This book presents an introduction to the mathematical basis of finite element analysis as applied to vibrating systems. Finite element analysis is a technique that is very important in modeling the response of structures to dynamic loads and is widely used in aeronautical, civil and mechanical engineering as well as naval architecture. Commercial computer programs based on this technique already exist. Nevertheless, a knowledge of the mathematical principles involved is necessary before they can be successfully used. Therefore, this book assumes no previous knowledge of finite element techniques by the reader. The author has taught courses on the subject at undergraduate and postgraduate levels. The book has been written in a modular style to make it suitable for use in courses of varying length and level.