Synchronization of Huygens' clocks and the Poincare method

TL;DR

This paper analyzes the synchronization of two coupled pendulum clocks using harmonic oscillator models with van der Pol terms, deriving conditions for stable in-phase and anti-phase regimes and their dependence on system parameters.

Contribution

It provides analytic conditions and expressions for synchronization regimes in coupled pendulum clocks, extending understanding of their stability and amplitude behaviors.

Findings

Anti-phase synchronization is always stable.

In-phase synchronization can be stable, unstable, or not exist depending on parameters.

Increasing damping reduces stable amplitude and period, leading to regime destabilization.

Abstract

We study two models of connected pendulum clocks synchronizing their oscillations, a phenomenon originally observed by Huygens. The oscillation angles are assumed to be small so that the pendulums are modeled by harmonic oscillators, clock escapements are modeled by the van der Pol terms. The mass ratio of the pendulum bobs to their casings is taken as a small parameter. Analytic conditions for existence and stability of synchronization regimes, and analytic expressions for their stable amplitudes and period corrections are derived using the Poincare theorem on existence of periodic solutions in autonomous quasi-linear systems. The anti-phase regime always exists and is stable under variation of the system parameters. The in-phase regime may exist and be stable, exist and be unstable, or not exist at all depending on parameter values. As the damping in the frame connecting the clocks is increased the in-phase stable amplitude and period are decreasing until the regime first destabilizes and then disappears. The results are most complete for the traditional three degrees of freedom model, where the clock casings and the frame are consolidated into a single mass.