Page 322 - 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 321
z

∆m
3 1
x 2 a
l
Figure 6.23 Pointlike mass deposited on a microbridge.

The first of Eqs. (6.54) is a third-degree equation which provides l , and
x
the third of Eqs. (6.54) gives the value of l y , provided ǻm and l x were
determined, namely,

2GI t
l = (6.56)
y (l –2l ) ǻmg
x
Resonant approach. A point mass which attaches to a microbridge of
length l is schematically illustrated in Fig. 6.23.
The bending resonant frequency of a microbridge was shown to be

EI y
Ȧ = 22.373 (6.57)
b,0 3
ml
where m is the total mass of the microbridge. The bending stiffness at
the microbridge midpoint is

192EI y
k = (6.58)
b
l 3
and the effective mass which is located at the same point is
128
m = m (6.59)
b
315
The bending resonant frequency in the presence of a deposited mass
ǻm is expressed in Eq. (6.20) that applied for a microcantilever
where ǻm e is the efficient deposited mass, whose significance was ex-
plained in the subsection treating mass deposition through resonance
by means of microcantilevers. For a microbridge, the efficient (or equiv-
alent) deposited mass ǻm e is located at the midpoint and is again
calculated by applying Rayleigh’s principle according to Eq. (6.21)
where

a 2 a 2
f (a) =16 2 ( 1– ) (6.60)
b l
l

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