Chaotic dynamics of an elastic beam resting on a winkler-type soil

Abstract

The statical and dynamical qualitative behaviour of a slender ‘elastica’, fixed at its base and free at the top, resting on an elastic foundation, axially loaded and subjected to a vibration at the base, is analysed. It has been shown that, depending on the foundation parameters, the static bifurcation may be supercritical or subcritical. Initially, the unperturbed dynamics is analysed, showing that there are four different behaviours for the four combinations, p ≶ p cr , supercritical/ subcritical bifurcation ( p is the loading parameter, p cr its buckling value). By considering the effects of nonlinear inertia, chaotic dynamics is then examined by means of Melnikov's method. In the case of supercritical bifurcation it has been shown that chaos occurs for p > p cr due to homoclinic transverse intersections, while in the other case chaos due to double transverse heteroclinic intersection occurs for p < p cr. These types of chaos furnish qualitatively different chaotic zone diagrams in the parameter space (Φ, α f ), where Φ is the frequency of the excitation, f its amplitude and α the damping parameter, which are illustrated and discussed. Emphasis is placed on the physical interpretation of the mathematical tools employed.