Page 241 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH05_240_08/30/05

Resonant Micromechanical Systems

240 Chapter Five
For a fixed-guided beam (as the root segment of the paddle microbridge
is), the distribution function is
(l í x) (l +2x)
2
f (x) = 1 1 (5.32)
b
l 1 3
By substituting the distribution function of Eq. (5.32) into Eq. (5.31),
the effective mass corresponding to a root segment becomes

13 13
m 1,e = 35 m = 35 ȡl w t (5.33)
1 1
1
and therefore the total mass undergoing free bending vibrations is

m II =2 13 m + m = ȡt(l w + 26 l w ) (5.34)
b,e 35 1 2 2 2 35 1 1
The bending resonant frequency can therefore be expressed as

k II Ew
II b,e t 1
Ȧ b,e = II = 70 (5.35)
m b,e l 1 ȡl (26l w +35l w )
1
1 1
2 2
The effective (equivalent) torsional mechanical moment of inertia of
one end microbar is
2 2
1 w + t 1
1
J t1,e = 3 m 1 12 = 3 J t1 (5.36)
and therefore the total torsion-related inertia fraction is

1 1
II ȡt 2 2 2l w 2 2
J t,e =2J t1,e + J t2,e = 12 l w (w + t ) + 3 (w + t ) (5.37)
2
1
2 2
The resulting torsional resonant frequency can be expressed as
k II Gw
II t,e 1
Ȧ t,e = =2 6t (5.38)
2
2
2
2
II
J t,e ȡl 2l w (w + t ) +3l w (w + t )
1
1
2
2 2
1 1
Figure 5.13 illustrates the relative errors in the bending resonant
frequency that are generated between the predictions of model I
[Eqs. (5.27)] and model II [Eqs. (5.35)] as a function of the parameters
defined in Eqs. (5.12). When the dimension parameters of the two dif-
ferent segments c and c are close to 1, the errors are larger; but they
l
w
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