Bi-state Control of Microelectromechanical Nonlinear and Parametric Resonance
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Parametric resonance provides a sharp jump in resonance amplitude at the bifurcation setpoint and is, therefore, ideal for ultrasensitive detection of mass or stress shifts that modify the resonance frequency and hence the bifurcation frequency. This work investigates a bi-state controller capable of servoing along the microelectromechanical (MEMS) parametric resonant bifurcation point at large displacement amplitudes. The control plant is a large-stroke (8 μm) parametric resonator excited by an in-plane “shaped-finger” electrostatic comb drive fabricated using a 15 μm-thick silicon-on-insulator MEMS (SOI-MEMS) process. The large-stroke parametric resonance and the ability to track the bifurcation jump point promote future implementation of ultrasensitive bifurcation-based chemical gravimetric sensors, strain gauges, and mode-matched gyroscope applications.
The controller feedback states are two levels of the electrostatic-drive voltage that shift the nonlinear resonance characteristics. The system is driven at a fixed frequency. The feedback states correspond to two steady-state conditions: the “on” state experiencing parametric resonance and the “off” state at zero amplitude. The states are cycled by 200 kHz pulse-width-modulation that circumvents the slow hysteretic latch-on that would occur in steady-state operation. The closed-loop control avoids the necessary setup and ring-down time that are required to settle between the “on” and the “off” states. The exemplary system demonstrated has 1.9 μm servo amplitude with a minimum normalized Allan deviation of 2.36×10–4. A second experimental embodiment of the bi-state bifurcation-based controller uses a crab-leg Duffing resonator to servo at the maximum “on” point (5 μm) prior to the bifurcation jump.
A quadratic capacitance-displacement response is synthesized by engineering a “shaped-finger” comb profile. The excitation of the nonlinear parametric resonance is realized by selecting an appropriate combination of the electrostatic stiffness coefficients through a specific varying-gap comb-finger design. A large frequency tuning range of 17% is demonstrated in an example crab-leg resonator.
A predictive simulation methodology for the nonlinear parametric resonance systems uses elemental behavioral models to compose the system. The mixed MEMS/analog behavioral composable modeling methodology is validated by the characterization of a canonical CMOS-MEMS non-interdigitated comb finger resonator.