Analysis of synchronization of n metronomes on a cart via describing function method: New results beyond two metronomes

This paper aims to analyze the synchronization phenomena in the system consisting of n identical metronomes hanging from an elastically fixed cart by using describing function (DF) method, which deals the driving torque as a discontinuous function of the angle and angular velocity of each metronome. This paper presents a unified DF solution of the system which shows that there exist two types of synchronization with the first harmonic of the angle of each metronome having the same amplitude. One is the out-of-phase synchronization; that is, there exist at least two of all metronomes having a nonzero phase shift. The other is the in-phase synchronization; that is, the phases of all metronomes are the same. Regarding the out-of-phase synchronization, this paper shows that the frequency and amplitude of its first harmonic of the angle of each metronome, are independent of n and the mechanical parameters related to the cart, and are exactly the same as those of the single metronome hanging from a fixed base. This paper shows that the out-of-phase synchronization is stable and presents a new result of the phase shifts of n ?? 1 metronomes with respect to a referenced metronome. The analytical results are validated by numerical simulation for three metronomes.