Nonlinearity-Induced Synchronization Enhancement in Micromechanical Oscillators

An autonomous oscillator synchronizes to an external harmonic force only when the forcing frequency lies within a certain interval—known as the synchronization range—around the oscillator’s natural frequency. Under ordinary conditions, the width of the synchronization range decreases when the oscillation amplitude grows, which constrains synchronized motion of micro- and nanomechanical resonators to narrow frequency and amplitude bounds. Here, we show that nonlinearity in the oscillator can be exploited to manifest a regime where the synchronization range increases with increasing oscillation amplitude. Experimental data are provided for self-sustained micromechanical oscillators operating in this regime, and analytical results show that nonlinearities are the key determinants of this effect. Our results provide a new strategy to enhance the synchronization of micromechanical oscillators by capitalizing on their intrinsic nonlinear dynamics.

Figure 1

Nonlinear MEMS device. (a) Schematic of the experimental setup. The closed-loop circuit, which amplifies and conditions the displacement signal read from the oscillator, maintains the system in self-sustained oscillation. An external synchronization signal is also fed to the oscillator. The scanning electron microscope image shows the clamped-clamped oscillator. (b) Measured frequency of the oscillator versus applied self-sustained drive voltage $V_0$ and no synchronization drive ($V_s=0$); blue curve shows the fit to Eq. (3) for $V_0\le 100\mathrm{mV}$. Inset shows a least-squares Lorentzian fit to the amplitude squared of a frequency sweep under weak excitation ($V=110\mu \mathrm{V}$), revealing $Q\gtrsim 20000$ [9].

Figure 2

Synchronization behavior. (a) Measured oscillation frequency versus synchronization frequency (${\mathrm{\Omega }}_s/2\pi$) for the oscillator, measured for $p=0.05$ and three values of the applied voltages. The corresponding pairs ($V_0$, $V_s$), measured in mV, are indicated by labels. Data over the graph’s diagonal correspond to synchronized oscillations. The arrows stand for the direction in which the synchronization frequency was changed during each run of the experiment. The (20,1) trace is shown in detail in the inset. (b) Measured synchronization range as a function of $V_0$ ($p=0.05$) for experimental data sets as shown in (a); uncertainty and repeatability in the measured synchronization ranges are smaller than the size of the data points. Red curve shows the predicted behavior from Eq. (4), where we have taken $Q=20000$. The green dotted line shows the prediction for a linear oscillator ($\beta =0$).

Figure 3

Contour plot of the synchronization range $\mathrm{\Delta }{\mathrm{\Omega }}^{\prime }$ predicted by Eq. (4) as a function of $|\beta {|}^{1/2}{f}_0$ and $Q$ for $p=0.05$. The various regions are labeled and separated by dotted lines.