
239Mechanical Nanosensors
The values for β
i
are given in Table 4.1. From Equations 4.22 and 4.21, the
natural resonant frequency and its harmonics (frequencies that are integral
multiples of the fundamental frequency) are calculated:
ω
β
ρ
i
z
i
l
EI
A
=
⎛
⎝
⎜
⎞
⎠
⎟
2
2
(4.30)
For i = 1, β
i
2
= 3.516; and for a rectangular cantilever
I
wt
z
=
3
12
(4.31)
Hence from Equation 4.30, we get
ω
ρ ρ
1
=
⎛
⎝
⎜
⎞
⎠
⎟
=
⎛
⎝
⎜
⎞
⎠
⎟
3 516
12
1 015
2
3
2
.
.
l
Ewt
tw
E t
l
(4.32)
The fundamental frequency ω
1
is a function of the material parameters
E and ρ as well as the beam dimensions t and l. High frequencies are
achieved by reducing the overall cantilever scale, by choosing st ...