Page 151 - Mechanical design of microresonators _ modeling and applications

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Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

150 Chapter Three
3.5 Sandwiched Microcantilevers
(Multimorphs)
Sandwiched components such as microcantilevers or microbridges are
often utilized in microtransduction (actuation and/or sensing) where
one layer achieves the structural and elastic recovery functions (most
often the silicon or polysilicon) and the other layers are active in the
sense that they deform under activation or environmental stimuli. The
resonant frequencies corresponding to axial deformation, torsion, and
loading are presented next for equal-length microcomponents and for
dissimilar-length ones.

3.5.1 Microcantilevers of equal-length
layers
In equal-length multimorphs, the component layers have identical
lengths. Figure 3.29 shows a microcantilever consisting of n layers. The
resonant frequencies are determined next for this micromember. The
resonant frequencies for a multimporph are very similar to those of a
homogeneous micromember having the same length.
In the case of axial vibrations, for instance, an equivalent rigidity
5
needs to be used in the form (see Lobontiu and Garcia ):
n E A
k a,e = ™ i i (3.128)
i =1 l
where l is the identical length and A is the cross-sectional area of
i
the ith component. In a similar manner, it can be shown that the
lumped-parameter mass which needs to be placed at the
microcantilever’s free tip, in order to generate the same inertial effect
as the distributed mass of the multimorph, according to Rayleigh’s
principle, is

layer # n layer # i layer # 1

l
Figure 3.29 Sandwiched microcantilever.

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