Analysis of synchronization of multiple identical metronomes on a cart via describing function approach

Abstract

Based on the describing function (DF) approach, we study how to obtain analytically the synchronization of n identical metronomes on a cart in this paper. To start with, by using the fact that the torque of each metronome is a discontinuous function of its angle and angular velocity, we obtain the formulae of its DF. Next, we present a unified DF solution of the two types of synchronization for which the amplitude of the first harmonic of the angle of each metronome is identical. One type is called the out‐of‐phase synchronization for which all metronomes do not have an identical phase, and the other type is called the in‐phase synchronization for which all metronomes have an identical phase. We reveal some properties of these two types of synchronization. We validate the filter hypothesis and show that it is indeed critical for guaranteeing the success of using the DF approach to analyze the synchronization modes of the metronomes on a cart. Finally, we report in detail our numerical simulations for two and three identical metronomes using the physical parameters of the metronomes in the literature to validate the obtained analytical results and show certain robustness regarding small parameter differences of the synchronization prediction.