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.
Vibration, Wave Propagation and Damping in One Space Dimension: Review of Lumped Parameter Control Systems. A First Example of DPS-A Vibrating String. Boundary Stabilization of a Vibrating String. Stabilization of a Vibrating Thin Beam. Vibrating Beam Continued: Boundary Feedback Stabilization. The Wave Propagation Method for a Vibrating String with Boundary Damping. The Wave Propagation Method for a Vibrating Beam with Boundary Damping. Mechanical Designs of Dampers Satisfying Stabilizing Boundary Conditions on a Beam. Point Controllers and Stabilizers Located in the Middle of the Span I: Vibrating Strings. Point Controllers and Stabilizers Located in the Middle of the Span II: Vibrating Beams. Further Examples of Damping and Vibration. Damping Devices in One-Dimensional Acoustic Systems. Other Structural Vibration Models. Functional Analysis: Metric Spaces and Vector Spaces. Bounded Linear Transformations. The Basic Principles of Linear Analysis. Some Important Properties in Hilbert Spaces. The Gram-Schmidt Orthonormalization Process and Orthonormal Basis. Dual Spaces. The Tri-Space Setting V c H c V*. The Weak and Weak* Convergence. Closable and Closed Linear Operators. Adjoint and Symmetric Operators. Resolvent and Spectrum. Compact Operators. Compact Symmetric Operators. The Lebesgue Measure and Integral. Distributions, Sobolev Spaces and Boundary Value Problems: Distributions. The Fourier Transform of Tempered Distributions. Sobolev Spaces and Imbedding Theorems. The Trace Theorems. Poincaré Type Inequalities. Regularity of Solutions for Boundary Value Problems. Strongly Continuous Semigroups of Evolution: Strongly Continuous Semigroups. Contraction Semigroups Generated by Dissipative Operators in a Hilbert Space. The Generation of Co-Semigroups in a Banach Space. The Fourier Inversion Formula. Adjoint Semigroups. The Lumer-Phillips Theorem for the Generation of Contraction Co-Semigroups in a Banach Space. Example of Co-Contraction Semigroups Corresponding to Conservative or Damped Distributed Parameter Systems. The Adjoint Operators Corresponding to BVP with Dissipative B.C. Asymptotic Stability and Exponential Decay of Energy: Weak Stability and Asymptotic Stability. Characterizing Conditions for Uniform Exponential Stability I: LP Summability Test. Characterizing Conditions for Uniform Exponential Stability II: Frequency Domain Test. Characterizing Conditions for Uniform Exponential Stability III: Eigenfunction Test. Applications to the Wave, Beam and Schrödinger Equations. Nonhomogeneous Evolution Equations. An Asymptotic Result. Russell's Exact Controllability Via Exponential Stabilizability. The Method of Energy Identities: Uniform Exponential Decay of Energy of the Wave Equation with Distributed Viscous Damping. Derivation of Energy Identities Using More Multipliers. Decay of Energy of the Wave Equation Exterior to a Star-Shaped Scatterer. Uniform Exponential Decay of Energy of the Wave Equation on a Bounded Domain with Partly Dissipative Boundary. The Loss of Energy Due to Expanding Boundary. Serially Connected Beams with Damping at Joints and the Boundary. Exponential Decay of Energy of a Thin Kirchhoff Plate with Boundary Damping. Holomorphic Semigroups Corresponding to Structures with Strong Damping: Holomorphic Semigroups and Differentiable Semigroups. Fractional Powers of Unbounded Linear Operators. Holomorphic Semigroups Corresponding to the Heat Equation. Holomorphic Semigroups Associated with Linear Elastic Systems with Structural Damping. Other Models of Structural Damping. Appendices: The Riesz Index of an Eigenvalue. The Fundamental Solution of a Time-Dependent Evolution Equation. Dispersive Waves. Bibliography. Index.