Page 160 - 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 159
which is the known relationship for a one-component microrod. By com-
bining Eqs. (3.161) and (3.163), the resonant frequency corresponding
to free axial vibrations of the patched microcantilever becomes
k a,e
Ȧ =
a,e m
a,e
(3.165)
p p /
1.73l E t (E t + E t ) [E t l + E t (l í l )]
1 1
p p
1 1
p
1 1
=
2
3
l t ȡ + l [3(l í l ) 2 í 3(l í l )l + l ] t ȡ
1 1 p 1 1 p p p p
In torsion, similar to the axial problem, the stiffness of the patched
microcantilever can be found by considering the series connection
between the three portions above-mentioned. The stiffness equation is
3
3
3
G t (G t + G t )w
1 1
1 1
p p
k = (3.166)
t,e 3 3
3 G lt + G (l í l )t p
p
1 1
p
For l p ĺ 0 and t p ĺ 0, Eq. (3.166) reduces to
G wt 1 3
1
k = (3.167)
t 3l
which is the stiffness of a homogeneous fixed-free bar.
The equivalent torsional mechanical moment of inertia is determined
again by means of Rayleigh’s principle, and its equation is
2
2
3
2
2
2
2
w{l ȡ t (t + w ) +3l l t 3ȡ t t + ȡ (3t + t + w )
1 1 1 p p 1 1 p p 1 p
2
2
2
෹3ll t (2l + l ) 3ȡ t t + ȡ (3t + t + w )
p p 1 p 1 1 p p 1 p
(3.168)
2
2
+l t (3l +3l l + l ) 3ȡ t t + ȡ (3t + t + w ) }
2
2
2
p p
1
p
1 1 p
1
p
1 p
p
J =
t,e 2
36l
When l p ĺ 0 and t p ĺ 0, Eq. (3.168) simplifies to
2
2
1 ȡ lwt (w + t )
1
1
1
J = 3 12 (3.169)
t
which is the equation expressing the equivalent mechanical moment of
inertia in torsion for a homogeneous fixed-free bar.
The torsional resonant frequency of the bar is calculated by
Eqs. (3.166) and (3.168), and its equation is
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