On the Bounds of Eigenvalues of a Clamped Plate in Tension

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Abstract
In this paper, use is made of Temple’s generalizations of Kato’s theorem for the determination of close lower bounds to the fundamental frequency of oscillation of a clamped square plate subjected to uniform biaxial edge tension. The success of the method depends upon the solution of an auxiliary problem governing the residual mode shape; and in the present case, this mode is determined by using the variational method of Ritz. The principle of computation of the number ϵV2 (which corresponds to the ratio of the elastic energies in the residual and the tentative mode) is explained, and the value of the lower bound is determined by using the inequality (1 − ρ/β)(ρ/α − 1) ≤ ϵV2. The results are compared with those obtained previously by Weinstein and Chien, and it is shown that the present method leads to much closer bounds.

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