Refined analytical solutions of the coupled problems on free and forced vibrations of a rectangular composite plate surrounded by acoustic media

V.N. Paimushin^a,b∗ , R.K. Gazizullin^a∗∗

^aKazan National Research Technical University named after A.N. Tupolev, Kazan, 420111 Russia

^bKazan Federal University, Kazan, 420008 Russia

E-mail: ^∗vpajmushin@mail.ru, ^∗∗gazizullin.rk@yandex.ru

Received April 27, 2020

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DOI: 10.26907/2541-7746.2020.2.160-179

For citation: Paimushin V.N., Gazizullin R.K. Refined analytical solutions of the coupled problems on free and forced vibrations of a rectangular composite plate surrounded by acoustic media. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2020, vol. 162, no. 2, pp. 160–179. doi: 10.26907/2541-7746.2020.2.160-179. (In Russian)

Abstract

Two problems of the monoharmonic sound wave transmission through a thin rectangular composite plate hinged in the opening of an absolutely stiff dividing wall were considered using the discrete layered damping model of a multilayer plate at small displacements and deformations, with account of the internal damping of layers according to the Kelvin–Voight model. In setting the first problem, it was assumed that the plate located between two semiinfinite spaces and a plane sound wave with a given amplitude value of the pressure is incident on it. In setting the second problem, it was considered that the plate is situated between two absolutely stiff barriers; one of them, owing to the harmonic vibration with a given displacement amplitude of the plate, forms an incident sound wave, while the other is stationary and coated by an energy-absorbing material with high damping properties. Behavior of the acoustic media was described by the classical wave equations based on the model of an ideal compressible fluid. Exact analytical solutions of the formulated problems were constructed. With their help, the sound insulation parameter of composite plate reinforced with carbon fiber textile was studied and the characteristics of its stress-strain state were investigated depending on the frequency of the incident sound wave. It was shown that the mechanics of the deformation of structural elements made of fiber-reinforced composites under high-frequency acoustic impact must be described by refined equations of motion, which have a high degree of accuracy and pithiness, because the stress-strain state formed in them are almost three-dimensional.

Keywords: multilayer plate, Timoshenko model, discrete structural model, two-dimensional equations of equilibrium and motion, analytical solution, Kelvin–Voight model, sound wave, sound insulation parameter, vibration frequency, stress-strain state

Acknowledgments. The study was supported by the Russian Science Foundation (project no. 19-19-00058).

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