Synchronization of pendulum clocks and metronomes: a qualitative treatment

Abstract Pendulum clocks and metronomes have in common the fact that they are self-excited oscillators, exhibiting stationary oscillations where an external energy input compensates losses due to damping. In the phase plane these oscillations are represented by closed trajectories called limit cycles, and the motion along them can be parameterized by a phase angle. In this work we investigate the dynamics resulting from the coupling between two phase oscillators, using average equations for the resonant cases. The results from this analysis are used to qualitatively describe phase locking and synchronization phenomena in phase oscillators, explaining results observed in the coupling of pendulum clocks and metronomes.

摆钟与节拍器的共性在于二者均为自激振荡器（self-excited oscillator），可产生稳态振荡——此时外部能量输入能够补偿阻尼造成的能量损耗。在相平面中，这类振荡以被称为极限环（limit cycle）的闭合轨迹来表征，沿轨迹的运动可通过相位角实现参数化。本研究采用谐振场景下的平均方程，探究两个相位振荡器耦合所产生的动力学行为。所得分析结果被用于定性描述相位振荡器中的锁相与同步现象，以此阐释摆钟与节拍器耦合场景下观测到的相关结果。