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.
In this article, the natural vibration investigation of a functionally graded material sandwich plate with initial geometrical imperfection of the exponential function is conducted. Two face sheets of the sandwich plate are composited by functionally graded material. Their material coefficients are affected by temperature and vary in the direction of thickness following the power law. The core layer is only the metal. With the aid of Reddy’s displacement fields, a quasi-three-dimensional model is used, in which the effects of transverse stretching on natural vibration are considered. The natural frequencies and model shapes of the system are calculated on the basis of Rayleigh–Ritz method and Chebyshev polynomials. A comparison of these results with those of the existing three-dimensional theory results shows the validity and computability of present method. The influences of various parameters on the natural frequencies are researched in detail.
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