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The study of vibration and damping is concerned with oscillatory motions of bodies and particles and the attenuation of those vibrations. Both engineered machines and natural physical systems have the property of being subject to a vibration all or part of the time. Dangerous vibrations are frequent practical problem. In the design of a bridge, for example, one must pay attention to the resonances of the bridge with the wind-induced oscillatory forces. Recent scientific and technological advances have generated strong interest in the field of vibrational systems governed by the partial differential equations. In this book a functional-analytic theory is developed for a particular type of vibrational systems, those which can be described by an abstract second order differential equation with symmetric non-negative forms as coefficients. Examples of such vibrational systems are mechanical systems. The main topics of the book are stability and optimal damping of these systems. The book is intended for mathematicians, engineers and students interested in the theory of vibrational systems and its practical use.