Motion of a parametrically driven damped coplanar double pendulum

TL;DR

This paper analyzes the linear stability and non-linear dynamics of a damped, parametrically driven coplanar double pendulum, revealing conditions for stable, periodic, and chaotic motions, including stabilization of inverted states.

Contribution

It provides a comprehensive analysis of the stability and non-linear behavior of a driven double pendulum with damping, including the effects of parametric excitation on stability and chaos.

Findings

Double pendulum exhibits both periodic and chaotic motion under parametric driving.

Stable inverted states can be achieved through parametric excitation.

Chaotic and multi-period swings occur at higher driving amplitudes.

Abstract

We present the results of linear stability of a damped coplanar double pendulum and its non-linear motion, when the point of suspension is vibrated sinusoidally in the vertical direction with amplitude $a$ and frequency $\omega $. A double pendulum has two pairs of Floquet multipliers, which have been calculated for various driving parameters. We have considered the stability of a double pendulum when it is in any of its possible stationary states: (i) both pendulums are either vertically downward or upward and (ii) one pendulum is downward, and the other is upward. The damping is considered to be velocity-dependent, and the driving frequency is taken in a wide range. A double pendulum excited from its stable state shows both periodic and chaotic motion. The periodic motion about its pivot may be either oscillatory or rotational. The periodic swings of a driven double pendulum may be either harmonic or subharmonic for lower values of $a$. The limit cycles corresponding to the normal mode oscillations of a double pendulum of two equal masses are squeezed into a line in its configuration space. For unequal masses, the pendulum shows multi-period swings for smaller values of $a$ and damping, while chaotic swings or rotational motion at relatively higher values of $a$. The parametric driving may lead to stabilization of a partially or fully inverted double pendulum.