Page 92 - Mechanical design of microresonators _ modeling and applications

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Basic Members: Lumped- and Distributed-Parameter Modeling and Design

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 91
As the situation was with the circular filleted microcantilever, for this
configuration, too, the influence of shearing on the bending resonant fre-
quency is reduced, as the ratio of Eq. (2.156) is slightly larger than 1, even
for very short and wide configurations (small values of Į and ȕ).
As shown in Chap. 3, microcantilevers and/or microhinges of more complex
configurations can be formed by serially combining the simple shapes that
have been presented thus far, such as constant-cross-section members, and
circularly or elliptically filleted designs. Finding the representative lumped-
parameter properties of those compound members will imply utilization of
the elastic and inertia properties of the basic designs presented thus far in
this chapter.

2.3 Distributed-Parameter Modeling and
Design

The distributed-parameter modeling approach describes the vibra-
tory motion of a mechanical system by formulating partial differential
equations (PDEs) that reflect the system’s behavior in both time and
space. As a consequence, it becomes possible to directly evaluate the
resonant characteristics of a system without having to separately cal-
culate stiffness and inertia fractions to obtain a specified resonant
frequency–as was the case with the lumped-parameter modeling
approach.
Microcantilevers and microbridges of constant cross section (gen-
erally rectangular) are analyzed first, and then the first resonant
frequencies are calculated. As shown when we examined the same
aspect by the lumped-parameter procedure, the first resonant fre-
quency corresponds to bending about the sensitive axis. The solution to
the partial differential equation (PDE) can be expressed uniquely for
both microcantilevers and microbridges. The different boundary
conditions will discriminate between the solutions particular to each of
the two components. The relevant resonant frequencies will also be
derived for circular rings, thin plates, and membranes by both exact
integration methods and approximate ones.

2.3.1 Line micromembers
Long configurations (treated by means of the Euler-Bernoulli model)
and short ones (modeled by means of Timoshenko’s assumption) are
analyzed in this section, and the bending resonant frequencies corre-
sponding to cantilevers and bridges are derived for constant-cross-
section members.

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