Qingfei Chen
Arizona State University
Collaboration with
Liang Huang
Ying-Cheng Lai
Arizona State University
MEMS device in array structure:
Micro resonator
Micro sensor
Micro mechanical
filter
Micro generator
E. Buks and M. L. Roukes, J. Microelectromech. Syst., 11, 802 (2002).
ILM
time
M. Sato et. al., Rev. Mod. Phys. 78, 137 (2006).

The dynamical mechanism for ILMs in MEMS
was not very clear.

We propose a dynamical analysis for better
understanding of ILMs’ creation mechanism.

We find that ILMs can be excited from
inherent spatiotemporal chaos.
Inherent spatiotemporal chaos:
Intrinsic localized mode:
ILM
a
b
a
b
a
b
6
MEM cantilever arrays equation:
mi xi + bi xi + k 2i xi + k 4i xi3 + k I (2 xi − xi + 1 − xi − 1 ) = miα cos(Ω t ),
where
xi (i = 1,2,..., N ) N
is even
(mi , bi , k 2i , k 4i ) = (ma , ba , k 2 a , k 4 a )
= (5.5 × 10 − 13 kg,6.2 × 10 − 11 kg/s,0.303N/m,5 × 108 N/m 3 )
if
i
is odd
if
i
is even
(mi , bi , k 2i , k 4i ) = (mb , bb , k 2b , k 4b )
= (5.0 × 10 − 13 kg,5.7 × 10 − 11 kg/s,0.353N/m,5 × 108 N/m 3 )
mi :
mass
bi :
damping coefficient
xi (t ) = U i (t ) cos(Ω t ) + Vi (t ) sin(Ω t )
dui
1
3k
Ω
k
= −
[(Ω 2 − Ω 02i )vi − 4i vi (ui2 + vi2 ) + oi Ω ui − I (2vi − vi + 1 − vi − 1 )],
dt
2Ω
4mi
Qi
mi
dvi
1
3k
Ω
k
= −
[(Ω 2 − Ω 02i )ui − 4i ui (ui2 + vi2 ) − oi Ω vi + α − I (2ui − ui + 1 − ui − 1 )],
dt
2Ω
4mi
Qi
mi
where Ω
oi
=
k 2i / mi , Qi =
mi k 2i / bi
ui and vi are the average functions of U and V over one period.
i
i
Amplitudes and phase angles can be approximated by:
Ai =
ui2 + vi2 ,θ i = arctan(vi / ui )
Bifurcation diagram of low energy state in averaged system (N=16):
k I = 0.0236 N/ m
k I = 0.0235 N/m
In the previous literature, people
considered that
1 Initial noise
2 Frequency chirping scheme
are necessary for generating ILMs in MEM
arrays.
However, due to inherent ST chaos, these
conditions can be relaxed.
Zero initial values
(LES)
Controller
Detected
position
Drive

An averaged model to study the dynamics of
driven mircocantilever arrays.

Spatiotemporal chaos serves as a natural
exciting platform for ILMs.

Forming different patterns of ILMs in MEM
cantilever arrays.
Creating ILMs from chaos
ILMs state
Low energy
state
Spatiotemporal
chaos
20
22
23
Controller
Detected
position
Drive