Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates

Abstract

This study aimed to develop series analytical solutions based on the Mindlin plate theory for the free vibrations of functionally graded material (FGM) rectangular plates. The material properties of FGM rectangular plates are assumed to vary along their thickness, and the volume fractions of the plate constituents are defined by a simple power-law function. The series solutions consist of the Fourier cosine series and auxiliary functions of polynomials. The series solutions were established by satisfying governing equations and boundary conditions in the expanded space of the Fourier cosine series. The proposed solutions were validated through comprehensive convergence studies on the first six vibration frequencies of square plates under four combinations of boundary conditions and through comparison of the obtained convergent results with those in the literature. The convergence studies indicated that the solutions obtained for different modes could converge from the upper or lower bounds to the exact values or in an oscillatory manner. The present solutions were further employed to determine the first six vibration frequencies of FGM rectangular plates with various aspect ratios, thickness-to-width ratios, distributions of material properties and combinations of boundary conditions.

Analytical Prediction for Nonlinear Buckling of Elastically Supported FG-GPLRC Arches under a Central Point Load

“…Hao et al [10] employed an asymptotic perturbation method to analyze the nonlinear oscillations, bifurcations, and chaotic motions of FGM plates. More relevant studies to investigate the significant performance of FGM structures could be found from the open literature [11][12][13][14][15][16][17].…”

Three-Dimensional Free Vibration Analyses of Preloaded Cracked Plates of Functionally Graded Materials via the MLS-Ritz Method

“…Most of the relevant studies in the literature have analyzed intact flat plates. Analytical solutions based on plate theories and the three-dimensional elasticity theory have been proposed for the vibration of rectangular plates with two simply supported opposite edges and four simply supported faces, respectively [5][6][7][8][9][10][11]. Moreover, solutions for the vibration of rectangular plates under different boundary conditions have been reported using various numerical approaches, such as the Ritz method [12][13][14], differential quadrature method [15][16][17], mesh-free method [18][19][20], and finite-element method (FEM) [21][22][23].…”

Mechanics and Analysis of Advanced Materials and Structures

“…It has many modes, and the dynamics depend on the boundary conditions and the material's characteristics. Huang et al [1] discuss an analytical solution based on the Mindlin plate theory that describes the free vibration of rectangular plates of functionally graded material. They analyzed various combinations of boundary conditions and validated the results via comparison with published results.…”