Page 190 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_189_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

Microbridges: Lumped-Parameter Modeling and Design 189
48țEI + GA(l í 2x)(l +4x) (l í 2x)
y
f b sh (x) = (4.75)
2
l(48țEI + GAl )
y
3
w(x)t
where I = 12 A = w(x)t (4.76)
y
The lumped-parameter mass, which is dynamically equivalent to the
distributed inertia of the half-microbridge undergoing free bending vi-
brations, is calculated as

/
l 2
2
m = ȡt ฒ f (x)w(x) dx (4.77)
b,e b
0
When the rectangular cross section is constant, Eq. (4.77) reduces to
Eq. (4.12), which qualifies the bending inertia fraction of a constant
rectangular cross-section half-bridge. For short microbridges, f (x) is
b
sh
substituted by f b (x) in Eq. (4.77).
The lumped-parameter resonant frequency which is associated with
free bending vibrations combines the stiffness of Eq. (4.67) and mass of
Eq. (4.77) for a relatively long microbridge in the form:
1
Ȧ =
b,e l/2
rь
2
ȡt(C í C 2 / C ) f (x)w(x) dx (4.78)
l c 0 b
sh
For short microbridge configurations, C needs to be used instead of
l
C and f b sh (x) instead of f (x) in Eq. (4.78).
l
b
In torsion, the stiffness of the half-microbridge is simply
1
k t,e = (4.79)
C
t
1 l/ 2 dx
where C = ь
t G I (4.80)
0 t
The lumped mechanical moment of inertia which is equivalent to the
distributed inertia of the torsionally vibrating half-microbridge is
ȡt
2
2
×
J = 12 ь l/2 f (x)w(x) w(x) + t 2 dx (4.81)
t,e 0 t
where the distribution function is

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