Accuracy and time are known to be conflicting factors in measurement. This paper re-evaluates the two-dimensional sampling problem for measuring the surface roughness and establishes that an optimal sampling strategy can be obtained by utilizing the point sequences developed in Number Theory. By modeling a machined surfaces as a Wiener process, the root-mean-square (RMS) error of measurement is shown to be equivalent to the L2-discrepancy of the complement of the sampling points. It is further shown that this relationship holds for more general surfaces, thus firmly linking the minimum RMS error of the measurement to the celebrated lower bound on L2-discrepancy asserted by Roth (1954), a 1958 Fields medalist.
Issue Section:
Research Papers
1.
Aoyama
H.
Kawai
M.
Kishinami
T.
1989
, “A New Method for Detecting the Contact Point Between a Touch Probe and a Surface
,” Annals of the CIRP
, Vol. 38
, No. 1
, pp. 517
–520
.2.
Cipra, B. A., Jan. 1991, “The Breaking of a Mathematical Curse,” SIAM Research News, pp. 165–165.
3.
Davenport
H.
1956
, “Note on Irregularities of Distribution
,” Mathematika
, Vol. 3
, pp. 131
–135
.4.
de Nijs
J. F. C.
Lammers
M. G. M.
Schellekens
P. H. J.
van der Wolf
A. C. H.
1988
, “Modeling of a Coordinate Measuring Machine for Analysis of Its Dynamic Behaviors
,” Annals of the CIRP
, Vol. 37
, No. 1
, pp. 507
–510
.5.
Dubbie
N. A.
Malmberg
S. J.
1987
, “Error Diagnosis and Compensation Using Kinematics Models and Position Error Data
,” Annals of the CIRP
, Vol. 36
, No. 1
, pp. 355
–358
.6.
Gorlenko
O. A.
1981
, “Assessment of Surface Roughness Parameters and Their Interdependence
,” Precision Engineering
, Vol. 3
, No. 2
, pp. 105
–108
.7.
Halton
J. H.
Zaremba
G. S. K.
1969
, “The Extreme and L2 Discrepancies of Some Plane Sets
,” Monatsh. Math.
, Vol. 73
, pp. 316
–328
.8.
Hammersley
J. M.
1960
, “Monte Carlo Methods for Solving Multivariable Problems
,” Ann. New York Acad. Sci.
, Vol. 86
, pp. 844
–874
.9.
Haykin, S., 1989, Modern Filters, Macmillan Publishing Company, New York.
10.
Kalpakjian, S., 1991, Manufacturing Processes for Engineering Materials, Addison Wesley.
11.
Kuipers, L., and Niederreiter, H., 1974, Uniform Distribution of Sequences, Wiley, New York.
12.
Kunzmann
H.
Waldele
F.
1988
, “Performance of CMMs
,” keynote paper, Annals of the CIRP
, Vol. 37
, No. 2
, pp. 633
–640
.13.
Meijer
J.
1990
, “Accuracy of Surface Plate Measurement—General Purpose Software for Flatness Measurement
,” Annals of the CIRP
, Vol. 39
, No. 1
, pp. 545
–548
.14.
Nawara
L.
Kowalski
M.
Sladek
J.
1989
, “The Influence of Kinematic Errors on the Profile Shapes by Means of CMM
,” Annals of the CIRP
, Vol. 38
, No. 1
, pp. 511
–516
.15.
Niederreiter
H.
1978
, “Quasi-Monte Carlo Methods and Pseudo-Random Numbers
,” Bull. Amer. Math. Soc.
, Vol. 84
–6
, pp. 957
–1041
.16.
Novak, E., 1988, “Deterministic and Stochastic Error Bounds in Numerical Analysis,” Lectures Notes in Math., Vol. 1349, Springer-Verlag, Berlin.
17.
Papageorgiou
A.
Wasilkowski
G. W.
1990
, “On the Average Complexity of Multi-Variate Problems
,” J. Complexity
, Vol. 6
, pp. 1
–23
.18.
Peklenik
J.
