Page 320 - 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 319
Layer-mass detection. When mass deposits as a thin layer over a vari-
able-cross-section microcantilever, the change in the bending resonant
frequency can be used as the quantifier indicator for calculation of the
deposited mass as well as of its position along the microcantilever. Only
the static approach is detailed here, as the resonant approach is iden-
tical to the one that applies to constant-cross-section microcantilevers,
and which has been dealt with in this section.
Figure 3.39 is used again as a supporting schematic for layerlike
mass deposition. In the case where the microcantilever has a variable
cross section and after we express the bending moments and their
corresponding derivatives, Eqs. (3.196) can be written in the form:

2
l
l 1 1 p( )
p
u = C í l C + C í l l + C + l C q
1z 2 3,c 1 3,l 2 3,d 1 2 4,c p 4,l z
2
p( )
l 1 1 l p
ș = C í l C + C í l l + C + l C q
1y 2 3,r 1 3,c 2 3,l 1 2 4,r p 4,c z
(6.52)
2
1(
íl l l
1
s p
í l + l C
u 2z = 2 C 3,r + l l + 2) ( 1 s) 3,l + 1 C 3,d
C
s
3,c
2
l p l p
+ l l l + C í l l + l + C + l C q
s p( 1 ) 4,r p( 1 s ) 4,c p 4,l z
2
2
where the compliances are calculated as
l +l l +l l +l
1 p 1 p 1 p 2
xd x
x dx
=
C 3,r ฒ dx C 3,c ฒ (EI ) C 3,l ฒ (EI )
=
=
l (EI ) l y e l y e
y e
1 1 1
l +l l l
1 p 3
dx
xd x
=
C 3,d ฒ x dx C 4,r ฒ (EI ) C 4,c ฒ (EI ) (6.53)
=
=
l (EI ) l +l y e l +l y e
y e
1 1 p 1 p
l
2
x dx
=
C 4,l ฒ (EI )
l +l y e
1 p
6.4 Mass Detection by Means of
Microbridges
Mass capture can also be realized by means of microbridges. Utilization
of microbridges in mass deposition detection is motivated in cases
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