Free and forced bending vibrations of a thin plate in a perfect compressible fluid with energy dissipation taken into account

Abstract

© 2020 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim A refined statement of stationary dynamic acoustoelasticity problems of thin isotropic plates surrounded from both sides by acoustic media assumed to be perfect compressible fluids has been studed. The problem statement takes into account the energy dissipation in the material of the plate and fluid on the basis of the Kelvin-Voigt hysteresis model. The refinement of the fluid behavior is based on the assumption that the pressure increment is proportional not only to volumetric strain, but also to the rate of its volumetric strain. This assumption allows us to obtain the generalized Helmholtz wave equation by introducing Skudrzyk's complex sound velocity into consideration to take account of the energy dissipation. The motion equations of the plate are based on the classical Kirchhoff-Love model and are obtained in two versions. In the first version unsimplified three-dimensional wave equations are used to determine the aerohydrodynamic load acting on a plate. The second version is based on simplifying the equations of fluid motion by the introduction of the well-known hypothesis of plane reflection and emission of sound waves. On the basis of the derived equations, exact analytical solutions of two types of problems are obtained. The first type is related to the problem of free vibrations of a rectangular plate hinged around the contour when complex eigenfrequencies were defined. The second type is related to the problem of forced vibrations of the plate under the action of a flat mono-harmonic incident sound wave when sound transmission loss, the stress-strain state parameters of the plate as well as the laws of change in sound pressure in a fluid were determined. It is shown that correct and more meaningful solutions of acoustoelasticity problems of the considered class are possible only in case of describing the behavior of a fluid with unsimplified three-dimensional wave equations with consideration of the energy dissipation.