Sub-harmonic Injection Locking in Metronomes

TL;DR

This paper demonstrates sub-harmonic injection locking in mechanical metronomes, combining modeling, analysis, and experimental verification to advance understanding of nonlinear oscillator synchronization.

Contribution

It introduces the first demonstration of SHIL in metronomes, including modifications to facilitate locking and insights into nonlinear oscillator behavior.

Findings

SHIL can be achieved in metronomes with specific modifications

Phase of metronome oscillation is highly immune to certain perturbations

Experimental verification confirms theoretical predictions

Abstract

In this paper, we demonstrate sub-harmonic injection locking (SHIL) in mechanical metronomes. To do so, we first formulate metronome's physical compact model, focusing on its nonlinear terms for friction and the escapement mechanism. Then we analyze metronomes using phase-macromodel-based techniques and show that the phase of their oscillation is in fact very immune to periodic perturbation at twice its natural frequency, making SHIL difficult. Guided by the phase-macromodel-based analysis, we are able to modify the escapement mechanism of metronomes such that SHIL can happen more easily. Then we verify the occurrence of SHIL in experiments. To our knowledge, this is the first demonstration of SHIL in metronomes; As such, it provides many valuable insights into the modelling, simulation, analysis and design of nonlinear oscillators. The demonstration is also suitable to use for teaching the subject of injection locking and SHIL.