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Determination of appropriate sampling conditions for three-dimensional microtopography measurement
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Int. J. Mich.ToolsManufict.Vol.36. No. 12. pp. 1347-1362,1996 Copyright© 1996ElsevierScienceLid Printedin GreatBritain.All rightsreserved 0890--6955/96515.00+ .00
Pergamon
PII:S0890-6955(96)00034-X DETERMINATION OF APPROPRIATE SAMPLING CONDITIONS FOR THREE-DIMENSIONAL MICROTOPOGRAPHY MEASUREMENT W. P. DONG,t E. MAINSAH:~§ and K. J. STOUT¶ (Original received 30 November 1995)
Abstract--Currently, there are no national or international standards available for the measurement of surface roughness in three-dimensions. The selection of measuring parameters (for example, the sampling interval and area) therefore relies largely on the experiences of users and is fraught with subjectivity. This is inconvenient, particularly for inexperienced users; moreover, this practice makes it difficult for inter-comparisons of measurement results to be conducted. This paper aims to develop an objective criterion for selecting appropriate sampling conditions. A philosophy of surface roughness measurement in terms of the Nyquist wavelength limit is discussed. A sampling interval selection approach, based on spectral analysis, is then proposed. The proposed approach is not only applicable for the three-dimensional measurement scenario, but is also useful for the traditional two-dimensional measurement approach. The proposed technique has the added advantage that it can also be used to check the suitability of the measurement probe tip for any particular application. Experimental results based on measurements carried out on a large range of engineering surfaces (using a stylus instrument) are presented to demonstrate the effectiveness of the proposed approach. Copyright © 1996 Elsevier Science Ltd
1. INTRODUCTION In recent years, three-dimensional (3-D) surface measurement techniques (and relevant instrumentation) have been gaining in scope, particularly in Europe. These techniques have been increasingly applied within a wide range of engineering disciplines, such as manufacturing, tribology and automobile production. Evidence of the wide application of 3-D surface measurement techniques is provided by a simple analysis of the technical papers presented at the last three triennial international conferences on the Metrology and Properties of Engineering Surfaces [1-3]; in each of them, over half of the papers presented rely on 3-D measurement techniques and in the latest of the conferences, more than 75% of the papers advocate or use 3-D topography techniques [3]. A common feature of 3-D topography instrumentation is the adoption of the digital processing techmque in one form or another. This has the effect of imposing additional measuring conditions and variables in addition to the cut-off parameter, which is usually defined in conventional analogue two-dimensional (2-D) systems. In 3-D systems, the sampling conditions consist of the sampling interval and the number of sampling points. These two variables are very important for determining the short wavelength and long wavelength limits of the measurement, as well as the bandwidth of functional interest and, hence, the numeric characterization parameters of the measured surface. Even though there are a wide variety of surfaces and functional requirements, no standardized guidelines have been produced or published in relevant standard documents [4, 5] for selecting appropriate measurement sampling conditions, and users have tended to choose sampling conditions based mainly on their own experiences. The process is, therefore, fraught with subjectivity. The result is that, on the one hand, the sampling interval
*Center for Grinding Research and Development, University of Connecticut, Longley Building, Route 44, Storrs, CT 06269-51~t9, U.S.A. ¢School of En$inecdng, Coventry University, Priory Street, Coventry, CVI 5FB, U.K. §Corresponding author. ¶School of Manufacturing and Mechanical Engineering, The University of Birmingham, Birmingham, Bi5 2TI', U.K. 1347
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W.P. Dong et al.
selected may be too small, with the result that data are highly correlated and a large number of data points are required to represent an area of functional interest. On the other hand, the choice of sampling interval may be so large that the resulting data are highly uncorrelated, resulting in the loss or deterioration of surface spatial information/relationships. There is no consistent selection criteria to ensure that appropriate sampling conditions are selected by different users at all times. This situation exists not only in 3-D measurement, but also in 2-D measurement. Investigations designed to lead to the determination of the most appropriate sampling conditions for surface measurement have been carried out ever since digital stylus instruments became widely used from the early 1980s. Tsukada and Sasajima [6] developed a method to determine an optimum sampling interval for digitizing surface profiles, but their method was based on the assumption of a random surface height distribution. They proposed an optimum sampling interval criterion based on the variance of the difference between the digitized surface and the true surface (continuous surface) being smaller than a certain pre-established threshold. Yim and Kim [7] later proposed another method; this determines the optimum sampling interval by ensuring that the absolute difference between the Ra calculated from the digitized surface and that from the true surface stays below a certain pre-set threshold. The main problem with these two methods is that only the consistence of the amplitude property of the digitized and true surfaces is considered, whilst the spatial distortion due to discrete sampling is largely ignored. In fact, for a given profile length, the amplitude property of the digitized profile is hardly sensitive to changes in the sampling interval, as many authors have demonstrated [8-10]. However, the spatial relationship of roughness components will certainly be distorted by a poor choice of sampling interval, owing to the well-known aliasing effect [ 11, 12]. Another drawback of the above techniques is that they make no proposals as to how the sampling length or the number of sampling points ought to be determined. Another method proposed by Lin et al. [13] took into consideration the aliasing effect. Lin's method is based on area/1 spectral analysis and the criterion for determining the proper frequency bandwidth for 3-D topography measurement is that the Nyquist folding frequency (half of the sampling rate) is set at the position where the cumulative spectral power reaches 95% of the total power. Although this method overcomes the problem encountered in Tsukada's and Yim's methods, a new problem is introduced, since the 95% cumulative power limit always exists, no matter how the sampling interval is selected (i.e. a 95% cumulative spectral power limit always exists regardless of the sampling interval used). Even if a significant amount of aliasing has already occurred, the 95% cumulative power limit can still be found. Another drawback of Lin's method is that a large area has to be mapped with a very fine sampling interval before the areal spectral analysis can be carried out. This is both time-consuming and unnecessary. Recently, Mainsah [14] improved Lin's method in two ways. Firstly, the preprocessing time was significantly reduced by using a profile instead of an area during the pre-measurement run (to determine the default initial sampling interval). Secondly, the recommended sampling interval is obtained by selecting the sampling interval at which 80% of the cumulative spectral power fails within 1/8 and 1/3 of the Nyquist folding frequency. As with Lin's method, the number of sampling points has to be chosen beforehand (depending on the functional interests of the data owner) and the sampling intervals predicted by this method were not always consistent for different numbers of data points (across a wide range of surfaces). The present paper seeks to overcome the problems specified above by developing a new, robust and consistent approach for the determination of the most appropriate sampling conditions for 3-D surface roughness measurement. This approach is based on profile spectral analysis, and takes into consideration the Nyquist sampling theorem. The proposed method is not only suitable for 3-D topography measurement, but would also be useful for 2-D profile measurement, when a standardized cut-off is given. Moreover, the proposed technique, as in Mainsah's [14] can also be used to check the appropriateness of probe tips when very fine surfaces are measured. Experimental results from measurements carded
Sampling Conditions for Three-dimensional Microtopography Measurement
! 349
out on a range of different engineering surfaces (and with a variety of manufacturing conditions) are presented and discussed. The measurements presented and analysed are carded out using a stylus instrument; however, the underlying principle of the technique is equally applicable to other scanning probe instruments, for example, optical probe devices. 2.
APPROPRIATE SAMPLING CONDITIONS FOR THREE-DIMENSIONAL MEASUREMENT
An appropriate sampling condition has two important components - the sampling interval A (which has a bearing on aliasing and determines the short wavelength limit, )ts), and the number of sampling points N, which is instrument land computer limited. A combination of the two components produces the functionally important measurement area and the long wavelength limit of the measurement 2to. The short wavelength limit is influenced by three factors--the frequency characteristics of the instrument (fl), mechanical filtering caused by the finite size of the stylus tip (f2) and the sampling interval A. The actual short wavelength: limit of the measurement is not obtained by a simple combination of the three factors. Rather, it is that which imposes the largest (longest) short wavelength limit of the three (fl, f2 and A) that determines the actual short wavelength limit As of the measurement. Since the design of the frequency characteristics of an electronic instrument is flexible, whereas the size of the stylus is more or less fixed, the. sampling interval is usually selected to match the size of stylus tips as well as the functional requirements of the short wavelength limit. The long wavelength limit, ~A, on the other hand, is related to the high-pass filter cutoff of the instruraent, a combination of the sampling interval and the number of sampling points and the size of the sample. The first two factors are usually considered to satisfy the functional requirements of the long wavelength limit. In addition to the physical considerations for the short and long wavelength limits, an appropriate sampling condition can also be considered from a spectral analysis perspective. Quite clearly, if a sampling condition is appropriate, the main power of a tested profile should be located well below the Nyquist folding frequency fc =l/2A. For a periodic surface, this condition would guarantee that the Nyquist sampling criterion is satisfied and the continuous tx~riodic surface can be faithfully represented by the discrete approximation. For a random surface, this ensures that the main wavelength components are not distorted. Although in this case it is possible for some very short wavelength components to be folded into the long wavelength region, the magnitude of these (short wavelength) components is very small and, therefore, the influence of these distorted components can in most cases be safely ignored. Conversely, if the assumption is not satisfied, that is, the main power is widely distribnted and close to the Nyquist folding frequency, then this would suggest that the sampling condition is inappropriate and/or the tip radius of the stylus is not fine enough to measure the given sample. (If the tip were not appropriate, reducing the sampiing interval would not be useful.) On the other hand, an accumulation of the main spectral power too close to the zero frequency axis suggests that the sampled data are highly correlated or that many of the data are redundant in terms of numeric parameter calculation and visual representation. In the case of a periodic surface, this would mean that there are more than sufficient data points (e.g. >100) within one period of the profile/surface.; In this case, it is then necessary to take a significantly larger amount of data in order to assess an area of functional interest/significance. Thus, there is a significant advantage to be gained from selecting sampling conditions such that the main spectral power not only satisfies the Nyquist sampiing criterion but also ensures that data are not highly Correlated. 3.
