Difference combination parametric resonance; Application to the garden hose problem

Abstract

This paper discusses combination resonance phenomena in parametric systems of two or more degrees of freedom starting from a theoretical result by Mailybayev & Seyranian. We present, to the best of our knowledge, the first example of difference combination resonance in a mechanical system. That is, where the system may exhibit significant response when forced with an external frequency that is the difference between its two internal resonant frequencies. The model system studied is a double pendulum with a follower load, a non-conservative force that would be described for example by an oscillating jet of fluid, like an idealized garden hose. For this example, after the inclussion of gravity, the difference combination frequency may be lower than the two individual resonant frequencies, a surprising effect that should be taken into account when analyzing the stability of other high-dimensional systems.