Page 218 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_217_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

Microbridges: Lumped-Parameter Modeling and Design 217
z

t1

l1 l1
Figure 4.28 Simplified paddle microbridge with linearly variable thickness over the end
segments.

3
4Gc wt 1 3
t
k = (4.166)
t,e 2
3(c + c + c )l 1
t
t
l
When c ඎ 1 (or l = l ) and c ඎ 1 (or t = t ) and l = l = l/3, Eq. (4.166)
l
2
2
1
t
1
2
1
transforms into Eq. (4.28), which corresponds to a constant-cross-sec-
tion microbridge of length l.
The effective mechanical moment of inertia corresponding to free
torsional vibrations is
m { 35 + 2c (22 + 7c ) +87c +6c (20 + 7c )c
1 l l t l l t
2
+3(47 + 4c (18 + 7c ))c + (185 + 2c (370 + 7c (70
l
l
t
l
l
3 2 2
+c (40 + c (10 + c )))))c t +14(10 + 14c +15c l (4.167)
1
l
t
l
l
l
+ (1+ c )(22 + c (4+ c )(11 + c (5+ c )))c )w }
2
l
t
l
l
l
l
J =
t,e 315(c +2) 4
l
When c l ඎ 1 (or l 2 = l 1 ) and c t ඎ 1 (or t 2 = t 1 ) and l 1 = l 2 = l/3, this equation,
too, reduces to Eq. (4.33), which stands for a constant-cross-section
microbridge. The torsion-related resonant frequency can be calculated
by Eqs. (4.166) and (4.167).
The configuration of Fig. 4.28 is a particular variant of the
microbridge shown in Fig. 4.27, which is obtained by eliminating the
middle portion. The relevant lumped-parameter resonant properties
result from those describing the microbridge of Fig. 4.28, by taking c l
to be zero (which sets l 2 to be zero).
The bending stiffness of a long member is
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