Homogenization Technique for Non-Linear Free Vibrations Analysis of FGM Rectangular Plates

Abstract

In the present study, the problem of geometrically nonlinear free vibrations of functionally graded rectangular plates (FGRP) is studied. A homogenization technique has been developed to reduce the FGRP problem under consideration to that of isotropic homogeneous rectangular plate. The material properties of the functionally graded composites examined herein are assumed to be graded in the thickness direction of the plate and estimated through the rule of mixture. The proposed theoretical model is based on the classical plate theory and the Von Karman relationships, and the amplitude equation is derived in the form of a set of non-linear algebraic equation using Hamilton’s principle and a multimode approach. The fundamental nonlinear frequency parameters and the bending stress are then calculated using the iterative and explicit methods of solution to show the effect of the vibration amplitudes and the material distributions. The results obtained in this study are found to be in a good agreement with the published ones dealing with the problem of large vibration of functionally graded plates.

Non-Linear Forced Vibration of Fully Clamped FGM Skew Plates using Homogenization Technique

“…2 shows the non dimensional non-linear maximum displacement W*max of FG square plate (N=1, =0° and =1) versus the nonlinear frequency ratio. The comparison made between the results obtained in reference [3] and the present ones shows a good agreement.…”

“…The aim of this section is to apply the theoretical model presented in ref [3] and [4] Fig. 2 shows the non dimensional non-linear maximum displacement W*max of FG square plate (N=1, =0° and =1) versus the nonlinear frequency ratio.…”

Geometrically Non-Linear Free Vibration of Fully Clamped FGM Skew Plates Using Homogenization Technique

“…The axial displacements u and v are neglected considering the analysis made in reference [13]. The expression for the bending strain energy V b , the membrane strain energy V m , the coupling strain energy V c and the kinetic energy T are given in rectangular coordinate in reference [9]. The skew co-ordinates are related to the rectangular co-ordinate (ξ,η) by: ξ=x-y tanθ ; η=y/cosθ.…”

“…The results obtained revealed that the temperature field and gradient index have significant effect on the nonlinear vibration of the FG plate. Boukhzer et al [9] presented a homogenization technique for the non-linear free vibrations analysis of FG rectangular plates. A homogenization technique has been developed to reduce the FGRP problem under consideration to that of an isotropic homogeneous rectangular plate.…”

Geometrically Nonlinear Free Vibration of Composite Materials: Clamped-Clamped Functionally Graded Beam with an Edge Crack Using Homogenisation Method

“…A homogenisation procedure already proposed in [6][7][8][9] for the case of rectangular plates and beams, is used to reduce the problem studied to that isotropic homogeneous cracked beams, in order to show the effect of the crack and the volume fraction on structural dynamic characteristics such as natural frequencies.…”