Page 312 - 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 311
The subscript 1 denotes the microcantilever (substrate for the
dissimilar-length sandwich microcantilever), and the subscript p indi-
cates the patch (deposited mass). The cross-sectional properties of the
two components are:

wt 1 3 wt 3 p
A = wt A = wt I = I = (6.30)
1 1 p p y1 12 yp 12
The coordinates z and z are
1
p
t 1 t p
z = 2 z = t + 2 (6.31)
1
p
1
whereas the position of the neutral axis is calculated, also previously
shown, as
z E A + z E A p
p
p
1 1
1
z N = (6.32)
E A + E A
1 1 p p
By considering that the uniformly distributed load q z (which is defined
as a force per unit length) is gravity-generated as
ǻmg
q = (6.33)
z l p
Eqs. (6.27) can be written into the following form after we perform
partial derivation and integration:

2
p /
l l (4l +3l ) (EI )e
p{ p 1 y
2
+2[2(l í l ) (2l + l )
1 1
2 2 3
í3l (l í l ) í l p/ ]( E I )] q
p 1 1 y1 } z
u 1z = 24
2
p /
l [ l p/ (EI ) +3(l í l )(l í l í l ) (E I )] q
p y e 1 1 1 y1 z (6.34)
ș =
1y 6
2
s /
l l (4l +3l í 4l ) (EI )e
p{ p 1 p y
2 2 2
+2(l í l í l )[ 4l í 2l í l l + l
1 p 1 1 p p
+l(l í 2l í 6l ) +6l l ]/ (E I )] } q
p 1 s 1 s 1 y1 z
u =
2z 24
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