Abstract
An Asymptotic-Numerical Method has been developed for large amplitude free vibrations of thin elastic plates. It is based on the perturbation method and the finite element method. This method eliminates the major difficulties of the classical perturbation methods, namely the complexity of the right hand sides and the limitation of the validity of the solution obtained. The applicability of this method to non-linear vibrations of plates is clearly presented. Based on the Von Karman theory and the harmonic balance method, a cubic non-linear operational formulation has been obtained. By using the mixed stress-displacement Hellinger–Reissner principle, a quadratic formulation is given. The displacement and frequency are expanded into power series with respect to a control parameter. The non-linear governing equation is then transformed into a sequence of linear problems having the same stiffness matrix, which can be solved by a classical FEM. Needing one matrix inversion, a large number of terms of the series can be easily computed with a small computation time. The non-linear mode and frequency are then obtained up to the radius of convergence. Taking the starting point in the zone of validity, the method is reapplied in order to determine a further part of the non-linear solution. Iteration of this method leads to a powerful incremental method. In order to increase the validity of the perturbed solution, another technique, called Padé approximants, is shrewdly incorporated. The solutions obtained by these two concepts coincide perfectly in a very large part of the backbone curve. Comprehensive numerical tests for non-linear free vibrations of circular, square, rectangular and annular plates with various boundary conditions are reported and discussed.