Page 184 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_183_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

Microbridges: Lumped-Parameter Modeling and Design 183
96E I
1 y1
k = (4.51)
b
l 3
when l ඎ 0 (no patch on the substrate).
p
The equivalent mass, which is placed at the guided end of the half-
model patched bimorph of Fig. 4.13, is calculated as

13
m b,e = 70 (m + f m ) (4.52)
p
p
1
where m 1 and m p are the masses of the substrate and patch, respec-
tively, and
6
4 2
2 4
3 3
5
2.69l í 5.38l l +2.69l l +4.85l l í 5.38ll +1.54l p 6
p
p
p
p
f = (4.53)
p
l 6
When m p ඎ 0, Eqs. (4.52) and (4.53) simplify to
13
m = m (4.54)
b 70 1
which is the known relationship for a one-component, half-model mi-
crobridge. The bending resonant frequency as provided by the half-
microbridge model is
22.736 E I (EI ) l E I + (l í l )(EI )
p
1 y1
p 1 y1
y e
y e
Ȧ =
b,e
2 2
4
2
E I l +2l (l í l )(2l í ll p (4.55)
1 y1 p
p
p
1
p
2 p / m + f m p
4
2
+l )E I (EI ) + (l Ì l ) (EI )
y e
p
1 y1
y e
Obviously, Eq. (4.55) simplifies to Eq. (4.5) when l p ඎ 0 and m p ඎ 0,
which is the known resonant frequency equation for a one-component
(homogeneous) microbridge. It can also be checked that when l p = l,
Eq. (4.55) changes to Eq. (4.42), which expresses the bending resonant
frequency of an equal-length layer bimorph.
In torsion, the stiffness of the half-model patched microbridge of
Fig. 4.13 is calculated by applying a torsional moment at the guided end
and by calculating the corresponding rotation angle at the same point
about the longitudinal direction. Under the assumption of very thin
layers for both the substrate and the patch, this stiffness is
2
k =
t,e (4.56)
p /
l p/ (GI ) + (l í l ) (G I )
1 t1
t e
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