Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates

Abstract

This paper deals with nonlinear free axisymmetric vibrations of functionally graded thin circular plates whose properties vary through its thickness. The inhomogeneity of the plate is characterized by a power law variation of the Young’s modulus and mass density of the material along the thickness direction, whereas Poisson’s ratio is assumed to be constant. The theoretical model is based on Hamilton’s principle and spectral analysis using a basis of admissible Bessel’s functions to yield the frequencies of the circular plates under clamped boundary conditions on the basis of the classical plate theory. The large vibration amplitudes problem, reduced to a set of non-linear algebraic equations, is solved numerically. The non-linear to linear frequency ratios are presented for various values of the volume fraction index n showing hardening type non-linearity. The distribution of the radial bending stress associated to the non-linear mode shape is also given for various vibration amplitudes, and is compared with those predicted by the linear theory.

A homogenization procedure for geometrically non-linear free vibration analysis of functionally graded circular plates involving the coupling between transverse and in-plane displacements

“…The comparison made between the results obtained here and those obtained in ref. [35], where in-plane displacements are omitted. It can be seen that the w-formulation leads to higher frequency parameters than those obtained from the u-w-formulation due to the mathematical stiffening that occurs once one neglects the in-plane displacements.…”

“…The plate thickness was supposed to be constant. The results are compared with those obtained when the in-plane displacements is neglected [35] and also with the published literature to demonstrate the applicability and the computational efficiency of the proposed method.…”