Synchronization of mechanical phase oscillators.

Abstract

The Kuramoto model of a large number of weakly interacting phase oscillators exhibits a phase transition to synchronization. The literature has seen analytic theory and numerical simulations for both the Kuramoto model and sundry generalizations. Most real‐world systems (e.g., flashing fireflies) imperfectly match the model, and their synchronization behaviors can therefore be taken to be in merely qualitative support of the theory. Here a mechanical system with well‐understood microphysics is presented. A number N (of order 50) of eccentrically weighted DC motors (cell phone vibrators) is mounted on a plate. Each motor radiates into the plate and responds to its motion. If the plate dynamics is dominated by a single normal vibration mode the system is well described by finite‐N Kuramoto equations. Transitions to synchronization are observed in accord with theoretical expectations based on the quality factor of the plate, the number of motors, the ratio of the motor mass to the plate mass, and the natural speed of the motors. Calculations and results on a rich generalized‐Kuramoto model are also presented, in which the plate dynamics entails many normal modes of vibration, and for which the governing equations resemble those of a laser.