Curved profile spectral moments

Measurement Vol. 18, No. 3, pp. 145-150, 1996

Copyright © 1996 Elsevier Science Ltd Printed in The Netherlands. All rights reserved 0263-2241/96 $15.00 +0.00

PII: S0263-2241 (96)00029-2

ELSEVIER

Curved profile spectral moments R.A.

Downs ~, J.L. Garbini

b

South Seattle Community College, 6000 16th Ave., SW-TC 140, Seattle, WA 98106-1499, USA t, Mechanical Engineering Department, FU-IO, University of Washington, Seattle, WA 98195. USA

Abstract For Gaussian surfaces, profile spectral moments can be used to parameterize a model describing the stochastic geometry of a surface. Instead of using straight profiles, the use of curved profiles is presented as an alternative method for measuring profile spectral moments in different directions. By obtaining curved profiles from a spiral profile, a surface can be sampled faster than is possible by taking five straight profiles and profiles in more directions are obtained so a least squares procedure can be used to estimate the surface spectral moments. An error analysis and experimental tests show that with the proper choice of sampling parameters, curved profiles can be used instead of straight profiles for estimating profile spectral moments. Copyright © 1996 Elsevier Science Ltd. Keywords: Spectral moments; Surface roughness; Stochastic geometry

The objective of this paper is to describe the use of spectral moments calculated from curved profiles instead of straight profiles for use with Nayak's Random Process Model. Curved profiles are obtained by sampling a surface in a spiral profile, then dividing the spiral profile into many different curved profiles. The direction of each curved profile is given by the tangent to the midpoint of the profile. The advantages of using curved profiles instead of straight profiles are that spiral profiling is faster than taking five straight profiles and the spectral moments in many different directions can be used to give a least squares solution for the surface spectral moments. Test results are presented of straight and curved profile spectral moments on a shot peened surface.

1. Introduction Surface topography measurement is important in many areas of engineering because of the effect surface topography has on the functional performance of a part. However, because of the time required for 3-D surface topography measurement, surface topography is not routinely measured in manufacturing. An alternative to measuring the complete 3-D surface topography over a region is to use Nayak's Random Process Model to describe a surface. Nayak l-l] showed that for Gaussian surfaces, many useful surface statistics can be found from a knowledge of the profile spectral moments from five nonparallel straight profiles. This model was experimentally evaluated by Sayles and Thomas [2] who found very good agreement between measured and theoretical values of the surface height distribution, summit height distribution, surface slope distribution, and the summit curvature distribution for ground, blasted and sparkeroded surfaces.

2. Error analysis There will be some error involved in calculating the higher order spectral moments from a curved profile. To estimate the error, we need to compare 145

146

R.A, Downs, J.L. Garbini

the spectral moments from a straight profile and a curved profile in the same region. We assume the angular variation of the profile spectral moments can be modeled as an ellipse [3]: 1

1 cos2(7 ) +

1

sin2(~),

(1)

where mr is the profile spectral moment in the radial direction at an angle ~ from the origin and a and b are the lengths of the semimajor and semiminor axes of the moment ellipse. If we take a circular profile at a radius R, then at an angle O, the direction of a line tangent to the circular profile is given by ~ - 0 + n/2 (Fig. 1). The value of the spectral moment of a profile tangent to the circle in the direction ~, will then be given by:

m,(~)Z-a2COS 2

+0

+~sin 2

+0

,

(2)

which reduces to

b£ a

m,(~)= /

(3)

bN/a5 ( 1 - cos(20)) + ( 1 + cos(20)) To compare the straight profile spectral moment, m,(~), with a curved profile spectral moment, th,(~), it is necessary to numerically integrate (3) over the arc length of the curved profile. 1 f0+4,/2

m(~j) d~,

(4)

where ~b is the sector angle of the arc. The spectral moment error is calculated from m,(~) - rhr(~),

metro r =

(5)

for different 0 values for the case where the major axis of the moment ellipse coincides with the x-axis. Figure 2 shows the percent spectral moment error for b/a = 0.4 and 0.6 with sector angle, q~, of 20 °. Figure 2 shows that the spectral moment error is a function of the angular direction 0 and the eccentricity of the moment ellipse. This error is caused because the curved profile is averaging the spectral moment over a region where the spectral moment value is changing. The greater the change that occurs over this region, the greater the expected error would be. The 2% error for b/a = 0.4 is still less than the experimental error typically found between skidded and non-skidded measured profiles (6%) E4]. The curved profile spectral moment error also depends on the size of the sector angle. The effect the sector angle has on the percent error can be seen from Fig. 3. In this figure, the magnitude of the maximum percent error found in any direction is plotted versus the sector angle, ~b, for a surface with b/a = 0.2, 0.4, 0.6 and 0.8. 3. Spectral moment estimation The accuracy of the estimated profile spectral moment in any direction is related to the number of data points that are used in making the estimate. The number of data points can be increased by

2 -

Segment of cicalar path

"~

A

/ ! ~\

o

b/a-.6 / I l k

"--'II" ........ li"

Straight profile

.

.

.

.

~-2 -3

0

i

I

50

100

~

l

I

t

I..

