all measurement and data-processing conditions satisfy the sampling theorem (the Nyquist criterion etc.) or, in other words, as long as the features which exist on a surface can be measured without much distortion, the characteristics of the parameter variation will be unaffected under certain ranges of the conditions. Basically, parameter variation is the product of inherent surface topography features and other external conditions but, in some cases, it is more likely to depend on manufacturing approaches and less on machining, measurement and signal-processing conditions. This paper illustrates this problem. Combined with the analysis of 3D surface topography, the parameter variation along two orthogonal directions is examined according to the features which exist on four kinds of surfaces: ground, end milled, shaped and turned. Some inherent properties of parameter variation will be discussed and the sensitivities of some moment parameters to the change of surface features on the surface will be analysed. This suggests why parameters vary so significantly and what format the variation trend is for some manufactured surfaces.
x direction is defined as the trace direction, i.e. this may be thought of as the direction which coincides with the movement of the pick-up if a gear box is used; the y direction (ortho-trace) is defined as being perpendicular to the (x,2) plane. The parameters used in this paper are calculated from points sampled in thex direction or points sampled in they direction. Note that no reference plane is used, and all parameters are calculated with data points relative to their own reference lines. For the uniqueness of each reference line and calculability using digital techniques the use of a least-squares line was adopted, as recommended by some national and international standards [23, 241. Thus, for each trace, the reference line of the ith trace is 2(X, yi) = U,i + &LX
(i=l,
2, . ..) s)
(I) where n,, is the number of traces, and a, and bxi are coefficients of the ith reference line. These coefficients are obtained by minimizing the error
(2)
k-1
where n, is the total number each trace, and Gk ‘z(xk,
Yi>
-f(Xk,
Yi)
of sampling points in (3)
Therefore
(44 2. Algorithm for calculating parameters For the logged surface shown in Fig. 1, the height of each point z(x, y) can be regarded as having a random height distribution over the whole area although it may be superimposed on some periodic components. The
It can be shown that the data series exik of the ith trace has zero mean. Thus the parameters of each trace are all based on the zero-mean data series. The parameters used correspond to the arithmetic mean height R,, the r.m.s. roughness R,, the skewness Rsk and the kurtosis Rku. Further, R$, R+*, RskiXand Rku; (where for example R$ is the R, of the ith trace in the x direction) can be expressed as
(5b)
Fig. 1. Coordinate
system for logged 3D surface.
W. P. Dong et al. / Three-dimensional
164
surface
topography
(54 Similarly, the parameters of the jth ortho-trace which correspond to eqns. (l)-(5) can be written as r&j, Y)=Q +byjY
(6)
a,= 2 Eyjk2
(7)
k-l
Eyjk’z(xj,
Yk)
-i($,
Yk)
(8)
(9a)
(9b)
(lOc> WW The advantages of these definitions of the parameters are that (i) their derivation from 2D parameters is straightforward, (ii) they are easy to comprehend, (iii) they are convenient to be calculated and (iv) the information in both orthogonal directions is included and, as we shall see in subsequent parts of the series, these parameters can represent surface quality with much higher confidence and reliability from a statistical point of view. However, such definitions also have obvious disadvantages in that some 3D information, especially functional properties, is lost owing to the adoption of least-squares reference lines and not a least-squares reference plane. 3. The effects of pits and troughs on parameter variation
To illustrate the inherent properties of parameter variation, a specific example, but one that is often encountered in ground, honed and lapped surfaces, is presented. Figure 2 shows the variations in four parameters R;, R,:,R,; and RktiX (i= 1, 2, . . ., n,,) of a ground surface along the y direction, i.e. the variations
Fig. 2. Parameter variation of a ground surface along the lay direction: (a) variation in R.“; (b) variation in RqX; (c) variation in Rskr; (d) variation in R,;‘.
of trace by trace. The data for the ground surface was obtained using an instrument developed at the University of Birmingham [25] and comprises 164 traces each containing 164 points with a sample interval of 8 pm in both directions. The choice of sample spacing and number of points was made by establishing suitable sampling conditions (those which determine a bandwidth to include all dominant features on the surface) for the surface discussed in this paper. From Fig. 2, it can be seen that (i) all parameters vary significantly from trace to trace, (ii) a small random variation is superimposed on a trended variation, (iii) RskXand RkuXshow an opposite variation trend in this instance and (iv) at approximately 0.76 mm in the orthotrace direction a “spike” may be observed in both the R, and the R, graphs but it may be seen that Rskand Rku may be insensitive when R, and R, change significantly. Table 1 lists the variation range of the parameters relative to the total averaged values. Obviously, it can be seen that even the smallest range of the four is larger than 70%.
W. P. Dong et al. / Three-dimensional surface topography TABLE 1. Variation range of parameters on the ground surface
Trace Orthotrace
- 26.3 +43.7
-47.3 + 69.8
-62 + 637.5
-61.1 + 560.3
-40.4 + 86.6 - 152100 + 52788
-86.1 + 47.5 - 56.7 + 1276
On the basis of these observations a number of hypotheses may be drawn. Does a form error exist? Is the variation caused by improper selection of measurement and signal-processing conditions? Does the surface really contain such a dramatic change which is evident over the whole area? In fact, the trend variation is nothing to do with all above-mentioned hypotheses but results from the special features on the surface. Figure 3 is the axonometric projection of the ground surface; it clearly shows the grinding marks caused by the cutting action of the grits attached to the periphery of the grinding wheel and the directional nature of the grinding process. Evidently, the values of R, or other
Fig. 3. Axonometric projection of the ground surface.
