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Five parametric resonances in a microelectromechanical system

Nature volume 396, pages 149–152 (1998)Cite this article

Abstract

The Mathieu equation1 governs the forced motion of a swing2, the stability of ships3 and columns4, Faraday surface wave patterns on water5,6, the dynamics of electrons in Penning traps7, and the behaviour of parametric amplifiers based on electronic8 or superconducting devices9. Theory predicts that parametric resonances occur near drive frequencies of 2ω0/n, where ω0 is the system's natural frequency and n is an integer ⩾1. But in macroscopic systems, only the first instability region can typically be observed, because of damping and the exponential narrowing10 of the regions with increasing n. Here we report parametrically excited torsional oscillations in a single-crystal silicon microelectromechanical system. Five instability regions can be measured, due to the low damping, stability and precise frequency control achievable in this system. The centre frequencies of the instability regions agree with theoretical predictions. We propose an application that uses parametric excitation to reduce the parasitic signal in capacitive sensing with microelectromechanical systems. Our results suggest that microelectromechanical systems can provide a unique testing ground for dynamical phenomena that are difficult to detect in macroscopic systems.

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Figure 1: Scanning electron microscope image of the torsional oscillator.
Figure 2: Comb-drive levitation of torsional oscillator.
Figure 3: Theoretical prediction and experimental measurements of the frequency ratio for the centre of the n th parametric resonance region.
Figure 4: Instability map for n = 1–4.

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Acknowledgements

We thank F. Bertsch for experimental assistance. This work was supported in part by the National Science Foundation and the Defense Advanced Research Projects Agency.

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Author notes

  1. Scott A. Miller

    Present address: Kionix, Inc., 22 Thornwood Drive, Ithaca, New York, 14850, USA

Authors and Affiliations

  1. Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, 14853-5401, New York, USA

    Kimberly L. Turner, Steven H. Strogatz & Scott G. Adams

  2. School of Applied and Engineering Physics, Cornell University, Ithaca, 14853-5401, New York, USA

    Scott A. Miller & Scott G. Adams

  3. School of Electrical Engineering and the Cornell Nanofabrication Facility, Cornell University, Ithaca, 14853-5401, New York, USA

    Peter G. Hartwell & Noel C. MacDonald

Authors
  1. Kimberly L. Turner
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Correspondence to Kimberly L. Turner.

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Turner, K., Miller, S., Hartwell, P. et al. Five parametric resonances in a microelectromechanical system. Nature 396, 149–152 (1998). https://doi.org/10.1038/24122

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