An iteration method for the large amplitude flexural vibration of antisymmetric cross-ply rectangular plates

Abstract

An analytical solution to the large amplitude free-vibration problem of antisymmetric cross-ply rectangular composite plates, having an additional quadratic nonlinear term in the modal equation of equilibrium is contained herein. It is shown that the classical two-term perturbation solution and further extension of the same for five-term fail to yield any meaningful results when the coefficients of nonlinear terms in the modal equation are large. Hence, an iteration method, used to solve the Duffing's equation for isotropic plates, has been extended to solve the present modal equation of equilibrium corresponding to the antisymmetric cross-ply rectangular plates with simply supported edges. It is observed that the Duffing's assumption of considering the first Fourier coefficient as amplitude results in erroneous estimation of nonlinear free-vibration behaviour of such plates. The nonlinear frequencies obtained from second iteration of the proposed iteration method compare well with those obtained from the direct numerical integration method which is found to yield accurate results. It is seen that the presence of an additional quadratic term in the modal equation results in different amplitudes in the positive and negative half cycles of vibration.