Page 330 - Mechanical design of microresonators _ modeling and applications

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Microcantilever and Microbridge Systems for Mass Detection

Microcantilever and Microbridge Systems for Mass Detection 329

a
w 2
w 1
∆m
w 1 / 2

l 2 l 1
Figure 6.30 Top view of paddle microcantilever with pointlike attached mass.

to check the divergence/convergence of results produced by the two
models. Two amounts are monitored: the resonant frequency ratio given
in Eq. (6.46) and the quantity of deposited mass which can be calculated
by means of Eq. (6.47).
The resonant frequency depends on the lumped-parameter mass of
the microcantilever and the quantity of deposited mass. While in a
previous example these amounts were determined for a fully compliant,
full-inertia model of the paddle microcantilever, the resonant frequency
ratio of the simplified model is

Ȧ b,0 ǻm s
rȦ = = 1+ (6.77)
s Ȧ ȡȦ tl
b 1 1
The resonant frequency ratio of the fully compliant, full-inertia paddle
microcantilever is

1.5a 0.5a 3 2
rȦ = 1+ 1– + f (6.78)
f l + l 2 (l + l ) 3 m
1
1
2
The plot of Fig. 6.31 shows the variation of the following ratio:
rȦ s
r = (6.79)
Ȧ
rȦ
f
The variables that appear in Fig. 6.31 are
l 2 w 2
c = l 1 c = w 1 (6.80)
w
l
and the subscripts f and s mean full and simplified, respectively, with
reference to the two models. The bending resonant frequency ratios
produced by the two models are quite close, as Fig. 6.31 shows. This
comparison can, however, be misleading, because it actually analyzes

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