Coupled metronomes on a moving platform with Coulomb friction

TL;DR

This study investigates the dynamics of two coupled metronomes on a moving platform, revealing that Coulomb dry friction, rather than viscous friction, governs the damping, leading to diverse synchronization behaviors analyzed through a new mathematical model.

Contribution

The paper introduces a novel model incorporating Coulomb friction for coupled metronomes on a moving platform, supported by theory, experiments, and simulations, advancing understanding of their complex dynamics.

Findings

Platform motion is damped by Coulomb friction, not viscous friction.

Various long-term behaviors including synchronization, phase locking, and metronome suppression.

Good agreement between theory, simulations, and experiments.

Abstract

Using a combination of theory, experiment, and simulation, we revisit the dynamics of two coupled metronomes on a moving platform. Our experiments show that the platform's motion is damped by a dry friction force of Coulomb type, not the viscous linear friction force that has often been assumed in the past. Prompted by this result, we develop a new mathematical model that builds on previously introduced models, but departs from them in its treatment of the friction on the platform. We analyze the model by a two-timescale analysis and derive the slow-flow equations that determine its long-term dynamics. The derivation of the slow flow is challenging, due to the stick-slip motion of the platform in some parameter regimes. Simulations of the slow flow reveal various kinds of long-term behavior including in-phase and antiphase synchronization of identical metronomes, phase locking and phase drift of non-identical metronomes, and metronome suppression and death. In these latter two states, one or both of the metronomes come to swing at such low amplitude that they no longer engage their escapement mechanisms. We find good agreement between our theory, simulations, and experiments, but stress that our exploration is far from exhaustive. Indeed, much still remains to be learned about the dynamics of coupled metronomes, despite their simplicity and familiarity.