Page 214 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_213_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

Microbridges: Lumped-Parameter Modeling and Design 213
y

l1 l2 l1
α α
w1 w1

w2 fixed
x w w

x
fixed x
Figure 4.25 Geometry of a reversed doubly trapezoid microbridge.

/ 1)
/
( 57.965t l G(c í 1) ȡ
w
Ȧ = (4.155)
t,e 2 2
28(11c +5)t + 35 + c (87 + c (141 + 185c )) w
w w w w 1
In Eqs. (4.150) through (4.155), l is actually equal to l/2, as shown in
l
Fig. 4.24.
The reversed doubly trapezoid microbridge of Fig. 4.25 is similar to
the design of Fig. 4.23, having its two trapezoid end portions that are
adjoining the constant-width middle portion of reversed inclination.
The maximum width of this design is w (it is also the width of the
2
middle segment), whereas the minimum width is w .
1
The lumped-parameter bending stiffness corresponding to the mid-
point of the microbridge pictured in Fig. 4.25 is
3 2
16Et w c (c Ì 1) c (c Ì 1) +2c ln c w
l
1 w
w
w
w
k b,e = (4.156)
3
l A
1
where
2
4
2
A = (c Ì 1) c (c Ì 1) Ì 48c c (c Ì 1) Ì 192c 2
l w
w
l
w
w
w
3
Ì 48c c (3c Ì 1) +8c w c (c Ì 1) 2 (4.157)
l
w
w
l w
2
2
+6c c (c Ì 1) +12c c +12c (c +1) ln c w
w
w
l w
l w
w
The stiffness expressed by Eqs. (4.156) and (4.157) reduces to Eq. (4.7),
which yields the stiffness of a constant-cross-section microbridge of
length l, when w 2 = w 1 and l 1 = l 2 = l/3. Similarly, the lumped-parameter
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