Page 132 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_131_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 131
y

fixed
b

w
w1
x
x
a

1
Figure 3.19 Elliptical corner-filleted microcantilever.

When a ĺ 0, this equation reduces to Eq. (2.45), which corr-
esponds to a constant-cross-section cantilever of length l. Similarly,
when a ĺ R and b ĺ R, Eq. (3.70) changes to Eq. (3.60), which describes
a long, circularly filleted microcantilever.
The effective mass in free axial vibrations is

3
a,e ( 0.036a b lw 1
m = ȡt 2 + 3 ) (3.71)
l
which again transforms to Eq. (2.49) for a ĺ 0 (constant rectangular
cross-section microcantilever) and to Eq. (3.61) when a ĺ R and
b ĺ R (long, circularly filleted microcantilever). The axial resonant
frequency is

3
2
/
E / ȡ(0.036a b l + lw 1/ 3)
Ȧ =
a,e
1 /
/
(l í a) w + a (2b + w ) w (4b + w ) (3.72)
1
1
1
1 / /
/
× arctan 1+4b w íʌ 4 b
Again, the torsional stiffness can be found from the axial one by means
of Eq. (3.20). The effective torsional moment of inertia is
2
2
2
3
ȡt{ a b 0.832b +4.567bw + 4.381(3w + t )
1
1
3
+40w l (w + t )} (3.73)
2
2
1
1
J =
t,e 2
1440l
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