Page 217 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_216_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

216 Chapter Four
The bending stiffness is

2 3 3 3
16E(c + c + c )(c í 1) c wt 1
l
t
t
t
t
k b,e =
3
2
2
4
2
l (c í 1)(c (c í 1) +24c (c í 1) c í 192c t 4
l
1
l
t
t
t
t
(4.163)
2
3
+4c (c í 1) (c +1)c í 48c c (3c í 1))
t
t
t
l
l t
t
2
3
+96k (c + c + c ) ln c t
t
t
t
l
with c and c being defined in Eqs. (4.133). This equation reduces to
t
l
Eq. (4.7) 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,
which proves that the microbridge of Fig. 4.27 transforms to a constant-
cross-section configuration when the above-mentioned conditions are
satisfied. In the case where the three portions compounding into the
microbridge of Fig. 4.20 are relatively short and shearing effects need
to be accounted for, the bending stiffness becomes
2
3
3
16EGc (c + c + c )(1 í c ) wt 3
t
t
t
l
t
1
sh
k b,e =
2
2
4
l G(1 í c )(c (c í 1) +24c c (c í 1) 2
t
l t
t
l
1
t
(4.164)
3
2
4
í192c +4c c (c í 1) (c +1) í 48c c (3c í 1))l 1 2
t
l t
t
l t
t
t
3 2 2 2 2
í8c (c + c + c )(12Gl + țE(c í 1) t ) ln c t
t
t
1
l
t
t
1
The effective mass in bending is
3
2
128{126 + 374c + 423c +216c +42c l 4
l
l
l
+(1+ c ) 386 + c (1544 + c (2641 + c (2519
l
l
l
l
(4.165)
+c (1417 + c (545 + c (127 + c (17 + c ))))))) c }m 1
t
l
l
l
l
l
m b,e =
315(2+ c ) 8
l
In Eq. (4.165), m is the mass of the middle constant-cross-section seg-
l
ment. Equation (4.165) transforms to the mass given in Eq. (4.12),
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, which proves the
validity of Eq. (4.165).
The bending resonant frequency can simply be found by combining
the lumped-parameter stiffness of Eq. (4.162) for long members or
Eq. (4.164) for short members with the mass of Eq. (4.165).
The lumped-parameter stiffness is
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