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 !
