Steady states of the parametric rotator and pendulum

TL;DR

This paper analyzes steady-state rotation and oscillation modes of a damped, non-linear parametric rotator and pendulum with elliptic excitation trajectories, extending classical results and providing accurate perturbative solutions.

Contribution

It introduces a perturbative approach to analyze steady states of the parametric rotator and pendulum with elliptic excitation, including new modes not previously discussed.

Findings

Identification of new steady-state modes

Extension of parametric resonance to elliptic trajectories

Provision of accurate perturbative solutions

Abstract

We discuss several steady-state rotation and oscillation modes of the planar parametric rotator and pendulum with damping. We consider a general elliptic trajectory of the suspension point for both rotator and pendulum, for the latter at an arbitrary angle with gravity, with linear and circular trajectories as particular cases. We treat the damped, non-linear equation of motion of the parametric rotator and pendulum perturbatively for small parametric excitation and damping, although our perturbative approach can be extended to other regimes as well. Our treatment involves only ordinary second-order differential equations with constant coefficients, and provides numerically accurate perturbative solutions in terms of elementary functions. Some of the steady-state rotation and oscillation modes studied here have not been discussed in the previous literature. Other well-known ones, such as parametric resonance and the inverted pendulum, are extended to elliptic parametric excitation tilted with respect to gravity. The results presented here should be accessible to advanced undergraduates, and of interest to graduate students and specialists in the field of non-linear mechanics.