Page 191 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_190_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

190 Chapter Four
C (x)
t
f (x) = (4.82)
t
C
t
l/ 2
1
×
with C (x) = G ฒ (dx I ) (4.83)
t / t
x
where the torsional moment of inertia is
w(x)t 3
I = 3 (4.84)
t
The lumped-parameter torsional resonant frequency is
3.46
Ȧ =
t,e
l/2
2
2
ȡ tC tฒ f (x)w(x) w(x) + t 2 dx (4.85)
t
0
A right elliptic microbridge is used as an example to illustrate the
generic formulation that has been developed here. Structurally, the
right elliptic microbridge is identical to the right elliptic microhinge
studied in Chap. 3 and sketched in Fig. 3.22.
Example: Find the lumped-parameter bending-related resonant frequency
of an elliptic microbridge by using the single-profile microbridge model.
The bending stiffness which is associated with the midpoint of the
microbridge is
3 3
4Eb t
k =
b, e 3 2 2
3a { 2(4+ ʌ)b +4(1+ ʌ)bw + ʌw
1 1
+4(2b + w ) w (4b + w ) arctan 1+4b/w
1 1 1 1 (4.86)
í2 (2b + w ) ln(1+2b / w ) í 2b 2
1
1
/ 2(2b + w ) arctan 1+ 4b/w / w (4b + w ) íʌ /2 }
1 1 1 1
The effective mass is
m = ȡta(14.28b +9.18w ) (4.87)
b,e 1
and the bending-related resonant frequency is simply determined by com-
bining Eqs. (4.86) and (4.87).

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