Page 133 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_132_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

132 Chapter Three
For a ĺ 0, Eq. (3.73) simplifies to Eq. (2.55), which is the effective tor-
sional mechanical moment of inertia of a constant-cross-section mem-
ber, whereas for a ĺ R and b ĺ R, the long, elliptically filleted
cantilever becomes a circularly filleted one, and indeed, Eq. (3.73)
changes to Eq. (3.63), as it should. The lumped-parameter torsional
resonant frequency is found by combining the corresponding stiffness
and inertia fractions, and its equation is
3
2
/
(G / ȡ) {a b 0.832b +4.56bRw 1
21.91lt
+4.381(3w + t ) +40w l (w + t )}
3
2
2
2
2
Ȧ = 1 1 1 (3.74)
t,e
/
(l í a) w + a (2b + w )
1
1
/ w (4b + w ) arctan 1+4b w íʌ 4 b
/ /
/
1
1
1
The bending stiffness for a long, elliptically filleted cantilever is found
by means of the series connection rule of Eq. (3.22) as
Et 3
k =
b,e
2
4(l í a) 3 / w +3a(l í a) 4(2b + w )
1
1
3
/ w (4b + w ) í ʌ b +0.75a 14.283b 2
/
1
1
(3.75)
2
+16.566bw +3.14w
1 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
When any of the two segments composing the elliptically filleted mi-
crocantilever of Fig. 3.19 is relatively short, and shearing effects need
be taken into account, the shearing-affected compliances have to be
used in formulating the corresponding bending stiffness. The effective
inertia fraction associated with the free bending vibrations is
2
2
5
ȡt (1208.75l í 498.31la + 54.959a )a b
7
+ 12,672l w (3.76)
1
m =
b,e 6
53,760l
Again, when shearing effects are important, the shearing-affected dis-
tribution function has to be utilized, instead of the normal one, and an
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.