Nonlinear vibrations of a composite laminated cantilever rectangular plate with one-to-one internal resonance

The nonlinear vibrations of a composite laminated cantilever rectangular plate subjected to the in-plane and transversal excitations are investigated in this paper. Based on the Reddy’s third-order plate theory and the von Karman type equations for the geometric nonlinearity, the nonlinear partial differential governing equations of motion for the composite laminated cantilever rectangular plate are established by using the Hamilton’s principle. The Galerkin approach is used to transform the nonlinear partial differential governing equations of motion into a two degree-of-freedom nonlinear system under combined parametric and forcing excitations. The case of foundational parametric resonance and 1:1 internal resonance is taken into account. The method of multiple scales is utilized to obtain the four-dimensional averaged equation. The numerical method is used to find the periodic and chaotic motions of the composite laminated cantilever rectangular plate. It is found that the chaotic responses are sensitive to the changing of the forcing excitations and the damping coefficient. The influence of the forcing excitation and the damping coefficient on the bifurcations and chaotic behaviors of the composite laminated cantilever rectangular plate is investigated numerically. The frequency-response curves of the first-order and the second-order modes show that there exists the soft-spring type characteristic for the first-order and the second-order modes.

[1]  S. A. Fazelzadeh,et al.  Aerothermoelastic Post-Critical and Vibration Analysis of Temperature-Dependent Functionally Graded Panels , 2010 .

[2]  Wei Zhang,et al.  Nonlinear oscillations and chaotic dynamics of an antisymmetric cross-ply laminated composite rectangular thin plate under parametric excitation , 2005 .

[3]  Marco Amabili,et al.  Nonlinear vibrations of functionally graded doubly curved shallow shells , 2011 .

[4]  M. Amabili,et al.  Nonlinear vibrations of FGM rectangular plates in thermal environments , 2011 .

[5]  Ji-Hwan Kim,et al.  Nonlinear vibration of viscoelastic laminated composite plates , 2002 .

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

[7]  Wei Zhang,et al.  Further Studies on Nonlinear oscillations and Chaos of a Symmetric Cross-ply Laminated Thin Plate under Parametric Excitation , 2006, Int. J. Bifurc. Chaos.

[8]  S. J. Lee,et al.  Non-linear response of laminated composite plates under thermomechanical loading , 2005 .

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

[10]  D. J. Gorman.  Free vibration analysis of cantilever plates by the method of superposition , 1976 .

[11]  Xinsheng Xu,et al.  A new symplectic approach for piezoelectric cantilever composite plates , 2008 .

[12]  H. F. Tiersten,et al.  Free vibrations of annular sector cantilever plates. Part 1: out-of-plane motion , 2004 .

[13]  Ali H. Nayfeh,et al.  A nonlinear composite plate theory , 1991 .

[14]  Jae-Hoon Kang,et al.  Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation , 2003 .

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

[16]  T. C. Kennedy,et al.  Nonlinear viscoelastic analysis of composite plates and shells , 1998 .

[17]  A. Mufti,et al.  Nonlinear statics and dynamics of antisymmetric composite laminated square plates supported on nonlinear elastic subgrade , 2006 .

[18]  Wei Zhang,et al.  Local and global nonlinear dynamics of a parametrically excited rectangular symmetric cross-ply laminated composite plate , 2005 .

[20]  Yukinori Kobayashi,et al.  Nonlinear dynamic behaviors of clamped laminated shallow shells with one-to-one internal resonance , 2007 .

[21]  K. M. Liew,et al.  Vibration of symmetrically laminated cantilever trapezoidal composite plates , 1992 .

[22]  P. Marzocca,et al.  A Review of Nonlinear Aero-Thermo-Elasticity of Functionally Graded Panels , 2011 .

[23]  H. Tiersten,et al.  Free vibrations of rectangular cantilever plates. Part 1: out-of-plane motion , 2004 .

[24]  Wei Zhang,et al.  Nonlinear oscillation of a cantilever FGM rectangular plate based on third-order plate theory and asymptotic perturbation method , 2011 .

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

[26]  鈴木 増雄.  A. H. Nayfeh and D. T. Mook: Nonlinear Oscillations, John Wiley, New York and Chichester, 1979, xiv+704ページ, 23.5×16.5cm, 10,150円. , 1980 .

[27]  Ali H. Nayfeh,et al.  Nonlinear combination resonances in cantilever composite plates , 1996 .

[28]  D. Yadav,et al.  Forced nonlinear vibration of laminated composite plates with random material properties , 2005 .

[29]  Tran Ich Thinh,et al.  Static behavior and vibration control of piezoelectric cantilever composite plates and comparison with experiments , 2010 .

[30]  H. F. Tiersten,et al.  Free vibrations of annular sector cantilever plates. , 2004 .

[31]  Zahit Mecitoğlu,et al.  Nonlinear dynamic behavior of simply supported laminated composite plates subjected to blast load , 2008 .

[32]  K. Shukla,et al.  Nonlinear dynamic response of laminated composite plates subjected to pulse loading , 2011 .