Non-linear finite element modal approach for the large amplitude free vibration of symmetric and unsymmetric composite plates

A theoretical analysis is presented for the large amplitude vibration of symmetric and unsymmetric composite plates using the non-linear finite element modal reduction method. The problem is first reduced to a set of Duffing-type modal equations using the finite element modal reduction method. The main advantage of the proposed approach is that no updating of the non-linear stiffness matrices is needed. Without loss of generality, accurate frequency ratios for the fundamental mode and the higher modes of a composite plate at various values of maximum deflection are then determined by using the Runge–Kutta numerical integration scheme. The procedure for obtaining proper initial conditions for the periodic plate motions is very time consuming. Thus, an alternative scheme (the harmonic balance method) is adopted and assessed, as it was employed to formulate the large amplitude free vibration of beams in a previous study, and the results agreed well with the elliptic solution. The numerical results that are obtained with the harmonic balance method agree reasonably well with those obtained with the Runge–Kutta method. The contribution of each linear mode to the maximum deflection of a plate can also be obtained. The frequency ratios for isotropic and composite plates at various maximum deflections are presented, and convergence of frequencies with the number of finite elements, number of linear modes, and number of harmonic terms is also studied