Page 211 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_210_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

210 Chapter Four

2
4
2
A = (c Ì 1) c (c Ì 1) Ì 48c c (c Ì 1)
w
l w
w
w
l
3
2
Ì 192c Ì 48c c (3c Ì 1) +8c w c (c Ì 1) 2 (4.141)
w
w
l w
w
l
2
2
+6c c (c Ì 1) +12c c +12c (c +1) ln c w
l w
w
l w
w
w
In Eqs. (4.140) and (4.141), the following notations have been used:
w = c w l = c l (4.142)
2 w 1 2 l 1
When w = w and l = l = l/3, Eq. (4.140) reduces to Eq. (4.47), which
1
2
1
2
gives the bending stiffness of a constant-cross-section microbridge of
length l.
By applying the inertia derivation of the generic microbridge model,
the lumped-parameter mass which needs to be placed at the midspan
and which is dynamically equivalent to the distributed-parameter
microbridge of Fig. 4.23 is
128ȡtw l B
1 1
m b,e = (4.143)
315(2+ c ) 8
l
where
B = 386 + 126c + c 1930 + 374c + c (9(465 + 47c )
w l w l w
+ c (24(215 + 9c ) + c (c (2016 + c (672 + c (144 (4.144)
l w l l l l
+ c (18 + c )))) +42(95 + c ))))
l l w
Again, Eq. (4.143) reduces to Eq. (4.12)–giving the lumped-parameter
mass of a constant-cross-section microbridge–when w 2 = w 1 and
l 1 = l 2 = l/3.
The resonant frequency corresponding to free bending vibrations is
found by combining Eqs. (4.140) and (4.143) according to the known
equation, namely,
4
6.275(2+ c ) (c í 1)t Ec w c (c í 1) +2c ln c w
l
l
w
w
w
Ȧ = (4.145)
b,e 2 ȡAB
l
1
In torsion, the lumped-parameter stiffness corresponding to the
midpoint of the microbridge sketched in Fig. 4.22 is
4Gc (c í 1)w t 3
1
w
w
k = (4.146)
t,e 3l c (c í 1) +2c ln c
1 l w w w
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies. All rights reserved.
Any use is subject to the Terms of Use as given at the website.