Page 90 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_89_08/30/05

Basic Members: Lumped- and Distributed-Parameter Modeling and Design

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 89
bG
3 2 2 2
ȡ [ 0.007b + 0.038b w + 0.333w (t + w
1 1 1
2 2 (2.149)
+0.036b(t +3w ) ]
4t 1
Ȧ t,e =
a
1 /
/
4(2b + w ) w (4b + w ) arctan 1+4b w íʌ
1
1
1
The lumped-parameter out-of-the-plane bending stiffness of a
relatively long, elliptically filleted microcantilever is
3 3
4Eb t
k =
b,e 3 2 2
3a 14.283b + 16.566bw +3.14w
1 1 (2.150)
/
í4(2b + w ) w (4b + w ) arctan 1+4b w 1
1
1
1
Equation (2.150) reduces to Eq. (2.135), which defines a circularly fil-
leted microcantilever, when a ĺ R and b ĺ R.
For short configurations, the bending stiffness of this design is
3 3
k sh = 4EGb t
b,e 2 2 2 2
a{ íʌțEb t +3Ga 2(4+ ʌ)b +4(1+ ʌ)bw
1
2 2
+ʌw 2 í 4(2b + w ) íțEb t (2.151)
1 1
2
/
+3Ga w (4b + w ) arctan 1+4b w 1 / w (4b + w )}
1 1 1 1
and it can be seen that for a ĺ R and b ĺ R, this equation becomes
Eq. (2.138), which characterizes a circularly filleted microcantilever.
The effective bending mass for a long microcantilever is
m = ȡat(0.014b +0.236w ) (2.152)
b,e 1
which transforms to Eq. (2.136) when a ĺ R and b ĺ R.
The resonant frequency of a long, elliptically filleted design is deter-
mined by means of Eqs. (2.150) and (2.152) as

bE
2bt ȡ(0.014b + 0.236w )
1
Ȧ =
b,e 2
2
3a 2(4+ ʌ)b +4(1+ ʌ)bw (2.153)
1
/
í4(2b + w ) w (4b + w ) arctan 1+ 4b w 1
1
1
1
The effective bending mass for a short microconfiguration is
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