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Five parametric resonances in a micromechanical system

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. Outline. Overview of Mathieu Equation Why is it important ?

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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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