Analysis of synchronization phenomena of two metronomes on a cart using describing function approach

This paper analyzes the synchronization phenomena in the system consisting two metronomes hanging from a cart by using describing function approach. Using the existing model of a metronome with the torque being a discontinuous function of angle and angular velocity of the pendulum, this paper derives the describing function for the discontinuous function and presents a method to obtain analytically the frequencies and amplitudes of the two types of synchronization occurred in the system. This paper studies the stability of these two types of synchronization. This paper shows via a numerical example that the obtained frequencies and amplitudes are very close to those obtained by solving numerically the nonlinear differential equations.