Page 85 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_84_08/30/05

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

84 Chapter Two
The lumped mass which needs to be placed at the free end of the mi-
crocantilever and which is equivalent to the distributed inertia of the
component during the free axial vibrations is

3 )
(
m = ȡRt 0.036R + w 1 (2.131)
a,e
The resonant frequency corresponding to free axial vibrations is

/
E {ȡR(0.036R + w 1/ 3)
1 /
/
Ȧ =2 × (1+2R w )(2R + w ) (4R + w ) (2.132)
1
1
a,e
× arctan 1+4R w íʌ 4 }
1 /
/
The torsional stiffness is expressed as a function of the axial one
according to Eq. (2.85). The equivalent mechanical moment of inertia
of the mass that is located at the microcantilever’s free end is
3 2
ȡRt 0.007R +0.038R w 1
2
2
2
2
+0.333w (w + t ) +0.036R(3w + t ) (2.133)
1
1
1
J =
t,e 12
The torsion-related resonant frequency is
2
3
/
G {ȡR 0.007R + 0.038R w 1
2t
+0.333w (w + t ) +0.036R(3w + t ) }
2
2
2
2
Ȧ t,e = 1 1 1 (2.134)
/
(1+2R w )(2R + w )
1
1
/ (4R + w ) arctan 1+4R w íʌ 4
1 /
/
1
In bending, two variants are treated corresponding to the long- and
short-beam models. For relatively long configurations, the lumped
stiffness and equivalent mass are calculated according to the Euler-
Bernoulli model, whereas for short configurations the lumped
parameters are determined by means of Timoshenko’s model and
consideration of the shearing effects.

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