Analogies between the classical problem on vibrations of two coupled bodies and the nonclassical problem on propagation of plane waves in theory of two-phase mixtures
Keywords:oscillations of two coupled bodies, propagation of elastic plane wave in two-phase mixture, analogies in procedures of analysis
In order to identify analogies in the procedures of analysis, two problems are described and commented on: the
classical problem (problem V) on harmonic vibrations of two interconnected rigid bodies suspended on the
springs, and the nonclassical problem (problem W) on the propagation of plane longitudinal waves in the twophase
elastic mixture. A number of analogies are constructed between these problems, which are described in the
form of 6 steps of the comparative analysis of problems V and W, each of which corresponds to a specific analogy.
Emphasis is placed on the usefulness of the observed analogies for the analysis of the problem of waves. Thus, it
is shown that the theory of two-phase mixtures in its development in the field of wave theory can successfully use
the analogies from the corresponding problems of the theory of vibrations of two coupled oscillating systems. The
more general fact follows from the analogies shown that the mechanism of mutual aid in the theory of vibrations
discovered almost 100 years ago by Academician Mandelstam extended its influence to the theory of waves
during this time.
Academician L.I.Mandelstam. (1979). To Centenary of Birthday K. Moscow: Nauka (in Russian).
Strelkov, S.P. (1964). Introduction in Theory of Vibrations. 2nd ed. Moscow: Nauka (in Russian).
Schmitz, T.L. & Smith, K.S. (2021). Mechanical Vibrations. Modeling and Measurement. 2nd ed. New York:
Rushchitsky, J.J. (1991). Elements of the theory of mixtures. Kiev: Naukova Dumka (in Russian).
Rushchitsky, J.J. & Tsurpal, S.I. (1998). Waves in Materials with Microstructure. Kyiv, S.P.Timoshenko
Institute of Mechanics (in Ukrainian).
Rushchitsky, J.J. (2011). Theory of waves in materials. Copenhagen: Ventus Publishing
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