Vibration and Coupling of Continuous Systems : Asymptotic Methods

Real problems concerning vibrations of elastic structures are among the most fascinating topics in mathematical and physical research as well as in applications in the engineering sciences. This book addresses the student familiar with the elementary mechanics of continua along with specialists. The authors start with an outline of the basic methods and lead the reader to research problems of current interest. An exposition of the method of spectra, asymptotic methods and perturbation is followed by applications to linear problems where elastic structures are coupled to fluids in bounded and unbounded domains, to radiation of immersed bodies, to local vibrations, to thermal effects and many more.

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.