Page 136 - Mechanical design of microresonators _ modeling and applications

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Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 135
studied in Chap. 2, it is possible to determine its axial, torsional, and
bending resonant frequencies by applying the algorithms developed for
two-segment configurations. It can be shown that the axial stiffness
4
(the axial compliance is given in Lobontiu, for instance), which is
connected to the free end 1 being derived with respect to the fixed end
2 and is found by using the series connection Eq. (3.22), is
Et
k a,e = (3.78)
/
1 /
/
2 (2R + w ) w (R + w ) arctan 1+4R w íʌ 4
1
1
1
The effective axial mass is
m a,e = ȡtR(0.352R +0.667w ) (3.79)
1
The corresponding axial resonant frequency is
0.707 E / ȡR(0.352R +0.667w )
1
Ȧ = (3.80)
a,e
1 /
1 /
(2R + w ) w (R + w ) arctan 1+4R/w í ʌ 4
1
1
The torsional stiffness is calculated by means of Eq. (3.20) in terms
of the axial stiffness expressed in Eq. (3.78). The effective torsional
mechanical moment of inertia is
ȡtR
J t,e =
3
2
2
2
12 0.403R +1.019R w +0.667w (w + t ) (3.81)
1
1
1
2
2
+0.352R(3w + t )
1
The torsion-related resonant frequency is
/
G {ȡR (2R + w )
1
1.414t
/ w (R + w ) arctan 1+4R w íʌ 4 }
/
/
1
1
1
Ȧ = (3.82)
t,e
2
2
2
3
0.403R +1.019R w +0.667w (w + t )
1
1
1
2
2
+0.352R(3w + t )
1
The direct linear (out-of-the-plane) bending stiffness of a long (Euler-
Bernoulli) configuration is
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