Page 209 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_208_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

208 Chapter Four
3
3
16EGc (c +2c )wt 1 3
t
t
l
k sh =
b,e 4 3 6 3
G{c +8c 4+ c (3+ c ) c +16c }l (4.135)
l l l l t t 1
3
3
+8țEc (c +2c )l t 2
t l t 1 1
The effective mass that corresponds to the out-of-the-plane free bending
vibrations is
0.4m{512 + 6c 279 + 2c (174 + 7c (14 + 3c ))
l l l l
+c 630 + c (2520 + c (4200 + c (3780 + c (2016
l l l l l
(4.136)
+c (672 + c (144 + c (18 + c ))))))) }
l
l
l
l
m b,e =
(2+ c ) 8
l
The effective mass of Eq. (4.136) reduces to the mass of Eq. (4.12) when
c ඎ 1, c ඎ 1, and l = l/3, which proves its validity. The resonant bending
l
t
1
frequency is the square root of the ratio of the bending stiffness given
in either Eq. (4.133) or Eq. (4.134) to the effective mass of Eq. (4.136).
The torsional stiffness at the microbridge midpoint is, according to
the generic algorithm,
3 3
4c t wG
t 1
k = (4.137)
t,e 3
3l (c +2c )
t
1 l
When c l ඎ 1, c t ඎ 1, and l 1 = l/3, Eq. (4.137) simplifies to Eq. (4.28) which
provides the torsional stiffness of a constant-cross-section microbridge
of length l.
The lumped-parameter (effective) mechanical moment of inertia is,
by way of utilizing the same generic model,
2 2
0.04{ 32 + 10c (5+2c ) (t + w ) + c (30 + c (60 + c (40
1
l
l
l
l
l
+c (10 + c )))) c (c t + w )} (4.138)
2
2 2
l
t
l
t 1
J =
t,e (2+ c ) 4
l
Again, for c l ඎ 1, c t ඎ 1, and l 1 = l/3, Eq. (4.138) reduces to Eq. (4.33)
which corresponds to the effective inertia of a constant-cross-section
microbridge, and this proves the validity of Eq. (4.138).
The resonant torsional frequency is found by combining Eqs. (4.137)
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