Page 137 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_136_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

136 Chapter Three
3
Et
k b,e =
2
2.572R + 24.849Rw +4.713w 2
1
1
2
2
+6(2R + w )(4R í 4Rw í w ) (3.83)
1
1
1
/ w (4R + w ) arctan 1+4R w 1
/
1
1
The effective mass associated with the free bending vibrations is
m b,e = ȡRt(0.308R +0.47w ) (3.84)
1
The resulting bending-related resonant frequency is
E
t
ȡR(0.308R + 0.471w )
1
Ȧ =
b,e
2
2.572R + 24.849Rw + 4.713w 2
1
1
(3.85)
2
2
+6(2R + w )(4R í 4Rw í w )
1
1
1
/ w (4R + w ) arctan 1+4R w 1
/
1
1
For relatively short configurations, the Timoshenko model needs to be
utilized, and the resulting bending stiffness is
3
sh 2EGt
k b,e =
2
2
3G(1.717R í 16.567Rw +3.142w ) íʌțEt 2
1
1
2
2
2
+4(2R + w ) țEt +3G(4R í 4Rw í w ) (3.86)
1
1
1
/ w (4R + w ) arctan 1+4R w 1
/
1
1
The effective mass generated by using the Timoshenko model with the
distribution function of Eq. (2.77) is
2 4
2
ȡRt ț E t (0.352R +0.667w )
1
2 2
+ țEGR t (10.46R +17.6w )
1
(3.87)
4
2
+ G R (78.941R +120.69w )
sh 1
m b,e =
2 2
2
(16GR + țEt )
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