Synchronization and stimulated emission in an array of mechanical phase oscillators on a resonant support

Inspired by recent laboratory observations, we propose a mechanical model for eccentrically weighted motors coupled through a vibrating plate. The equations are found to generalize those of Kuramoto to frequency- and position-dependent coupling. The behavior of the model, as determined analytically for no disorder and numerically for systems with and without quenched disorder, shows the key features observed in the laboratory, including hysteresis, bistability, and a spectral gap. The model exhibits stimulated emission and discontinuous lasinglike transitions.

Figure 1

Photograph of a motor and an illustration. The mechanical model describes an eccentrically weighted dc motor on a compliant foundation. (The foundation, a thin aluminum plate, is not shown in the photograph.) The motor has total mass $m+\mu$. It has a rotor of mass $\mu$, first moment of inertia $\lambda$, and second moment of inertia $I$. The rotor is subject to an applied torque ${\Gamma }_i$ and a viscous drag (not illustrated).

Figure 2

Representative cases for $\Omega$, the speed of the synchronized state, as a function of the natural speeds $\omega$ of the motors, according to Eq. (18). The bold lines are the solutions that we expect to be stable because the imaginary part of $\tilde{\mathbf{g}}$ is positive. The thin lines are solutions for which the imaginary part of $\tilde{\mathbf{g}}$ is negative and should be unobservable in practice. The dotted line $\Omega =\omega$ is provided for reference. For all four curves, the resonant frequency ${\omega }_r$ is 0.08 rad/s. (a) Curves for two different couplings $X$ and the same loss tangent $\eta$ of 0.06. (b) Curves for two different loss tangents $\eta$ but identical coupling $X=2$ s/rad. The star, circle, and square in (a) are related to Figs. 4 and 5 and are discussed in Sec. 4a.

Figure 3

Solution to Eq. (12) for the case in which the Green function $\mathbf{g}$ has two resonances. For the meaning of the line weights, see Fig. 2. The resonances have identical coupling strengths $X_1$ and $X_2$ of 2.0 s/rad and identical loss tangents ${\eta }_1$ and ${\eta }_2$ of 0.038, but different resonant frequencies ${\omega }_1=0.06$ rad/s and ${\omega }_2=0.09$ rad/s.

Figure 4

Order parameter $R$ and discrepancy ${\Delta }_i\equiv {\mathrm{\theta ̇}}_i-\omega$ between instantaneous motor speeds and the natural motor speed for 100 identical motors. The initial phases are distributed randomly. For these plots, the natural speed of the motors $\omega$ is 0.11 rad/s. There is a single resonance with frequency ${\omega }_r$ of 0.08 rad/s, a loss tangent $\eta$ of 0.06, and a coupling strength $X$ of 2 s/rad. The inset shows how the instantaneous speeds differ from the natural speed of 0.11 rad/s for 11 arbitrarily chosen motors. The final state corresponds to the star in Fig. 2a.

Figure 5

Order parameter $R$ and representative speeds ${\beta }_i\equiv {\mathrm{\theta ̇}}_i$ for a simulation of 100 identical motors with initial phases distributed randomly. In contrast to Fig. 4, the coupling strength $X$ of the resonance is 4 s/rad; all other parameters are identical. The final state corresponds to the open square in Fig. 2a.

Figure 6

Order parameter $R$ and representative motor speeds ${\beta }_i\equiv {\mathrm{\theta ̇}}_i$ for a disordered population of 100 motors, started from initially random phases. The resonance parameters are the same as in Fig. 4, except the coupling $X$ is 2.9 rad/s. The natural frequencies are chosen from a Gaussian distribution with mean $\mathrm{\omega ̄}$ of 0.1 rad/s and standard deviation $\sigma$ of 0.01 rad/s (i.e., 10% of $\mathrm{\omega ̄}$).

Figure 7

Order parameter $R$ and average motor speeds $\Omega$ for sweeps of mean natural motor speed $\mathrm{\omega ̄}$ from 0.03 to 0.2 rad/s (dash-dotted curves) and then back to 0.03 rad/s (dashed curves), for disordered populations. In (a) the standard deviation of the natural motor speeds ${\omega }_i$ is $\sigma =0.05\mathrm{\omega ̄}$, whereas in (b) it is $\sigma =0.1\mathrm{\omega ̄}$. These results correspond to $X_1=X_2=2.0$ s/rad, ${\omega }_{1r}=0.06$ rad/s, ${\omega }_{2r}=0.09$ rad/s, and ${\eta }_1={\eta }_2=0.038$. The solid curves indicate stable solutions for no disorder from Eq. (12) and plotted in Fig. 3. The marked locations in (a) correspond to the spectral power densities of Fig. 8.

Figure 8

Spectral power density vs frequency. The data for these figures are from the same upward run as the data in Fig. 7a and correspond to the labels in that figure. The means of the natural speeds are indicated. Note that the differences in natural speeds among (d)–(f) are small—the transition is abrupt.