A note on parametric resonance induced by a singular parameter modulation

TL;DR

This paper simplifies the analysis of parametric resonance in a pendulum with oscillating pivot by using non-smooth forcing functions, resulting in straightforward stability criteria expressed with elementary functions.

Contribution

It introduces a novel approach using non-smooth pivot motions to derive simple analytical stability boundaries for parametric resonance.

Findings

Stability boundaries are obtained with elementary functions.

Non-smooth forcing simplifies the Mathieu equation analysis.

The approach clarifies the resonance phenomenon compared to classical methods.

Abstract

We investigate the classical problem of motion of a mathematical pendulum with an oscillating pivot. This simple mechanical setting is frequently used as the prime example of a system exhibiting the parametric resonance phenomenon, which manifests itself by surprising stabilisation/destabilisation effects. In the classical case the pivot oscillations are described by a cosine wave, and the corresponding stability analysis requires one to investigate the behaviour of solutions to the Mathieu equation. This is not a straightforward procedure, and it does not lead to exact and simple analytical results expressed in terms of elementary functions. Consequently, the explanation of the parametric resonance phenomenon can be in this case obscured by the relatively involved technical calculations. We show that the stability analysis is much easier if one considers the pivot motion described by a non-smooth function -- a triangular or a nearly rectangular wave. The non-smooth pivot motion leads to the presence of singularities (Dirac distributions) in the corresponding Mathieu type equation, which seemingly further complicates the analysis. Fortunately, this is only a minor technical difficulty. Once the mathematical setting for the non-smooth forcing is settled down, the corresponding stability diagram is indeed straightforward to obtain, and the stability boundaries are, unlike in the classical case, given in terms of simple analytical formulae involving only elementary functions.