165
parameters calculated across the surface lay or along the lay would be quite different, but no obvious features indicate that the parameter variation would have the features shown in Fig. 2, because no waveform is present on the surface. In theory, the grinding process produces a gaussian surface; however, it is unavoidable that superimposed on this is the resultant topography caused by the action of single sharp grits which plough into the ground surface or by the adhesion of particles pulled out of the surface, thus some troughs or pits might be formed. This may be observed in Fig. 4 which is a representation of the same surface shown in Fig. 3 truncated by 30% R,of the deepest valley of the ground surface (the R, was calculated relative to the least squares plane). It can be seen that the majority of the topography of the original surface has been removed and that all that remains are two deep troughs, A and B, one deep pit, C, and some other smaller troughs and pits on the surface. It is the deep troughs and pits that dominate the parameter variation although they occupy only a small area in the total logged surface. It is worth noting how we may distinguish between a pit and a trough. A trough is clearly elongated in one direction whilst a pit is not. One advantage of 3D analysis is that it can detect troughs whilst 2D cannot.
W. P. Dong et al. I Three-dimensional
166
surface
topography
3.20
1 brn I
0.00 1
\ Fig. 4. Surface
shown
in Fig. 3 truncated
by 30% R,.
Comparing Figs. 2 and 4, it is readily realized that the parameters are greatly influenced by the troughs and the pit. The absolute values of the parameters of several initial traces are small, they just coincide with the area where no apparent troughs or pits exist, i.e. they are in an ideal ground surface area, one of the profiles in this area is shown in Fig. 5(a). A profile which crosses the longest trough, A, on the surface is shown in Fig. 5(b). Since the trough was created by a sharp grit attached to the surface of the grinding wheel, it would be shallow at both extremes, and deep in the middle part. Figure 5(d) shows a transverse profile of trough A. Therefore, the variation of the parameters calculated from the profiles which cross the trough has a similar shape. It can be seen that towards the end of the longest trough, A, the shorter trough, B, starts (Fig. 4). Thus, as a consequence of this the absolute values of the parameters increase again. The peculiar “spike” of R: and Rq"in Figs. 2(a) and (b) may be attributed to the contribution of the pit C, as one more deep valley in the profile is observed in Fig. 5(c). Figure 6 shows the variation in the profile parameters in the ortho-trace direction for R,Y, RqY, RskY and RkuY
the same surface shown in Fig. 3. Obviously, the influence of the pit C will have the same effect on parameters calculated in both the x and they directions. However, the trough A does not have much effect on Rsky and RkuYin this direction, although it does significantly influence Rayand RqY. It should be noted from Fig. 2 and Fig. 6 that high moment relative parameters (such as & and RkU) are not always as sensitive as low moment parameters (such as R, and RJ. In some instances the former change more significantly than the latter, and sometimes the former remain almost stable while the latter exhibit large variation. The question of parameter sensitivity will be discussed later in the paper. 4. The effect of machining variation
process on parameter
The inherent phenomenon of parameter variation not only is produced by troughs and pits on ground, honed or lapped surfaces but also may result from the machining process itself. A typical example is from the end-milling process which produces a non-homogeneous surface, i.e. the surface on which the microscopic struc-
W. P. Dong et al. / Three-dimensional surface topography
Fig. 5. Typical profiles across the surface lay: (a) a normal profile; (b) a profile which crosses the trough A; (c) a profile which crosses the trough A and the pit C; (d) a profile of the trough A.
ture is not unique. As seen in Fig. 7, the paths of an end-milling cutter form two zones: one is non-overlapping, and the other is overlapped. These two zones exhibit totally different characteristics. Figure 8 shows an axonometric plot of the top portion (30% R,) for an overlapped part of an end-milled surface [25]. As a consequence of the overlapping cut, flat areas are shown on the surface at a zero datum which represent areas where no metal exists above this height. Clearly the distribution of remaining metal is nearly regular and the topography of these features will have a large influence on parameters measured. Thus, as can be seen in Fig. 9, the parameters along both trace and ortho-trace directions show distinctive variation. Evidently, these variation features could not be affected readily by measurement and signal-processing conditions. It is interesting to see that the variation in R,Yin Fig. 9(b) shows periodicity because of the overlapping cut. However, for a shaped surface, as shown in
167
Fig. 6. Parameter variation of the ground surface across the lay direction: (a) variation in R,Y; (b) variation in Rz; (c) variation in RSky;(d) variation in RtuY.
non-overlapped
overlapped
Fig. 7. Schematic
orea
area
diagram of end-milling
paths.
168
Fig. 8. Axonometric
W P. Dong et al. I Three-dimensional
plot of a top portion
of 30% R, truncation
z a
surface topography
of an end-milled
surface.
surface, However, it is possible for surfaces produced by these machining processes to show some tendency towards a deterministic trend on the parameter variation owing to some synthetic effects of machining conditions and material characteristics on the machined surface. Figure 12 gives an example of a face-turned surface for which the variation in R,"is plotted. It is clear that the random variation is superimposed on a deterministic increasing trend. In many cases it is possible to explain the trend in the parameter variation through examination of the 3D topography of the machined surface. 5. Sensitivity analysis
Fig. 9. Variation in R, in an overlapped portion of an end-milled surface: (a) variation in R,“; (b) variation in R,Y.
