Page 207 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_206_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

206 Chapter Four

12.5
6
ω t / ω b
2.5
1 1
cw

c l
2
3 3
Figure 4.21 Torsion-to-bending resonant frequency ratio in terms of length and width
parameters.

9 5 4 3 2 2
128ȡt 512l w + l w (2016l +672l l +144l l
1 1 2 2 1 1 2 1 2
3
4
5 4
+18l l + l ) +252l l (w +15w )
1
1 2
1 2
2
2
6 3
7 2
+168l l (7w +25w ) +72l l (29w +35w ) (4.132)
1 2
1
2
1 2
1
2
8
+18l l (93w +35w )
2
1
1 2
m =
b,e 8
315(2l + l )
1 2
For l = l = l/3 and w = w , Eq. (4.132) simplifies to Eq. (4.12), which de-
2
1
2
1
fines the bending effective mass of a constant-cross-section microbridge
of length l. The bending resonant frequency is calculated by Eqs. (4.131)
and (4.132) and is too complex to be included here. It can be shown,
however, that when l 1 = l 2 = l/3 and w 1 = w 2 , this bending frequency
reduces to Eq. (4.13), which defines the resonant frequency of a con-
stant-cross-section microbridge of length l.
Example: Compare the torsional and bending resonant frequencies of
the paddle microbridge sketched in Fig. 4.20. Known are the thickness
t = 1 Ím, width of root segment w 1 = 10 Ím, and length of the same segment
l 1 = 100 Ím. By using the nondimensional parameters c l = l 2 /l 1 and c w = w 2 /w 1 ,
the ratio of the torsional resonant frequency [Eq. (4.122)] to the bending one
[Eq. (4.127)] is pictured in Fig. 4.21, which shows that the torsional frequency
is higher than the bending frequency for the chosen parameter ranges.
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