Page 129 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_128_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

128 Chapter Three
circularly filleted microcantilever. The resonant frequency of the free
axial vibration of the microcantilever shown in Fig. 3.17 is

4 2
/
E {ȡ(0.036R /l +lw /3)
1
Ȧ =
a,e
/
(l í R) w + (2R + w ) (3.62)
1
1
/ w (4R + w ) arctan 1+4R w íʌ/4
/
1
1
1
In free torsional vibrations the lumped-parameter stiffness is
expressed by means of Eq. (3.20) in terms of the axial stiffness given in
Eq. (3.60). The lumped torsional mechanical moment of inertia is
6
5
2
2
4
ȡt 0.832R +4.567R w + 4.381R (3w + t )
1 1
2
2
3
+40w l (w + t ) (3.63)
1 1
J =
t,e 2
1440l
For R ĺ 0, Eq. (3.63) reduces to Eq. (2.55), which expresses the lumped-
parameter inertia corresponding to torsional vibrations of a constant
rectangular cross-section microcantilever. Also, when l ĺ R, Eq. (3.63)
changes to Eq. (2.133), which yields the effective torsional mechanical
moment of inertia of a circularly filleted microcantilever. The torsion-
related resonant frequency of the microcantilever sketched in Fig. 3.17
is
6
5
/
G 0.832R +4.567R w 1
21.91lt
2
2
2
3
4
2
+4.381R (3w + t ) +40w l (w + t )
1
1
1
Ȧ t,e = (3.64)
/
(l í R) w + (2R + w )
1
1
/ w (4R + w ) arctan 1+4R w íʌ/4]
/
1
1
1
The lumped-parameter stiffness in bending is
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