The theoretical analysis of vibrating elastic structures
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University of Pretoria
Abstract
This dissertation is a literature study on linear vibration problems. The
research forms part of an ongoing research project Vibration analysis. It
deals with modelling, theoretical analysis and fi nite element computation.
In this project the aim is to obtain results of a theoretical as well as practical
nature.
A signifi cant part of this dissertation deals with the theory for existence
of solutions and the application thereof to model problems. The existence
results in a 2002 article of Van Rensburg and Van der Merwe are presented
and proved in greater detail. Additional results were formulated for this
purpose. Semigroup theory is used to obtain existence results.
The theory of existence is applied to various models with different types
of damping. Examples with weak damping as well as boundary damping
are presented. The application to each model problem is rigorous and more
complete than any publication before on this topic.
In a recent article on hyperbolic heat conduction a model problem that is
not well posed, is given. In one section of this dissertation we do a thorough
analysis and prove that the problem does not even admit a mild solution.
We also prove rigorously that solution methods in the article are valid and
provide estimates for errors (not finite element method errors).
Two recent articles on error estimates for the semi-discrete and fully discrete
Galerkin approximations of the general weak variational problem are a paper
by Basson and Van Rensburg published in 2013 and another by Basson,
Stapelberg and Van Rensburg published in 2017. Different types of damping
are considered since the properties of the solutions as well as the numerical
approximations of these solutions depend on the damping. We consider weak
damping as well as general damping.
In one chapter of this dissertation the fi nite element method (FEM) is applied
to the vibration of the Timoshenko beam. The objective was to compare the
standard finite element method using Hermite cubic basis functions
to the mixed fi nite element method using piecewise linear basis
functions. It should however be noted that the method used in this dissertation
to compare the models is naive since we do not have the means
to calculate the computational effort of each method. More numerical experiments and detailed analysis are required, but that is considered to be a
project in its own right.
A modest contribution was made regarding the two-dimensional beam model.
The derivation of the matrices for the mixed finite element method should be useful for future work.
Tracking a sharp crested wave front in hyperbolic heat transfer, an article
by Sieberhagen and Van Rensburg published in 2012, is studied in one of
the chapters of this dissertation. The article was written for an audience
not particularly interested in mathematical analysis; the contribution of this
dissertation was to conduct a serious analysis of the problem and methods
used.
Description
Dissertation (MSc)--University of Pretoria, 2018.
Keywords
UCTD
Sustainable Development Goals
Citation
van den Brink, S, 2018, The theoretical analysis of vibrating elastic structures, MSc dissertation, University of Pretoria, Pretoria