Extension of Poincaré's nonlinear oscillation theory to continuum mechanics. I: Basic theory and method (Q1087359)

We extend Poincaré's nonlinear oscillation theory for discrete systems to continuum mechanics. First we investigate the existence conditions of periodic solution for linear continuum system in the states of resonance and non-resonance. By applying the results of linear theory, we prove that the main conclusion of Poincaré's nonlinear oscillation theory can be extended to continuum mechanics. Besides, in this paper a new method is suggested to calculate the periodic solution in the states of both resonance and nonresonance by means of the direct perturbation of partial differential equation and weighted integration.