Page 178 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_177_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

Microbridges: Lumped-Parameter Modeling and Design 177
With the distribution function of Eq. (4.31), the lumped-parameter
mechanical moment of inertia at the midspan is
8J t
J t,e = 15 (4.33)

The torsional resonant frequency results from Eqs. (4.28) and (4.33),
and its equation is

GI t
Ȧ =2.74 (4.34)
t,e lJ
t
It can be seen that the resonant frequency corresponding to the full-
length microbridge is not identical to that yielded by the half-length
model. It can be shown by using distributed-parameter modeling, and
16
also as indicated by Rao, that the exact torsional frequency is
GI t
Ȧ = S (4.35)
t lJ
t
Comparison of the exact torsional frequency to those produced by the
half- and full-length lumped-parameter models indicates that the
lumped-parameter half-length model prediction overevaluates the ex-
act value of the resonant frequency, whereas the full-length model
result underevaluates the exact value, and this is similar to the situa-
tion corresponding to bending.

Example: Compare the bending resonant frequency to the torsional one by
analyzing a rectangular constant-cross-section microbridge in terms of the
defining geometry parameters.
The following resonant frequency ratio can be formed:
Ȧ t,e
rȦ = (4.36)
Ȧ
b,e
By using the nondimensional parameters

w l
c w = t c = t (4.37)
l
the frequency ratio of Eq. (4.36) is plotted in Fig. 4.8. The torsional resonant
frequency is higher than the bending one, and the frequency ratio increases
with increasing length-to-thickness parameter c l and decreasing width-to-
thickness ratio c t .

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.