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Three-dimensional vibration analysis of functionally graded material rectangular plates by Chebyshev polynomials
Q. Li; V. P. Iu; K. P. Kou
2006
Conference NameEnhancement and promotion of computational methods in engineering and science X (EPMESC X)
Source PublicationProceedings of “Enhancement and promotion of computational methods in engineering and science X (EPMESC X)
Conference DateAugust 21–23, 2006
Conference PlaceSanya, China
Abstract

Functionally graded materials (FGMs) are generally two-phase composites with continuously varying volume fractions. This advantage of continuousness can eliminate interface problems of composite materials. In this paper, threedimensional vibration analysis of FGM rectangular plates has been investigated.

Consider the case of a uniform thickness, FGM rectangular plate simply supported along the four edges. The volume fraction of the FGM properties is assumed to obey a power-law function along the thickness direction. However, the Poisson’s ratio is assumed to be constant for the effect of Poisson’s ratio on the deformation is much less than that of Young’s modulus. The three mechanical displacements components of the plate are expanded in triplicate series of Chebyshev polynomials multiplied by the boundary R-function which makes expansions satisfy the essential boundary conditions along the edges. Chebyshev polynomial series are chosen as admissible functions for they are a set of complete and orthogonal series in the interval [−1;1]. The Lagrangian function L of the plate is expressed in terms of Chebyshev polynomial series. By Ritz method, the partial differential of Lagrangian function L with respect to independent coefficients leads to a set of linear equations in form of eigenvalue matrix for natural vibration frequencies. The mode shapes corresponding to each eigenvalue may be obtained by back substitution of the eigenvalues one by one in the usual manner.

In present study, rectangular FGM plates with four simply supported edges are taken as examples for the convergence study. The trial plates are of different volume fraction exponents and thickness-side ratios. The convergence studies reveal the rapid convergence rate and high efficiency. The frequencies monotonically decrease and approach certain values with the increase in the number of terms of admissible functions. Three terms are sufficient for the requirement of expansion in the thickness direction to obtain the first twenty natural frequencies. It is also found that the convergence rate is independent of volume fraction exponents k values of FGMs.

In the last part of this paper, two comparisons have been carried out to validate the present method. The numerical results of the first eight natural frequencies of simply supported square FGM plates are compared with the solutions of the higher-order shear and normal deformable plate theory for two special cases of isotropy; the natural fundamental frequency parameters of simply supported square FGM plates with different thickness-side ratios are compared with the solutions of finite element method. In the modeling of finite element analysis, there is an simplification of material definition for the material properties variation along the thickness of the plate. The relative thick plate and thin plates are meshed by solid and shell element types according to their thickness-side ratios, respectively. It is seen that both comparisons show good agreements. The method presented in this paper is a good and valid approach in analyzing the rectangular FGM plates.

DOIhttps://doi.org/10.1007/978-3-540-48260-4_172
Language英语
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Document TypeConference paper
CollectionFaculty of Science and Technology
DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING
AffiliationDepartment of Civil and Environmental Engineering,University of Macau,Taipa, Macao SAR,China
First Author AffilicationUniversity of Macau
Recommended Citation
GB/T 7714
Q. Li,V. P. Iu,K. P. Kou. Three-dimensional vibration analysis of functionally graded material rectangular plates by Chebyshev polynomials[C],2006.
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