  Various oscillatory phenomena occur in the world. Because some oscillations
are related to abnormal states (e.g., particular diseases), establishing
state-transition methods from an oscillatory to a resting state is important.
In this study, we construct a simple metronome model and analyze the
oscillation-quenching phenomenon of metronomes on a platform as an example of
such state transitions. Although numerous studies were conducted on the
metronome dynamics, most of them focused on the synchronization, and few
studies treated the oscillation quenching because of the difficulty in
analysis. To facilitate the analysis, we model a metronome as a linear spring
pendulum with an impulsive force (escapement mechanism) described by a
fifth-order polynomial. By performing an averaging approximation, we obtain a
phase diagram for the in-phase synchronization, anti-phase synchronization, and
oscillation quenching. We also numerically integrate the equation of motion and
confirm the agreement between the analytical and numerical results. Despite the
simplicity, our model successfully reproduces essential phenomena in
interacting mechanical clocks, such as the bistability of in-phase and
anti-phase synchrony and oscillation quenching occurring for a large mass ratio
between the oscillator and the platform. We believe that our simple model will
contribute to future analyses of other dynamics observed in metronomes.
