Page 221 - Mechanical design of microresonators _ modeling and applications

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Microbridges: Lumped-Parameter Modeling and Design

220 Chapter Four

1.15
2
r m b
1
1.001 ct

c1
0.5
2
Figure 4.31 Comparison between the microbridges of Figs. 4.27 and 4.29 by means of the
effective bending mass ratio.

the bending stiffness of the configuration shown in Fig. 4.27 [Eq. (4.164)] to
the bending stiffness of the configuration sketched in Fig. 4.29 [Eq. (4.175)].
As Fig. 4.30 indicates, the stiffness corresponding to the microbridge of
Fig. 4.27 can be 1.6 times larger than the stiffness describing the configura-
tion of Fig. 4.29.
The effective mass corresponding to free bending vibrations is
2 3 4
128m { 386 + 1300c +1665c +960c +210c
1 l l l l
+(1+ c ) 126 + c (2+ c )(439 + c (813
l
l
l
l
(4.177)
+c (769 + c (351 + c (97 + c (15 + c )))))) c }
l
l
l
l
t
l
m =
b,e 8
315(2+ c )
l
Another plot, shown in Fig. 4.31, which is similar to the previous ones, com-
pares the two similar microbridge configurations in terms of their effective
masses.
It can be seen that the differences between the effective masses of the two
configurations are less marked, and that the maximum ratio is only 1.15. The
resonant bending frequency, which is not given here due to its complex
form, can simply be calculated by using the lumped-parameter stiffness
[Eq. (4.175)] and mass [Eq. (4.177)].
The torsion stiffness, which is associated with the midspan of the
microbridge shown in Fig. 4.29, is identical to the torsional stiffness of
the microbridge sketched in Fig. 4.27, as expected, because the order in
which the three component portions are interconnected is irrelevant in
torsion. The same is not true with regard to the effective torsional
inertias of the two compared configurations, which are different. The
effective mechanical moment of inertia of the microbridge shown in
Fig. 4.29 is

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