Fig. 10, although it exhibits distinctive periodic&y across the lay direction, the variation in Raydoes not have apparent periodicity (Fig. 11). This phenomenon indicates that the waveform in one direction does not affect the parameter variation along the same direction, as the waveform is removed owing to the adoption of a least-squares reference line for each profile. For some surfaces such as turned, bored or slab milled, the effect of the machining process does not have such a significant determinate trend on the parameter variation as seen in Fig. 9 for the end-milled
of parameter
variation
An accepted “rule of thumb” about parameter variation is that the higher order relative moment parameters such as R,,and Rkuare subject to larger variation than the lower order parameters and are therefore more sensitive. However, in some cases the lower order moment parameters such as R,and R,are more sensitive to the variation in surface features. As seen from Fig. 2 and Fig. 6, some peculiar spikes arise in R,", RqX, R,Yand RqY, whilst R,;, RkuX, Rsky and RkuY change little at the same area. In order to investigate the relative variation in these parameters we wish to analyse this behaviour theoretically. Let zl, z2, . . . . z, be the discrete heights of each profile point and suppose that zi (i= 1, 2, . . ., n) is an independent random variable with zero mean;
PP
\ \
170
K P. Dong et al. I Three-dimensional surface topography
Wb) (13c)
Hence the relative variations in the parameters
become
Figure 2 and Fig. 6 clearly demonstrate this phenomenon; this is why some peculiar spikes appear in the variations in R, and R, whereas R,, and R,, vary less. (3) The relative variations dRsk/Rsk and dR,,/R,, may have opposite variation trends when
Generally, in ground, honed and lapped surfaces where troughs and pits exist, this situation is true, whilst in some surfaces produced by other machining processes where predominantly the peaks are larger than the valleys, it is the other way around.
04a) (14b)
(14c)
(14d) It is clearly seen from eqns. (14a) and (14b) that R, and R, possess the same variation trends, as their relative variations have the same sign. Since the ratio rj of each element of dR,/R, to the corresponding element of dR,/R, is
(15) their magnitude relationship is not known, i.e. in some cases R, is subject to larger relative variation than R,, and in other cases it is vice versa. From eqns. (14~) and (14d), it can be seen that the second terms in the square brackets are just the same as the term in dR,/R,. Thus, some characteristics of the relative variation can be drawn. (I) R, and R,, may not have the same variation trends as R,, as the signs of eqns. (14~) and (14d) may be opposite to that of eqn. (14b). This situation may also be observed in Fig. 2 and Fig. 6. (2) The relative variations in dRst/Rsk and dRku/Rku may be smaller than dRqlRq and dR,IR, when
6. Conclusions This paper has discussed the problem of 2D parameter variation through the analysis of 3D surface topography. Special attention is paid to the nature of surfaces which possess inherent properties of parameter variation. Experimental results and theoretical analysis have been presented to demonstrate (i) why parameters vary significantly, (ii) why the variation shows a determinate trend in some cases, (iii) what surface features play an important role in parameter variation, (iv) how interrelationships exist between relative variations of some parameters and (v) how the sensitivities of some parameters may be affected by changes in surface features. The following conclusions may be drawn. (1) The causes of parameter variation may be classified into two categories: cause A, inherent variation in the surface geometry; cause B, variation due to the measurement and data processing. (2) Parameter variation is mainly due to the nature of surface features (cause A). The inherent characteristics of surfaces play an important role in the parameter variation rather than the measuring and data processing conditions. (3) Greater variation (above 50%) is possible for most parameters on the surface where troughs and pits exist, e.g. ground, honed and lapped surfaces. (4) Although troughs and pits are randomly distributed on a surface and they may only occupy a small portion in the logged area, the parameter variation may still be significant and may show a deterministic trend. (5) A predictable variation trend of the parameters may be obtained from a non-homogeneous surface; it is the machining process that results in such a trend. (6) R, and R, have similar variation trends, but the relationship between their relative variation magnitudes is unknown.
W. P. Dong et al. / Three-dimensional surface topography
(7) 4, and Rku may or may not have a similar variation trend and depend on the relationship shown in eqn. (17). For surfaces which have predominantly larger valleys than peaks, their variation trends are opposite. (8) In some cases, lower order moment parameters are more sensitive to the change in surface topography features than higher order relative moment parameters, and in other cases it is vice versa. (9) The use of 3D topography measurement and visualization techniques enables the true nature of the surface features to be established. This information can then be used, in many instances, to explain 2D parameter variation.
Acknowledgment The authors would like to thank the Commission of the European Communities for providing the funds for the work under its Programme for Applied Metrology and Chemical Analysis (BCR).
References T. R. Thomas, Rough Surfaces, Longmans Green, London, 1982. T. R. Thomas, Characterization of surface roughness, Prec. Eng., 2 (1981) 97-104. B. Nowicki, Multiparameter representation of surface roughness, Wear, 102 (1985) 161-176. D. J. Whitehouse, The parameter rash - is there cure?, Wear, 83 (1982) 75-78. T. R. Thomas and G. Charlton, Variation of roughness parameters on some typical manufactured surfaces, Prec. Eng., 2 (2) (1981) l-96. T. A. Spedding, The machined surface: statistics and characterization, Ph.D. Thesis, Coventry Polytechnic, 1983. T. G. King, Rms skew and kurtosis of surface profile height distributions: some aspects of sample variation, Prec. Eng., 1 (3) (1980) 207-215.
