Page 145 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_144_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

144 Chapter Three
When R ĺ 0, Eq. (3.107) simplifies to Eq. (2.55), which gives the effec-
tive mechanical moment of inertia for a constant rectangular cross-
section cantilever. The lumped-parameter torsional resonant frequency
is

Gw
1
ȡ 4l í ʌw +4w (2R + w )
[ 1 1 1 2
/ 2]
/ w (4R + w ) arctan 1+4R w
2
2
Ȧ =4t
t,e (3.108)
2
4
2
2
2
0.44R + l w (l +6l R +12R )(w + t )
1 1 1 1 1
2
3
2
/ 3(l +2R) 2 +1.15R w + Rw (w + t )
2
2
1
2
2 2
2
+0.43R (t +3w )
2
For long configurations, the bending stiffness is
3
2Et w 1
k b,e =
3
8l í 37.7l w (l +2R) +5.15w R 2
1
1 1 1
1
2
+ 49.7Rw w +9.42w w
1 2
1 2
(3.109)
+12w (2R + w )(l (l +2R){ 2 arctan 2R
1 1
2
1
2
2
/ w (4R + w ) + ʌ} + (4R í 4Rw í w )
2 ]
2
2
2
arctan 1+ 4R w )/ w (4R + w )
/
2
2
2
When R ĺ 0, Eq. (3.109) simplifies to Eq. (2.61), which corresponds to
a constant-cross-section cantilever of length l . The effective mass that
1
is located at the free end of the microcantilever and is dynamically
equivalent to the distributed-parameter inertia of the bending vibrat-
ing member is
2
2
3
3
4
ȡt l w (0.236l +3.3l R +19.8l R +66l R 3
1 1 1 1 1 1
4
5
2
+132R ) + l R (10.22R +144w
1 1
7
+13.95w ) + R (19.66R + 28.84w ) (3.110)
2 2
6
+l R (28.34R +64w +40.05w )
2
1
1
m =
b,e 6
(l +2R)
1
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.