Suppressing Frequency Fluctuations of Self-Sustained Vibrations in Underdamped Nonlinear Resonators

We study frequency fluctuations in self-sustained oscillators based on nonlinear underdamped resonators. Important types of such resonators are nano- and microelectromechanical systems. Various noise sources are considered, with the emphasis on the fundamentally unavoidable noise that comes along with dissipation from the coupling to a thermal reservoir. The formulation in terms of the action-angle variables of the resonator allows us to analyze a deeply nonlinear regime. In this regime the vibration frequency as a function of the action can have an extremum. We show that frequency fluctuations can be strongly reduced by choosing the operation point at this extremum. We suggest a practical implementation of a nanoresonator that has the appropriate property and show explicit results for the corresponding model.

Figure 1

Sketch of the types of noise that lead to frequency fluctuations and the ways of suppressing them.

Figure 2

Frequency dispersion $\omega (I)$ for the biased Duffing resonator with potential energy (26) as a function of the scaled action $\tilde{I}=\gamma I/M{\omega }_e^3$ for several values of the scaled bias parameter $\lambda$, Eq. (27). From top to bottom, $\lambda =2.5$ (brown), 2 (purple), 1.5 (orange), 1 (green), 0.5 (yellow), and 0 (blue). The black dashed line is the locus of zero dispersion points, $d\omega /dI=0$.

Figure 3

Scaled fractional frequency fluctuations ${\tilde{S}}_y$ versus the scaled action $\tilde{I}$ calculated at the operating point ${\tilde{I}}_{\mathrm{st}}=I\gamma /M{\omega }_e^3$ given by Eq. (22) for the biased Duffing oscillator; $\mathrm{\Gamma }/{\omega }_e=0.01$. The curve with a single minimum at relatively small ${\tilde{I}}_{\mathrm{st}}$ corresponds to $\lambda =0$, whereas the curves that display the increasingly smaller minimal ${\tilde{S}}_y$ correspond to $\lambda =0.5,1.0,1.5,2.0,2.5$. The color coding is the same as in Fig. 2.