The aim of this book is to impart a sound understanding, both physical and mathematical, of the fundamentals of the theory of vibration and its applications. It presents in a simple and systematic manner techniques that can be easily applied to the analysis of vibration of mechanical and structural systems. In this book, an attempt has been made to provide the rational development of the methods of vibration from their foundations and develop the techniques in clearly understandable stages. This is the first volume, entitled "An Introduction", intended for an introductory semester course in the theory of vibration. The solution procedures are explained in details easily understandable by students. The second volume, "Discrete and Continuous Systems", is planned for publication in the fall of 1990.