Page 234 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH05_233_08/30/05

Resonant Micromechanical Systems

Resonant Micromechanical Systems 233

0.85
I- III 0.1
rω t
0.65
0.1
1
0.
c w
c l
1 1 0.01
Figure 5.6 Torsion resonant frequency ratio: model I predictions against model III
predictions (paddle microcantilever of constant thickness).

predictions by the fully compliant, full-inertia model III are larger than those
yielded by the simplified model I.

Paddle microcantilever of constant width. A study similar to that per-
formed for the paddle microcantilever of constant thickness is now
carried out for a paddle microcantilever of constant width, as the one
introduced in Chap. 3 and sketched in Fig. 3.9. Resonant frequencies
are again derived for bending and torsion according to the assumptions
corresponding to models I and II defined in Fig. 5.2.
Model I. According to model I, inertia comes entirely from the free-end
segment, whereas stiffness is furnished by only the thinner root seg-
ment. The lumped-parameter bending stiffness is therefore
3
Ewt 2
I
k b,e = 3 (5.14)
4l
2
The mass of the free-end segment is identical to the bending mass, ac-
cording to the model I assumptions:

I
m = ȡl wt (5.15)
b,e 1 1
The lumped-parameter bending resonant frequency can therefore be
expressed as

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies. All rights reserved.
Any use is subject to the Terms of Use as given at the website.