Page 152 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_151_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 151
n ȡ A m
m = l ™ i i = (3.129)
a,e 3 3
i =1
where ȡ i is the material density of the ith component. As a consequence,
the axial resonant frequency of a multimorph can be expressed as
n
™ E A
k a,e 3 i =1 i i
Ȧ = = (3.130)
a,e m a,e l n
™ ȡ A i
i
i =1
In the case of a two-component sandwiched microcantilever (bimorph),
the axial resonant frequency becomes

3 E t + E t
2 2
1 1
Ȧ = (3.131)
a,e l ȡ t + ȡ t
1 1 2 2
By considering that
t = ct (3.132)
2 1
Eq. (3.131) changes to

3 E + cE 2
1
Ȧ = (3.133)
a,e l ˮ + cˮ 2
1
When c ĺ 0 (corresponding to the situation where the microcantilever
is homogeneous and made up of a single layer), Eq. (3.133) reduces to

E A 1
1
Ȧ = 3 (3.134)
a m l
1
which is the known equation for the axial resonant frequency of a ho-
mogeneous microcantilever.
A similar reasoning is now applied to the free torsional vibrations of
a multimorph whose equivalent rigidity is

n G I
k t,e = ™ i ti (3.135)
i =1 l
where G i is the shear modulus and I ti is the torsional moment of inertia
for the ith component and was defined previously for a homogeneous,
constant rectangular cross-section microcantilever. For very thin

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