Page 231 - Mechanical design of microresonators _ modeling and applications

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Resonant Micromechanical Systems

230 Chapter Five
The torsional resonant frequency according to model I is therefore

I
k t,e Gw 2
Ȧ I = =2t (5.6)
t,e I 2 2
m ȡl l w (w + t )
t,e 1 2 1 1
Model II. As suggested in Fig. 5.2, the next step in modeling complexity
is model II, which considers that, in addition to the assumptions of the
previous (simpler) model 1, inertia is produced by the root segment, and
as a consequence, this is a full-inertia model.
For the paddle microcantilever of Fig. 3.8, the lumped-parameter
bending-related inertia is
2 2)
33
33
II
m b,e = m + 140 m = ȡt l w + 140 l w (5.7)
2 ( 1 1
1
The bending stiffness remains that given in Eq. (5.1). By combining
Eqs. (5.1) and (5.7), the bending resonant frequency becomes
Ȧ II =0.5 t Ew 2 (5.8)
b,e l 2 ȡl (l w + (33 140)l w )
/
2 1 1
2 2
In torsion, the full inertia corresponding to free torsional vibrations
is expressed as
2
2
ȡt
2
2 2
II
2
2
J t,e = J + 1 J = 12 l w (w + t ) + l w (w + t ) (5.9)
1
t1
1 1
t2
3
3
The resulting torsional resonant frequency can be determined by com-
bining Eqs. (5.4) and (5.9) as
Gw 2
Ȧ II =2t
t,e 2 2 2 2 (5.10)
/
ȡl l w (w + t ) + l w (w + t ) 3
2 2
2
2 1 1
1
Example: Compare the bending and torsional resonant frequencies provided
by the three models based on the example of a paddle microcantilever of con-
stant thickness.
The following frequency ratios are formulated:
Ȧ I b,e Ȧ I b,e
rȦ I – II = rȦ I – III = (5.11)
b II b III
Ȧ Ȧ
b,e b,e
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