On the motion of a rigid block, tethered at one corner, under harmonic forcing

A rigid block, tethered at one corner, is subjected to harmonic forcing. The motion is shown to be equivalent to that of the inverted pendulum impacting one of a pair of asymmetrically placed side-walls. The dynamics of the problem contain subharmonic responses, multiple solutions, period-doubling bifurcations, etc. Stability boundaries are given for a wide range of parameters and orbits are shown to be possible for a large range of forcing amplitudes. Some orbits are possible at forcing amplitudes larger than those in the untethered case. A period- and impact-doubling sequence is shown explicitly for the first time, using digital simulations. Evidence is offered for the existence of more than one type of multiple-impact solution. Large amplitude subharmonic orbits are found.