Stability and local bifurcation analysis of functionally graded material plate under transversal and in-plane excitations☆

Abstract In this paper, stability and local bifurcation behaviors for a simply supported functionally graded material (FGM) rectangular plate subjected to the transversal and in-plane excitations in the uniform thermal environment are investigated using both analytical and numerical methods. Three kinds of degenerated equilibrium points of the bifurcation response equations are considered, which are characterized by a double zero eigenvalues, a simple zero and a pair of pure imaginary eigenvalues as well as two pairs of pure imaginary eigenvalues in nonresonant case, respectively. With the aid of Maple and normal form theory, the explicit expressions of transition curves are obtained, which may lead to static bifurcation, Hopf bifurcation and 2-D torus bifurcation. Finally, the numerical solutions obtained by using fourth-order Runge–Kutta method agree with the analytic predictions.

[1]  Wei Zhang,et al.  Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate , 2008 .

[2]  A. Maccari,et al.  Approximate Solution of a Class of Nonlinear Oscillators in Resonance with a Periodic Excitation , 1998 .

[3]  Pei Yu,et al.  Bifurcations associated with a three-fold zero eigenvalue , 1988 .

[4]  K. Huseyin,et al.  On the analysis of hopf bifurcations , 1983 .

[5]  Hui-Shen Shen,et al.  Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels , 2003 .

[6]  K. Huseyin,et al.  Static and dynamic bifurcations associated with a double-zero eigenvalue , 1986 .

[7]  P. Yu,et al.  On bifurcations into nonresonant quasi-periodic motions , 1988 .

[8]  Katsuhiko Ogata,et al.  Modern Control Engineering , 1970 .

[9]  Hui-Shen Shen,et al.  VIBRATION CHARACTERISTICS AND TRANSIENT RESPONSE OF SHEAR-DEFORMABLE FUNCTIONALLY GRADED PLATES IN THERMAL ENVIRONMENTS , 2002 .

[10]  Pei Yu,et al.  Symbolic computation of normal forms for semi-simple cases , 1999 .

[11]  Pei Yu,et al.  COMPUTATION OF NORMAL FORMS VIA A PERTURBATION TECHNIQUE , 1998 .

[12]  Pei Yu,et al.  Analysis on Double Hopf Bifurcation Using Computer Algebra with the Aid of Multiple Scales , 2002 .

[13]  Pei Yu,et al.  A perturbation analysis of interactive static and dynamic bifurcations , 1988 .

[14]  Rubens Sampaio,et al.  Vibrations of axially moving flexible beams made of functionally graded materials , 2008 .

[15]  A. Maccari,et al.  The Asymptotic Perturbation Method for Nonlinear Continuous Systems , 1999 .

[16]  Wei Zhang,et al.  Vibration analysis on a thin plate with the aid of computation of normal forms , 2001 .

[17]  Hui-Shen Shen,et al.  Nonlinear vibration and dynamic response of functionally graded plates in thermal environments , 2004 .

[18]  H. Yuda,et al.  Bifurcation and chaos of thin circular functionally graded plate in thermal environment , 2011 .

[19]  Pei Yu,et al.  Analysis of Non-Linear Dynamics and Bifurcations of a Double Pendulum , 1998 .

[20]  P. Yu,et al.  Bifurcations associated with a double zero and a pair of pure imaginary eigenvalues , 1988 .

[21]  P. Yu,et al.  SYMBOLIC COMPUTATION OF NORMAL FORMS FOR RESONANT DOUBLE HOPF BIFURCATIONS USING A PERTURBATION TECHNIQUE , 2001 .