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Five parametric resonances in a micromechanical system

This presentation delves into the significance of the Mathieu Equation in micromechanical systems, particularly highlighting its application in analyzing parametric resonances. We discuss the theoretical foundations and experimental results using laser vibrometry for instability analysis. Notably, the findings align closely with theoretical calculations (within 0.7%), showcasing how the resonance effects can be leveraged to enhance sensitivity in MEMS devices. Furthermore, we explore strategies to reduce parasitic signals in capacitive sensing while operating within the first instability region.

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Five parametric resonances in a micromechanical system

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  1. Five parametric resonances in a micromechanical system Turner K. L., Miller S. A., Hartwell P. G., MacDonald N. C., Strogatz S. H., Adams S. G., Nature, 396, 149-152 (1998). Journal Club Presentation 10/06/05 Onur Basarir

  2. Outline • Overview of Mathieu Equation • Why is it important ? • Nature Paper

  3. for small Stable equilibrium Simple Pendulum

  4. m for small g l P Unstable equilibrium Inverted Pendulum

  5. y m g l P x X(t) Y(t) Hill’s Equation If There is a way to make it stable !

  6. Time-dependent The Mathieu Equation • Can not be solved analytically. • Solutions found using Floquet Theorem. • In solid state it is known as Bloch Theorem. • ME is Schrödinger eq. of an electron in a spatially periodic potential.

  7. Stability Regions of ME

  8. Stability Regions of ME

  9. Mathieu Equation, n=1 case

  10. x * Rugar D., Grütter P., PRL, 67, 699 (1991). What is the importance? • It can be used as a parametric amplifier.

  11. Parametric amplifier

  12. Nature Paper (Turner et al.)

  13. Fabrication * Cleland A.N., Foundations of Nanomechanics, Springer, 2003.

  14. Comb-Drive Levitation * *Tang, JMEMS,1992

  15. Torsional Simulation Results Linear approximation

  16. Non-dimensionalizing Equation of Motion

  17. Experiment Laser vibrometer mounted on an optical microscope is used. Instabilities centered at The instability frequencies match theoretical values within 0.7%.

  18. Instability map for n=1-4

  19. Given device with Driving with Parasitic signal at The device will vibrate at Seperating the drive and sense signals Filter out high frequency  left with 57kHz

  20. Conclusion • 4 Instability resonances • To reduce parasitic signals in capacitive sensing MEMS. • To increase sensitivity when operated in the first instability region.

  21. References • Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, 1981. • Stoker, J.J., Nonlinear Vibrations in Mechanical and electrical Systems, Interscience,1950. • Rand, R., Nonlinear Vibrations. • Cleland A.N., Foundations of Nanomechanics, Springer, 2003. • Rugar D., Grütter P., PRL, 67, 699 (1991). • Tang. W.C.,et al.,JMEMS,170-178,1992.

  22. Thank You !

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