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Various oscillatory phenomena occur in the world. Because some oscillationsare related to abnormal states (e.g., particular diseases), establishingstate-transition methods from an oscillatory to a resting state is important.In this study, we construct a simple metronome model and analyze theoscillation-quenching phenomenon of metronomes on a platform as an example ofsuch state transitions. Although numerous studies were conducted on themetronome dynamics, most of them focused on the synchronization, and fewstudies treated the oscillation quenching because of the difficulty inanalysis. To facilitate the analysis, we model a metronome as a linear springpendulum with an impulsive force (escapement mechanism) described by afifth-order polynomial. By performing an averaging approximation, we obtain aphase diagram for the in-phase synchronization, anti-phase synchronization, andoscillation quenching. We also numerically integrate the equation of motion andconfirm the agreement between the analytical and numerical results. Despite thesimplicity, our model successfully reproduces essential phenomena ininteracting mechanical clocks, such as the bistability of in-phase andanti-phase synchrony and oscillation quenching occurring for a large mass ratiobetween the oscillator and the platform. We believe that our simple model willcontribute to future analyses of other dynamics observed in metronomes.

Synchronization and oscillation quenching in interacting metronomes on a movable platform: simple model and bifurcation analysis