1963
, “Contribution to the Theory of Surface Characterization
,” Annals of the CIRP
, Vol. 12
, No. 1
, pp. 173
–176
.19.
Roth
K. F.
1954
, “On Irregularities of Distribution
,” Mathematika
, Vol. 1
, pp. 73
–79
.20.
Roth
K. F.
1980
, “On Irregularities of Distribution
,” Acta Arith.
, Vol. 37
, pp. 67
–75
.21.
Stout, K. J., Davis, E. J., and Sullivan, P. J., 1990, Atlas of Machined Surfaces, Chapman and Hall, London.
22.
Sherrington
I.
Smith
E. H.
1987
, “Parameters for Characterizing the Surface Topography of Engineering Components
,” Proc. Institute of Mechanical Engineers
, Part C, Vol. 201
(C4
), pp. 297
–306
.23.
Stark, H., and Woods, J. W., 1986, Probability, Random Processes, and Estimation Theory for Engineers, Prentice-Hall, New Jersey.
24.
Stout
K. J.
Sullivan
P. J.
1989
, “The Analysis of the Three Dimensional Topography of the Grinding Process
,” Annals of the CIRP
, Vol. 38
, No. 1
, pp. 545
–548
.25.
Thomas
T. R.
1981
, “Characterization of Surface Roughness
,” Precision Engineering
, Vol. 3
, No. 2
, pp. 97
–104
.26.
Thomas, T. R., 1982, Rough Surfaces, Longman, London.
27.
Traub, J. F., Wasilkowski, G. W., and Wozniakowski, H., 1988, Information-based Complexity, Academic Press, New York.
28.
Treib
T.
1987
, “Error Budgeting—Applied to the Calculation and Optimization of the Volumetric Error Field of Multiaxis Systems
,” Annals of the CIRP
, Vol. 36
, No. 1
, pp. 365
–368
.29.
Tsukada
T.
Sasajima
K.
1982
, “An Optimal Sampling Interval for Digitizing Surface Asperity Profiles
,” Wear
, Vol. 83
, pp. 119
–128
.30.
Vanmarcke, E., 1983, Random Fields: Analysis and Synthesis, The MIT Press, Cambridge, Massachusetts.
31.
van der Corput, J. G., 1935, “Verteilungsfunktionen. I, II,” Nederl. Akad. Wetensch. Proc., Vol. 38, pp. 813–821, pp. 1058–1066.
32.
Whitehouse, D. J., 1971, “The Properties of Random Surfaces of Significance in Their Contact,” Ph.D. thesis, the University of Leicester.
33.
Whitehouse
D. J.
1978
, “The Digital Measurement of Peak Parameters on Surface Profiles
,” J. Mech. Engineering Science
, Vol. 20
, No. 4
, pp. 221
–227
.34.
Whitehouse
D. J.
1988
, “Comparison Between Stylus and Optical Methods for Measuring Surfaces
,” Keynote Paper, Annals of the CIRP
, Vol. 37
, No. 2
, pp. 649
–653
.35.
Whittle
P.
1954
, “On Stationary Processes in the Plane
,” Biometrika
, Vol. 41
, pp. 434
–449
.36.
Wozniakowski
H.
1991
, “Average Case Complexity of Multivariate Integration
,” Bull. Amer. Math. Soc. (N.S.)
, Vol. 24-1
, pp. 185
–194
.37.
Yim
D. Y.
Kim
S. W.
1991
, “Optimal Sampling Interval for Ra Roughness Measurement
,” Proc. Ins. Mech. Eng.
, Vol. 205
, pp. 139
–142
.38.
Zhang
G.
Ouyang
R.
Lu
B.
Hocken
R.
Veale
R.
Donmez
A.
1988
, “A Displacement Method for Machine Geometry Calibration
,” Annals of the CIRP
, Vol. 37
, No. 1
, pp. 515
–518
.39.
Zhang
G. M.
Kapoor
S. G.
1991
, “Dynamic Generation of Machined Surfaces, Part 2: Construction of Surface Topography
,” ASME Journal of Engineering for Industry
, Vol. 113
, pp. 145
–153
.
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