Page 142 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_141_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 141
3
4
2
2
3
ȡt(5.15l R ෹ 2.3lR +0.88R +4l w )
1
m = (3.98)
a
12l 2
The axial resonant frequency is
Ew
1
2 2
3
4
3
ȡ(5.15l R í 2.3lR +0.88R +4l w )
1
Ȧ =4.9l
a,e (3.99)
2l í 4R íʌw +4(2r + w )
1 1
/
/ 1+4r w arctan 1+4r w 1
/
1
The torsional stiffness is related to the axial stiffness of Eq. (3.97)
according to Eq. (3.20). The effective mechanical moment of inertia
which is dynamically equivalent to the distributed inertia of the hinge
shown in Fig. 3.23 is
0.001ȡt 3 2 2 3 2
J = 20l w (w + t ) í lR (4.57R
t,e 2 1 1
l
2 2 4 2
+11.5t + 18.05rw + 34.51w ) + R (0.83R
1
1
(3.100)
2 2 2 2 2
+4.38t +4.57Rw +13.14w ) + l R (26.28R
1
1
2
2
+ 25.75t + 69.03Rw + 77.26w )
1
1
The torsional resonant frequency is
Gw
1
ȡ[2l í 4R íʌw +4(2r + w )
1 1
/ 1+4r w arctan 1+4r w
/ 1
/ 1
Ȧ = 21.91lt
t,e
3
2
3
2
2
20l w (w + t ) í lR (4.57R +11.5t 2
1
1
(3.101)
2
4
+ 18.05rw + 34.51w ) + R (0.83R 2
1
1
2
2
+4.38t +4.57Rw + 13.14w )
1
1
2
2
2
+ l R (26.28R + 25.75t 2
2
+ 69.03Rw + 77.26w )
1 1
In bending, the lumped-parameter stiffness corresponding to a long
configuration is
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