The precise determination of the frequencies of elastic plates of arbitrary geometry and boundary conditions involves considerable difficulties in the integration of the fourth order partial differential equation. From the point of view of practical application, approximate methods of determination of the fundamental mode could be helpful.
Jones [1] examined the applicability of a frequency-static deflection relation,
for plates of various geometry and boundary conditions. This relation is obtained from the expression for the fundamental frequency (v) of a clamped elliptical plate [2], and the maximum deflection (W max ) of the same plate under a uniformly distributed load, q [3].
Here r is the mass per unit area and h is the plate thickness. Jones cautioned that this relation may be inappropriate if any portion of the plate boundary is freely supported. Maurizi, Belles and Laura [4] have made a comparative study of various existing expressions [1,4,5] to obtain the fundamental frequency for the case of a clamped elliptical plate. Sundararajan [6] also presented a similar type of relation, which is based on Rayleigh's method, where the fundamental mode of a rectangular plate is approximated by the deflection functions of beams subjected to uniform distributed loads. The constant C in equation ( 1) for rectangular plates having all edges clamped, all edges simply supported, two opposite edges simply supported and the others clamped, and two opposite edges simply supported with the third edge clamped and the fourth edge free, examined by him, was found to be 1β’723, 1β’613, 1β’667 and 1β’978, respectively. This variation in the values of the constant indicates that its value changes with the geometry of the plate and the boundary conditons. Although the simple approximate expressions suggested by Jones [1] and Sundararajan [6] are good estimates for various plate configurations, there is no formal derivation of the frequency-static deflection relation for a plate of arbitrary shape and complex boundary conditions. The purpose of this study is to examine the possibility of such frequency-static displacement relations, and to propose a methodology for estimating the fundamental frequency of a plate through its static deflections under a uniformly distributed load.
Consider a linear elastic plate occupying an area A, inside the boundary S, undergoing free harmonic vibration. According to the Rayleigh-Ritz method, the fundamental frequency (v) can be obtained by selecting a function c(x,y) for the lateral deflection (w)