Page 230 - Mechanical design of microresonators _ modeling and applications

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Resonant Micromechanical Systems

Resonant Micromechanical Systems 229
Two paddle microcantilevers that were introduced in Chap. 3 are
analyzed next by comparing the free bending and torsional responses
provided by the three models. It should be mentioned that model III,
the fully compliant, full-inertia model, was derived also in Chap. 3, and
therefore only models I and II are formulated here.

Paddle microcantilever of constant thickness. The paddle microcan-
tilever whose top view is shown in Fig. 3.8 and which is defined by two
segments of constant thickness, and different widths and lengths, is
studied now.
Model I. According to model I of Fig. 5.2, the wider segment which is
located at the free end in Fig. 3.8 is considered rigid whereas the root
segment is the one providing compliance to the system. Likewise, the
inertia contribution of only the free-end segment is taken into account.
The lumped-parameter bending stiffness comes from only the root
segment and is
Ew t 3
I 2
k = (5.1)
b,e 3
4l 2
The mass of the free-end segment that corresponds to out-of-the-plane
bending is

I
m b,e = ȡl w t (5.2)
1 1
and therefore the lumped-parameter bending resonant frequency
yielded by model I becomes

I
k b,e t Ew 2
Ȧ I = = (5.3)
b,e I 2l ȡl l w
m b,e 2 1 2 1
Similarly, the torsional stiffness of the root segment is identical to
that of the whole microcantilever, and therefore
Gw t 3
2
k I = (5.4)
t,e 3l
2
The torsional mechanical moment of inertia produced by the free-end
segment is

2
2
ȡl w t(w + t )
I
1
1 1
J t,e = 12 (5.5)

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