DESCRIPTION OF TIlE APPROACH
In the proposed approach, the major spectral power is represented by 80% of the cumulative power (see Fig. 1). Fig. l(a) is a measured profile, denoted by x(n), n--O, 1 . . . . . N - 1 ; Fig. l(b) is the corresponding power spectrum and is represented by [11, 12]
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(2)
i=0
The Nyquist folding frequency (Fig. 1 (c)) is denoted by f~=l/2A such that fb=fJ2. As previously discussed, if a sampling interval is appropriate, the main spectral power is located well below the Nyquist folding frequency. The threshold is set at one-eighth of f~, i.e. f~=fJ8 (see Fig. 1 (c)). The reason for choosing this position as a reference is that if 80% of the cumulative power is close to this position, then (1) aliasing can be properly controlled during the measurement of random surfaces; (2) there will be about 16 discrete data points within one period--for periodic surfaces-which is satisfactory in a variety of engineering applications. Thus an appropriate sampling condition may be determined such that the 80% cumulative power level is closest to the reference position fa. If the 80% cumulative power level is assumed to be at fu for various combinations of A" and Nj, an appropriate sampling condition can be obtained from min [ICij} :
fJ-fa]forallUo(fk)=80% fc
(3)
Sampling Conditions for Three-dimensional Microtopography Measurement
1351
where CUis a normalized proximity factor, which indicates the extent of the closeness of f~ to f~. The smaller the absolute magnitude o f the proximity factor Ic d, the closer the position of the 80% cumulative power level is to f,. Let us assume that fp is the value off~j which satisfies Equation (3). When the stylus used in the measurement is suitable, then it is always possible to find a small value of min[ICu[, i.e. min[ICuI]<1/16. However, when the stylusitip radius is too large to measure the given sample or if the sampling interval is too large, fp would normally be around fJ2, i.e. fb--fJ2. In this case, the proximity factor CU is always positive with a min[ICuI] value around 5'/16. If the sampling interval is too small, fp tends to zero (0) and the proximity factor, Ci~ is always negative and approaches -1/8. These conditions suggest two things--firstly, that an appropriate sampling condition can be determined when Equation (3) is satisfied; secondly, that this technique can be used to check the suitability of the stylus/probe for the measurement. Theoretically, min[[C#l] is mainly a function of the sampling interval A, and is independent of the number of sampling points N. This character illustrates that, for differing numbers of sampling points, smaller values of the proximity factor Ic#l can always be found for any fixed sampling interval determined by Equation (3). (This is borne out by experiment as ,can be observed from Tables 2-8.) However, it is worth noting that the number of sampling points determined purely by Equation (3) may be unstable and may not meet other requirements, e.g. functional and instrument requirements. The number of sampling points should, therefore, be selected only after considering other conditions-with other priorities taken into consideration. One option of the order of priorities is, for example,, (1) functional considerations (e.g. a combination of N and A forms a sampling area and the user needs to be satisfied that the sample area is functionally significant); (2) instrument and computational considerations (e.g. the selected number of points should be such that the resulting data should fall within the memory capabilities of the computer and be processed by the computer at an acceptable speed); (3) a sampling interval purely determined by Equation (3). By taking real engineering conditions into consideration, a practical procedure for the implementation of the technique advocated in this paper is as follows. (1) An assessment length, Ao is selected by the user depending on functional interests. (2) A profile of length Ao is measured across the lay direction, using a very small default sampling interval Ao. As previously discussed (and as suggested by other authors [8, 15-21]), it is appropriate to ensure that the pre-measurement default sampling interval is ,equal to the stylus tip radius r. However, this is based on an ideal shape of the tip and, in practice, it is the local outline of the stylus tip (with a radius typically smaller than the tip radius) that plays an actual role in the measurement. Thus, a practical smallest sampling interval should be smaller than the tip radius. Since the tip radius of commercial instrmnents is usually larger than 2/zm as specified in the standard [5], a choice such as Ao 0.5/~m would be Suitable as the smallest sampling interval for most measurements. (3) Using different increments of Ao and corresponding idata points from the original profile (that in Equation (2) above) the power spectrum, G U(fk), and the cumulative power Uu (fk) of sub-sections of the profile are calculated. As suggested by Stout et al. [22], in order to cover a wide range of measurement spectra and to facilitate signal processing, e.g. the use of the FFT algorithm, a sampling interval series can be based on increments within the range 1, 2, 5, 10, ..., and a value for the number of data points N can be based on an integer power of 2 starting from N=27=128. Thus, for the smallest sampling interval Ao--0.5/~m, the successive sampling intervals A~ (i=0, 1, 2,...) used in the calculation are 0.5, 1, 2, 5, 10, 20, 50, ... Izm and the successive numbers of sampling points, Nj axe 128, 256, 512, .... It should be emphasized that, since an estimation of the power spectrum has a large error if only a single data set is used
1352
W.P. Dong et aL
[ 11 ], an improvement in the estimation can be obtained by carrying out an ensemble average of the power spectra of several data sets. (4) By comparing the absolute magnitude of the proximity factor, ICijl of all Uij~), an appropriate sampling interval A as well as fp are then determined when Equation (3) is satisfied. Then the number of sampling points N is selected, taking into consideration either the functional requirements or instrument and computer capability or Equation (3). The number of data points and the sampling interval in the direction perpendicular to the measured profile can be the same as N and A. Note that, currently, it takes a lot of computing time and memory space for a personal computer to process a digital surface signal with a data matrix larger than or equal to 1024x1024 [23]. Those wishing to process data sizes of this order might wish to consider the merits of using larger computers or mainframes. 4.
APPROPRIATE SAMPLING CONDITIONS FOR TWO-DIMENSIONAL MEASUREMENT
In essence, 2-D measurement is just a simplification of 3-D measurement with obvious differences. • Firstly, a high-pass filter is required in 2-D measurement, thus a 2-D profile is generally restricted to a defined bandwidth. • Secondly, the 2-D cut-off length (which is equal to the sampling length) has been clearly defined in relevant standards [4, 5]; therefore, if a standardized cut-off length has been selected, only A or N needs to be determined. • Thirdly, the computer memory restrictions on the number of sampling points is not a critical problem in 2-D measurement. Because of these differences, it is more important in 2-D measurement to determine objectively an appropriate sampling interval and, hence, ensure the choice of consistent sampling conditions. Following the 3-D measurement procedure discussed above, a practical procedure for selecting appropriate sampling conditions for 2-D measurement can be summarized as follows. (1) Choose a long measuring length Ao such that it is equal to an assessment length which is several times one of the cut-off lengths, At=(0.08, 0.25, 0.8, 2.5, 8) mm [4, 5, 24]. (2) Measure a profile of length )to with a very small sampling interval Ao (/1o--0.5/zm for stylus instruments). (3) Calculate the ensemble average of Gi; (fk) of all Uij (fk), different increments of Ao and from the numbers of data points listed in Table 1. It is worth noting that, since the number of data points used in the spectral calculation (see Table 1) are not powers of 2, the FFT algorithm cannot be used, hence the processing speed will be considerably reduced. (4) By comparing values of the absolute magnitude of the proximity factor IC~jl for all Uij (fk), appropriate sampling conditions A and N are then determined when Equation (3) is satisfied. Table 1. The number of data points used in spectral calculation Cut-off hc (mm)
0.08 0.25 0.8 2.5 8
Ai (mm)
0.0005
0.001
0.002
0.005
0.01
0.02
0.05
161 501 1601 5001 16001
251 801 2501 8001
126 401 1251 4001
161 501 1601
251 801
126 401
161
Sampling Conditions for Three-dimensional Microtopography Measurement 5.
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EXPERIMENTAL VFJtIFICATION AND! DISCUSSION
In order to velify the proposed approach, various surfaces were tested. The instrument used in the test Jis a Form Talysuff Series 3-D topography instrument employing a laser transducer and with a stylus tip radius of about 2/zm. The minimum sampling interval Ao used is 0.5/zm in all cases.
