Existence theory for linear vibration models of elastic bodies

Abstract

The vibration of elastic bodies or systems of elastic bodies are modeled by partial differential equations or systems of partial differential equations. In this dissertation we consider the wave equation (in one and three dimensions), different partial differential equations for the vibration of beams and other complex structures. Various forms of damping are included in the models. It is well known that solutions of partial differential equations are often not possible to determine explicitly. To further complicate matters, solutions often do not exist if it is required that the partial differential equation be satisfied in every point of the relevant domain. (An example is provided in the introduction.) In such cases a solution may exist if the definition of a solution is “relaxed" and it turns out that satisfactory numerical results can be obtained. However, an alternative definition of a solution must be unambiguous and the solutions must be useful in applications. The weak variational form for vibration problems meet these requirements. In this dissertation we consider only linear vibration problems. The aim is to provide a general framework for existence theory. This framework is based on an article by Van Rensburg and Van der Merwe in 2002 where an existence theory for a general linear vibration problem in variational form, is presented. In this dissertation the pre-knowledge for the existence theory (Sobolev spaces and semigroup theory) is provided and a detailed version of the exis- tence theory is provided. More auxiliary results are also included to clarify the exposition. Great care was taken in the formulation of results that link the existence results to semigroup theory. It is shown through examples that all linear vibration problems have the same variational form. (A proof is not possible but the examples are convincing.) The general theory is then applied to the model problems in weak variational form. Six of the problems are standard (but not trivial), e.g. the three- dimensional wave equation with boundary damping. However, two of the problems are from recent articles and are complex mathematical models. Copyright