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. Simple Pendulum for small Stable equilibrium

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

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

6. 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

7. Stability Regions of ME

9. Mathieu Equation, n=1 case

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

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. Equation of Motion Non-dimensionalizing

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

18. Instability map for n=1-4

19. 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

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 !