Free Vibrations of Isotropic Cubes and Nearly Cubic Parallelepipeds

Abstract
Free vibrations of isotropic cubes and nearly cubic parallelepipeds are derived by a modification of the Ritz method. The trial functions are chosen to be similar to actual modes, but simple enough to be manageable. The semidirect method of variational calculus is used to obtain some of them, others are simple solutions of somewhat different elastic problems. The resultant secular determinant of 15th degree is reduced (factored) by the use of group theory, so that the frequency equation can be solved analytically. The modes are classified according to symmetry properties. Some unexpected degeneracies are found which are not predicted by group theory.

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