Consolidation of Basics of Dimensional Analysis
Publication: Journal of Engineering Mechanics
Volume 110, Issue 9
Abstract
The echelon matrix procedure is introduced, as a new base procedure for dimensional analysis, by its application to examples selected by previous authors. It is thereby shown that Buckingham's actual and very perceptive enumeration of the number of nondimensional groups is never in error, despite repeated claims to the contrary. The achievement or the forcing of the echelon matrix formation is shown also to be the natural entry to the stepwise and the proportionality‐stepwise procedures, which, like the basic echelon matrix procedure, offer advantages in straightforward execution, over the traditional, nonintegrated, procedures for dimensional analysis which are ascribed to Rayleigh and to Buckingham. The echelon matrix procedure is also shown to be valuable in examination of various initiatives and claimed improvements to dimensional analysis of the past and present. Vector lengths or “directional analysis” and such concepts are reviewed, and their supposed advantages are shown to be based on misconceptions. It is then shown that the stepwise‐proportionality procedure alone offers an infallible check in the matter of dimensional homogeneity of the variables. An outline is given of the logical sequence of arrangement of variables and execution of analysis by the alternative procedures. Overall, it is shown that dimensional analysis is still a meaningful method for the future provided it is dissociated from past and current misconceptions and some supposed improvements, and is taught and executed in a manner which relates it to the mainstream of basic modern mathematics.
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References
1.
Barr, D. I. H., “Echelon Matrices in Dimensional Analysis,” International Journal of Mechanical Engineering Education, Vol. 7, No. 2, Apr., 1979, pp. 85–89, and linked letter to Editor, “The 'Pi' Theorem,” pp. 108–110.
2.
Barr, D. I. H., “Consolidation of the Webb and Fryer Procedure for Dimensional Analysis,” International Journal of Mechanical Engineering Education, Vol. 10, No. 4, Oct., 1982, pp. 279–284.
3.
Barr, D. I. H., “A Survey of Procedures for Dimensional Analysis,” International Journal of Mechanical Engineering Education, Vol. 11, No. 3, July, 1983, pp. 147–159.
4.
Bridgman, P. W., Dimensional Analysis, Harvard University Press, Cambridge, Mass., 1922 (Revised Edition 1931).
5.
Buckingham, E., “On Physically Similar Systems: Illustrations of the Use of Dimensional Equations,” Physical Review, Vol. 4, 1914, pp. 345–376.
6.
Buckingham, E., “Model Experiments and the Forms of Empirical Equations,” Transactions, American Society of Mechanical Engineers, Vol. 37, 1915, pp. 263–296.
7.
Bowman, C. C., and Hansen, V. E., “Simplification of Dimensional Analysis,” Journal of the Engineering Mechanics Division, ASCE, Vol. 85, No. EM1, Proc. Paper 1898, Jan., 1959, pp. 67–73.
8.
Chick, A. C., “The Principle of Similitude,” Hydraulic Laboratory Practice, J. R. Freeman, ed., American Society of Mechanical Engineers, 1929, pp. 796–827.
9.
Costa, F. V., “Directional Analysis in Model Design,” Journal of the Engineering Mechanics Division, ASCE, Vol. 97, No. EM2, Proc. Paper 8053, Apr., 1971, pp. 519–539.
10.
Fourier, Baron J. B., Théorie Analytique de la Chaleur, 3rd ed., Paris, France, 1822, (English translation by A. Freeman, Cambridge Univ. Press, London, 1878).
11.
Gessler, J., “Discussion of Reference 9,” Journal of the Engineering Mechanics Division, ASCE, Vol. 97, No. EM6, Proc. Paper 1746, Dec., 1971, pp. 1746–1748.
12.
Gessler, J., “Vectors in Dimensional Analysis,” Journal of the Engineering Mechanics Division, ASCE, Vol. 99, No. EM1, Proc. Paper 9561, Feb., 1973, pp. 121–129.
13.
Gibbings, J. C., “Directional Attributes of Length in Dimensional Analysis,” International Journal of Mechanical Engineering Education, Vol. 9, No. 3, July, 1981, pp. 263–272.
14.
Gibbings, J. C., “A Logic of Dimensional Analysis,” Journal of Physics A: Mathematical and General, Vol. 15, 1982, pp. 1991–2002.
