The parametrically excited upside-down rod: an elastic jointed pendulum model

A model is studied which consists of a chain of N identical pendulums coupled by damped elastic joints subject to vertical sinusoidal forcing of its base. Particular attention is paid to the stability of the upright equilibrium configuration with a view to understanding recent experimental results on the stabilization of an unstable stiff column under parametric excitation. It is shown via an appropriate scaling argument how the continuum rod model arises by taking the limit N→∞.

The effect of the inclusion of bending stiffness is first studied via asymptotics and numerics for the case N=1, showing how the static bifurcation of the pendulum varies with the four dimensionless parameters of the system; damping, bending stiffness and amplitude and frequency of excitation. For the multiple pendulum system, the bifurcation behaviour of the upright position as a function of the same four parameters is studied via numerical methods applied to the linearized equations. The damping term is found to be crucial in destroying many of the resonant instabilities that occur in the limit as N→∞. At realistic damping levels only a few instabilities remain, which are shown to be largely independent of N. These instabilities agree qualitatively with the experiments.