Page 144 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_143_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 143
y

R

2 1
w1
x
w2

1 2 = 2R 11

Figure 3.24 Circularly notched microcantilever.

2
2
4
3
ȡt(16.88R +4l w +24l Rw +48l R w 1
1
1
1
1
1
3
+32R w ) (3.105)
2
m a,e =
12(l +2R) 2
1
This equation, too, reduces to that corresponding to a constant-cross-
section cantilever of length l [Eq. (2.49)] for R ĺ 0. The axial resonant
1
frequency is
Ew
1
[
ȡ 4l íʌw +4w (2R + w )
1
2
1
1
/ 2]
/ w (4R + w ) arctan 1+4R w (3.106)
2
2
Ȧ =6.93(l +2R)
a,e 1 4 3 2
16.88R +4l w +24l Rw
1 1 1 1
2 3
+48l R w +32R w
1 1 2
The torsional stiffness is related to the axial stiffness, according to
Eq. (3.20). The effective mechanical moment of inertia is calculated as
ȡt 4 2 2 2 2
J t,e = 12 { 0.44R + l w (l +6l R +12R )(w + t )
1
1
1 1 1
2
2
3
/ 3(l +2R) 2 +1.15R w + Rw (w + t ) (3.107)
2
2
1
2
+0.43R (t +3w )}
2 2
2
2
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