The Vibration of Rectangular Plates

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Abstract
In this paper are considered the free transverse vibrations of rectangular plates with all possible boundary conditions obtained by combining free, freely-supported, and fixed edges. The Rayleigh method, assuming waveforms similar to those of beams, is used to derive a simple approximate frequency expression for all modes of vibration. The terms in this expression depend on the nodal pattern and the boundary conditions; they are tabulated for fifteen boundary conditions—all four edges free, freely-supported, or fixed and the twelve cases in which some of the edges have one condition and the rest another. The expression can also be used to obtain frequencies for a plate which has a combination of all three boundary conditions. The effect on frequency of an edge being supported and partially restrained is discussed. For some boundary conditions it is possible to compare derived frequencies with those obtained by various methods of accurate analysis and by experiment; except for a few cases, the results from the approximate expression and accurate analysis agree closely. For rectangular plates and for most boundary conditions and modes of square plates, the nodal pattern consists of lines approximately parallel to the sides of the plate. The exceptions are discussed and the gradual transition from these non-parallel patterns characteristic of a square plate to those of a rectangular plate is traced. The frequencies of extensional vibrations of rectangular plates are derived for two boundary conditions.