Page 215 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_214_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

214 Chapter Four
mass which is equivalent to the distributed inertia of the microbridge
undergoing free bending vibrations is expressed by Eq. (4.143) where

2
B = c {386 + 1300c +1665c +960c 3
w l l l
4
+210c + (c +1){126 + c (c +2) 439 (4.158)
l l l l
+ c (813 + c (769 + c (351 + c (97 + c (15 + c ))))) }}
l l l l l l
Again, this equation simplifies to Eq. (4.12), expressing the effective
mass of a constant-cross-section microbridge of length l, when w = w 1
2
and l 1 = l 2 = l/3. The resonant frequency describing the free bending
vibrations is
4
6.275(2+ c ) (c í 1)t Ec w c (c í 1) +2c ln c w
l
w
w
w
l
Ȧ = × (4.159)
b,e 2 ȡAB
l
1
The torsional stiffness at the microbridge midpoint is identical to that
of the microbridge shown in Fig. 4.23.
3
4Gc (c Ì 1)w t
w
w
1
k t,e = (4.160)
3l c (c Ì 1) +2c ln c
1 l w w w
It reduces to Eq. (4.28), which expresses the torsional stiffness of a
constant-cross-section microbridge of length l, when w 2 = w 1 and
l 1 = l 2 = l/3.
The mechanical moment of inertia which is placed at the microbridge
midpoint and is dynamically equivalent to the distributed-parameter
inertia of the microbridge undergoing free torsional vibrations is
expressed again by means of Eq. (4.147) where D goes instead of D :
2
1
2
D =14{22 + 36c +15c + (c +1){10 + c (c +2) 17
l
l
l
l
l
2
2
+ c (c +7) }c }l w t +4{185 + 20c (7c +16)
l
l
l
l
w 1 1
(4.161)
+141c +2c 318 + 7c (66 + c (40 + c (10 + c ))) c w
l
w
l
l
l
l
3
+3 29 + 2c (7c +20) c + 35 + 2c (7c +22) c }l w }
2
3
l
l
l
w 1 1
l
w
When w = w and l = l = l/3, this inertia fraction simplifies to that of
2
1
1
2
Eq. (4.33) which defines a constant-cross-section microbridge of length
l. The torsional resonant frequency is therefore again:
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