Asymptotic analysis of geometrically nonlinear vibrations of long plates

B. Affane^∗ , A.G. Egorov^∗∗

Kazan Federal University, Kazan, 420008 Russia

E-mail: ^∗boudkhil.affane@gmail.com, ^∗∗aegorov0@gmail.com

Received August 4, 2020

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DOI: 10.26907/2541-7746.2020.4.396-410

For citation : Affane B., Egorov A.G. Asymptotic analysis of geometrically nonlinear vibrations of long plates. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2020, vol. 162, no. 4, pp. 396–410. doi: 10.26907/2541-7746.2020.4.396-410. (In Russian)

Abstract

In this paper, we performed an asymptotic analysis for equations of the classical plate theory with the von K´arma´n strains under the assumption that the width of the plate is small compared with its length. A system of one-dimensional equations, which describes the nonlinear interaction of flexural and torsional vibrations of beams, was derived. This enables the possibility of exciting torsional vibrations by flexural vibrations. This possibility was analyzed for a model problem, when flexural vibrations occur in normal modes.

Keywords: asymptotic analysis, flexural vibrations, torsional vibrations, parametric resonance, resonance gaps, Mathieu equation.

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