Basic principles and theorems of dimensional analysis and the theory of model similarity of physical phenomena

Andrzej Flaga


The paper concerns dimensional analysis and the theory of model similarity of physical phenomena. At the beginning of the considerations, basic notions, definitions, relationships and fundamental principles of the issues to be analysed has been presented. Next, Buckingham’s Π-theorem of dimensional analysis and theory of similarity as well as authorial generalized theorems Π of this field of interest have been derived. A separate chapter has been devoted to the problem of the nature of physical phenomena occurring in the mechanics of continuous or discrete material mediums. At the end of the paper, an example of the determination of similarity numbers in the case of a system with one degree of freedom at mechanical and kinematic excitation have been given.

Słowa kluczowe: dimensional analysis, model similarity, physical phenomena, principles and theorems

[1] Barenblatt G.I., Scaling, self-similarity, and intermediate asymptotics, Cambridge University Press, Cambridge, UK 1996.

[2] B aker W.E., Westine P.S., Dodge F. T., Similarity methods in engineering dynamics, Hayden, Rochelle Park, NY 1973.

[3] B irkhoff G., Hydrodynamics, Princeton University, 1950.

[4] Bridgman P.W., Dimensional analysis, 2nd edition (the first edition appeared in 1922), Yale University Press, New Haven, 1931.

[5] B uckingham E., On physically similar systems, Illustrations of the use of dimensional analysis, Physical Review, 4,1914, 345-376.

[6] D avid F.W. Nolle H., Experimental modelling in engineering, Butterworths, London 1982.

[7] de Jong F.J., Dimensional analysis for economists, Nort Holland, Amsterdam 1967.

[8] D uncan W.J., Physical similarity and dimensional analysis, Arnold, London 1955.

[9] Flaga A., Wind vortex-induced excitation and vibration of slender structures – single structure of circular cross-section normal to flow, Monograph No. 202, Cracow University of Technology, Cracow 1996.

[10] Flaga A., Inżynieria wiatrowa, Podstawy i zastosowania, Arkady, Warszawa 2008.

[11] Flaga A., Siłownie wiatrowe, Wydawnictwo Politechniki Krakowskiej, Kraków 2012.

[12] Focken C.M., Dimensional methods and their applications, Arnold, London 1953.

[13] Gryboś R., Podstawy mechaniki płynów, PWN , Warszawa 1998.

[14] Huntley H. E., Dimensional analysis, MacDonald & Co., London 1952.

[15] I psen D. C., Units, dimensions and dimensional numbers, McGraw Hill, New York 1960.

[16] I saacson E. de St.Q. Isaacson M. de St.Q., Dimensional methods in engineering and physics: Reference sets and the possibility of their extension, Wiley, New York 1975.

[17] Kline S.J., Similitude and approximation theory, McGraw-Hill, New York 1965.

[18] Kolouek V., Pirner M., Fischer O., Naprstek J., Wind effects on civil engineering structures, Academia, Praha 1983.

[19] Kurth R., Dimensional analysis and group theory in astrophysics, Pergamon Press, Oxford 1972.

[20] L anghaar H.L., Dimensional analysis and theory of models, Wiley, New York 1951.

[21] L okarnik M., Dimensional analysis and scale-up in chemical engineering, Springer Verlag, Berlin 1991.

[22] M acagno E.O., Historico-critical review of dimensional analysis, Journal of the Franklin Institute, 292, 1971,391-402.

[23] M artynow A.K., Ekspriminetalnaja aerodinamika, Moskwa 1958.

[24] M axwell J.C., A treatise on electricity and magnetism, 3rd Edition, Clarendon Press, Cambridge 1891, republished by Dover, New York, 1954.

[25] M cMahon T.A. and Bonner J.T., On size and life. Scientific American Library, New York 1983.

[26] M urphy G., Similitude in engineering, Ronald Press, New York 1950.

[27] M ller L., Zastosowania analizy wymiarowej w badaniach modelowych, PWN, Warszawa 1983.

[28] P alacios J., Dimensional analysis, MacMillan, London 1964.

[29] P ankhurst R.D., Dimensional analysis and scale factors, Reinhold, New York 1964.

[30] S achs P., Wind forces in engineering, Pergamon Press, 1972.

[31] S churing D.J., Scale models in engineering: Bases and applications, Pergamon, New York 1977.

[32] S iedov L.I., Similarity and dimensional analysis in mechanics, Academic Press, New York 1959.

[33] S imiu E. Scanlan R., Wind effects on structures, An introduction to wind engineering, Fundamentals and applications to the design, Third edition, John Wiley & Sons, New York 1996.

[34] S koglund V.J., Similitude: Theory and applications, International Textbook Co. Scranton, 1967.

[35] S onin A.A., The physical basis of dimensional analysis, Department of Mechanical Engineering MIT, Cambridge 2001.

[36] S zucs E., Similitude and modeling, Elsevier, Amsterdam 1980.

[37] T aylor E.S., Dimensional Analysis for Engineers, Clarendon Press, Oxford 1974.

[38] Z ierep J., Similarity and Modeling, Marcel Dekker, New York 1971.

[39] Z ierep J., Kryteria podobieństwa i zasady modelowania w mechanice płynów, PWN, Warszawa 1978.