Vibration and Coupling of Continuous Systems

I Classical Theory of Vibration for Systems with Infinitely Many Degrees of Freedom.- 1. Introduction.- 2. Elements of Vibration Theory for Systems with n Degrees of Freedom.- 3. Infinite-Dimensional Separable Hilbert Spaces.- 4. A Class of Compact Self-Adjoint Operators.- 5. Introduction of the Spaces V and H Associated with the Elastic and Kinetic Energies.- 6. The Standard Vibration Problem for a System with Discrete Spectrum.- 7. Variational Properties of Eigenvalues. Rayleigh Principle. Minimax Principle and Comparison Theorem.- II Some Classical Vibration Problems.- 1. Introduction.- 2. Distributions and Sobolev Spaces.- 3. Vibrating Membrane.- 4. Examples and Remarks about Strings and Membranes. A Form of the.- Saint-Venant Principle.- 5. Linear Shallow-Water Oscillations. Neumann Boundary Condition.- 6. Complement. A Problem without Boundary Conditions.- 7. Vibration of a Three-Dimensional Elastic Body. Application of the Comparison Theorem and Particular Cases.- 8. Small Oscillations of a Compressible Fluid in a Vessel with or without Free Surface.- 9. Exercises.- III Elements of Operator Theory.- 1. Generalities on Banach Spaces and Operators.- 2. Unbounded Linear Operators. Closed Operators.- 3. Resolvents and Spectra.- 4. Singularities of the Resolvent. Fredholm Alternative.- 5. Spectra of Compact and Anticompact Operators.- 6. Symmetric and Self-Adjoint Operators.- 7. Spectral Families.- 8. Semigroups.- 9. Some General Remarks on the Regularity Theory for Elliptic Equations and the System of Elasticity.- 10. Trace Theorems for Solutions of Elliptic Equations. The Elements of the Lions-Magenes Theory.- 11. Comments and Exercises.- IV Examples of Nonstandard Vibrations and Coupling.- 1. The Thermoelasticity System.- 2. Vibration of a Viscoelastic Solid.- 3. Essential Spectrum. First Example of a Vibrating System without Compactness.- 4. An Example of Compact-Noncompact Coupled Vibrating System.- 5. Bloch Waves and Related Topics.- 6. Systems Containing a Part without Kinetic Energy.- 7. Plates—Coupling of Flexion and Traction Modes.- 8. A Problem where the Part without Kinetic Energy Is Unbounded.- 9. Comments and Problems.- V Spectral Perturbation.- 1. Generalities. The Implicit Function Theorem, the Weierstrass Preparation Theorem, and Holomorphic Functions with Values in a Banach Space.- 2. Eigenvalues of Matrices Depending Holomorphically on a Parameter.- 3. Power Series Expansions for Eigenvalues and Eigenvectors.- 4. Spectral Perturbations for Anticompact Operators Associated with a Holomorphic Sesquilinear Family.- 5. Complements and Generalizations.- 6. First Example: Smooth Perturbation of the Boundary.- 7. Some Implicit Holomorphic Eigenvalue Problems.- 8. Perturbation of an Eigenvalue of Multiplicity Two of a Self-Adjoint Operator Depending on Two Parameters z1, z2.- 9. Eigenvalue Problem for Families Depending Nonanalytically on a Parameter.- 10. Some Implicit Nonholomorphic Eigenvalue Problems.- 11. Perturbation of Spectral Families. Rellich’s Theorem.- 12. Remarks on Time-Dependent Solutions of Standard Vibration Problems.- 13. Numerical Computation of Spectral Families.- 14. Complements and Problems.- VI Formal Perturbation Methods.- 1. Introduction.- 2. The Order Symbols o and O. Gauge Functions.- 3. Singular Perturbation. Asymptotic Expansion of the Explicit Solution for a Model Boundary Value Problem.- 4. Asymptotic Study of the Solution to the Model Problem from the Equation and the Boundary Conditions.- 5. Comments and Heuristic Ideas for Other Problems.- 6. Matching Rule of Kaplun and Lagerstrom.- 7. An Interpretation of the Matching.- 8. Matching by Intermediate Variables.- 9. Extension Theorem of Kaplun.- 10. Introduction to Two-Scale Problems. Linear Oscillator with Small Damping.- 11. Second Example. Van der Pol Oscillator.- 12. Van der Pol’s Transformation and Average Method.- 13. Integral Continuity. Error Estimate for the Average and Two-Scale Methods.- 14. Moment Expansion of a Function with Shrinking Support.- 15. Exercises.- VII Perturbation of Vibrating Systems.- 1. A Model Stiff Problem. Expansions for Eigenvalues and Eigenvectors.- 2. Justification of the Preceding Expansions.- 3. Elastic Body Coupled with a Gas of Small Density (Bounded Domains).- 4. Vibration of an Almost Incompressible Elastic Body.- 5. Spectral Families in Large Domains. Application to High Frequency Homogenization.- 6. A New Class of Stiff Problems. Low and High Frequencies.- 7. Plate with Small Rigidity.- 8. Vibrations of a Slightly Viscous Gas.- 9. Thermoelastic Body with Small Thermal Conductivity.- 10. General Considerations on Vibrations of Systems with Concentrated Masses.- 11. Concentrated Masses. Local Vibrations in the Case N = 3, m > 2.- 12. Concentrated Masses. Global Vibrations for Space Dimension N = 2 or 3.- 13. Concentrated Masses. Global Vibrations for Space Dimension N = 1 and m = 1.- 14. Comments and Problems.- VIII The Helmholtz Equation in Unbounded Domains.- 1. Generalities on the Helmholtz Equation in a Neighborhood of Infinity. Radiation Condition.- 2. Some Properties of the Wave Equation in the Space-Time.- 3. Existence, Uniqueness, and Scattering Frequencies for the Dirichlet Problem in an Outer Domain with Smooth Boundary.- 4. Dirichlet, Neumann, and Transmission Problems in an Outer Domain with not Necessarily Smooth Boundary. Examples and Complements.- 5. Spectrum and Spectral Family of the Laplacian in an Outer Domain. Limiting Absorption.- 6. Local Decay of Solutions as t ? ?.- 7. Limiting Amplitude.- 8. The Rudiments of the Lax and Phillips Theory of Scattering.- 9. Comments and Exercises.- IX Scattering Problems Depending on a Parameter. Elastic Structure-Fluid Interaction in Unbounded Domains.- 1. Introduction.- 2. Elastic Body Surrounded by a Compressible Fluid.- 3. Asymptotics for a Fluid of Small Density. Resonance of Two Bodies Across Air.- 4. Asymptotics for a Fluid with Small Compressibility. Low and High Frequencies.- 5. The Helmholtz Resonator. Asymptotic Expansion for the Solution.- 6. Complements and Problems.- References.