Page 197 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_196_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

196 Chapter Four

6 4 3 2 2
ȡt 2R (2.56l +14.14l R + 29.71l R
2 2 2
3 4 9
+28.14l R +10.13R ) +0.4(l +2R) w (4.97)
2 2 2
m =
b,e 8
(l +2R)
2
In torsion, the stiffness associated to the midpoint of the microbridge is
4Gt 3
k =
t,e
3 l w +2(2R + w ) arctan 1+ 4R / w
2 2 2 2 (4.98)
/ w (4R + w ) íʌ /2
2 2
The torsional mechanical moment of inertia is
ȡt{ (23.3l + 80.86l R + 70.58R )R 6
2
2
2 2
2
2
+8 3(5.33l +17.59l R +14.66R )R 5
2 2
5 2 2
+7(l +2R) t w +42(8.76l + 26.51l R
2 2 2 2
(4.99)
2
5 3
4 2
+20.38R )R w +56(l +2R) w
2 2 2
4 2
+14 8.76l + (26.51l +20.38R)R R t }
2
2 2
J =
t,e 4
1260(l +2R)
2
The torsional resonant frequency equation, which is too long and is not ex-
plicitly given here, can be found by combining Eqs. (4.98) and (4.99).
Three-segment microbridges. We now consider microbridge formed of
three segments of which two that are placed at the fixed ends are iden-
tical and mirrored with respect to the midpoint of the microbridge.
Their cross section is variable whereas the middle portion of the
microbridge which connects the two end segments is of constant cross
section, as sketched in Fig. 4.18. The bending and torsional resonant
frequencies are now derived by first calculating the lumped-parameter
stiffness and inertia fractions corresponding to bending and torsional
free vibrations.
Bending resonant frequency. The bending stiffness of the microbridge is
calculated as
F Cz
k b,e = u Cz (4.100)

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