Post-buckling longterm dynamics of a forced nonlinear beam: A perturbation approach

The aim of this paper is to determine by a singular perturbation approach the dynamic response of a harmonically forced system experiencing a pitchfork bifurcation. The model of an extensible beam forced by a harmonic excitation and subject to an axial static buckling is space-discretized by a Galerkin approach and studied by the Normal Form Method for different values of equation parameters influencing the nonlinear dynamic behavior like damping coefficient, load amplitude and frequency. A relevant issue in the perturbation methods is the concept of small and zero divisors which are related to the possibility to build a transformation that simplifies the original studied problem, i.e. to obtain the Normal Form, by eliminating as much as possible nonlinearities in the equations. For nonconservative systems, like structural damped systems, there are no conditions in the prior literature that define what “small” means relatively to a divisor. In the present paper some conditions about the order of magnitude of the divisors with respect to the perturbation entity are given and related to some physical parameters in the governing equations in order to estimate the relevance of some nonlinear effects.