171
8 E. J. Davis and K. J. Stout, Stylus measurement techniques: a contribution to the problem of parameter variation, Wear, 83 (1982) 49-60. 9 T. Y. Lin, P. J. Sullivan and K. J. Stout, Measurement variation of surface profile parameters, to be published. 10 0. A. Gorlenko, Assessment of surface roughness parameters and their interdependence, Prec. Eng., 2 (1) (1981) 105-108. 11 T. G. King and T. A. Spedding, Towards a rational surface profile characterization system, Prec. Eng., 4 (3) (1983) 153-160. 12 T. S. R. Murth and G. C. Reddy, Different functions and computations for surface topography, Wear, 83 (1982) 203-214. 13 I. Sherrington, Parameters for characterizing the surface topography of engineering components, Proc. In.st Mech. Eng., London, 201 (C4) (1987) 297-306. 14 A. Bengtsson and A. Ronnberg, Wide range three-dimensional roughness measuring system, Prec. Eng., 6 (3) (1984) 141-147. 15 B. Snaith, M. J. Edmonds and S. D. Probert, Use of a profilometer for surface mapping, Prec. Eng., 3 (2) (1981) 87-90. 16 E. C. Teague, F. E. Scire, S. M. Baker and S. W. Jensen, Three-dimensional stylus protilometry, Wear, 83 (1982) l-12. 17 J. R. T. Lewis and T. Sopwith, Three-dimensional surface measurement by microcomputer, Image mien Complcl., 4 (3) (1986) 159-166. 18 P. W. O’Callaghan, R. F. Babus’Haq, S. D. Probert and G. N. Evans, Three-dimensional surface topography assessment using a stylus/computer system, Int. J. Comp. Appl. Technol., 2 (2) (1989) 101-107. 19 S. R. Lange and B. Bhushan, Use of two- and three-dimensional non-contact surface profile for tribology applications, Sur$ Topogr., I (3) (1988) 277-289. 20 N. Ldrus, An integrated digital system for three-dimensional surface scanning, Prec. Eng., 3 (3) (1981) 37-43. 21 R. S. Sayles and T. R. Thomas, Mapping a small area of a surface, L Phys. E, 9 (1976) 855-861. 22 T. Tsukada and K. Sasajima, A three-dimensional measuring technique for surface asperities, Wear, 71 (1) (1981) l-14. 23 Surface roughness terminology - Part 1: surface and its parameters, IS0 Stand. 428711, 1984 (International Standards Organization). 24 Assessment of surface texture. Part 2: General information and guidance, Br. Stand. 1134, 1972 (British Standards Institution). 25 K. J. Stout, E. J. Davis and P. J. Sullivan, Atlas of machined surfaces, Chapman and Hall, London, 1990.
161
Comprehensive study of parameters for characterizing dimensional surface topography I: Some inherent properties of parameter variation
three-
W. P. Dong, P. J. Sullivan and K. J. Stout School of Manufacturing
and Mechanical
Engineering
The University of Birmingham,
Edgbaston,
Birmingham,
BI5
2TT
(UK)
(Received August 28, 1991; revised and accepted January 8, 1992)
Abstract The characterization of surface topography is critically important in many disciplines and industrial applications including triboiogy. Surfaces are often characterized through the use of parameters (usually using statistical techniques) which attempt to provide an indication of some attribute of the surface such as amplitude variation. However, many questions regarding the use of such parameter characterization remain unanswered such as (i) the degree to which the parameters are effective at representing the geometry of the surface and (ii) the dependence of the parameters on measurement conditions and characterization procedures. These and many other questions still need to be considered in some detail before an effective characterization of surface topography is possible. Through the analysis of surfaces, which have been measured using three-dimensional surface topographic techniques, we present the first of a series of reports on parameters for characterizing surface topography. This paper addresses inherent properties of two-dimensional parameter variation for the case of two orthogonal directions for some typical machined surfaces. Some experimental results are presented to illustrate that parameter variation is determinate rather than random in some cases, and that the variation is unable to be affected by measurement and signal-processing conditions. The factors which produce determinate and significant parameter variation are proposed through the analysis of surface topography features. The sensitivity of parameter variation to changes in surface topography is also discussed in the paper.
1. Introduction It is well known [l] that surface roughness greatly influences the mechanical and physical properties of contacting parts. The understanding of this behaviour is important in many applications such as wear, friction, lubrication, sealing tightness of joint, contact rigidity, contact stress, loaded area and thermal conductivity. Therefore engineers are very concerned about the surface roughness of their products. As early as the 193Os,researchers started to use profilometer techniques to characterize surface roughness. These characterization techniques included a visual display of the profile; however, in order to quantio and standardize measurements, the use of statistical parameters developed. Although the subject of surface topography has developed over the last 60 years, some important questions about the significance of parameter characteristics still arise. For example (i) which of the many parameters available may be used to fully characterize surface topography and (ii) how effective are the parameters calculated from a profile at representing the surface
0043-1648192/$5.00
character? For the first question, many researchers [14] have considered the proliferation of parameters (the “parameter rash”) for characterizing a surface. For an answer to the second question, engineers and scholars all agreed that some parameters vary significantly even with a very small area [5-131. For example Thomas and Charlton [5] pointed out that, on machined surfaces, variations of 50% are not uncommon. In recent years, coinciding with the development of microcomputers, digital instrumentation and other measurement techniques, three-dimensional (3D) surface topography measurement has become realistic. Many 3D systems have been proposed and developed [14-221, with the view that the topography of a surface could be represented by area data, as long as the sampling interval is small enough to satisfy the sampling theorem. However, although the surface data can be obtained relatively easily using these systems, the problem still remains of how to characterize such data from surfaces manufactured in different ways, both mathematically and functionally. Although some analysis of 3D surface topography has been carried out, very few parameters
0
1992 - Elsevier Sequoia. All rights reserved
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U! P. Dong Ed al. / Three-dimensional
