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, (1998). Journal Club Presentation 10/06/05 Onur Basarir

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

Simple Pendulum for small Stable equilibrium

Inverted Pendulum Unstable equilibrium P l g m for small

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

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. Time-dependent

Stability Regions of ME

Mathieu Equation, n=1 case

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

Parametric amplifier

Nature Paper (Turner et al.)

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

Comb-Drive Levitation *Tang, JMEMS,1992 *

Torsional Simulation Results Linear approximation

Equation of Motion Non-dimensionalizing

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

Instability map for n=1-4

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

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

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

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