Synchronization and avalanching amongst coupled phase oscillators, mechanical lasing

We present theory and numerical simulations for two nonlinear systems consisting of a large number of eccentrically-weighted DC motors on mechanical bases. These are generalizations of the Kuramoto model for synchronizing phase oscillators, one chief difference being that the coupling is now frequency dependent. In one system the base is resonant and has a single degree of freedom. This system exhibits the expected second order phase transformation: for sufficient coupling strength the motors synchronize with a power output that grows with the distance above criticality. In the synchronized state the base oscillates at a single frequency (below its nominal resonance) with an amplitude that rises superluminescently with the number of motors. The degree of synchronization fluctuates intermittently, with statistics similar to those of universal crackling noise and avalanches. In our other system the motors are placed densely on a tensioned membrane, with sound speed such that wavelength is large compared to motor size and spacing. As such the structure is an active nonlinear metamaterial. As a function of coupling strength we observe a lasing transition from a near-quiescent state to a state in which the membrane is dominated by a single wavelength, and acoustic power emission is high.