# High-Frequency Free Vibrations of Plates in the Reissner’s Type Theory

• Elena A. Ivanova
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 15)

## Abstract

The classic plate theory by Kirchhoff allows to accurately describe the processes slowly varying by time. To solve the problems of the plate vibrations in the case of the external loads quickly varying by time the Reissner’s type plate theory should be used. The Reissner’s type plate theory includes three eigenfrequency spectra: one low-frequency spectrum whose asymptotic order is $$O(h)$$, and two high-frequency spectra whose asymptotic order is $$O(h^{-1})$$. Solving the problems of plate vibrations under the action of the quickly varying by time loads it is necessary to take into account vibrations with eigenfrequencies from the high-frequency spectra. That is why the problem of plate free vibrations with eigenfrequencies whose asymptotic order is $$O(h^{-1})$$ is interesting and practically important. In this paper asymptotic analysis of the equations of the Reissner’s type plate theory for high-frequency free vibrations is carried out and the approximate equations of plate vibrations with eigenfrequencies of the asymptotic order $$O(h^{-1})$$ are proposed. Asymptotic analysis of the equations of plate free vibrations shows that behavior of the functions defining the stress-strain state of the plate for high-frequency free vibrations differs from it for low-frequency free vibrations. For high-frequency free vibrations the solution includes the functions which quickly vary along the space coordinates but which are not the boundary layer type functions. Because of that using of the exact equations of the Reissner’s type theory in numerical procedures is difficult. Approximate equations of high-frequency free vibrations of plates independent of quickly varying along the space coordinates functions are formulated in this paper. These equations describe vibrations with eigenfrequencies from the high-frequency spectra only, like the classic plate theory describes vibrations with eigenfrequencies from the low-frequency spectrum only.

## Keywords

Reissner plate High-frequency vibration

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