Dynamics of a double pendulum with distributed mass

TL;DR

This paper explores the complex dynamics of a double pendulum with square plates, revealing how its behavior transitions from regular to chaotic and back, with chaos occurring at lower energies than in the simple double pendulum.

Contribution

It introduces a detailed analysis of a novel asymmetric double pendulum with distributed mass, including equilibrium states, normal modes, and numerical chaos analysis.

Findings

Behavior transitions from regular to chaotic and back with energy variation

Chaos onset occurs at lower energy levels than in simple double pendulum

Normal modes and equilibrium configurations are characterized

Abstract

We investigate a variation of the simple double pendulum in which the two point masses are replaced by square plates. The double square pendulum exhibits richer behavior than the simple double pendulum and provides a convenient demonstration of nonlinear dynamics and chaos. It is also an example of an asymmetric compound double pendulum, which has not been studied in detail. We obtain the equilibrium configurations and normal modes of oscillation and derive the equations of motion, which are solved numerically to produce Poincar\'{e} sections. We show how the behavior varies from regular motion at low energies, to chaos at intermediate energies, and back to regular motion at high energies. We also show that the onset of chaos occurs at a significantly lower energy than for the simple double pendulum.