5.1. Surfaces fi'om different manufacturing processes Experimental :results based on measurements carded out on three kinds of surfaces are presented. The fi:rst is a ground surface with a very fine random topography; the second is milled and with a rougher topography; the third is a shaped comparison specimen (Rubert) with a ,distinctive periodic character. Profiles (about 8.2 nun in length) from the three samples are shown in Fig. 2. Tables 2-4 list all the values of the normalized proximity factor C o calculated from the three profiles. Ensemble averages of at least two subcomponents of the profiles have been carded out through spectral analysis. The maximum number of data points used in the tables is 8192. The three tables prompt a number of observations : similar values of the absolute magnitude of the proximity factor Icu] are clustered in the same column, implying the same choice of At for different numbers of data points, Nj; for surfaces with a fine topography, the smallest absolute values of the proximity factor
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Table 2. The proximity factor C~j calculated from the ground surface Ni
Ai(~m) 0.5
128 256 512 1024 2048 4196 8192
0.0519 -0.0566 -0.0640 -0.0419 -0.0268 -0.0548 -0.0561
1 0.0119" 0.0042 0.0360 0.0614 0.0108 0.0078
2
5
0.1168 0.1879 0.2341 0.1401 0.1380
0.4792 0.5698 0.4698 0.4343
10 0.6547 ~ 0.5504 0.4959
20
50
0.5832 0.5946
0.7130
Table 3. The proximity factor Cijcalcula~d from themilled surface
Ni
~ (~m) 0.5
128 256 512 1024 2048 4196 8192
-0.0703 -0.1100 -0.1172 -0.1199 -0.1206 0.1222 -0.1224
1
2
-0.0953 -0.1095 -0.1147 -0.1162 -0.1194 -0.1197
-0.0939 -0.1044 -0.1075 -0.1139 -0.1144
5 -0.0865 ~ -0.0820 -0.0946 -0.1003
10 -0.0394* -0.0641 -0.0756
20 -0.0034" -0.0258
50 0.1168
Table 4. The proximity factorCijcalculated from the shaped comparison specimen Ni
Ai (~m)
128 256 512 1024 2048 4196 8192
0.5
1
2
-0.0901 -0.1179 -0.1189 -0.1230 -0.1231 -0.1231 -0.1231
-O.ll09 -0.1129 -0,1210 -0.1213 -0.1212 -0.1211
-0.1010 -O. ll71 -0.1176 -0.1174 -0.1172
5 -0.0965# -0.0924 -0.1058 -0.1055
10 -0.0598 -0.0866 -0.0861
20 -0.0483* -0.0471
50 0.0618
[C~i[ are clustered towards the smaller sampling interval region, e.g. A=I /xm for the ground surface (Table 2); for surfaces with a rougher topography, the smallest absolute values of the proximity factor Ic,jl are clustered towards the larger sampling interval region, e.g. A=20/xm for the milled and the shaped surfaces (Tables 3 and 4). For example, from Equation (3) and taking into consideration the other requirements discussed in Section 3, appropriate sampling conditions (denoted by * in the tables) for each surface are as follows: • A=I/xm, N=128 (ground surface); • A=20/xm, N=128 (milled surface); • A=20/xm, N=128 (shaped surface). The profiles, power spectra, cumulative power spectra and 3-D maps obtained using the chosen appropriate conditions are presented in (a), (c), (e) and (g) of Figs 4-6. In Figs
Sampling Conditions for Three-dimensional Microtopography Measurement
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3-5(e), the 80% cumulative power spectra are indicated:by a small circle at position f~j, the left vertical ,dashed line is at the position f,--fd8, and: the right vertical dashed line is at the position jS,--fd2. A visual comparison of the effectiveness of the technique can be carded out by examining surfaces measured using the recommended sampling conditions as well as those measured using "unsuitable" sampling conditions. Surfaces measured using some unsuitable conditions (denoted by # in the tables) are presented in (b), (d), (f) and (h) of Figs 4-6. It is clear from these figures that the sampling interval used to measure the ground surface is too large, and that used to measure the other two surfaces (milled and shaped) is too small--the 80% cumulative power level f0 is close to f~ in the former and close to the zero frequency axis in the latter two cases.
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1356
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5.2. Same process--different manufacturing conditions To verify further the proposed approach, a set of electro-discharge machining (EDM) samples prepared using different conditions were measured using the Form Talysurf Series topography instrument. Figure 6 shows profilometric measurements taken from three of the EDM samples, which show clearly that the three specimens have distinctive spatial and amplitude characteristics--which is to be expected as their manufacturing conditions are different. Tabular readings of the proximity factor C U of measurements from the three specimens are presented in Tables 5-7. A simple comparison of the three measurements reveals that a small sampling interval (2 txm) is required to measure sample 1 properly, a sampling interval of 10/xm is required to measure sample 2 properly, and a much larger sampling interval of 20/xm is required to measure sample 3 properly. The correctness of the choice of appropriate sampling intervals (indicated by * in Tables 5-7) is borne out by a visual examination of the topographic features of the measured samples as depicted in Fig. 6.
Sampling Conditions for Three-dimensional Microtopography Measurement
! 357
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(g) A-3-D map of the selected condition
(h) A 3-D map of an unsuccessful condition Fig. 5.
5.3.
The two-dimensional measurement scenario
A profilometric measurement (assessment length Ao=8 mm) was taken from a honed engine bore surface (Fig. 7(a)). The proximity factors Co obtained from the cumulative power spectra of at least two ensemble averages are presented in Table 8. Based on this proposed approach, it can be seen that a sampling interval A=2/~m is appropriate for the measurement. In addition, if the cut-off length selected is At=0.8 mm (which is the usual default cut-off length for 2-D measurement), then the sampling conditions A=2/.~m and N=401 are autornatically determined by the proposed approach. Figures 7(b) to 8(e) show a sub-profile plot obtained using four sampling conditions (see Table 1), all with the cutoff At=0.8 mm. From a visualization perspective, the difference between Fig. 7(b)-(d) is insignificant; but the difference between Fig. 7(e) and the three others is clearly visible. Table 9 lists some amplitude parameters calculated from the four profiles. (Ra is the arithmetic mean average, Rq is the root-square-average, Sk is the skewness and Ku is the kurtosis.) The relative differences • in Table 9 are calculated with respect to the sampling
1358
W.P. Dong et al.
< 0
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2 4 6 (a) Profile of sample 1
20
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0
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r
.
,
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Profile length (c) A profile of sample 3
Fig. 6. Table 5. The proximity factor Cij calculated from EDM sample 1 Ai (/zm)
Ni
128 256 512 1024 2048 4196 8192
0.5
1
2"
5
10
20
50
-0.0847 -0.1032 -0.1037 -0.1016 -0.1012 -0.1037 -0.1072
-0.0813 -0.0824 -0.0783 -0.0775 -0.0824 -0.0895
-0.0404 -0.0316 -0.0301 -0.0399 -0.0539
0.1224 0.1080 0.0807 0.0528
0.3276 0.2841 0.2262
0.5143 0.4867
0.4502
Table 6. The proximity factor CU calculated from EDM sample 2 Ni
128 256 512 1024 2048 4196 8192
A, (/zm) 0.5
1
2
5
10"
20
50
-0.1053 -0.0991 -0.1163 -0.1165 -0.1177 -0.1198 -0.1212
-0.0732 -0.1076 -0.1081 -0.1105 -0.1146 -0.1174
-0.0899 -0.0912 -0.0959 -0.1042 -0.1098
-0.0494 -0.0691 -0.0740 -0.0865
-0.0135 -0.0230 -0.0479
0.0798 0.0302
0.2046
conditions, A--0.5/xm and N=1601. Quite clearly, from a parametric point of view, the conclusions about the measurements are consistent with those arrived at through visual characterization (see above). These comparisons suggest that the selected sampling condition, A=2/xm and N--401, is appropriate. 5.4. Checking the suitability of the stylus As discussed above, if the stylus tip radius is so large that it cannot effectively measure the intended surface (in the case of very fine surfaces), then the 80% cumulative power
Sampling Conditions for Three-dimensional Microtopography Measurement
1359
Table 7. The proximity factor Co calculated from EDM sample 3 i
Ni
128 256 512 1024 2048 4196 8192
At (/~m) 0.5
1
2
5
10
20"
50
-0.(g)91 -0.1095 -0.1177 -0.1178 -0.1228 -0.1220 -0.1219
-0.0941 -0.1104 -O.ll07 -0.1207 -0.1190 -0.1188
-0.0959 -0.0965 -0.1163 -0.1130 -0.1126
-0.0886 -0.0973 -0.0953 --0.0941
-0.0698 -0.0695 -0.0632
-0.0078 -0.0091
0.1842
<
m
0
,
i
i
2 4 6 (a) A long profile
1
"
0
0.2
"
"
0.4
0.6
8
0.8
(b) A profile obtained when A = 0.5 izm and N = 1601
0
0.2
0.4
0.6
0,8
(e) The profile obtained when A = 1 t~m and N = 801 |
....
.
.
.
i < 0 0.2 0.4 0.6 0.8 (d) The profile obtained when A = 2 Izm and N = 401
l
'"
0
"
"
"
i¸
0.2 0.4 0.6 Profile length (ram)
0.8
(e) The profile obtained when A = 5 ttm and N = 161 i
Fig. 7.
1360
W. P. Dong et al.
Table 8. The proximity factor C~/calculated from the honed engine surface Cut-off hZmm)
Ai (/xm)
0.08 0.25 0.8 2.5 8
0.0005
0.001
0.002
0.005
0.01
0.02
0.05
-0.0776 -0.0784 -0.0898 -0.0901 -0.0925
-0.0317 -0.0545 -0.0552 -0.0601
0.0604 0.0162" 0.0145 0.0044
0.2290 0.2167 0.1882
0.4974 0.4760
0.6163 0.6185
0.6353
0.02 0.02 -0.02~
' tl
' [
< -0.041
'
1~
t
,
0.2
0.4
Profile length (ram) (a) A measured profile
.~ ~
004 I°L 0.5
~
. . . . . . . ~ J.J.~Jo.k~.x . . . . . 500 1000 Frequency (I/mm)
0
(b) Power spectrum of the profile
/
50F/
t 500 Frequency (1/mm)
1000
(c) Cumulative power of the profile Fig. 8.
