Page 181 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_180_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

180 Chapter Four
n
(GI ) = ™ G I (4.44)
t e i ti
i =1
The mechanical moment of inertia which is equivalent to the dis-
tributed inertia of the half multimorph microbridge can be calculated
as

2
1 n w + t i 2
J t,e = 6 wl ™ ȡ t 12 (4.45)
i i
i =1
which, according to the example analyzed in Chap. 3, ignored the terms
in the individual inertias that were calculated in terms of an axis pass-
ing through the symmetry center of the compound cross section, and
which were shown to be negligibly small. For a bimorph, Eq. (4.45) re-
duces to
1 2 2 2 2
J t,e = 72 wl ȡ t (t + w ) + ȡ t (t + w ) (4.46)
2 2 2
1 1 1
When t 2 ඎ 0, Eq. (4.46) further simplifies to

1 2 2
J = m (w + t ) (4.47)
t,e 72 1 1
which is the known relationship for a single-component, homogeneous
bar.
The resonant frequency of a multimorph microbridge is found by
combining Eqs. (4.43) and (4.45) in the form:

n
(3.46/l) ™ G I
k t,e i =1 i ti
Ȧ t,e = = (4.48)
J
t,e n 2 2
w ™ ȡ t (w + t )
i i
i
i =1
For a bimorph, Eq. (4.48) changes to
3 3
G t + G t
6.93 1 1 2 2
Ȧ =
t l 2 2 2 2 (4.49)
ȡ t (t + w ) + ȡ t (t + w )
1 1 1 2 2 2
Equation (4.49) simplifies to Eq. (4.27) when t 2 ඎ 0 (one-component
microbridge).

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