Analysis of synchronization of n metronomes on a hanging plate via describing function method without assumption on amplitudes of metronomes

This paper aims to analyze the synchronization phenomena in the system consisting of n identical metronomes on a hanging plate by using the describing function (DF) method, which deals with the driving torque as a discontinuous function of the angle and angular velocity of each metronome. Different from a previous study on the synchronization of the system with the first harmonic of the angle of each metronome having the same amplitude, this paper analyzes the synchronization without any assumption on the amplitudes of n metronomes. By investigating the relation between the motion of the hanging plate (common base) and n metronomes, this paper characterizes the synchronization phenomena by the following two types of synchronization: One is the inherent synchronization, whose frequency and amplitude of all synchronized metronomes are the same as those of a single metronome on a fixed base. The other is the non-inherent synchronization, which includes the in-phase synchronization with the same amplitude, and the synchronization with different amplitudes. The analytical results are validated by numerical simulation for three metronomes about the synchronization with the same or different amplitudes.