Page 139 - Mechanical design of microresonators _ modeling and applications

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Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

138 Chapter Three
characterizes a right circular hinge, and this proves the validity of
Eq. (3.89).
The effective axial mass which is dynamically equivalent to the
distributed-parameter inertia of the right elliptic hinge is

m = ȡta(0.352b + 0.667w ) (3.90)
a,e 1
Again, when a ĺ R and b ĺ R, and therefore when the elliptic regions
become circular, Eq. (3.90) simplifies to Eq. (3.79), which defines the
axial effective stiffness of a right circular microhinge. The correspond-
ing axial resonant frequency is
Eb
0.707 0.352b + 0.667w 1
Ȧ = (3.91)
a,e a
1 /
1 /
/
(2b + w ) w (4b + w ) arctan 1+4b w í ʌ 4
1
1
The torsional stiffness is found by means of Eq. (3.20) in terms of the
axial stiffness. The effective torsion-related mechanical moment of
inertia is
ȡta
J t,e =
2
2
2
3
12 0.403b +1.019b w + 0.667w (w + t ) (3.92)
1
1
1
2
2
+0.352b(3w + t )
1
Equation (3.92) transforms to Eq. (3.81) for a ĺ R and b ĺ R, which
characterizes a right circular microhinge. The lumped-parameter tor-
sion-related resonant frequency is
bG
3 2 2 2
ȡ [0.403b + 1.019b w + 0.667w (w + t )
1 1 1
2 2
+ 0.352b(3w + t )]
1.414t 1 (3.93)
Ȧ =
t,e a
1 /
/
(2b + w ) w (4b + w ) arctan 1+4b w 1
1
1
íʌ / 4
The bending stiffness of a relatively long, right elliptic hinge is

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