Page 199 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_198_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

198 Chapter Four

2
2
2
3
3
12 3{ (c + c )l ( l 2) 2 2 c [. (l + l 2/ 2) í l .]
1
1
2
3 2
2
1
c = í c c l + í l 1 +
1
1 2
Ewt 2 3
(4.102)
2
2
3
2
1
4
1
2 ( l 2) c [. (l + l ) 3 í (l + l 2/ 2) .] }
2
í c c (l + l ) í l + + 3
1
2
3 4
1
2 ( 1 )
c = c í l + l 2 2 c = c í 1 (4.103)
4
1
3
a b + a b a b + a b
21 1
12 2
11 2
22 1
c = c = (4.104)
2
1
a a
11 22
11 22
12 21
12 21 í a a a a í a a
The coefficients defining c 1 and c 2 in Eqs. (4.104) are
2
6 (l + l ) í l 2
ƍ 1 2 1 Ǝ
a 11 = a 22 = C + + C c (4.105)
c
Ewt 3
12l
ƍ 2 Ǝ
a = C + + C (4.106)
12 r 3 r
Ewt
3
4 (l + l ) í l 1 3
2
1
ƍ
a 21 = C + 3 + C l Ǝ (4.107)
l
Ewt
2
(
6 l (l + l /2) í (l + l ) + (l + l /2) 2 l 2 Ǝ Ǝ
1
2
2
2 1
1
2
b = + l + 2) C í C (4.108)
1 3 1 r c
Ewt
3
2/
2{2 (l + l ) í (l + l 2) 3
2
1
1
2
í3(l + l 2) (l + l ) í (l + l 2) } (4.109)
2
2/
2/
1 1 2 1 l 2 Ǝ Ǝ
b = í l + C + C l
( 1 ) c
2
Ewt 3 2
A check can be performed on this model by considering that all seg-
ments are identical with constant rectangular cross section (with w and
t as cross-sectional dimensions) and length l/3. Equations (4.101)
through (4.109) yield the stiffness corresponding to a fixed-fixed homo-
geneous beam of length l and cross section defined by w and t.
The lumped mass which is dynamically equivalent to the distributed
mass microbridge undergoing free bending vibrations, and which needs
to be placed at the midspan, is calculated by equating the kinetic energy
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