An Asymptotic-Numerical Method has been developed for large amplitude free
vibrations of thin elastic plates. It is based on the perturbation method a
nd 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 formul
ation is given. The displacement and frequency are expanded into power seri
es with respect to a control parameter. The non-linear governing equation i
s then transformed into a sequence of linear problems having the same stiff
ness matrix, which can be solved by a classical FEM. Needing: one matrix in
version, 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 obtain
ed up to the radius of convergence. Taking the starting point in the zone o
f 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 incr
emental method. In order to increase the validity of the perturbed solution
, another technique, called Pade approximants, is shrewdly incorporated. Th
e solutions obtained by these two concepts coincide perfectly in a very lar
ge part of the backbone curve. Comprehensive numerical tests for non-linear
free vibrations of circular, square, rectangular and annular plates with v
arious boundary conditions are reported and discussed. (C) 1999 Academic Pr
ess.