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 Presentation10/06/05Onur Basarir

Outline

• Overview of Mathieu Equation

• Why is it important ?

• Nature Paper

Simple Pendulum

2

20

d g

dt l

for small

Stable equilibrium

Inverted Pendulum

Unstable equilibrium

P

l

g

m

2

20

d g

dt l

for small

There is a way to make it stable !

sinx l

mx Xsin cos 0Yl Xl

0ml Y

( ) ( )Y t mg mp t

P

l

g

m

Y(t)X(t)x

y

( ) ( )p t p nT t If Hill’s Equation

2

2

1( ) 0

d gp t

dt l l

The Mathieu Equation

( ) cosp t z

2

2cos 0

d wz w

dz

• 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

2

, 0,1,2,3,...4

nn

Stability Regions of ME

2

2

1( ) 0

d gp t

dt l l

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

0( ) ( )pk t k k t

2

2

2

1( ) ( )

2e

p

F Ck t V t

x x

20

02( ) ( )p

md x dxm k k t x F t

dt Q dt

0

4

500

10

33.57

10

1 /

l m

w m

kHz

Q

k N m

Parametric amplifier

0 0( ) cos( )F t F t

0 0( ) sin 2PV t V V t

Nature Paper (Turner et al.)

2150A m

82.75 10k Nm 19 22.12 10I kgm

3000(@18 )Q mTorr

Fabrication

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

Comb-Drive Levitation

*Tang, JMEMS,1992

2 21 1

2 2Stat Subs Rot SubsdC dCW

F V Vz dz dz

*

Torsional Simulation Results

2( , ) ( )M t V

( )

Linear approximation

12 21.216 10 NmV

1

20

10

2

5Electodes

Elec Subs

w m

l m

t m

d m

d m

Equation of Motion

( , )I c k M t

1/ 2cosDC ACV A A t 2( , ) ( ) cosDC ACM t V A A t

cos 0DC ACI c k A A t

Non-dimensionalizing t

2

2 2

1cos 0DC AC

ck A A

I I

c

aI

2

DCk Ab

I

2ACA

dI

cos 0a b d

Experiment

2

4

nb 02

n

Instabilities centered at 20 /k I

The instability frequencies match theoretical values within 0.7%.

Laser vibrometer mounted on an optical microscope is used.

Instability map for n=1-4

2DCk A

bI

2ACA

dI

Seperating the drive and sense signals

0 57kHz Given device with

02 114kHzn

Driving with

02 114kHz Parasitic signal at

Filter out high frequency left with 57kHz

The device will vibrate at 0 57kHz

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, 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.

Thank You !