Vibration and Damping in Distributed Systems, Volume I provides a comprehensive account of the mathematical study and self-contained analysis of vibration and damping in systems governed by partial differential equations. The book presents partial differential equations techniques for the mathematical study of this subject. A special objective of establishing the stability theory to treat many distributed vibration models containing damping is discussed. It presents the theory and methods of functional analysis, energy identities, and strongly continuous and holomorphic semigroups. Many mechanical designs are illustrated to provide concrete examples of damping devices. Numerical examples are also included to confirm the strong agreements between the theoretical estimates and numerical computations of damping rates of eigenmodes.