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Research Papers

# On Asymptotic Analysis for Large Amplitude Nonlinear Free Vibration of Simply Supported Laminated Plates

[+] Author and Article Information
S. K. Lai, Y. Xiang

School of Engineering, University of Western Sydney, Locked Bag 1797, Penrith South DC, NSW 1797, Australia

C. W. Lim1

Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, P.R.C.bccwlim@cityu.edu.hk

W. Zhang

College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P.R. China

1

Corresponding author.

J. Vib. Acoust 131(5), 051010 (Sep 21, 2009) (8 pages) doi:10.1115/1.3142881 History: Received November 15, 2007; Revised January 28, 2009; Published September 21, 2009

## Abstract

An analytical approximation is developed for solving large amplitude nonlinear free vibration of simply supported laminated cross-ply composite thin plates. Applying Kirchhoff’s hypothesis and the nonlinear von Kármán plate theory, a one-dimensional nonlinear second-order ordinary differential equation with quadratic and cubic nonlinearities is formulated with the aid of an energy function. By imposing Newton’s method and harmonic balancing to the linearized governing equation, we establish the higher-order analytical approximations for solving the nonlinear differential equation with odd nonlinearity. Based on the nonlinear differential equation with odd and even nonlinearities, two new nonlinear differential equations with odd nonlinearity are introduced for constructing the analytical approximations to the nonlinear differential equation with general nonlinearity. The analytical approximations are mathematically formulated by combining piecewise approximate solutions from such two new nonlinear systems. The third-order analytical approximation with better accuracy is proposed here and compared with other numerical and approximate methods with respect to the exact solutions. In addition, the method presented herein is applicable to small as well as large amplitude vibrations of laminated plates. Several examples including large amplitude nonlinear free vibration of simply supported laminated cross-ply rectangular thin plates are illustrated and compared with other published results to demonstrate the applicability and effectiveness of the approach.

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## Figures

Figure 1

Geometry of simply supported cross-ply laminated thin plate with immovable edges

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