Page 46 - Mechanical design of microresonators _ modeling and applications

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Source: Mechanical Design of Microresonators

Chapter
2

Basic Members: Lumped- and

Distributed-Parameter Modeling

and Design

2.1 Introduction
In this chapter we study the methods of determining the resonant
frequencies of basic micromembers such as one-dimensional or two-
dimensional ones by using the lumped-parameter modeling and the
distributed-parameter technique. Microhinges, microcantilevers, and
microbridges, in their most common configurations, can be modeled as
line elements of either constant or variable cross section. More specifi-
cally, microhinges and microcantilevers can be characterized in terms
of their resonant behavior by means of lumped-parameter elastic and
inertia properties defined about 6 degrees of freedom that are associ-
ated to the free endpoint, namely, three translations (u x , u y , and u z ) and
three rotations (ș x , ș y , and ș z ), as suggested in Fig. 2.1. These degrees
of freedom are physical deformations (either linear or rotary) of the
member itself and are produced by actuation or interaction with sup-
ports and/or adjacent members through bending, torsion, and/or axial
loading. As shown in the following, these degrees of freedom are related
to the corresponding loads (the forces F x , F y , F z and the moments
M x , M y , M z ) in the static domain by means of stiffnesses or by means of
compliances.
The stiffness and mass can be lumped at the free end of a fixed-free
(microcantilever) member, such as the one in Fig. 2.1. The lumped
stiffness k i will be evaluated as the ratio of an applied force/moment

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