A new look on the stabilization of inverted pendulum with parametric excitation and large random frequencies: analytical and numerical approaches

TL;DR

This paper investigates the stabilization of an inverted pendulum using parametric excitation with multiple frequencies, demonstrating that high-frequency excitation can stabilize the pendulum under realistic gravity conditions through analytical and numerical methods.

Contribution

It introduces a novel analysis of inverted pendulum stabilization with superimposed multiple frequencies and provides a critical amplitude criterion validated by numerical simulations.

Findings

High-frequency parametric excitation stabilizes the inverted pendulum.

Critical amplitude depends on frequency distribution and system parameters.

Numerical results confirm analytical predictions across various configurations.

Abstract

In this paper we explore the stability of an inverted pendulum with generalized parametric excitation described by a superposition of $N$ sines with different frequencies and phases. We show that when the amplitude is scaled with the frequency we obtain the stabilization of the real inverted pendulum, i.e., with values of $g$ according to planet Earth ($g\approx 9.8$m/s$^{2}$) for high frequencies. By randomly sorting the frequencies, we obtain a critical amplitude in light of perturbative theory in classical mechanics which is numerically tested by exploring its validity regime in many alternatives. We also analyse the effects when different values of $N$ as well as the pendulum size $l$ are taken into account.