Coupling Finite Element Method and Perturbation Techniques for Non Linear Vibrations of Plates

The effectiveness of the coupling of the perturbation techniques and the finite element method has been demonstrated using a method called Asymptotic-Numerical Method (ANM). This concept 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. In this paper we present the development of this method and its applicability for large amplitudes free vibrations of plates. The displacement and the frequency are expanded into power series with respect to a control parameter. The nonlinear governing equation is transformed into a sequence of linear problems having the same stiffness matrix. Needing one matrix inversion, a large number of terms can be computed with a small computation time. Taking the starting point in the zone of validity, the method is reapplied in order to determine a further part of the nonlinear solution. In order to increase the zone of validity, the Pade approximants are incorporated. Iterations of this method lead to a powerful incremental method. Numerical tests for large amplitudes free vibrations of plates with various shapes and boundary conditions are reported. Recent improvements in the basic ANM algorithm as well as applications to various structural problems are added in order to exhibit the effectiveness and the applicability of this method.