The aim of this book is to address important practical aspects of vibration analysis. It presents cases rarely discussed in the existing literature on vibration that are problems of considerable interest for researchers and practical engineers, such as rotor dynamics and torsional vibration of engines. The book can be used not only as a reference, but also as a textbook for advanced graduate and undergraduate students, as it develops the subject from its foundations and contains problems and solutions for each chapter. The book begins with a discussion of vibrations in linear systems with one degree of freedom, providing a mathematical and physical basis for the sub-sequent chapters. Linear systems with many degrees of freedom serve to introduce the modal analysis of vibrations as well as some useful computational procedures. The book then turns to continuous linear systems, discussing both analytical solutions that provide physical insights as well as discretization techniques that supply tools for actual computation. The discussion of nonlinear vibrations includes a treatment of chaotic vibrations and other new insights. The book concludes with detailed discussions of the dynamics of rotating and reciprocating machinery. In this new edition, the notation has been modernized, the classical approach to vibration and the modern approach through dynamical systems theory have been integrated, the material on control and active systems has been completely rewritten, and material relevant to mechatronics has been added.