Resonance Tongue Interaction in the Parametrically Excited Column

Abstract

This paper concerns a codimension-two analysis of the interaction between various resonances that occur in an upright flexible rod subject to sinusoidal parametric excitation. Particular attention is paid to rods that are just longer than their critical length for self-weight buckling, and their possible stabilization by the excitation. Previous work has identified three small dimensionless parameters in this problem: the closeness of the length (divided by the cube root of bending stiffness) to the critical one, the amplitude of excitation, and the reciprocal of the frequency of excitation. Multiple timescale analysis is used to show how the asymptotics of resonance tongues in the amplitude-versus-bending-stiffness plane becomes of lower order at certain special values of the frequency ratio where two resonances interact. In particular, an O(1) change in the shape of the parameter region of the stabilized supercritical rod occurs through interaction with the pure harmonic resonance of some other mode of ...