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Analysis of synchronization of two metronomes hanging from a plate via describing function approach

This paper concerns the synchronization phenomena in the system consisting of two metronomes hanging from a plate, by using describing function (DF) approach, which deals with the driving torque as a discontinuous function of the angle and angular velocity of each metronome. This paper presents a DF solution of the system and 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 anti-phase synchronization; that is, the phase shift of two metronomes is π. The other is the in-phase synchronization; that is, the phase shift of two metronomes is zero. This paper shows analytically the phase shifts of two types of synchronization of two metronomes rather than showing via numerical simulation in a previous result. Moreover, this paper demonstrates numerically that there are two subtypes of in-phase synchronization for some mechanical parameters of the systems. The numerical investigation indicates that the obtained frequencies and amplitudes are close to those obtained by solving numerically the nonlinear differential equations.