Table 9. Parameters of a profile from a honed surface calculated using different sampling conditions (At=0.8 ram) A (/xm) 0.5 1 2 5
N
1601 801 401 161
Ra (gm)
E (%)
0.584 0.583 -0.17 0.582 -0.34 0.572 -2.05
R, (/zm) 0.734 0.735 0.735 0.731
~ (%)
Sk
E (%)
-1.032 0.14 -1.033 -0.097 0.14 -1.028 0.39 0 . 4 1 -1.077 -4.36
KI, (/zm)
~ (%)
3.777 3.778 0.026 3.767 -0.26 3.823 1.22
RD (/xm)
e (%)
R7 (/xm)
E (%)
1.126 3.2 1.093 -2.931 3.117 -2.593 1.058 -6.039 3.077 -3.843 1 . 0 5 -6.749 3.032 -5.25
level fj will tend towards (or beyond) fb. This may be an indication of aliasing. This phenomenon can be observed by measuring an optical flat (clearly, the roughness cannot be effectively measured by any standardized stylus). Table 10 shows tabulated values of the proximity factor CU, calculated from optical flat measurements. Clearly, all the proximity factor values CUare positive, with the smallest proximity factor value C0=0.4572, which is so large (relatively) that it cannot possibly represent an appropriate sampling condition.
Sampling Conditions for Three-dimensional Microtop0graphy Measurement
1361
Table 10. The proximity factor CU calculated from optical flat data Ni
Ai (/~m) 0.5
1
2
5
10
20
0.6358 0.6652
0.6682
i
128 256 512 1024 20.48 4196
0.5267 0.5255 (I.4714 (I.4572" (I.4823 0.4955
0.5711 0.6061 0.5521 0.5176 0.5202
0.6355 0.6781 0.6677 0.6635
0.5869 0.6016 0.6403
Figure 8 shows a profile, its power spectrum and the corresponding cumulative power spectral plot, obtained using the smallest proximity factor (C 0) prediction from Table 10 (i.e. A=2/xm and N=401). It can be seen that the power spectrum is broadly distributed (spread out) and the smallest value off0 is beyond fa. 6. CONCLUSIONS
This paper has proposed a new philosophy of surface r~ughness measurement based on an objective definition of surface topography measuring iconditions. It is recognized that the short wavelength measurement limit depends on whichever is the largest of the three factors--the fre~quency characteristics of instruments, tile size of the stylus tip and the sampling intervalmwhilst the long wavelength limit is linked to the functional requirements of the stufface as well as instrument capability. A new approach for the determination of appropriate sampling conditions for 2-D and 3-D surface roughness measurement has been proposed. The approach is based on spectral analysis and takes into consideration the Nyquist sampling criterion. It has been advanced that an appropriate sampling condition can be obtained when 80% of the cumulative power of a pre-measured trace lies around one-eighth of the Nyquist folding frequency. In addition, the proposed approach can also be used to check the suitability of the stylus for any particular measurement--this will help ensure that out-of-specification styli are not used. Procedures for the implementation of the technique in both 2-D and 3-D measurement practice have been advanced and developed. Experiments have been conducted on a range of different surfaces--including random and periodic ones--and for different manufacturing process conditions, to demonstrate the effectiveness of the technique. The results verify the effectiveness of the approach. Although the measurements presented in this paper were all carried out using a stylus instrument, the principle of the approach can be readily extended to other scanning probe instruments (including optical probe-based systems). REFERENCES [1] K. J. Stout (F~t.), Proc. 4th Int. Conference on Metrology and Properties of Engineering Surfaces, Washington. Kogan Page (1988). [2] K. J. Stout (EdL.),Proc. 5th Int. Conference on Metrology and Properties of Engineering Surfaces, Leicester. Pergamon Press, Oxford (1991). [3] K. J. Stout (Ed.), Proc. 6th Int. Conference on Metrology and Properties of Engineering Surfaces, Birmingham. ]Pergamon Press, Oxford (1994). [4] Rules and pr¢~edures for the measurement of surface roughness using stylus instruments, International Standard ISO 4288 (1985). [5] Instruments for the measurement of surface roughness by the profile method---contact (stylus) instruments of consecutiw~ profile transformation contact profile meters, System M, International Standard ISO 3274 (1975). [6] T. Tsukada and K. Sasajima, An optimum sampling interval for digitising surface asperity profiles, Wear 83, 119-128 (1982). [7] D. Y. Yim and S. W. Kim, Optimum sampling intervals for Ra roughness measurement, Proc. Inst. Mech. Engrs C205, 139-142 (1991). [8] D. D. Chetwynd, The digitisation of surface profiles, Wear 57, 137-145 (1979). [9] T. R. Thomas and G. Charlton, Variation of roughness parameters on some typical manufactured surfaces, Precision Engng 3, 91-96 (1981).
1362
W.P. Dong et al.
[10] W. P. Dong, P. J. Sullivan and K. J. Stout, The significance of surface features in characterising 2-D and 3-D surface topography, ASME PED 62, 115 0992). [l l] J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures. Wiley-lnterscience, New York (1971 ). [12] J. G. Proakis and D. G. Manolakis, Digital Signal Processing, Principles, Algorithms, and Applications. Macmillan, Basingstoke (1992). [13] T. Y. Lin, L. Blunt and K. J. Stout, Determination of proper frequency bandwidth for 3-D topography measurement using spectral analysis Part l: Isotropic surfaces, Wear 166, 221-232 (1993). [14] E. Mainsah, Investigation ofpre-characterisation aspects of three-dimensional micro-topography. Ph.D. Thesis, University of Birmingham, U.K. (1994). [15] T. R. Thomas, Roughness Surfaces. Longman, London (1982). [16] T. Nakamura, On deformation of surface roughness curves caused by finite radius of stylus tip and tilting of stylus holder arm, Bull. Japan Soc. Prec. Engng 1, 240-249 (1966). [17] J. L. Guerrero and J. T. Black, Stylus tracer resolution and surface damage as determined by scanning electron microscopy, Trans. ASME J. Engng Ind. 94, 1087-1093 (1972). [18] I. Sherrington and E. H. Smith, A quantitative study of the influence of stylus shape and load on the fidelity of data recorded by stylus instruments, Proc. Second National Conference on Production Research, Edinburgh, pp. 762-783. University of Edinburgh Press (1986). [19] T. R. Thomas, Some examples of the versatility of stylus instruments, Mecanique, Materiaux, Electricite 337, 17-25 (1978). [20] D. J. Whitehouse, Some ultimate limits on the measurement of surfaces using stylus techniques, Measurement Control 8, 147-151 (1975). [21] V. Radhakrishnan, Effect of stylus radius on the roughness values measured with tracing stylus instruments, Wear 16, 325-335 (1970). [22] K. J. Stout, P. J. Sullivan, W. P. Dong, E. Mainsah, N. Luo, T. Mathia and H. Zahouani, The Development of Methodsfor the Characterisation of Roughness in Three-dimensions. Commission of the European Communities, ISBN 0 7044 1313 2 (1993). [23] W. P. Dong, E. Mainsah, P. J. Sullivan and K. J. Stout, Instruments and measurement techniques of 3dimensional surface topography, Three-DimensionalSurface Topography; Measurement, Interpretation and Applications, A Survey and Bibliography (edited by K. J. Stout). Penton Press, London (1994). [24] Metrological characterisation of phase corrected filters and transmission bands for use in contact (stylus) instruments, Draft International Standard ISO/DIS ! 1562 (1994).
Int. J. Mich.ToolsManufict.Vol.36. No. 12. pp. 1347-1362,1996 Copyright© 1996ElsevierScienceLid Printedin GreatBritain.All rightsreserved 0890--6955/96515.00+ .00
Pergamon
PII:S0890-6955(96)00034-X DETERMINATION OF APPROPRIATE SAMPLING CONDITIONS FOR THREE-DIMENSIONAL MICROTOPOGRAPHY MEASUREMENT W. P. DONG,t E. MAINSAH:~§ and K. J. STOUT¶ (Original received 30 November 1995)
Abstract--Currently, there are no national or international standards available for the measurement of surface roughness in three-dimensions. The selection of measuring parameters (for example, the sampling interval and area) therefore relies largely on the experiences of users and is fraught with subjectivity. This is inconvenient, particularly for inexperienced users; moreover, this practice makes it difficult for inter-comparisons of measurement results to be conducted. This paper aims to develop an objective criterion for selecting appropriate sampling conditions. A philosophy of surface roughness measurement in terms of the Nyquist wavelength limit is discussed. A sampling interval selection approach, based on spectral analysis, is then proposed. The proposed approach is not only applicable for the three-dimensional measurement scenario, but is also useful for the traditional two-dimensional measurement approach. The proposed technique has the added advantage that it can also be used to check the suitability of the measurement probe tip for any particular application. Experimental results based on measurements carried out on a large range of engineering surfaces (using a stylus instrument) are presented to demonstrate the effectiveness of the proposed approach. Copyright © 1996 Elsevier Science Ltd
1. INTRODUCTION In recent years, three-dimensional (3-D) surface measurement techniques (and relevant instrumentation) have been gaining in scope, particularly in Europe. These techniques have been increasingly applied within a wide range of engineering disciplines, such as manufacturing, tribology and automobile production. Evidence of the wide application of 3-D surface measurement techniques is provided by a simple analysis of the technical papers presented at the last three triennial international conferences on the Metrology and Properties of Engineering Surfaces [1-3]; in each of them, over half of the papers presented rely on 3-D measurement techniques and in the latest of the conferences, more than 75% of the papers advocate or use 3-D topography techniques [3]. A common feature of 3-D topography instrumentation is the adoption of the digital processing techmque in one form or another. This has the effect of imposing additional measuring conditions and variables in addition to the cut-off parameter, which is usually defined in conventional analogue two-dimensional (2-D) systems. In 3-D systems, the sampling conditions consist of the sampling interval and the number of sampling points. These two variables are very important for determining the short wavelength and long wavelength limits of the measurement, as well as the bandwidth of functional interest and, hence, the numeric characterization parameters of the measured surface. Even though there are a wide variety of surfaces and functional requirements, no standardized guidelines have been produced or published in relevant standard documents [4, 5] for selecting appropriate measurement sampling conditions, and users have tended to choose sampling conditions based mainly on their own experiences. The process is, therefore, fraught with subjectivity. The result is that, on the one hand, the sampling interval
*Center for Grinding Research and Development, University of Connecticut, Longley Building, Route 44, Storrs, CT 06269-51~t9, U.S.A. ¢School of En$inecdng, Coventry University, Priory Street, Coventry, CVI 5FB, U.K. §Corresponding author. ¶School of Manufacturing and Mechanical Engineering, The University of Birmingham, Birmingham, Bi5 2TI', U.K. 1347
1348
W.P. Dong et al.