15.
Gibson, A. H., “The Principle of Dynamic Similarity with Special Reference to Model Experiments,” Engineering, London, England, Vol. 117, 1924, pp. 325, 357, 391, 422.
16.
Groat, B. F., “Ice Diversion for St. Lawrence River Power Co.,” The Canadian Engineer, Vol. 39, Nov. 25, 1920, pp. 545–552.
17.
Groat, B. F., “Ice Diversion, Hydraulic Models, and Hydraulic Similarity,” Transactions, ASCE, Paper 1419, Vol. 32, 1918, pp. 1138–1190.
18.
Huntley, H. E., Dimensional Analysis, MacDonald & Co., London, England, 1953.
19.
Ipsen, D. C., Units, Dimensions and Dimensionless Numbers, McGraw‐Hill, New York, N.Y., 1960.
20.
Isaacson, E. de St. Q., and Isaacson, M. de St. Q., Dimensional Methods in Engineering and Physics, Arnold, London, 1975.
21.
Kline, S. J., Similitude and Approximation Theory, McGraw‐Hill Book Co., New York, 1965.
22.
Langhaar, H. L., Dimensional Analysis and Theory of Models, John Wiley & Sons, Inc., New York, N.Y., 1951 (and reprinted with minor corrections, Krieger, Huntington, N.Y., 1980).
23.
Macagno, E. O., “Discussion of Reference 7,” Journal of the Engineering Mechanics Division, ASCE, Vol. 85, No. EM3, Proc. Paper, 1959, p. 149.
24.
Massey, B. S., Units, Dimensional Analysis and Physical Similarity, Van Nostrand Reinhold, London, England, 1971.
25.
Massey, B. S., “Directional Analysis?,” International Journal of Mechanical Engineering Education, Vol. 6, No. 1, Jan., 1978, pp. 33–36.
26.
Newton, Sir Isaac, Mathematical Principles of Natural Philosophy, translated by Andrew Motte, revised by F. Cajori, University of California Press, 1934 (Reprinted 1952 in Vol. 34, of Great Books of Western World—Encyclopedia Brittanica).
27.
Pedoe, D., A Geometric Introduction to Linear Algebra, John Wiley and Sons, Inc., New York, N.Y., 1963.
28.
Palacios, J., Dimensional Analysis, MacMillan and Co., London, England, 1964.
29.
Riabouchinski, D. P., “Methode des Variables des Dimension Zéro, et son Application en Aérodynamique,” L'Aérophile, Sept., 1911, pp. 407–408.
30.
Riabouchinski, D. P., “Thirty Years of Theoretical and Experimental Research in Fluid Mechanics,” Journal, Royal Aeronautical Society, Vol. 39, 1935, pp. 282–348 and 377–444.
31.
Schenk, H., Theories of Engineering Experimentation, 2nd ed., McGraw‐Hill Book Co., New York, N.Y., 1968.
32.
Staicu, C. I., “General Dimensional Analysis,” Journal of the Franklin Institute, Vol. 292, No. 6, Dec., 1971, pp. 433–439.
33.
Staicu, C. I., “Restricted and General Dimensional Analysis,” Abacus Press, Tunbridge Wells, Kent, England, 1982.
34.
Stevens, J. C., et al., Hydraulic Models, American Society of Civil Engineers Manual of Engineering Practice, No. 25, 1942.
35.
Taylor, E. S., Dimensional Analysis for Engineers, Clarendon Press, Oxford, England, 1974.
36.
Van Driest, E. R., “On Dimensional Analysis and the Presentation of Data in Fluid‐Flow Problems,” Journal of Applied Mechanics, American Society of Mechanical Engineers, Vol. 13, No. 1, Mar., 1946, pp. 1–40.
37.
Williams, W., “On the Relations of the Dimensions of Physical Quantities to Directions in Space,” Philosophical Magazine, London, England, Ser. 5, Vol. 34, No. 208, 1892, p. 234.
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Published In
Journal of Engineering Mechanics
Volume 110 • Issue 9 • September 1984
Pages: 1357 - 1376
Copyright
Copyright © 1984 ASCE.
History
Published online: Sep 1, 1984
Published in print: Sep 1984
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