Analysis of the Elastica with Applications to Vibration Isolation
Date
2007-05-02
Author
Advisors
Virgin, Lawrence N
Dolbow, John
Franzoni, Linda
Knight, Josiah D.
Witelski, Thomas
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Abstract
Linear theory is useful in determining small static and dynamic deflections. However,
to characterize large static and dynamic deflections, it is no longer useful or accurate,
and more sophisticated analysis methods are necessary. In the case of beam deflections,
linear beam theory makes use of an approximate curvature expression. Here, the exact
curvature expression is used to derive the governing partial differential equations
that describe the in-plane equilibrium and dynamics of a long, thin, inextensible
beam, where the self-weight of the beam is included in the analysis. These beam equations
are expressed in terms of arclength, and the resulting equilibrium shape is called
the elastica. The analysis gives solutions that are accurate for any deflection size,
and the method can be used to characterize the behavior of many structural systems.
Numerical and analytical methods are used to solve or to approximate solutions to
the governing equations. Both a shooting method and a finite difference, time-stepping
algorithm are developed and implemented to find numerical solutions and these solutions
are compared with some analytical approximation method results. The elastica equations
are first used to determine both linear and nonlinear equilibrium configurations for
a number of boundary conditions and loading types. In the case of a beam with a significant
self-weight, the system can exhibit nonlinear static behavior even in the absence
of external loading, and the elastica equations are used to determine the weight corresponding
to the onset of instability (or self-weight buckling). The equations are also used
to characterize linear and nonlinear vibrations of some structural systems, and experimental
tests are conducted to verify the numerical results. The linear vibration analysis
is applied to a vibration isolator system, where a postbuckled clamped-clamped beam
or otherwise highly-deformed structure is used (in place of a conventional spring)
to reduce system motion. The method is also used to characterize nonlinear dynamic
behavior, and the resulting frequency-response curves are compared with those in the
literature. Finally, the method is used to investigate the dynamics of subsea risers,
where the effects of gravity, buoyancy, and the current velocity are considered.
Type
DissertationPermalink
https://hdl.handle.net/10161/180Citation
Santillan, Sophia Teresa (2007). Analysis of the Elastica with Applications to Vibration Isolation. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/180.Collections
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