Page 86 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_85_08/30/05

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 85
For long microcantilevers, the bending-related stiffness is

4Et 3
k =
b,e 2 2
{3 14.283R + 16.566Rw + 3.141w
1 1 (2.135)
í4(2R + w ) w (4R + w ) arctan 1+4R / w }
1
1
1
1
The lumped mass which is equivalent to the distributed inertia of the
microcantilever undergoing free bending vibrations is
m b,e = ȡRt(0.014R +0.236w ) (2.136)
1
The bending-related resonant frequency of the long circularly filleted
microcantilever is

1.15t E / ȡR(0.014R + 0.236w )
1
Ȧ b,e =
2
14.283R + 16.566Rw +3.141w 2 (2.137)
1
1
í4(2R + w ) w (4R + w ) arctan 1+4R / w 1
1
1
1
For short microcantilevers, the Timoshenko model with consider-
ation of the shearing effects has to be utilized, as detailed in previous
sections. It has been shown that the linear bending compliance C l as
well as the axial compliance C is needed to enable determination of the
a
linear shear-dependent compliance according to Eq. (2.28). These two
compliances can be calculated by means of their definition Eqs. (2.25)
and (2.27); their explicit forms, as mentioned previously, are given in
3
Lobontiu and Garcia and are not reproduced here.
It can be shown that the lumped-parameter stiffness for the
Timoshenko model is

3
sh 4EGt
k b,e =
2
2
2
í3.14țEt + G(42.85R +49.7Rw +9.42w )
1
1
2
2
+ 4țEt (2R + w ) í 12(8R +6Rw 1 (2.138)
1
2
/
+w )w G arctan 1+4R w 1/ w (4R + w )
1
1
1
1
and that the lumped mass is
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