Natural Frequencies of Parallelogrammic Plates Using Characteristic Orthogonal Polynomials in Rayleigh-Ritz Method

Abstract

Abstract Natural frequencies of parallelogrammic plates are obtained by employing a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The orthogonal polynomials are generalted by using a Gram-Schmidt process, after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems. The strain energy functional and kinetic energy functionals are transformed from Cartesian coordinate system to a skew coordinate system. The natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods with all four edges clamped boundary conditions and greet agreements are found between them. The natural frequencies for parallelogrammic plates with other boundary conditions, such as four edges simply supported, clamped-free and simply supported-free, are also obtained. This method is considered as a better and accurate comprehensive treatment for this type of problems.