and approaches to characterizing 3D surfaces have been proposed because of the complexity of surfaces and mathematical description difficulties. Some problems such as the following still have to be solved. (1) What set of parameters are able to characterize surface topography? (2) How do the parameters vary with sampling area and sampling interval? (3) How may the parameters correlate with a functional sense? (4) How may we define and identify lay structure or randomness of a surface? (5) How might we perform data pre-processing such as filtering and levelling for logged area data? Owing to the existence of these problems, a comprehensive and thorough study for 3D surface topography seems to be very necessary and urgent. This paper is the first in a series of reports towards carrying out a comprehensive study of parameters for characterizing 3D surface topography. It addresses the serious problem of parameter variation encountered in two-dimensional (2D) analysis through the analysis of 3D surface topography data. In analysing the variation of digital profile parameters it is helpful to classily the causes of parameter variation. The degree of variability due to these causes is different for different parameters. Cause A: variation in the geometry of the real machined surfaces. There are many possible causes for a variation in the geometry of a machined surface. Some causes are (i) manufacturing effect where factors such as tool wear and damage, and the presence of cutting debris will result in a variation in the geometry of the surface being produced and this variation will be superimposed on the rougher machining process which may be more or less random and (ii) surface geometric integrity with some features of surfaces described in terms of homogeneity, isotropy, stationarity and normality. The geometric variation of a machined surface is the main interest of metrologists in investigating profile measurement variation, Cause B: variation due to the measurement and data processing of real machined surfaces. There are many factors which determine a digital measurement system’s ability to obtain information about the real continuous surface. The inability of an instrument to record information about a real surface will result in variation in the parameters derived from such data. These factors may be classified into (i) inherent limitations of the measurement process such as finite stylus geometry and load, temperature and vibration effects, signal drift and quantization effects and (ii) selectable measurement conditions such as the influence of the profile length, sample spacing, filter type and cut-off, reference line or reference surface. The measurement
surface
topography
orientation may also cause profile measurement variation. Previous research work has been mainly concerned about whether parameter variation exists, or whether it is affected by machining and measurement conditions. To the authors’ knowledge no work has been carried out to date on the behaviour of the variation characteristics, and the degree of variation for a given machined surface. Thomas and Charlton [5] examined ten profiles for different surfaces obtained with different manufacturing methods. Attention was paid to the external factors which affect the parameter variation, e.g. how the variation changes with the measurement conditions such as cut-off, magnification range and skid. Spedding [6] developed statistical models for illustrating the variation which might be expected when estimates of skew and kurtosis were made from uncorrelated or correlated observations of gaussian or non-gaussian surfaces. The scatter of skew and kurtosis for measured surface data and simulated data was also addressed in his paper. King [7] also conducted theoretical and experimental examination of the distribution of skew and kurtosis of normal and non-normal populations, and suitable sample sizes for required parameter confidence limits were suggested. Davis and Stout [S] explored several sources which result in parameter variation. These sources are considered to be tracking inaccuracy, filtering of data by the choice of skid or datum, and surface damage caused by stylus and skid, or by particles attaching themselves to the surface under assessment. Recently Lin et al. [9] assessed joint effects of sample length and sample interval on the variation of roughness parameters. Multiple-regression analysis was carried out to infer statistical properties of parameter variation. Previous researchers all recognized that significant parameter variation exists, and they attempted to analyse the problem from different perspectives. Generally these investigations may be categorized into three areas. (i) Some researchers [l, 6, 71 have attempted to describe surface geometry theoretically and thus have developed random data models of surfaces such as normal or non-normal, and correlated or non-correlated. This approach is limited by the accuracy of the modelling process which is necessarily based upon many assumptions and it is impossible to model all engineering surfaces completely. (ii) Other workers [5, 81 examined the effects of measurement conditions on parameter variation such as the cut-off, the range of magnification, and the effects of the use of a skid. (iii) A fu r th er group [7, 9, 121 investigated the effects of (post-measurement) signal-processing conditions on parameter variation such as sampling interval, sampling length and filtering conditions.
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Clearly, in (ii) and (iii) the parameter variation is affected by sampling and processing conditions with different parameters being more or less affected by different conditions. However, all these considerations are subject to measurement choices as mentioned above in cause B (ii), and revealed just one aspect of parameter variation. Another aspect to be considered is the inherent variation in the surface geometry of manufactured surfaces, cause A. This variation results in surfaces that cannot be simply modelled by normal or nonnormal distributions and, as long as all measurement and data-processing conditions satisfy the sampling theorem (the Nyquist criterion etc.) or, in other words, as long as the features which exist on a surface can be measured without much distortion, the characteristics of the parameter variation will be unaffected under certain ranges of the conditions. Basically, parameter variation is the product of inherent surface topography features and other external conditions but, in some cases, it is more likely to depend on manufacturing approaches and less on machining, measurement and signal-processing conditions. This paper illustrates this problem. Combined with the analysis of 3D surface topography, the parameter variation along two orthogonal directions is examined according to the features which exist on four kinds of surfaces: ground, end milled, shaped and turned. Some inherent properties of parameter variation will be discussed and the sensitivities of some moment parameters to the change of surface features on the surface will be analysed. This suggests why parameters vary so significantly and what format the variation trend is for some manufactured surfaces.