selected may be too small, with the result that data are highly correlated and a large number of data points are required to represent an area of functional interest. On the other hand, the choice of sampling interval may be so large that the resulting data are highly uncorrelated, resulting in the loss or deterioration of surface spatial information/relationships. There is no consistent selection criteria to ensure that appropriate sampling conditions are selected by different users at all times. This situation exists not only in 3-D measurement, but also in 2-D measurement. Investigations designed to lead to the determination of the most appropriate sampling conditions for surface measurement have been carried out ever since digital stylus instruments became widely used from the early 1980s. Tsukada and Sasajima [6] developed a method to determine an optimum sampling interval for digitizing surface profiles, but their method was based on the assumption of a random surface height distribution. They proposed an optimum sampling interval criterion based on the variance of the difference between the digitized surface and the true surface (continuous surface) being smaller than a certain pre-established threshold. Yim and Kim [7] later proposed another method; this determines the optimum sampling interval by ensuring that the absolute difference between the Ra calculated from the digitized surface and that from the true surface stays below a certain pre-set threshold. The main problem with these two methods is that only the consistence of the amplitude property of the digitized and true surfaces is considered, whilst the spatial distortion due to discrete sampling is largely ignored. In fact, for a given profile length, the amplitude property of the digitized profile is hardly sensitive to changes in the sampling interval, as many authors have demonstrated [8-10]. However, the spatial relationship of roughness components will certainly be distorted by a poor choice of sampling interval, owing to the well-known aliasing effect [ 11, 12]. Another drawback of the above techniques is that they make no proposals as to how the sampling length or the number of sampling points ought to be determined. Another method proposed by Lin et al. [13] took into consideration the aliasing effect. Lin's method is based on area/1 spectral analysis and the criterion for determining the proper frequency bandwidth for 3-D topography measurement is that the Nyquist folding frequency (half of the sampling rate) is set at the position where the cumulative spectral power reaches 95% of the total power. Although this method overcomes the problem encountered in Tsukada's and Yim's methods, a new problem is introduced, since the 95% cumulative power limit always exists, no matter how the sampling interval is selected (i.e. a 95% cumulative spectral power limit always exists regardless of the sampling interval used). Even if a significant amount of aliasing has already occurred, the 95% cumulative power limit can still be found. Another drawback of Lin's method is that a large area has to be mapped with a very fine sampling interval before the areal spectral analysis can be carried out. This is both time-consuming and unnecessary. Recently, Mainsah [14] improved Lin's method in two ways. Firstly, the preprocessing time was significantly reduced by using a profile instead of an area during the pre-measurement run (to determine the default initial sampling interval). Secondly, the recommended sampling interval is obtained by selecting the sampling interval at which 80% of the cumulative spectral power fails within 1/8 and 1/3 of the Nyquist folding frequency. As with Lin's method, the number of sampling points has to be chosen beforehand (depending on the functional interests of the data owner) and the sampling intervals predicted by this method were not always consistent for different numbers of data points (across a wide range of surfaces). The present paper seeks to overcome the problems specified above by developing a new, robust and consistent approach for the determination of the most appropriate sampling conditions for 3-D surface roughness measurement. This approach is based on profile spectral analysis, and takes into consideration the Nyquist sampling theorem. The proposed method is not only suitable for 3-D topography measurement, but would also be useful for 2-D profile measurement, when a standardized cut-off is given. Moreover, the proposed technique, as in Mainsah's [14] can also be used to check the appropriateness of probe tips when very fine surfaces are measured. Experimental results from measurements carded
Sampling Conditions for Three-dimensional Microtopography Measurement
! 349
out on a range of different engineering surfaces (and with a variety of manufacturing conditions) are presented and discussed. The measurements presented and analysed are carded out using a stylus instrument; however, the underlying principle of the technique is equally applicable to other scanning probe instruments, for example, optical probe devices. 2.
APPROPRIATE SAMPLING CONDITIONS FOR THREE-DIMENSIONAL MEASUREMENT
An appropriate sampling condition has two important components - the sampling interval A (which has a bearing on aliasing and determines the short wavelength limit, )ts), and the number of sampling points N, which is instrument land computer limited. A combination of the two components produces the functionally important measurement area and the long wavelength limit of the measurement 2to. The short wavelength limit is influenced by three factors--the frequency characteristics of the instrument (fl), mechanical filtering caused by the finite size of the stylus tip (f2) and the sampling interval A. The actual short wavelength: limit of the measurement is not obtained by a simple combination of the three factors. Rather, it is that which imposes the largest (longest) short wavelength limit of the three (fl, f2 and A) that determines the actual short wavelength limit As of the measurement. Since the design of the frequency characteristics of an electronic instrument is flexible, whereas the size of the stylus is more or less fixed, the. sampling interval is usually selected to match the size of stylus tips as well as the functional requirements of the short wavelength limit. The long wavelength limit, ~A, on the other hand, is related to the high-pass filter cutoff of the instruraent, a combination of the sampling interval and the number of sampling points and the size of the sample. The first two factors are usually considered to satisfy the functional requirements of the long wavelength limit. In addition to the physical considerations for the short and long wavelength limits, an appropriate sampling condition can also be considered from a spectral analysis perspective. Quite clearly, if a sampling condition is appropriate, the main power of a tested profile should be located well below the Nyquist folding frequency fc =l/2A. For a periodic surface, this condition would guarantee that the Nyquist sampling criterion is satisfied and the continuous tx~riodic surface can be faithfully represented by the discrete approximation. For a random surface, this ensures that the main wavelength components are not distorted. Although in this case it is possible for some very short wavelength components to be folded into the long wavelength region, the magnitude of these (short wavelength) components is very small and, therefore, the influence of these distorted components can in most cases be safely ignored. Conversely, if the assumption is not satisfied, that is, the main power is widely distribnted and close to the Nyquist folding frequency, then this would suggest that the sampling condition is inappropriate and/or the tip radius of the stylus is not fine enough to measure the given sample. (If the tip were not appropriate, reducing the sampiing interval would not be useful.) On the other hand, an accumulation of the main spectral power too close to the zero frequency axis suggests that the sampled data are highly correlated or that many of the data are redundant in terms of numeric parameter calculation and visual representation. In the case of a periodic surface, this would mean that there are more than sufficient data points (e.g. >100) within one period of the profile/surface.; In this case, it is then necessary to take a significantly larger amount of data in order to assess an area of functional interest/significance. Thus, there is a significant advantage to be gained from selecting sampling conditions such that the main spectral power not only satisfies the Nyquist sampiing criterion but also ensures that data are not highly Correlated. 3.
DESCRIPTION OF TIlE APPROACH
In the proposed approach, the major spectral power is represented by 80% of the cumulative power (see Fig. 1). Fig. l(a) is a measured profile, denoted by x(n), n--O, 1 . . . . . N - 1 ; Fig. l(b) is the corresponding power spectrum and is represented by [11, 12]
W . P . Dong et al.
1350 80 60
'
"
40
,< _40[
~/
IN- . . . .
~
-
v
2 4 Profile length ( m m ) (a) A measured profile
.:~ 10 2'0 Frequency ( l / m m )
30
(b) Power spectrum of the profile
100
.2
~
50
i
I
I
[ [
I I
i
i
I
e~
fafij
II II
fl
Zl II
r
r
0
"
10 fb 20 Frequency (1/mm)
fc
30
(c) Cumulative power of the profile Fig. 1. A N-I
k
2
G(fk)=~ ~x(n)e-J27r~" . .k.=. 0 ,
1.
n =0
NI2;fk--NA k
(1)
and Fig. l(c) is the normalized cumulative power along the frequency axis and is given by k
U(f~)- i=0 u/2
.100%, k = 0 , 1 . . . .
NI2.
