Page 158 - Mechanical design of microresonators _ modeling and applications

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Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 157

30
0.00015
r ω
10
0.0008 w [m]

1 [m]
0.00005
0.0015
Figure 3.34 Torsional-to-bending resonant frequency ratio in terms of layer length and
width.

The equivalent bending rigidity (EI ) is expressed in Eq. (3.148). The
y e
shear-related rigidity (GA) e can be found in a similar manner to the
axial and torsional rigidities, namely,
n
(GA) = ™ G A (3.157)
e i i
i =1
The equivalent mass is similar to the one expressed in Eq. (2.80) for a
homogeneous microcantilever, namely,

2 4
2
2
2
3m 140ț (EI ) +77ț(EI ) (GA) l +11(GA) l
e
e
y e
y e
m b,e = (3.158)
2
140 (GA) l +3ț(EI ) 2
e y e
The first bending-related resonant frequency of a short multimorph is
therefore
2
y e /
11.832 (EI ) (GA) (GA) l +3ț(EI ) (lm)
e
e
y e
Ȧ = (3.159)
b,e 2 2 2 4
2
140ț (EI ) +77ț(EI ) (GA) l +11(GA) l
y e y e e e
The resonant frequency of a bimorph can be obtained from Eq. (3.159)
by employing only two components in the bending and shearing rigidi-
ties, but the final equation is too complicated and is not provided here.
3.5.2 Microcantilevers of dissimilar-length
layers
Bimorphs can be microfabricated for use in transduction by attaching
a patch over a substrate such that the lengths of the two microcompo-
nents are not equal, although their widths w are equal. Figure 3.35 is
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