Resonance tongue interaction in the parametrically excited column

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 time-scale 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 stabilised supercritical rod occurs through interaction with the pure harmonic resonance of some other mode of vibration of the rod. It is also shown how to include material damping within the analysis. The results help explain why earlier theories failed to qualitatively explain experimental observation, and are also likely to be of relevance in other 3-parameter parametric resonance problems for continuous structures.