Online proceedings - EDA Publishing Association

11-13
May 2011, Aix-en-Provence, France
Amplitude Enhancement Using Vibration Mode
Localization with A Single Micro-mechanically
Coupled Beam-shaped Resonator Array
Keisuke Chatani 1 , Dong F. Wang 1 , Tsuyoshi Ikehara 2 , and Ryutaro Maeda 2
1
Micro Engineering & Micro Systems Laboratory, Ibaraki University (College of Eng.), Hitachi, Ibaraki 316-8511, Japan
(Tel: +81-294-38-5024; Fax: +81-294-38-5047; E-mail: dfwang@mx.ibaraki.ac.jp)
2
Ubiquitous MEMS and Micro Engineering Research Center (UMEMSME), AIST, Tsukuba, Ibaraki 305-8564, Japan
Abstract- The use of vibration mode localization in arrays of
micro-mechanically coupled, nearly identical beam-shaped resonators
has been studied for ultrasensitive mass detection and analyte
identification. Eigenstate shifts that are 3 to 4 times (compared to
single resonator), and orders (compared to resonator array) of
magnitude greater than corresponding shifts in resonant frequency for
an induced mass perturbation are theoretically analyzed, from the view
points of geometrical design of the coupling overhang, cantilever
length, as well as number of the identical coupled cantilevers.
Furthermore, the shifts in eigenstates are unique to the resonator to
which the stiffness or mass perturbation is induced, therefore
providing a characteristic “fingerprint” that identifies the particular
resonator where the stiffness or mass perturbation is induced.
Keywords- Vibration mode localization, Eigenstate shifts,
Amplitude enhancement, Ultrasensitive mass detection, Analyte
identification, Coupled resonator array, Coupling overhang
I. VIBRATION MODE LOCALIZATION
In resonant frequency based sensors the output corresponds
to a shift in the resonant frequency of a vibrating
micromechanical structure when subjected to small
perturbations in either its stiffness or mass. The most sensitive
micro cantilever based mass detection experiments using the
frequency-shift approach have reported attogram level
detection in ultrahigh vacuum environment [1-3] and
femtogram level detection under ambient conditions [4-5].
In contrast, the concept of using Anderson or vibration
mode localization [6-13] in any array of nearly identical
coupled resonators has also been proposed as a eigenstate-shift
based sensing mechanism in recent years in coupled micro
cantilevers under ambient conditions [6, 14-15].
Some advantages of mode localized sensing can be listed
below. Firstly, times or orders of magnitude in parametric
sensitivity of micromechanical mass detection compared to the
conventional frequency-shift approach can be obtained.
Secondly, such sensors can offer the important advantages to
intrinsic common mode rejection that renders it less susceptible
to false-positive readings that frequency-shift based sensors.
Thirdly, both the ultra sensitive detection and analyte
identification of small perturbation can be achieved at same
time with a single coupled resonator array.
While many studies of mode localization in coupled
structures and arrays of coupled resonators have been
performed, the question of whether this phenomenon can be
used in a sensing capacity has not been examined
systematically.
This work however, first theoretically studies the effects of
geometrical design of the coupling overhang, cantilever length,
as well as number of the identical coupled cantilevers on the
magnitude enhancement by means of hypothesizing a small
mass perturbation, which binds to cantilever surface due to
molecule specific interactions. A preliminary evaluation has
been then carried out by using microfabricated coupled
beam-shaped resonator arrays.
II.
PHYSICS OF THE AMPLITUDE ENHANCEMENT
A. Vibration localization in coupled two-resonator
array
A schematic and a discretized model of two identical
beam-shaped cantilevers coupled by an overhang are shown in
Fig. 1(a) and 1(b), respectively. Each cantilever is modeled as a
damped simple harmonic oscillator, while the effect of the
overhang coupling is modeled as spring connecting the two
oscillators.
Considering first the case of two initially identical
cantilevers, the eigenvalue governing the undamped free
oscillations of the system can be written as follows [6]:
⎡
⎢
⎣
−
+
/1/
1
− KK
1 ⎤ cKK c
= λuu
+ KK + δ )1
⎥
(1)
2
2 ⎦ cKK c /
where K 1 (=K), M 1 (=M) and K 2 (=K), M 2 (=M) are, respectively
the bending stiffness and suspended mass of the two cantilevers,
while δ represents the ratio of the effect mass ( Δ M) being
detected to the single cantilever mass (M) . Kc is the stiffness of
the overhang coupling the two cantilevers.
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