Page 130 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_129_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 129
3
Et
k b,e =
2
1 /
4(l í R) 3 / w +3(l í R) 4(2R + w ) w (4b + w ) í ʌ
1
1
1
2
+0.75 14.283R + 16.566Rw +3.14w 2
1 1 (3.65)
/
í 4(2R + w ) w (4R + w ) arctan 1+4R w 1
1
1
1
/
+6(l í R) (2R + w )ln(1+2R w ) í 2R
1
1
Equation (3.65) reduces to Eq. (2.61) when R ĺ 0, which demonstrates
that the long, circularly filleted configuration of Fig. 3.17 transforms to
a constant rectangular cross-section microcantilever. Also, when
l ĺ R, the same Eq. (3.65) changes to Eq. (2.135), which defines the
bending stiffness of a circularly filleted configuration. The lumped-
parameter mass, which is equivalent to the distributed inertia of the
microcantilever of Fig. 3.17 during free bending vibrations, is
2 2 6 7
ȡt (1208.75l í 498.31lR + 54.959R )R +12.672l w 1
m b,e = 6 (3.66)
53.760l
Equation (3.66) simplifies to Eq. (2.66) when R ĺ 0, as expected,
whereas for l ĺ R the same equation simplifies to Eq. (2.136), which
gives the effective bending mass of a circularly filleted microcantilever.
The resonant frequency corresponding to the free bending vibrations is
2
2
/
E {ȡ (1208.75l –498.31lR + 54.959R )R 6
3
66.933l t
7
+12672l w 1
Ȧ =
b,e
2
4(l – R) 3 / w +3(l – R) 4(2R + w )
1
1
(3.67)
2
/ w (4b + w ) – ʌ +0.75 14.283R + 16.566Rw 1
1
1
2
/
+3.14w –4(2R + w ) w (4R + w ) arctan 1+4R w 1
1
1
1
1
/
+6(l – R) (2R + w ) ln (1+ 2R w ) –2R
1 1
Example: Compare the resonant frequencies in bending and torsion of a
long, circularly filleted microcantilever, which is defined by l = 500 “m and
t = 1 “m when Poisson’s ratio is “ = 0.25.
The following nondimensional parameters are considered:
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.