x direction is defined as the trace direction, i.e. this may be thought of as the direction which coincides with the movement of the pick-up if a gear box is used; the y direction (ortho-trace) is defined as being perpendicular to the (x,2) plane. The parameters used in this paper are calculated from points sampled in thex direction or points sampled in they direction. Note that no reference plane is used, and all parameters are calculated with data points relative to their own reference lines. For the uniqueness of each reference line and calculability using digital techniques the use of a least-squares line was adopted, as recommended by some national and international standards [23, 241. Thus, for each trace, the reference line of the ith trace is 2(X, yi) = U,i + &LX
(i=l,
2, . ..) s)
(I) where n,, is the number of traces, and a, and bxi are coefficients of the ith reference line. These coefficients are obtained by minimizing the error
(2)
k-1
where n, is the total number each trace, and Gk ‘z(xk,
Yi>
-f(Xk,
Yi)
of sampling points in (3)
Therefore
(44 2. Algorithm for calculating parameters For the logged surface shown in Fig. 1, the height of each point z(x, y) can be regarded as having a random height distribution over the whole area although it may be superimposed on some periodic components. The
It can be shown that the data series exik of the ith trace has zero mean. Thus the parameters of each trace are all based on the zero-mean data series. The parameters used correspond to the arithmetic mean height R,, the r.m.s. roughness R,, the skewness Rsk and the kurtosis Rku. Further, R$, R+*, RskiXand Rku; (where for example R$ is the R, of the ith trace in the x direction) can be expressed as
(5b)
Fig. 1. Coordinate
system for logged 3D surface.
W. P. Dong et al. / Three-dimensional
164
surface
topography
(54 Similarly, the parameters of the jth ortho-trace which correspond to eqns. (l)-(5) can be written as r&j, Y)=Q +byjY
(6)
a,= 2 Eyjk2
(7)
k-l
Eyjk’z(xj,
Yk)
-i($,
Yk)
(8)
(9a)
(9b)
(lOc> WW The advantages of these definitions of the parameters are that (i) their derivation from 2D parameters is straightforward, (ii) they are easy to comprehend, (iii) they are convenient to be calculated and (iv) the information in both orthogonal directions is included and, as we shall see in subsequent parts of the series, these parameters can represent surface quality with much higher confidence and reliability from a statistical point of view. However, such definitions also have obvious disadvantages in that some 3D information, especially functional properties, is lost owing to the adoption of least-squares reference lines and not a least-squares reference plane. 3. The effects of pits and troughs on parameter variation
To illustrate the inherent properties of parameter variation, a specific example, but one that is often encountered in ground, honed and lapped surfaces, is presented. Figure 2 shows the variations in four parameters R;, R,:,R,; and RktiX (i= 1, 2, . . ., n,,) of a ground surface along the y direction, i.e. the variations
Fig. 2. Parameter variation of a ground surface along the lay direction: (a) variation in R.“; (b) variation in RqX; (c) variation in Rskr; (d) variation in R,;‘.
of trace by trace. The data for the ground surface was obtained using an instrument developed at the University of Birmingham [25] and comprises 164 traces each containing 164 points with a sample interval of 8 pm in both directions. The choice of sample spacing and number of points was made by establishing suitable sampling conditions (those which determine a bandwidth to include all dominant features on the surface) for the surface discussed in this paper. From Fig. 2, it can be seen that (i) all parameters vary significantly from trace to trace, (ii) a small random variation is superimposed on a trended variation, (iii) RskXand RkuXshow an opposite variation trend in this instance and (iv) at approximately 0.76 mm in the orthotrace direction a “spike” may be observed in both the R, and the R, graphs but it may be seen that Rskand Rku may be insensitive when R, and R, change significantly. Table 1 lists the variation range of the parameters relative to the total averaged values. Obviously, it can be seen that even the smallest range of the four is larger than 70%.
W. P. Dong et al. / Three-dimensional surface topography TABLE 1. Variation range of parameters on the ground surface
Trace Orthotrace
- 26.3 +43.7
-47.3 + 69.8
-62 + 637.5
-61.1 + 560.3
-40.4 + 86.6 - 152100 + 52788
-86.1 + 47.5 - 56.7 + 1276
On the basis of these observations a number of hypotheses may be drawn. Does a form error exist? Is the variation caused by improper selection of measurement and signal-processing conditions? Does the surface really contain such a dramatic change which is evident over the whole area? In fact, the trend variation is nothing to do with all above-mentioned hypotheses but results from the special features on the surface. Figure 3 is the axonometric projection of the ground surface; it clearly shows the grinding marks caused by the cutting action of the grits attached to the periphery of the grinding wheel and the directional nature of the grinding process. Evidently, the values of R, or other
Fig. 3. Axonometric projection of the ground surface.
165
parameters calculated across the surface lay or along the lay would be quite different, but no obvious features indicate that the parameter variation would have the features shown in Fig. 2, because no waveform is present on the surface. In theory, the grinding process produces a gaussian surface; however, it is unavoidable that superimposed on this is the resultant topography caused by the action of single sharp grits which plough into the ground surface or by the adhesion of particles pulled out of the surface, thus some troughs or pits might be formed. This may be observed in Fig. 4 which is a representation of the same surface shown in Fig. 3 truncated by 30% R,of the deepest valley of the ground surface (the R, was calculated relative to the least squares plane). It can be seen that the majority of the topography of the original surface has been removed and that all that remains are two deep troughs, A and B, one deep pit, C, and some other smaller troughs and pits on the surface. It is the deep troughs and pits that dominate the parameter variation although they occupy only a small area in the total logged surface. It is worth noting how we may distinguish between a pit and a trough. A trough is clearly elongated in one direction whilst a pit is not. One advantage of 3D analysis is that it can detect troughs whilst 2D cannot.
W. P. Dong et al. I Three-dimensional
166
surface
topography
3.20
1 brn I
0.00 1
\ Fig. 4. Surface
shown
in Fig. 3 truncated
by 30% R,.
