Flexural Vibrations of Elliptical Plates When Transverse Shear and Rotary Inertia are Considered

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Abstract
In this paper, the author solves the partial-differential equations governing the motion of plates, which were derived by Mindlin, Reissner, and Uflyand, by using product solutions and finds, by taking an infinite series of product solutions, that the boundary conditions are satisfied. The elimination of the arbitrary constants occurring in each boundary problem leads to the frequency equation for the normal modes of vibration. The frequency equation for each problem is an infinite determinant and each element in it is an infinite series of Mathieu functions containing as unknown the frequency. A method is described permitting one to find the roots of the infinite determinant. These roots represent the normal modes of vibration for the elliptical plate. Since the classical (Lagrange) theory of plate is good only for plate when the wavelength is large in comparison with the thickness of the plate, it is restricted to low-frequency vibrations. The present theory gives good results for high-frequency vibrations, essentially because it includes coupling between flexural and shear motions.

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