(2)
i=0
The Nyquist folding frequency (Fig. 1 (c)) is denoted by f~=l/2A such that fb=fJ2. As previously discussed, if a sampling interval is appropriate, the main spectral power is located well below the Nyquist folding frequency. The threshold is set at one-eighth of f~, i.e. f~=fJ8 (see Fig. 1 (c)). The reason for choosing this position as a reference is that if 80% of the cumulative power is close to this position, then (1) aliasing can be properly controlled during the measurement of random surfaces; (2) there will be about 16 discrete data points within one period--for periodic surfaces-which is satisfactory in a variety of engineering applications. Thus an appropriate sampling condition may be determined such that the 80% cumulative power level is closest to the reference position fa. If the 80% cumulative power level is assumed to be at fu for various combinations of A" and Nj, an appropriate sampling condition can be obtained from min [ICij} :
fJ-fa]forallUo(fk)=80% fc
(3)
Sampling Conditions for Three-dimensional Microtopography Measurement
1351
where CUis a normalized proximity factor, which indicates the extent of the closeness of f~ to f~. The smaller the absolute magnitude o f the proximity factor Ic d, the closer the position of the 80% cumulative power level is to f,. Let us assume that fp is the value off~j which satisfies Equation (3). When the stylus used in the measurement is suitable, then it is always possible to find a small value of min[ICu[, i.e. min[ICuI]<1/16. However, when the stylusitip radius is too large to measure the given sample or if the sampling interval is too large, fp would normally be around fJ2, i.e. fb--fJ2. In this case, the proximity factor CU is always positive with a min[ICuI] value around 5'/16. If the sampling interval is too small, fp tends to zero (0) and the proximity factor, Ci~ is always negative and approaches -1/8. These conditions suggest two things--firstly, that an appropriate sampling condition can be determined when Equation (3) is satisfied; secondly, that this technique can be used to check the suitability of the stylus/probe for the measurement. Theoretically, min[[C#l] is mainly a function of the sampling interval A, and is independent of the number of sampling points N. This character illustrates that, for differing numbers of sampling points, smaller values of the proximity factor Ic#l can always be found for any fixed sampling interval determined by Equation (3). (This is borne out by experiment as ,can be observed from Tables 2-8.) However, it is worth noting that the number of sampling points determined purely by Equation (3) may be unstable and may not meet other requirements, e.g. functional and instrument requirements. The number of sampling points should, therefore, be selected only after considering other conditions-with other priorities taken into consideration. One option of the order of priorities is, for example,, (1) functional considerations (e.g. a combination of N and A forms a sampling area and the user needs to be satisfied that the sample area is functionally significant); (2) instrument and computational considerations (e.g. the selected number of points should be such that the resulting data should fall within the memory capabilities of the computer and be processed by the computer at an acceptable speed); (3) a sampling interval purely determined by Equation (3). By taking real engineering conditions into consideration, a practical procedure for the implementation of the technique advocated in this paper is as follows. (1) An assessment length, Ao is selected by the user depending on functional interests. (2) A profile of length Ao is measured across the lay direction, using a very small default sampling interval Ao. As previously discussed (and as suggested by other authors [8, 15-21]), it is appropriate to ensure that the pre-measurement default sampling interval is ,equal to the stylus tip radius r. However, this is based on an ideal shape of the tip and, in practice, it is the local outline of the stylus tip (with a radius typically smaller than the tip radius) that plays an actual role in the measurement. Thus, a practical smallest sampling interval should be smaller than the tip radius. Since the tip radius of commercial instrmnents is usually larger than 2/zm as specified in the standard [5], a choice such as Ao 0.5/~m would be Suitable as the smallest sampling interval for most measurements. (3) Using different increments of Ao and corresponding idata points from the original profile (that in Equation (2) above) the power spectrum, G U(fk), and the cumulative power Uu (fk) of sub-sections of the profile are calculated. As suggested by Stout et al. [22], in order to cover a wide range of measurement spectra and to facilitate signal processing, e.g. the use of the FFT algorithm, a sampling interval series can be based on increments within the range 1, 2, 5, 10, ..., and a value for the number of data points N can be based on an integer power of 2 starting from N=27=128. Thus, for the smallest sampling interval Ao--0.5/~m, the successive sampling intervals A~ (i=0, 1, 2,...) used in the calculation are 0.5, 1, 2, 5, 10, 20, 50, ... Izm and the successive numbers of sampling points, Nj axe 128, 256, 512, .... It should be emphasized that, since an estimation of the power spectrum has a large error if only a single data set is used
1352
W.P. Dong et aL
[ 11 ], an improvement in the estimation can be obtained by carrying out an ensemble average of the power spectra of several data sets. (4) By comparing the absolute magnitude of the proximity factor, ICijl of all Uij~), an appropriate sampling interval A as well as fp are then determined when Equation (3) is satisfied. Then the number of sampling points N is selected, taking into consideration either the functional requirements or instrument and computer capability or Equation (3). The number of data points and the sampling interval in the direction perpendicular to the measured profile can be the same as N and A. Note that, currently, it takes a lot of computing time and memory space for a personal computer to process a digital surface signal with a data matrix larger than or equal to 1024x1024 [23]. Those wishing to process data sizes of this order might wish to consider the merits of using larger computers or mainframes. 4.
APPROPRIATE SAMPLING CONDITIONS FOR TWO-DIMENSIONAL MEASUREMENT
In essence, 2-D measurement is just a simplification of 3-D measurement with obvious differences. • Firstly, a high-pass filter is required in 2-D measurement, thus a 2-D profile is generally restricted to a defined bandwidth. • Secondly, the 2-D cut-off length (which is equal to the sampling length) has been clearly defined in relevant standards [4, 5]; therefore, if a standardized cut-off length has been selected, only A or N needs to be determined. • Thirdly, the computer memory restrictions on the number of sampling points is not a critical problem in 2-D measurement. Because of these differences, it is more important in 2-D measurement to determine objectively an appropriate sampling interval and, hence, ensure the choice of consistent sampling conditions. Following the 3-D measurement procedure discussed above, a practical procedure for selecting appropriate sampling conditions for 2-D measurement can be summarized as follows. (1) Choose a long measuring length Ao such that it is equal to an assessment length which is several times one of the cut-off lengths, At=(0.08, 0.25, 0.8, 2.5, 8) mm [4, 5, 24]. (2) Measure a profile of length )to with a very small sampling interval Ao (/1o--0.5/zm for stylus instruments). (3) Calculate the ensemble average of Gi; (fk) of all Uij (fk), different increments of Ao and from the numbers of data points listed in Table 1. It is worth noting that, since the number of data points used in the spectral calculation (see Table 1) are not powers of 2, the FFT algorithm cannot be used, hence the processing speed will be considerably reduced. (4) By comparing values of the absolute magnitude of the proximity factor IC~jl for all Uij (fk), appropriate sampling conditions A and N are then determined when Equation (3) is satisfied. Table 1. The number of data points used in spectral calculation Cut-off hc (mm)
0.08 0.25 0.8 2.5 8
Ai (mm)
0.0005
0.001
0.002
0.005
0.01
0.02
0.05
161 501 1601 5001 16001
251 801 2501 8001
126 401 1251 4001
161 501 1601
251 801
126 401
161
Sampling Conditions for Three-dimensional Microtopography Measurement 5.
1353
EXPERIMENTAL VFJtIFICATION AND! DISCUSSION
In order to velify the proposed approach, various surfaces were tested. The instrument used in the test Jis a Form Talysuff Series 3-D topography instrument employing a laser transducer and with a stylus tip radius of about 2/zm. The minimum sampling interval Ao used is 0.5/zm in all cases.
5.1. Surfaces fi'om different manufacturing processes Experimental :results based on measurements carded out on three kinds of surfaces are presented. The fi:rst is a ground surface with a very fine random topography; the second is milled and with a rougher topography; the third is a shaped comparison specimen (Rubert) with a ,distinctive periodic character. Profiles (about 8.2 nun in length) from the three samples are shown in Fig. 2. Tables 2-4 list all the values of the normalized proximity factor C o calculated from the three profiles. Ensemble averages of at least two subcomponents of the profiles have been carded out through spectral analysis. The maximum number of data points used in the tables is 8192. The three tables prompt a number of observations : similar values of the absolute magnitude of the proximity factor Icu] are clustered in the same column, implying the same choice of At for different numbers of data points, Nj; for surfaces with a fine topography, the smallest absolute values of the proximity factor
0 :3. o "O
-I
=a.-2 -3 '2 (a) A long profile of a ground surface
'
10
~0 o
-10 t~
< -20
(b) A long profile of a milled surface
4O
O
~.
20 0
B .<
-20 i
i
:1
!
2
4 Profile length (mm)
6
8
(c) A long profile of a shaped comparison specimen Fig. 2. HTM 36-12-0
W. P. Dong
1354
et
al.
