Rayleigh–Ritz method

Abstract: Exact expressions for rates of change of eigenvalues and eigenvector to facilitate computerized design of complex structures

Abstract: Functionally gradient materials (FGMs) have attracted much attention as advanced structural materials because of their heat-resistance properties In this paper, a study on the vibration of cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of configurations of the constituent materials on the frequencies The properties are graded in the thickness direction according to a volume fraction power-law distribution The results show that the frequency characteristics are similar to that observed for homogeneous isotropic cylindrical shells and the frequencies are affected by the constituent volume fractions and the configurations of the constituent materials The analysis is carried out with strains–displacement relations from Love’s shell theory and the eigenvalue governing equation is obtained using Rayleigh–Ritz method The present analysis is validated by comparing results with those in the literature

Abstract: Approximate eigenvalues given by the Rayleigh-Ritz variation method for handling linear differential equations are examined and relations are established between the discrete eigenvalues obtained in successive approximations. These relations should be of use in practical computations. A method for fixing upper bounds to eigenvalues is given and a procedure previously employed by the writer to simplify determinant calculations is adapted for use in the present theory.

Abstract: Natural frequencies of rectangular plates are obtained by employing a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The orthogonal polynomials are generated 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. Natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods. The method yields superior results for lower modes, particularly when plates have some of the edges free.

Abstract: Present investigation is concerned with the free vibration analysis of functionally graded material (FGM) beams subjected to different sets of boundary conditions. The analysis is based on the classical and first order shear deformation beam theories. Material properties of the beam vary continuously in the thickness direction according to the power-law exponent form. Trial functions denoting the displacement components of the cross-sections of the beam are expressed in simple algebraic polynomial forms. The governing equations are obtained by means of Rayleigh–Ritz method. The objective is to study the effects of constituent volume fractions, slenderness ratios and the beam theories on the natural frequencies. To validate the present analysis, comparison studies are also carried out with the available results from the existing literature.