Biomedical Engineering Reference
In-Depth Information
Fig. 2.27
The quantized
displacement of a
double-clamped CNT
y
max
continuous
behaviour if no
electrons are
injected in the
CNT
discrete steps of the
double-clamped
oscillator
V
G
y
max
D
0:24.
ne
/
2=3
L
2=3
=.E
1=3
R
2
h
1=3
/;
(2.55)
otherwise. Thus, the CNT acquires a quantized displacement as a function of the
actuation voltage, illustrated in Fig.
2.27
,wheny is measured with respect to the
position of the clamps.
If the distance between an actuation electrode and a mechanical oscillator
represented by a cantilever, a double-clamped oscillator, or other types of
nanomechanical oscillators, is of few nanometers, the weird Casimir force appears.
A recent review about the Casimir force and its implication in NEMS is found in
(Lamoureaux 2005). So, a wealth of quantum effects manifest in NEMS, despite
their simple geometrical structure.
Taking into consideration all the above-mentioned properties, it results that
MEMS are very sensitive sensors, which change their deflection due to any
external excitation. It is necessary to introduce some additional parameters to fully
understand the cantilever-based devices. One such parameter is the fundamental
frequency of oscillation of a rectangular cantilever,
!
0
D
2 f
0
D
1:015.t=L
2
/.E=/
1=2
;
(2.56)
and the superior oscillation frequencies labeled by a positive integer n,
!
n
D
.c
n
=L/
2
.
EI
=m
B
/
1=2
;
(2.57)
where is the density of the cantilever material, m
B
D
A is the cantilever mass
per unit length with A the cantilever area, and c
n
is constant, for instance, c
1
D
1:9.
On the other hand, for a cylindrical cantilever consisting of a multiwalled CNT, the
resonance frequencies are
!
n
D
2ˇ
n
Œ.E=/.d
out
C
d
in
/
1=2
=8L
2
;
(2.58)
where n
0, d
in
and d
out
are the inner and outer diameters of the CNT, and ˇ
n
is constant, for example, ˇ
0
D
1:875 and ˇ
1
D
4:694. For single-walled CNTs, the
term .d
out
C
d
in
/ in (
2.58
) must be replaced by d
2
,whered is the diameter of
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