The `Indian rope trick' for a parametrically excited flexible rod : nonlinear and subharmonic analysis

Recently Mullin has demonstrated experimentally that an upright column that is longer than its critical length for self-weight buckling can be stabilized by subjecting it to vertical harmonic excitation. This paper extends an earlier linearized analysis of a rod-mechanics model of this set up, to include three-dimensionality, and geometric nonlinearity. The stability of the upright state is then analysed using weakly nonlinear asymptotic expansions in the limit of small-amplitude excitation. First, the unforced problem is treated, extending the classical result by Greenhill to show that all bifurcations are supercritical. The main results are for the forced problem near the simplest among the potential infinity of dynamic instabilities. These correspond to pure bending modes, and to resonances between a vibration mode of the column and the first harmonic or subharmonic of the drive. The result is to produce an asymptotic description of these instabilities, including information on the stability of dynamically bifurcating states, in terms of the three dimensionless parameters of the problem (bending stiffness and the amplitude and frequency of excitation). A qualitative explanation is offered of why the earlier linearised analysis fails to quantitatively match the experiments.