Page 134 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_133_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 133
effective mass, different from that of Eq. (3.76) will be obtained. The
same limit calculations that have been performed before, namely, by
considering that a ĺ 0 and a ĺ R and b ĺ R, respectively, have been
performed on Eqs. (3.75) and (3.76). The expected equations corre-
sponding to constant rectangular cross-section and long, circularly
filleted microcantilevers have been obtained. The bending-related res-
onant frequency can be found by combining Eqs. (3.75) and (3.76), and
its equation is

5
2
2
/
E ȡ (1208.75l í 498.31la + 54.959a )a b
3
66.933l t
7
+12,672l w 1
Ȧ =
b,e
2
1 /
4(l í a) 3 / w +3a(l í a) 4(2b + w ) w (4b + w )
1
1
1
(3.77)
2
3
/
íʌ b +0.75a 14.283b + 16.566bw +3.14w 1 2
1
/
í4(2b + w ) w (4b + w ) arctan 1+4b w 1 / b 3
1
1
1
2 2
/
+6a (l í a) (2b + w ) ln(1+2b w ) í 2b / b
1
1
Example: A long, elliptically filleted microcantilever is defined by the ge-
ometry parameters l = 300 “m, w 1 = 10 “m, and t = 2 “m and material
3
properties E = 150 GPa and ȡ = 2300 kg/m . Study the bending resonant
frequency of this microcantilever in terms of the semiaxis lengths a and b.
When the lengths of the semiaxes are allowed to range within the interval
[10 “m, 300 “m], the bending resonant frequency of Eq. (3.77) varies accord-
ing to the three-dimensional plot of Fig. 3.20.
As expected, the resonant frequency increases with both a and b increas-
ing, as illustrated in Fig. 3.20, which indicates that stiffness dominates over
300000
ω [rad/s] 0.0003
200000
0.00001 b [m]
a [m]
0.00001
0.0003
Figure 3.20 Bending resonant frequency of long, elliptically filleted microcantilever in
terms of semiaxis lengths a and b.

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.