Comparing Figs. 2 and 4, it is readily realized that the parameters are greatly influenced by the troughs and the pit. The absolute values of the parameters of several initial traces are small, they just coincide with the area where no apparent troughs or pits exist, i.e. they are in an ideal ground surface area, one of the profiles in this area is shown in Fig. 5(a). A profile which crosses the longest trough, A, on the surface is shown in Fig. 5(b). Since the trough was created by a sharp grit attached to the surface of the grinding wheel, it would be shallow at both extremes, and deep in the middle part. Figure 5(d) shows a transverse profile of trough A. Therefore, the variation of the parameters calculated from the profiles which cross the trough has a similar shape. It can be seen that towards the end of the longest trough, A, the shorter trough, B, starts (Fig. 4). Thus, as a consequence of this the absolute values of the parameters increase again. The peculiar “spike” of R: and Rq"in Figs. 2(a) and (b) may be attributed to the contribution of the pit C, as one more deep valley in the profile is observed in Fig. 5(c). Figure 6 shows the variation in the profile parameters in the ortho-trace direction for R,Y, RqY, RskY and RkuY
the same surface shown in Fig. 3. Obviously, the influence of the pit C will have the same effect on parameters calculated in both the x and they directions. However, the trough A does not have much effect on Rsky and RkuYin this direction, although it does significantly influence Rayand RqY. It should be noted from Fig. 2 and Fig. 6 that high moment relative parameters (such as & and RkU) are not always as sensitive as low moment parameters (such as R, and RJ. In some instances the former change more significantly than the latter, and sometimes the former remain almost stable while the latter exhibit large variation. The question of parameter sensitivity will be discussed later in the paper. 4. The effect of machining variation
process on parameter
The inherent phenomenon of parameter variation not only is produced by troughs and pits on ground, honed or lapped surfaces but also may result from the machining process itself. A typical example is from the end-milling process which produces a non-homogeneous surface, i.e. the surface on which the microscopic struc-
W. P. Dong et al. / Three-dimensional surface topography
Fig. 5. Typical profiles across the surface lay: (a) a normal profile; (b) a profile which crosses the trough A; (c) a profile which crosses the trough A and the pit C; (d) a profile of the trough A.
ture is not unique. As seen in Fig. 7, the paths of an end-milling cutter form two zones: one is non-overlapping, and the other is overlapped. These two zones exhibit totally different characteristics. Figure 8 shows an axonometric plot of the top portion (30% R,) for an overlapped part of an end-milled surface [25]. As a consequence of the overlapping cut, flat areas are shown on the surface at a zero datum which represent areas where no metal exists above this height. Clearly the distribution of remaining metal is nearly regular and the topography of these features will have a large influence on parameters measured. Thus, as can be seen in Fig. 9, the parameters along both trace and ortho-trace directions show distinctive variation. Evidently, these variation features could not be affected readily by measurement and signal-processing conditions. It is interesting to see that the variation in R,Yin Fig. 9(b) shows periodicity because of the overlapping cut. However, for a shaped surface, as shown in
167
Fig. 6. Parameter variation of the ground surface across the lay direction: (a) variation in R,Y; (b) variation in Rz; (c) variation in RSky;(d) variation in RtuY.
non-overlapped
overlapped
Fig. 7. Schematic
orea
area
diagram of end-milling
paths.
168
Fig. 8. Axonometric
W P. Dong et al. I Three-dimensional
plot of a top portion
of 30% R, truncation
z a
surface topography
of an end-milled
surface.
surface, However, it is possible for surfaces produced by these machining processes to show some tendency towards a deterministic trend on the parameter variation owing to some synthetic effects of machining conditions and material characteristics on the machined surface. Figure 12 gives an example of a face-turned surface for which the variation in R,"is plotted. It is clear that the random variation is superimposed on a deterministic increasing trend. In many cases it is possible to explain the trend in the parameter variation through examination of the 3D topography of the machined surface. 5. Sensitivity analysis
Fig. 9. Variation in R, in an overlapped portion of an end-milled surface: (a) variation in R,“; (b) variation in R,Y.
Fig. 10, although it exhibits distinctive periodic&y across the lay direction, the variation in Raydoes not have apparent periodicity (Fig. 11). This phenomenon indicates that the waveform in one direction does not affect the parameter variation along the same direction, as the waveform is removed owing to the adoption of a least-squares reference line for each profile. For some surfaces such as turned, bored or slab milled, the effect of the machining process does not have such a significant determinate trend on the parameter variation as seen in Fig. 9 for the end-milled
of parameter
variation
An accepted “rule of thumb” about parameter variation is that the higher order relative moment parameters such as R,,and Rkuare subject to larger variation than the lower order parameters and are therefore more sensitive. However, in some cases the lower order moment parameters such as R,and R,are more sensitive to the variation in surface features. As seen from Fig. 2 and Fig. 6, some peculiar spikes arise in R,", RqX, R,Yand RqY, whilst R,;, RkuX, Rsky and RkuY change little at the same area. In order to investigate the relative variation in these parameters we wish to analyse this behaviour theoretically. Let zl, z2, . . . . z, be the discrete heights of each profile point and suppose that zi (i= 1, 2, . . ., n) is an independent random variable with zero mean;
PP
\ \
170
K P. Dong et al. I Three-dimensional surface topography
Wb) (13c)
Hence the relative variations in the parameters
become
Figure 2 and Fig. 6 clearly demonstrate this phenomenon; this is why some peculiar spikes appear in the variations in R, and R, whereas R,, and R,, vary less. (3) The relative variations dRsk/Rsk and dR,,/R,, may have opposite variation trends when
Generally, in ground, honed and lapped surfaces where troughs and pits exist, this situation is true, whilst in some surfaces produced by other machining processes where predominantly the peaks are larger than the valleys, it is the other way around.
