||S. Graham Kelly
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Advanced Vibration Analysis is a result of three graduate classes that I have regularly taught at The University of Akron: Vibrations of Discrete Systems, Vibrations of Continuous Systems, and Engineering Analysis. The latter teaches a framework for the development of mathematical solutions for models of engineering systems, relying on knowledge of the physics of the system to guide the mathematical solution. The foundation upon which this framework is built is linear algebra, focusing on vector spaces and linear operators whose domain is a subspace of a larger vector space. While teaching the vibrations courses, it became clear that the analysis of vibration of a multi-degree-of-freedom system or a continuous system is best performed in this framework. This book ties all three courses together. The objectives of this book are to develop a general mathematical framework for the analysis of a model of a physical system undergoing vibration and to illustrate how the physics of a problem is used to develop a more specific framework for the analysis of that problem. Such an analysis includes the determination of an exact solution for a linear problem and approximate solutions for problems in which an exact solution is difficult to obtain. A general theory is developed that is applicable to both discrete and continuous systems. Presentation of the theory includes proofs of important results, especially proofs that are themselves instructive for a comprehensive understanding of the result. The application of the theory to discrete systems is straightforward, and its understanding requires little addition to what is developed in this book. A thorough understanding of the application of the theory to continuous systems requires additional discussion regarding convergence of sequences and series in infinite dimensional vector spaces. A basic discussion of the required theory is presented, but proofs are lengthy and not contained within this book.