150

200

250

300

350

Angle

(degrees)

K

Fig. 1. Circular path coordinate system.

Fig. 2. Curved profile spectral moment error for various values of 0.

147

R,A. Downs, J.L. Garbini

20

-

/

-

'i

/

15 10

-~

/

/~b/, = .2

./"

/'/"

b/a--.4

5

.a.

0

0

10

20

30 40 50 60 70 Sector angles (degrees)

80

90

Fig. 3. Variation of maximum % error with sector angle.

taking a larger sector angle or taking several revolutions - - so an average spectral moment can be obtained from the different revolutions. For a single curved profile, a 95% confidence interval is given by: 2s

m,-~
2s

(6)

where s is the standard deviation of the profile and N is the number of data points in the profile. Figure 4 shows how the size of the confidence interval varies with sector angle by plotting 2/x/-N versus sector angle. This occurs because decreasing the sector angle reduces the number of points in a profile. When Np curved profiles are used to estimate m,, a 95% confidence interval is given by: m,-

S p i N - 1.025

~


Spin- 1.025 ~-~p

,

(7)

where sp is the standard deviation between profiles, t is the t-statistic and Np is the number of profiles. Figure 5 shows how the size of the confidence interval varies with the number of revolutions. As an example, consider a circular profile taken at a radius of 3 mm with a sample spacing of 15 p.m. For a sector angle of 20 °, there are 70 data points in each curved profile. From Eq. (6) we have the confidence interval for each curved profile as plus or minus 0.24s, where s is the profile standard deviation. A 95% confidence interval of the spectral moment is then given from Eq. (7) as plus or minus 0.63sp, where sp is the standard deviation between the 10 profiles.

4. Experimental procedure The accuracy of using curved profiles for spectral moment estimation was determined by comparing curved and straight profile spectral moments measured with a stylus profilometer. The procedure for taking curved profiles is: (i) From several straight profile spectrums, find the minimum and maximum frequencies of interest. The maximum frequency is the Nyquist sampling frequency, f~-- 1/2Ax. The minimum frequency resolution available from the data is given by Af = 1/NAx, where N is the number of data points and Ax is the sample spacing. The longest wavelength of interest will be )~= NAx; (ii) The radius, R, is selected based on the required sample spacing, Ax and the number of samples/ revolution, Ne,R = AxN~/2n.

0.4 2.5 ,-, 0.35 0.3

~_~ 1.5

0.5 0.15

5

10

15 20 Sector angle (degrees)

25

Fig. 4. Variation of confidence interval with sector angle.

30

0

0

I 5

I L 1 10 15 20 Number of revolutions

I 25

J 30

Fig. 5. Moment variation with number of revolutions.

R.A. Downs, J.L. Garbini

t48

The confidence interval for the spectral moments can be controlled by selecting: (iii) A sector angle so that the sector length is greater than the longest wavelength of interest, q~> 2/R; (iv) The variation in the moment estimation (+__2 s / ~ , , ) desired by selecting the number of revolutions, Nr. For this experiment, the curved profiles were obtained from one spiral profile that consisted of 10 revolutions. The outer radius of the spiral profile was 8.2 mm and the distance between spirals was 140 ~tm. The distance between spirals was chosen to be the correlation distance for the sample (the distance where the autocorrelation function is 1/10 its original value), so each profile revolution would be statistically independent. At 30 ° intervals around a circle, straight profiles were taken to coincide with the curved profiles taken in the same region. As an example, at an angle of 30 ° from an arbitrary axis on a surface, there are 10 curved profiles, each with a sector angle of 25 °. The outer curved profile is at a radius of 8.2 rnm and each successive curved profile is 140~m closer to the origin. At the same 30 ° location, 10 straight profiles are taken so that each straight profile is approximately tangent to one of the curved profiles at the midpoint of the curved profile. This gives 10 curved and straight profiles of approximately the same length, with the same sample spacing, in the same direction, and over approximately the same portion of the surface. The profile spectral moments are defined to be too, the variance of the surface heights, m2, the mean square surface slope, and m4, the mean square surface curvature. mo = o-~ =

G(co) dco,

(dz~ z z mz = E k d x / = a s =

m 4 = E k-~x2x2j

= o ' ~2 =

f)

(8) coZG(co) dco'

co4G(CO)dco,

(9)

(10)

where ah is the standard deviation of the surface heights, ~rs is the standard deviation of the surface slope, o-c is the standard deviation of the

surface curvature, and G(co) is the profile power spectrum at angular frequency co. The slope and curvature are estimated using the seven point formulas: dz dx

1 -

-

60Ax -45z

(Z+ 3 -- 9Z+2 + 45Z+I l+9Z-2-z-3),

(11)

d2z 1 dx 2 - 180Ax2 (2z+ 3 - 27z+ z + 270z+L - 490z0 + 270z_ 1 - 27z_ 2 - 2z- 3),

