The theoretical analysis of vibrating elastic structures

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.