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

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.