Analysis of Synchronization of Mechanical Metronomes

Abstract

Synchronization phenomena are ubiquitous around us and are observed in various real systems, for example, hands clapping (rhythmic applause) at the concert hall, light emission of fireflies, callings of frogs, circadian rhythms, pendulum clocks, mechanical metronomes placed on a plate, pedestrians on a suspension bridge, water flowing out of plastic bottles connected by hoses, candle flames fluctuation. In this paper, we focused on the synchronization phenomena observed in a mechanical system: mechanical metronomes on a plate. In particular, we discussed how to construct a mathematical model, or the equations of motion, which describe dynamical behavior of synchronization of mechanical metronomes put on a plate hung by strings. We also investigated their dynamical behavior by solving the equations of motions numerically. In the numerical experiments, parameter values of the equations of motion are experimentally obtained from the experimental equipment. We found that synchronization behavior of mechanical metronomes depends on the following two factors: relation between the frequencies of the metronomes and the plate, and initial angles of the metronomes. We also found the individual difference of the metronomes strongly affects the final behavior. In addition, the results also indicate that if the number of mechanical metronomes increases, it becomes extremely harder to observe the in-phase synchronization until energy applied to the metronomes through spiral springs is exhausted.