(12)

where A x is the sample spacing and zl is the surface height of the i th point from the location of interest. Curved profile spectral moments were calculated for a sector angle of 25 ° at 11 locations around a circle. At each angular location, each revolution was treated as a separate profile, so an average value for too, m2, and m4 was determined by averaging the spectral moments estimated from each of the 10 revolutions. Straight profile spectral moments were calculated at the same 11 locations as the curved profiles. Since each straight profile was statistically independent, an average value for too, m2, and m4 at each angular position was determined by averaging the spectral moments from each of the individual profiles. The spectral moments from straight and curved profiles on two different shot peened surfaces were compared by: (i) a t-test on the mean value; (ii) a Kolmogorov-Smirnov Goodness of Fit test on the distribution; (iii) calculating a 95% confidence interval; and (iv) calculating the % error. The results for both surfaces were similar, so only one of the surfaces is presented here. t-Test The t-test is used to determine if there is a significance in the difference between the mean spectral moment values. The probability that the Itl could be this large or larger just by chance for each of the spectral moments is plotted as a function of angular position in Fig. 6.

R.A. Downs, J.L. Garbini 100

straight and distributions.

A .....\

... # % ..." .... ¢.., ,~

80 -

/.

60

curved

profile

spectral

Confidence interval m2 v

•f~mO------

20

m4 0

t

I

50

100

v

.... I I I 150 200 250 Angle (degrees)

I ..... i 300 350

Fig. 6. t-Test.

Small probability values indicate that the straight and curved profile spectral moment values have significantly different means. At the 5% level, the hypothesis that the mean value of the spectral moments are the same can not be rejected. So we conclude that there is not a significant difference between the mean value of straight and curved profile spectral moments. K olmogorov- Smirnov test

Figure 8 shows a 95% confidence interval (dashed line) and the m0 value (solid line} at different angular positions. Figure 9 shows a 95% confidence interval and the mz value at different angular positions. Figure 10 shows a 95% confidence interval and the m4 value at different angular positions. These figures show that the spectral moments from the curved profiles lie within the 95% confidence interval. Percent error

Figure 11 shows the percent error m4 as a function of position around surface. The shape of the curve is predicted error (5.6), although the

in mo, mz, and a circle on the similar to the magnitude of

5.5

The Kolmogorov-Smirnov test [5] is used to determine if the spectral moment distributions are statistically different. Figure 7 shows for different locations on a surface, the probability that the two data sets are drawn from the same distribution. Small probability values indicate that the cumulative distribution functions for the straight and curved profiles are significantly different. At the 5% level, the hypothesis that the two distributions are the same can not be rejected. So we conclude that there is not a significant difference between

5.0 ¢##

"$

i ~"

4.5

4.0

3.5

0

50

100

150 200 Angle (degrees)

250

l 350

300

Fig, 8. mo a n d c o n f i d e n c e interval,

a,

100

i

w~

ss

80

i

•

2',.Uli

20 0

•

I 100

I

,.I

I

150 200 250 Angle (degrees)

Fig. 7. K o l m o g o r o v - S m i r n o v

; ~'

o~'

\

~

i

.

~•

~,

i

'

.....

50

,

s

%i I

',....,,

°oV1

4o

¢ 1 ,~ ~

•

b9

60

0

moment

,~

_

0

149

test.

I

I

300

350

6

~"l 0

i

50

100

L

i

",-.'f'"

150 200 Angle (degrees)

250

Fig. 9. m 2 a n d c o n f i d e n c e interval.

L

I

300

350

150

R.A. Downs, 3.L. Garbini

5. Conclusion

4.0

,.--.,

3.0 2.5

1

..,..

3.5-

m

t

-

-

2.01.5

•

0

[

50

I

100

I

I

150 200 Angle (degrees)

I

__1

250

I

300

350

Fig. 10. m4 and confidence interval. 15 10 5

Curved profiles have been presented as an alternative sampling pattern for the measurement of spectral moments for use with Nayak's Random Process Model. With the proper selection of measurement parameters, curved profile spectral moments are statistically equivalent to straight profile spectral moments. The error involved in using a curved profile instead of a straight profile depends on the degree of anisotropy of the surface and the sector angle of the curved profile. Error criteria, relating instrument parameters to the expected error were developed. Comparisons were made of curved and straight profile spectral moments measured on shot peened surfaces.

0 d~-5

References

-10 -15 -20 - m 4 / I 0 50

I 100

I 1 ! 50 200 Angle (degrees)

• 250

1 300

350

Fig. 11. Percent error.

the error is larger than predicted. The larger error can be accounted for because in calculating the theoretical error (5.18), it was assumed that a spectral moment could be measured exactly. But for any real measurement, there will be some error associated with the measurement.

[ I ] P.R. Nayak, Random process model of rough surfaces, Trans. ASME J. Lubrication Technology 93F (1971) 398-407. [2] R,S. Sayles and T.R. Thomas, Measurements of the statistical microgeometry of engineering surfaces, Trans. ASME d. Lubrication Technology 101 (1979)409-418. [3] R,S. Sayles and T.R. Thomas, Thermal conductance of a rough elastic contact, Applied Energy 2 (1976) 249-267. [4] E.J. Davis and K.J. Stout, Stylus measurement techniques: a contribution to the problem of parameter variation, Wear 83 (19821 49-60. [5] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Receipts - - The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.