Page 108 - 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 107
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from rectangular regions are presented by Lobontiu and Garcia and
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Garcia, Lobontiu, and Nam who used an analytical model to evaluate
both the static and the modal behavior of these members.
As Figs. 3.1 and 3.2 do indicate, microhinges and microcantilevers
can be constructively and structurally identical, the only difference
consisting in the boundary conditions: While microcantilevers are
physically free at one end, microhinges are often connected at both ends
to either the substrate or other rigid members of the micromechanism.
Both flexible members are directly amenable to the same lumped-
parameter modeling as fixed-free members (this is natural and direct
for microcantilevers, but also valid for microhinges, which can be
considered as fixed-free members). As a consequence, microhinges and
microcantilevers are treated unitarily in this chapter.
Microhinges and microcantilevers are fully determined by means of
lumped-parameter properties through 6 degrees of freedom associated
to the free endpoint, namely, three translations (u , u y , and u z ) and three
x
rotations (ș x , ș y , and ș z ), as mentioned in Chap. 2 and shown in Fig. 2.1.
Determining the relevant stiffness properties of the various
geometric configurations presented here often requires one to express
compliances as a simpler calculation route. On occasion, compliances
need to be determined in terms of reference frames that are not placed
at one end of the member. This subsection addresses such designs by
formulating the necessary quantitative rules. Several microcantilevers,
including trapezoid, paddle, and circularly and elliptically filleted
designs, are studied based on a generic approach which treats them as
two-segment members to derive their lumped-parameter resonant
frequencies. Circularly and elliptically filleted microhinges are also
thoroughly presented by applying the same generic calculation tool.
Hollow microcantilevers such as rectangular and trapezoidal are
further presented in this chapter. The resonant frequencies of
multimorph (sandwiched) line members are also derived by focusing on
two categories: the equal-length structures and the dissimilar-length
ones. The chapter concludes by presenting the microcantilever arrays.

3.2 Compliance Transforms by Reference
Frame Translation
Microhinges and microcantilevers can be designed as serial connections
of elementary flexible segments which can be defined geometrically by
a unique mathematical function (line, circle, or other curve), and which
have been presented in Chap. 2 as basic units. In such cases a global
reference frame is associated to all different segments making up the
compound configuration. Different segments are, however, defined in

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