The free-vibration behavior of rectangular plates constitutes an important field in applied mechanics, and the natural frequencies are known to be primarily affected by the boundary conditions as well as aspect and thickness ratios. Any one of the three classical edge conditions, i.e., free, simply supported, and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations the present paper introduces the Polya counting theory in combinatorial mathematics. Formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three classical edge conditions and is used to numerically verify the numbers. In this numerical study the number of combinations in the free-vibration behavior is determined for some plate models by using the derived formulas. Results are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the modified Ritz method. [S0021-8936(00)02203-0]

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