Page 163 - Mechanical design of microresonators _ modeling and applications

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Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

162 Chapter Three

7250
0.0001
ω [rad/s]
6500
0.0008
1p [m]

11 [m] 0.00005
0.002
Figure 3.36 Bending resonant frequency of a dissimilar-length sandwich microcantilever.

The bending resonant frequency of Eq. (3.177) is plotted in Fig. 3.36 for
the parameters given in this example. Figure 3.36 indicates that the bending
resonant frequency of a bimorph increases with a larger patch length, and
this is due to the fact that the patch stiffness increases more compared to the
patch mass with length (third power and first power, respectively). Also,
Fig. 3.36 suggests that for a constant patch length, the resonant frequency
is higher for patch positions that are closer to the microcantilever root (which
means larger l 1 ), and this also makes physical sense since the effective mass
of the patch (the mass transformed into an equivalent mass which is placed
at the microcantilever’s free end) is smaller when the patch gets closer to the
fixed end, which produces higher resonant frequencies.

3.6 Resonant Microcantilever Arrays

Microcantilevers of identical cross sections and various lengths can be
used in a resonant array formation as tools of detecting a range of
bending resonant frequencies. In Chap. 4, similar arrays constructed of
microbridges are presented, and more details are provided with respect
to the precision of this spatial and frequency-domain discretization pro-
cess. Figures 3.37 and 3.38 show two microcantilever architectures with
members having rectangular and circular cross sections, respectively.
It is of interest here to determine the lengths of the component
microcantilevers in such a way that each microcantilever matches a
bending resonant frequency in a prescribed range. It can be shown that
for a constant rectangular cross-section member, the length is related
to a frequency f as
2
4 Et
l =0.405 2 (3.179)
ȡ f

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