Nonlinear oscillation of a cantilever FGM rectangular plate based on third-order plate theory and asymptotic perturbation method

Abstract Nonlinear dynamic analysis of a cantilever functionally graded materials (FGM) rectangular plate subjected to the transversal excitation in thermal environment is presented for the first time in this paper. Material properties are assumed to be temperature dependent. The nonlinear governing equations of motion for the FGM plate are derived based on Reddy’s third-order plate theory and Hamilton’s principle. The first two vibration mode functions satisfying the boundary conditions of the cantilever FGM rectangular plates are chosen to be the admissible displacement functions. Galerkin’s method is utilized to convert the governing partial differential equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms under combined external excitations. The present study focuses on resonance case with 1:1 internal resonance and subharmonic resonance of order 1/2. The asymptotic perturbation method is employed to obtain four nonlinear averaged equations which are then solved by using Runge–Kutta method to find the nonlinear dynamic responses of the plate. It is found that chaotic, periodic and quasi-periodic motions of the plate exist under certain conditions and the forcing excitations can change the form of motions for the FGM rectangular plate.

[1]  Jie Yang,et al.  Semi-analytical solution for nonlinear vibration of laminated FGM plates with geometric imperfections , 2004 .

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

[3]  D. J. Gorman.  Accurate free vibration analysis of the orthotropic cantilever plate , 1995 .

[4]  Wei Zhang,et al.  Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory , 2010 .

[5]  P. M. Ciancio,et al.  Approximate study of the free vibrations of a cantilever anisotropic plate carrying a concentrated mass , 2007 .

[6]  K. M. Liew,et al.  Large amplitude vibration of thermo-electro-mechanically stressed FGM laminated plates , 2003 .

[7]  A. Bhimaraddi,et al.  Large Amplitude Vibrations Of Imperfect Antisymmetric Angle-ply Laminated Plates , 1993 .

[8]  Chun-Sheng Chen.  Nonlinear vibration of a shear deformable functionally graded plate , 2005 .

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

[10]  J. W. ALIAGA,et al.  Nonlinear Thermoelastic Analysis of Functionally Graded Plates Using the Third-order Shear Deformation Theory , 2004, Int. J. Comput. Eng. Sci..

[11]  Hongjun Xiang,et al.  Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction , 2008 .

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

[13]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[14]  Liqun Chen,et al.  Bifurcation and chaos of an axially accelerating viscoelastic beam , 2004 .

[15]  J. N. Reddy,et al.  Large deformation analysis of functionally graded shells , 2007 .

[16]  Xiaolin Huang,et al.  Nonlinear transient response of functionally graded plates with general imperfections in thermal environments , 2007 .

[17]  J. N. Reddy,et al.  Three-dimensional thermomechanical deformations of functionally graded rectangular plates , 2001 .

[18]  市川 洌.  Functionally graded materials in the 21st century : a workshop on trends and forecasts , 2001 .

[19]  A. Maccari,et al.  Modulated motion and infinite-period homoclinic bifurcation for parametrically excited Liénard systems , 2000 .

[20]  J. Reddy.  Analysis of functionally graded plates , 2000 .

[21]  M. Chiba,et al.  Vibration characteristics of a cantilever plate with attached spring–mass system , 2003 .

[22]  K. M. Liew,et al.  Vibration of mindlin plates using boundary characteristic orthogonal polynomials , 1995 .

[23]  Y. Liu,et al.  Free vibration of stepped cantilever Mindlin plates , 2004 .

[24]  A. Maccari.  Dissipative bidimensional systems and resonant excitations , 1998 .

[25]  J. N. Reddy,et al.  A study of non-linear dynamic equations of higher-order shear deformation plate theories , 1991 .

[26]  Wei Zhang,et al.  Nonlinear dynamic response of a functionally graded plate with a through-width surface crack , 2010 .

[27]  F. Bakhtiari-Nejad,et al.  Nonlinear vibration analysis of isotropic cantilever plate with viscoelastic laminate , 2009 .

[28]  Qiusheng Li.  Flexural free vibration of cantilevered structures of variable stiffness and mass , 1999 .

[29]  J. N. Reddy,et al.  Frequency of Functionally Graded Plates with Three-Dimensional Asymptotic Approach , 2003 .

[30]  Hui-Shen Shen,et al.  Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments , 2002 .

[31]  J. N. Reddy,et al.  Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates , 1998 .

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

[33]  J. Reddy.  A Simple Higher-Order Theory for Laminated Composite Plates , 1984 .

[34]  A. Maccari,et al.  Multiple Resonant or Non-Resonant Parametric Excitations for Nonlinear Oscillators , 2001 .