Page 180 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_179_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

Microbridges: Lumped-Parameter Modeling and Design 179
half-microbridge are one-half the corresponding properties of the full
microbridge (exactly for stiffnesses and approximately for mass frac-
tions), which results in the natural frequencies predicted by the two
models being approximately equal. It can simply be shown that the
lumped-parameter stiffness for the half-microbridge (the midpoint is
guided, as indicated in Fig. 4.4) is
96(EI )
y e
k b,e = (4.38)
l 3
with the equivalent bending rigidity (EI y ) e being calculated according
to Eq. (3.148). The lumped mass for the half-microbridge is
n
13
m b,e = 70 lw ™ (ȡ A ) (4.39)
i
i
i =1
such that the bending natural frequency is
n
(EI ) / (w ™ ȡ t )
k b,e y e i =1 i i (4.40)
Ȧ = = 22.736
b,e m b,e l 2

Equation (4.40) reduces to the following for a bimorph (two-component)
microbridge:

2
2 4
2
2 4
6.563 E t + E t +2E E t t (2t +3t t +2t )
1 2 1 2
2
1 2
1
1 1
2 2
Ȧ b,e = 2 (4.41)
l (E t + E t )(ȡ t + ȡ t )
2 2
1 1
1 1
2 2
When t 2 ඎ 0, Eq. (4.41) simplifies to
E I
1 y1
Ȧ = 22.736 (4.42)
b 3
m l
1
which is the equation providing the bending resonant frequency of a
homogeneous (one-component) microbridge.
In torsion, the stiffness pertaining to the half-microbridge is
expressed as
2(GI )
t e
k t,e = l (4.43)
where the equivalent torsional rigidity is
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