Stability and Bifurcation for a Simply Supported Functionally Graded Material Plate with One-to-One Internal Resonance

Dongmei Zhang,Fangqi Chen

doi:10.1155/2014/414903

Abstract

Stability and bifurcation behaviors for a model of simply supported functionally graded materials rectangular plate subjected to the transversal and in-plane excitations are studied by means of combination of analytical and numerical methods. The resonant case considered here is 1 : 1 internal resonances and primary parametric resonance. Two types of degenerated equilibrium points are studied in detail, which are characterized by a double zero and two negative eigenvalues, and a double zero and a pair of pure imaginary eigenvalues. For each case, the stability regions of the initial equilibrium solution and the critical bifurcation curves are obtained in terms of the system parameters which may lead to Hopf bifurcation and 2D torus. With both analytical and numerical methods, bifurcation behaviors on damping parameters and detuning parameters are studied, respectively. A time integration scheme is used to find the numerical solutions for these bifurcation cases, and numerical results agree with the analytic predictions.

Highlights

Graded materials (FGMs) are extremely excellent engineering composite materials
The objective of this paper is to investigate the local dynamic behaviors of a supported rectangular plate subjected to the transversal and in-plane excitations in a thermal environment
The paper focuses on the stability and bifurcation behaviors of a functionally graded material rectangular plate subjected to the in-plane and transversal excitations

Summary

Introduction

Graded materials (FGMs) are extremely excellent engineering composite materials. Ye et al [7] investigated the nonlinear vibrations and chaotic dynamics of an ant symmetric crossply laminated composite rectangular thin plate under parametric excitation. Zhang et al [10] studied the nonlinear dynamics of a parametrically excited supported laminated composite plate and found that there exist the multiple steady bifurcation solutions under the certain conditions. Hao et al [17] analyzed the nonlinear dynamics of a supported FGM plate subjected to the transverse and in-plane excitations in a thermal environment. The objective of this paper is to investigate the local dynamic behaviors of a supported rectangular plate subjected to the transversal and in-plane excitations in a thermal environment.

Formulation of the Problem

Stability and Bifurcation Behaviors

Stability and Bifurcation Behaviors on the Detuning

Stable region for zero solution ζ1

Conclusions