04a) (14b)
(14c)
(14d) It is clearly seen from eqns. (14a) and (14b) that R, and R, possess the same variation trends, as their relative variations have the same sign. Since the ratio rj of each element of dR,/R, to the corresponding element of dR,/R, is
(15) their magnitude relationship is not known, i.e. in some cases R, is subject to larger relative variation than R,, and in other cases it is vice versa. From eqns. (14~) and (14d), it can be seen that the second terms in the square brackets are just the same as the term in dR,/R,. Thus, some characteristics of the relative variation can be drawn. (I) R, and R,, may not have the same variation trends as R,, as the signs of eqns. (14~) and (14d) may be opposite to that of eqn. (14b). This situation may also be observed in Fig. 2 and Fig. 6. (2) The relative variations in dRst/Rsk and dRku/Rku may be smaller than dRqlRq and dR,IR, when
6. Conclusions This paper has discussed the problem of 2D parameter variation through the analysis of 3D surface topography. Special attention is paid to the nature of surfaces which possess inherent properties of parameter variation. Experimental results and theoretical analysis have been presented to demonstrate (i) why parameters vary significantly, (ii) why the variation shows a determinate trend in some cases, (iii) what surface features play an important role in parameter variation, (iv) how interrelationships exist between relative variations of some parameters and (v) how the sensitivities of some parameters may be affected by changes in surface features. The following conclusions may be drawn. (1) The causes of parameter variation may be classified into two categories: cause A, inherent variation in the surface geometry; cause B, variation due to the measurement and data processing. (2) Parameter variation is mainly due to the nature of surface features (cause A). The inherent characteristics of surfaces play an important role in the parameter variation rather than the measuring and data processing conditions. (3) Greater variation (above 50%) is possible for most parameters on the surface where troughs and pits exist, e.g. ground, honed and lapped surfaces. (4) Although troughs and pits are randomly distributed on a surface and they may only occupy a small portion in the logged area, the parameter variation may still be significant and may show a deterministic trend. (5) A predictable variation trend of the parameters may be obtained from a non-homogeneous surface; it is the machining process that results in such a trend. (6) R, and R, have similar variation trends, but the relationship between their relative variation magnitudes is unknown.
W. P. Dong et al. / Three-dimensional surface topography
(7) 4, and Rku may or may not have a similar variation trend and depend on the relationship shown in eqn. (17). For surfaces which have predominantly larger valleys than peaks, their variation trends are opposite. (8) In some cases, lower order moment parameters are more sensitive to the change in surface topography features than higher order relative moment parameters, and in other cases it is vice versa. (9) The use of 3D topography measurement and visualization techniques enables the true nature of the surface features to be established. This information can then be used, in many instances, to explain 2D parameter variation.
Acknowledgment The authors would like to thank the Commission of the European Communities for providing the funds for the work under its Programme for Applied Metrology and Chemical Analysis (BCR).
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8 E. J. Davis and K. J. Stout, Stylus measurement techniques: a contribution to the problem of parameter variation, Wear, 83 (1982) 49-60. 9 T. Y. Lin, P. J. Sullivan and K. J. Stout, Measurement variation of surface profile parameters, to be published. 10 0. A. Gorlenko, Assessment of surface roughness parameters and their interdependence, Prec. Eng., 2 (1) (1981) 105-108. 11 T. G. King and T. A. Spedding, Towards a rational surface profile characterization system, Prec. Eng., 4 (3) (1983) 153-160. 12 T. S. R. Murth and G. C. Reddy, Different functions and computations for surface topography, Wear, 83 (1982) 203-214. 13 I. Sherrington, Parameters for characterizing the surface topography of engineering components, Proc. In.st Mech. Eng., London, 201 (C4) (1987) 297-306. 14 A. Bengtsson and A. Ronnberg, Wide range three-dimensional roughness measuring system, Prec. Eng., 6 (3) (1984) 141-147. 15 B. Snaith, M. J. Edmonds and S. D. Probert, Use of a profilometer for surface mapping, Prec. Eng., 3 (2) (1981) 87-90. 16 E. C. Teague, F. E. Scire, S. M. Baker and S. W. Jensen, Three-dimensional stylus protilometry, Wear, 83 (1982) l-12. 17 J. R. T. Lewis and T. Sopwith, Three-dimensional surface measurement by microcomputer, Image mien Complcl., 4 (3) (1986) 159-166. 18 P. W. O’Callaghan, R. F. Babus’Haq, S. D. Probert and G. N. Evans, Three-dimensional surface topography assessment using a stylus/computer system, Int. J. Comp. Appl. Technol., 2 (2) (1989) 101-107. 19 S. R. Lange and B. Bhushan, Use of two- and three-dimensional non-contact surface profile for tribology applications, Sur$ Topogr., I (3) (1988) 277-289. 20 N. Ldrus, An integrated digital system for three-dimensional surface scanning, Prec. Eng., 3 (3) (1981) 37-43. 21 R. S. Sayles and T. R. Thomas, Mapping a small area of a surface, L Phys. E, 9 (1976) 855-861. 22 T. Tsukada and K. Sasajima, A three-dimensional measuring technique for surface asperities, Wear, 71 (1) (1981) l-14. 23 Surface roughness terminology - Part 1: surface and its parameters, IS0 Stand. 428711, 1984 (International Standards Organization). 24 Assessment of surface texture. Part 2: General information and guidance, Br. Stand. 1134, 1972 (British Standards Institution). 25 K. J. Stout, E. J. Davis and P. J. Sullivan, Atlas of machined surfaces, Chapman and Hall, London, 1990.