Non-Floquet oscillations of a parametrically driven rigid planar pendulum

TL;DR

This paper investigates the complex oscillatory behaviors of a damped rigid pendulum with a vibrating pivot, revealing new non-Floquet oscillations with unique spectral properties.

Contribution

It identifies a new class of nonlinear oscillations occurring where Floquet analysis indicates stability, characterized by periods longer than twice the driving period.

Findings

Discovery of non-Floquet oscillations with periods four, six, eight, or twelve times the driving period.

Spectral analysis shows the two dominant frequencies sum to the driving frequency.

Floquet analysis maps stability regions for different oscillation types.

Abstract

The linear and nonlinear motions of a damped rigid planar pendulum, driven by vibrating its pivot sinusoidally, are reexamined. The pendulum is known to exhibit periodic, quasiperiodic, and chaotic motions. Floquet analysis identifies regions of instability and stability within the driving parameter space. A new type of nonlinear oscillation may occur at driving parameters where Floquet analysis predicts a stable stationary state. Such non-Floquet oscillations always have periods longer than twice the period of the vibrating pivot. The possible periods of these oscillations may be four, six, eight, or twelve times the driving period. The power spectrum of the pendulum's angular velocity during these oscillations reveals a novel feature: the two dominant response frequencies sum to the driving frequency.