Page 143 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_142_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

142 Chapter Three
3
2Et
k b,e =
2
2
/
8(l í 2R)(l í lR + R ) w í 6(3.14l 2
1
2
í2.28lR í 8.86R ) +49.7Rw +9.42w 2
1 1
(3.102)
/
+12(l í 2R)(2R + w ) ln (1+2R w )
1
1
2 2
í12(2R + w )(w +4w R í 4R
1
1
1
2
/
+4lR í 2l ) / w (4R + w ) arctan 1+ 4R w 1
1
1
The effective bending mass is
2 6 5
0.002R (209.92l í 140.67l R
4
2
3
3
2
+40.17l R +8.7l R +3.67l R 4
6
5
í4.53lR + R )
m b,e = ȡt +0.24lw 1 (3.103)
l 6
The bending-related resonant frequency is too complex to be explicitly
given, but it can readily be found by means of Eqs. (3.102) and (3.103).
3.3.4 Circularly notched microcantilevers
The circularly notched microcantilever design, which was first pre-
19
sented by Garcia et al., is formed of a right circular portion connected
in series to a constant-cross-section segment, as shown in Fig. 3.24.
The axial stiffness is
4Etw 1
k a,e =
4l íʌw +4w (2R + w )
1 1 1 2 (3.104)
/ w (4R + w ) arctan 1+4R w 2
/
2
2
Equation (3.104) reduces to Eq. (2.45) when R ĺ 0, which shows that
the compound configuration reduces to a constant rectangular cross-
section microcantilever when the circularly notched segment vanishes.
The effective axial inertia is

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.