Synchronization of a double pendulum with moving pivots: a study of the spectrum

TL;DR

This paper analyzes the synchronization behavior of a double pendulum system with moving pivots and spring coupling, using eigenvalue analysis to understand conditions for synchronization.

Contribution

It introduces a mathematical model for coupled double pendulums with moving pivots and applies eigenvalue analysis to study synchronization conditions.

Findings

Eigenvalues determine synchronization stability.

Analytical methods locate eigenvalues for identical pendula.

Model encompasses various physical configurations.

Abstract

The model we consider consists in a double pendulum set, where the pivot points are free to shift along a horizontal line. Moreover, the two pendula are coupled by means of a spring whose extremities connect two points of each pendulum, at a fixed distance from the corresponding pivot. The mathematical model is first written encompassing a large class of setting for the device (different sizes, different physical properties, ...). In order to carry on the problem of synchronization via analytical me\-thods, we focus on the circumstance of identical pendula: in that case, some classical theorems concerning the zeroes of polynomial equations are used in order to locate the eigenvalues governing the process, so that the possibility of synchronization of the device can be better understood.