Table 2. The proximity factor C~j calculated from the ground surface Ni
Ai(~m) 0.5
128 256 512 1024 2048 4196 8192
0.0519 -0.0566 -0.0640 -0.0419 -0.0268 -0.0548 -0.0561
1 0.0119" 0.0042 0.0360 0.0614 0.0108 0.0078
2
5
0.1168 0.1879 0.2341 0.1401 0.1380
0.4792 0.5698 0.4698 0.4343
10 0.6547 ~ 0.5504 0.4959
20
50
0.5832 0.5946
0.7130
Table 3. The proximity factor Cijcalcula~d from themilled surface
Ni
~ (~m) 0.5
128 256 512 1024 2048 4196 8192
-0.0703 -0.1100 -0.1172 -0.1199 -0.1206 0.1222 -0.1224
1
2
-0.0953 -0.1095 -0.1147 -0.1162 -0.1194 -0.1197
-0.0939 -0.1044 -0.1075 -0.1139 -0.1144
5 -0.0865 ~ -0.0820 -0.0946 -0.1003
10 -0.0394* -0.0641 -0.0756
20 -0.0034" -0.0258
50 0.1168
Table 4. The proximity factorCijcalculated from the shaped comparison specimen Ni
Ai (~m)
128 256 512 1024 2048 4196 8192
0.5
1
2
-0.0901 -0.1179 -0.1189 -0.1230 -0.1231 -0.1231 -0.1231
-O.ll09 -0.1129 -0,1210 -0.1213 -0.1212 -0.1211
-0.1010 -O. ll71 -0.1176 -0.1174 -0.1172
5 -0.0965# -0.0924 -0.1058 -0.1055
10 -0.0598 -0.0866 -0.0861
20 -0.0483* -0.0471
50 0.0618
[C~i[ are clustered towards the smaller sampling interval region, e.g. A=I /xm for the ground surface (Table 2); for surfaces with a rougher topography, the smallest absolute values of the proximity factor Ic,jl are clustered towards the larger sampling interval region, e.g. A=20/xm for the milled and the shaped surfaces (Tables 3 and 4). For example, from Equation (3) and taking into consideration the other requirements discussed in Section 3, appropriate sampling conditions (denoted by * in the tables) for each surface are as follows: • A=I/xm, N=128 (ground surface); • A=20/xm, N=128 (milled surface); • A=20/xm, N=128 (shaped surface). The profiles, power spectra, cumulative power spectra and 3-D maps obtained using the chosen appropriate conditions are presented in (a), (c), (e) and (g) of Figs 4-6. In Figs
Sampling Conditions for Three-dimensional Microtopography Measurement
1355
3-5(e), the 80% cumulative power spectra are indicated:by a small circle at position f~j, the left vertical ,dashed line is at the position f,--fd8, and: the right vertical dashed line is at the position jS,--fd2. A visual comparison of the effectiveness of the technique can be carded out by examining surfaces measured using the recommended sampling conditions as well as those measured using "unsuitable" sampling conditions. Surfaces measured using some unsuitable conditions (denoted by # in the tables) are presented in (b), (d), (f) and (h) of Figs 4-6. It is clear from these figures that the sampling interval used to measure the ground surface is too large, and that used to measure the other two surfaces (milled and shaped) is too small--the 80% cumulative power level f0 is close to f~ in the former and close to the zero frequency axis in the latter two cases.
0
E
0.05
l.O
0.10
t
0
0.5 1.0 Profile length (rain)
Profile length (a) A selected profile
unsuccessful profile
(b) A n
E
1o[ ,
0.5
0
200
400
600
0
I i
Frequency (l/ram) (c) Power spectrum of profile (a)
20 40 Frequency (1/mm)
60
(~) Power spectrum of profile (b) i
,0o i
. Ii/
.o
50 a,
0
I
_
_
I , I 200
400 Frequency (l/ram)
600
0
1.5
, I
20
40
60
Frequency (1/mm) (f) Cumulative power of profile (b)
(e) Cumulative power of profile (a)
-4
.4
VI
(g) A - 3 - D map of the selected condition
1.5
(h) A 3-D map of an unsuccessful condition Fig. 3.
1356
W.P. Dong et al.
0
-~ 0 ~,-5
-~ -10 <
1
2
<-I0
0
Profile length (a) Aselected profile 4000
~10°° I
~
0.2 0.4 0.6 Profile length (mm) (b) An unsuccessful profile
o:1"2000
5000
t~
30 10 20 Frequency (1/mm) (c) Power spectrum of profile (a)
0
~ 1oo~[~=--~ 5o[/j
~
~l r 0
50
o
I00
Frequency (l/ram) (d) Power spectrum of profile (b)
- -
I
I
t
j
]
f
10 20 30 Frequency (l/mm) (e) Cumulative power of profile (a)
0
50 100 Frequency (1/mm) (f) Cumulative power of profile (b)
E=k34.0
34.C ::L
(g) A-3-D map of the selected condition
(h) A 3-D map of an unsuccessful condition Fig. 4.
5.2. Same process--different manufacturing conditions To verify further the proposed approach, a set of electro-discharge machining (EDM) samples prepared using different conditions were measured using the Form Talysurf Series topography instrument. Figure 6 shows profilometric measurements taken from three of the EDM samples, which show clearly that the three specimens have distinctive spatial and amplitude characteristics--which is to be expected as their manufacturing conditions are different. Tabular readings of the proximity factor C U of measurements from the three specimens are presented in Tables 5-7. A simple comparison of the three measurements reveals that a small sampling interval (2 txm) is required to measure sample 1 properly, a sampling interval of 10/xm is required to measure sample 2 properly, and a much larger sampling interval of 20/xm is required to measure sample 3 properly. The correctness of the choice of appropriate sampling intervals (indicated by * in Tables 5-7) is borne out by a visual examination of the topographic features of the measured samples as depicted in Fig. 6.
Sampling Conditions for Three-dimensional Microtopography Measurement
! 357
20
0 "~t~ -20 <
1 2 Profile length (a) A selected profile
0
0.2 0.4 0.6 Profile length (mm) (b) An unsuccessful profile
0
50 100 Frequency (l/mm) (d) Power spectrum of profile (b)
.-. 100 O
~ 50
I
I I I I
O
o
10
20
30
Frequency (1/mm) (c) Power spectrum of profile (a) 100
o
I I
I t
i
I
i
~
.... I
I
50 ~lt'q
^ 0
10 20 30 Frequency (l/mm) (e) Cumulative power of profile (a)
0
100 50 Frequency (l/mm) (f) 12 umulative power of profile (b)
E 60"301 ~01t
2
nm
L
(g) A-3-D map of the selected condition
(h) A 3-D map of an unsuccessful condition Fig. 5.
5.3.
The two-dimensional measurement scenario
A profilometric measurement (assessment length Ao=8 mm) was taken from a honed engine bore surface (Fig. 7(a)). The proximity factors Co obtained from the cumulative power spectra of at least two ensemble averages are presented in Table 8. Based on this proposed approach, it can be seen that a sampling interval A=2/~m is appropriate for the measurement. In addition, if the cut-off length selected is At=0.8 mm (which is the usual default cut-off length for 2-D measurement), then the sampling conditions A=2/.~m and N=401 are autornatically determined by the proposed approach. Figures 7(b) to 8(e) show a sub-profile plot obtained using four sampling conditions (see Table 1), all with the cutoff At=0.8 mm. From a visualization perspective, the difference between Fig. 7(b)-(d) is insignificant; but the difference between Fig. 7(e) and the three others is clearly visible. Table 9 lists some amplitude parameters calculated from the four profiles. (Ra is the arithmetic mean average, Rq is the root-square-average, Sk is the skewness and Ku is the kurtosis.) The relative differences • in Table 9 are calculated with respect to the sampling
1358
W.P. Dong et al.
< 0
~
2 4 6 (a) Profile of sample 1
20
o
~-20 2 4 6 (b) A profile o f sample 2 "~
0
2
r
.
,
4
6
8
Profile length (c) A profile of sample 3
Fig. 6. Table 5. The proximity factor Cij calculated from EDM sample 1 Ai (/zm)
Ni
128 256 512 1024 2048 4196 8192
0.5
1
2"
5
10
20
50
-0.0847 -0.1032 -0.1037 -0.1016 -0.1012 -0.1037 -0.1072
-0.0813 -0.0824 -0.0783 -0.0775 -0.0824 -0.0895
-0.0404 -0.0316 -0.0301 -0.0399 -0.0539
0.1224 0.1080 0.0807 0.0528
0.3276 0.2841 0.2262
0.5143 0.4867
0.4502
Table 6. The proximity factor CU calculated from EDM sample 2 Ni
128 256 512 1024 2048 4196 8192
A, (/zm) 0.5
1
2
5
10"
20
50
-0.1053 -0.0991 -0.1163 -0.1165 -0.1177 -0.1198 -0.1212
-0.0732 -0.1076 -0.1081 -0.1105 -0.1146 -0.1174
-0.0899 -0.0912 -0.0959 -0.1042 -0.1098
-0.0494 -0.0691 -0.0740 -0.0865
-0.0135 -0.0230 -0.0479
0.0798 0.0302
0.2046
conditions, A--0.5/xm and N=1601. Quite clearly, from a parametric point of view, the conclusions about the measurements are consistent with those arrived at through visual characterization (see above). These comparisons suggest that the selected sampling condition, A=2/xm and N--401, is appropriate. 5.4. Checking the suitability of the stylus As discussed above, if the stylus tip radius is so large that it cannot effectively measure the intended surface (in the case of very fine surfaces), then the 80% cumulative power
Sampling Conditions for Three-dimensional Microtopography Measurement
1359
Table 7. The proximity factor Co calculated from EDM sample 3 i
Ni
128 256 512 1024 2048 4196 8192
At (/~m) 0.5
1
2
5
10
20"
50
-0.(g)91 -0.1095 -0.1177 -0.1178 -0.1228 -0.1220 -0.1219
-0.0941 -0.1104 -O.ll07 -0.1207 -0.1190 -0.1188
-0.0959 -0.0965 -0.1163 -0.1130 -0.1126
-0.0886 -0.0973 -0.0953 --0.0941
-0.0698 -0.0695 -0.0632
-0.0078 -0.0091
0.1842
<
m
0
,
i
i
2 4 6 (a) A long profile
1
"
0
0.2
"
"
0.4
0.6
8
0.8
(b) A profile obtained when A = 0.5 izm and N = 1601
0
0.2
0.4
0.6
0,8
(e) The profile obtained when A = 1 t~m and N = 801 |
....
