Nonlinear dynamics of a microelectromechanical oscillator with delayed feedback

We study the dynamics of a nonlinear electromechanical oscillator with delayed feedback. Compared to their linear counterparts, we find that the dynamics is dramatically different. The well-known Barkhausen stability criterion ceases to exist, and two modes of operation emerge: one characterized by hysteresis in combination with a bistable frequency and amplitude; the other, by self-stabilization of the oscillation frequency and amplitude. The observed features are captured by a model based on a Duffing equation with delayed force feedback. Nonlinear oscillators with delayed force feedback are exemplary for a large class of dynamic systems.

Figure 1

(a) Scanning electron microscope image of the silicon nitride beam with embedded piezoelectric actuator. (b) Schematic of the experimental setup. PD, two-segment position-sensitive photodetector; NA, network analyzer. (c) Open-loop network analyzer measurements (magnitude and phase response) of the resonator susceptibility at weak and strong driving.

Figure 2

(a) Schematic of the feedback loop. The delay time ($\tau$), feedback gain ($G$), and limiter are implemented in a digital signal processor. SA, spectrum analyzer. (b) Closed-loop measurement of the displacement noise power spectral density at the indicated feedback gain.

Figure 3

(a) Oscillation frequency for different values of $G$, for $f_0=126.2$ kHz and $f_0=127.3$ kHz. The arrow indicates the sweep direction. (b) Full measurement from which the cross sections (a) are taken. The color scale indicates $\Delta f$, as obtained from the noise spectrum. $f_0$ is swept by applying a constant voltage to the piezo actuator. (c), (d) Calculated oscillator frequency with parameters similar to those in the experiment. The dashed black line in (b) and (d) indicates the region of hysteresis.

Figure 4

Switching and transient response in a bistable oscillator. (a) Monostability for $G=102$ (top) and $G=106$ (bottom). Bistability for $G=104$ (middle). Insets: State of the oscillator and procedures to “set” and “reset” the switch. (b) Temporary amplitude (top) and frequency (bottom) during a switch from a “low” to a “high” state. (c) Relation between amplitude and frequency during the transient. Solid (red) line: quadratic fit.