Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope

Abstract

Parametric amplification, resulting from intentionally varying a parameter in a resonator at twice its resonant frequency, has been successfully employed to increase the sensitivity of many micro- and nano-scale sensors. Here, we introduce the concept of self-induced parametric amplification, which arises naturally from nonlinear elastic coupling between the degenerate vibration modes in a micromechanical disk-resonator, and is not externally applied. The device functions as a gyroscope wherein angular rotation is detected from Coriolis coupling of elastic vibration energy from a driven vibration mode into a second degenerate sensing mode. While nonlinear elasticity in silicon resonators is extremely weak, in this high quality-factor device, ppm-level nonlinear elastic effects result in an order-of-magnitude increase in the observed sensitivity to Coriolis force relative to linear theory. Perfect degeneracy of the primary and secondary vibration modes is achieved through electrostatic frequency tuning, which also enables the phase and frequency of the parametric coupling to be varied, and we show that the resulting phase and frequency dependence of the amplification follow the theory of parametric resonance. We expect that this phenomenon will be useful for both fundamental studies of dynamic systems with low dissipation and for increasing signal-to-noise ratio in practical applications such as gyroscopes.

Figure 1. Disk Resonator Gyroscope (DRG).

a) SEM of DRG and drawing showing DRG shape, with inset SEM of rings. b) Orthogonal elliptical mode shapes, with color indicating displacement. Red corresponds to maximum displacement, while blue corresponds to zero displacement.

Figure 2. Amplitude and phase response of the two axes of the DRG.

Shown before mode-matching (pale lines) and after mode-matching (dark lines and inset figures). The initial frequency mismatch of 90 Hz is reduced to <50 mHz.

Figure 3. Lumped element model of gyroscope.

Due to geometric nonlinearity, displacement of the drive axis (q_A) modulates the stiffness of the sense axis (k_B) at twice the resonant frequency, thus parametrically amplifying Coriolis force and electrostatic inputs to the sense axis. The relative phases of these signals are shown.

Figure 4. Observed response to rate.

As the drive mode's amplitude is increased, the rate sensitivity increases nonlinearly, and quality factor Q is artificially increased, both resulting from self-induced parametric amplification of the Coriolis force. Inset shows the measured frequency response at small and large amplitudes, indicating the reduced bandwidth observed at large amplitude due to the artificial increase in Q.

Figure 5. Electrostatically-probed parametric amplification.

a) Parametric gain measured at various vibration amplitudes, , versus phase shift between the sense axis excitation force and the drive mode's vibration, with theoretical fit superposed. b) Measured and theoretical parametric gain at ±90° and 0° phase shifts versus change in stiffness extracted by fitting data in Fig. 5a) to Equation (2). c) Measured parametric gain at amplitude plotted as the tuning voltage is varied from V_T, the voltage required for the degenerate (mode-matched) condition. At ΔV_T = 0 the modes are degenerate (red dashed line) and the parametric gain curve is symmetric about 0° phase shift, as shown in red in the top inset. Non-degenerate operation (blue dashed line), decreases the maximum gain and shifts the phase at which this occurs, shown in blue in the inset.

Figure 6. Theoretical parametric amplification of the Coriolis force versus frequency offset from 2ω.

The measured amplification, black circles, is calculated by dividing the measured S_Ω by the linear prediction obtained from the first three data points. The measured points are plotted at the offset frequency required to obtain the observed amplification. a) Shows perspective view, and b) shows top-down view.