.
.
.
i < 0 0.2 0.4 0.6 0.8 (d) The profile obtained when A = 2 Izm and N = 401
l
'"
0
"
"
"
i¸
0.2 0.4 0.6 Profile length (ram)
0.8
(e) The profile obtained when A = 5 ttm and N = 161 i
Fig. 7.
1360
W. P. Dong et al.
Table 8. The proximity factor C~/calculated from the honed engine surface Cut-off hZmm)
Ai (/xm)
0.08 0.25 0.8 2.5 8
0.0005
0.001
0.002
0.005
0.01
0.02
0.05
-0.0776 -0.0784 -0.0898 -0.0901 -0.0925
-0.0317 -0.0545 -0.0552 -0.0601
0.0604 0.0162" 0.0145 0.0044
0.2290 0.2167 0.1882
0.4974 0.4760
0.6163 0.6185
0.6353
0.02 0.02 -0.02~
' tl
' [
< -0.041
'
1~
t
,
0.2
0.4
Profile length (ram) (a) A measured profile
.~ ~
004 I°L 0.5
~
. . . . . . . ~ J.J.~Jo.k~.x . . . . . 500 1000 Frequency (I/mm)
0
(b) Power spectrum of the profile
/
50F/
t 500 Frequency (1/mm)
1000
(c) Cumulative power of the profile Fig. 8.
Table 9. Parameters of a profile from a honed surface calculated using different sampling conditions (At=0.8 ram) A (/xm) 0.5 1 2 5
N
1601 801 401 161
Ra (gm)
E (%)
0.584 0.583 -0.17 0.582 -0.34 0.572 -2.05
R, (/zm) 0.734 0.735 0.735 0.731
~ (%)
Sk
E (%)
-1.032 0.14 -1.033 -0.097 0.14 -1.028 0.39 0 . 4 1 -1.077 -4.36
KI, (/zm)
~ (%)
3.777 3.778 0.026 3.767 -0.26 3.823 1.22
RD (/xm)
e (%)
R7 (/xm)
E (%)
1.126 3.2 1.093 -2.931 3.117 -2.593 1.058 -6.039 3.077 -3.843 1 . 0 5 -6.749 3.032 -5.25
level fj will tend towards (or beyond) fb. This may be an indication of aliasing. This phenomenon can be observed by measuring an optical flat (clearly, the roughness cannot be effectively measured by any standardized stylus). Table 10 shows tabulated values of the proximity factor CU, calculated from optical flat measurements. Clearly, all the proximity factor values CUare positive, with the smallest proximity factor value C0=0.4572, which is so large (relatively) that it cannot possibly represent an appropriate sampling condition.
Sampling Conditions for Three-dimensional Microtop0graphy Measurement
1361
Table 10. The proximity factor CU calculated from optical flat data Ni
Ai (/~m) 0.5
1
2
5
10
20
0.6358 0.6652
0.6682
i
128 256 512 1024 20.48 4196
0.5267 0.5255 (I.4714 (I.4572" (I.4823 0.4955
0.5711 0.6061 0.5521 0.5176 0.5202
0.6355 0.6781 0.6677 0.6635
0.5869 0.6016 0.6403
Figure 8 shows a profile, its power spectrum and the corresponding cumulative power spectral plot, obtained using the smallest proximity factor (C 0) prediction from Table 10 (i.e. A=2/xm and N=401). It can be seen that the power spectrum is broadly distributed (spread out) and the smallest value off0 is beyond fa. 6. CONCLUSIONS
This paper has proposed a new philosophy of surface r~ughness measurement based on an objective definition of surface topography measuring iconditions. It is recognized that the short wavelength measurement limit depends on whichever is the largest of the three factors--the fre~quency characteristics of instruments, tile size of the stylus tip and the sampling intervalmwhilst the long wavelength limit is linked to the functional requirements of the stufface as well as instrument capability. A new approach for the determination of appropriate sampling conditions for 2-D and 3-D surface roughness measurement has been proposed. The approach is based on spectral analysis and takes into consideration the Nyquist sampling criterion. It has been advanced that an appropriate sampling condition can be obtained when 80% of the cumulative power of a pre-measured trace lies around one-eighth of the Nyquist folding frequency. In addition, the proposed approach can also be used to check the suitability of the stylus for any particular measurement--this will help ensure that out-of-specification styli are not used. Procedures for the implementation of the technique in both 2-D and 3-D measurement practice have been advanced and developed. Experiments have been conducted on a range of different surfaces--including random and periodic ones--and for different manufacturing process conditions, to demonstrate the effectiveness of the technique. The results verify the effectiveness of the approach. Although the measurements presented in this paper were all carried out using a stylus instrument, the principle of the approach can be readily extended to other scanning probe instruments (including optical probe-based systems). REFERENCES [1] K. J. Stout (F~t.), Proc. 4th Int. Conference on Metrology and Properties of Engineering Surfaces, Washington. Kogan Page (1988). [2] K. J. Stout (EdL.),Proc. 5th Int. Conference on Metrology and Properties of Engineering Surfaces, Leicester. Pergamon Press, Oxford (1991). [3] K. J. Stout (Ed.), Proc. 6th Int. Conference on Metrology and Properties of Engineering Surfaces, Birmingham. ]Pergamon Press, Oxford (1994). [4] Rules and pr¢~edures for the measurement of surface roughness using stylus instruments, International Standard ISO 4288 (1985). [5] Instruments for the measurement of surface roughness by the profile method---contact (stylus) instruments of consecutiw~ profile transformation contact profile meters, System M, International Standard ISO 3274 (1975). [6] T. Tsukada and K. Sasajima, An optimum sampling interval for digitising surface asperity profiles, Wear 83, 119-128 (1982). [7] D. Y. Yim and S. W. Kim, Optimum sampling intervals for Ra roughness measurement, Proc. Inst. Mech. Engrs C205, 139-142 (1991). [8] D. D. Chetwynd, The digitisation of surface profiles, Wear 57, 137-145 (1979). [9] T. R. Thomas and G. Charlton, Variation of roughness parameters on some typical manufactured surfaces, Precision Engng 3, 91-96 (1981).
1362
W.P. Dong et al.
[10] W. P. Dong, P. J. Sullivan and K. J. Stout, The significance of surface features in characterising 2-D and 3-D surface topography, ASME PED 62, 115 0992). [l l] J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures. Wiley-lnterscience, New York (1971 ). [12] J. G. Proakis and D. G. Manolakis, Digital Signal Processing, Principles, Algorithms, and Applications. Macmillan, Basingstoke (1992). [13] T. Y. Lin, L. Blunt and K. J. Stout, Determination of proper frequency bandwidth for 3-D topography measurement using spectral analysis Part l: Isotropic surfaces, Wear 166, 221-232 (1993). [14] E. Mainsah, Investigation ofpre-characterisation aspects of three-dimensional micro-topography. Ph.D. Thesis, University of Birmingham, U.K. (1994). [15] T. R. Thomas, Roughness Surfaces. Longman, London (1982). [16] T. Nakamura, On deformation of surface roughness curves caused by finite radius of stylus tip and tilting of stylus holder arm, Bull. Japan Soc. Prec. Engng 1, 240-249 (1966). [17] J. L. Guerrero and J. T. Black, Stylus tracer resolution and surface damage as determined by scanning electron microscopy, Trans. ASME J. Engng Ind. 94, 1087-1093 (1972). [18] I. Sherrington and E. H. Smith, A quantitative study of the influence of stylus shape and load on the fidelity of data recorded by stylus instruments, Proc. Second National Conference on Production Research, Edinburgh, pp. 762-783. University of Edinburgh Press (1986). [19] T. R. Thomas, Some examples of the versatility of stylus instruments, Mecanique, Materiaux, Electricite 337, 17-25 (1978). [20] D. J. Whitehouse, Some ultimate limits on the measurement of surfaces using stylus techniques, Measurement Control 8, 147-151 (1975). [21] V. Radhakrishnan, Effect of stylus radius on the roughness values measured with tracing stylus instruments, Wear 16, 325-335 (1970). [22] K. J. Stout, P. J. Sullivan, W. P. Dong, E. Mainsah, N. Luo, T. Mathia and H. Zahouani, The Development of Methodsfor the Characterisation of Roughness in Three-dimensions. Commission of the European Communities, ISBN 0 7044 1313 2 (1993). [23] W. P. Dong, E. Mainsah, P. J. Sullivan and K. J. Stout, Instruments and measurement techniques of 3dimensional surface topography, Three-DimensionalSurface Topography; Measurement, Interpretation and Applications, A Survey and Bibliography (edited by K. J. Stout). Penton Press, London (1994). [24] Metrological characterisation of phase corrected filters and transmission bands for use in contact (stylus) instruments, Draft International Standard ISO/DIS ! 1562 (1994).