Vibration in Continuous Media

Three aspects are developed in this book: modeling, a description of the phenomena and computation methods. A particular effort has been made to provide a clear understanding of the limits associated with each modeling approach. Examples of applications are used throughout the book to provide a better understanding of the material presented.

Table of Contents:
Preface. Chapter 1. Vibrations of Continuous Elastic Solid Media. 1.1 Objective of the Chapter. 1.2 Equations of Motion and Boundary Conditions of Continuous Media. 1.3 Study of the Vibrations: Small Movements Around a Position of Static, Stable Equilibrium. 1.4 Conclusion. Chapter 2. Variational Formulation for Vibrations of Elastic Continuous Media. 2.1 Objective of the Chapter. 2.2 Concept of the Functional, bases of the Variational Method. 2.3 Reissner's Functional. 2.4 Hamilton's Functional. 2.5 Approximate Solutions. 2.6 Euler Equations Associated to the Extremum of a Functional. 2.7 Conclusion. Chapter 3. Equation of Motion for Beams. 3.1 Objective of the Chapter. 3.2 Hypotheses of Condensation of Straight Beams. 3.3 Equations of Longitudinal Vibrations of Straight Beams. 3.4 Equations of Vibrations of Torsion of Straight Beams. 3.5 Equations of Bending vibrations of Straight Beams. 3.6 Complex Vibratory Movements: Sandwich Beam with Felxible Inside. 3.7 Conclusion. Chapter 4. Equation of Vibration for Plates. 4.1 Objective of the Chapter. 4.2 Thin Plate Hypotheses. 4.3 Equations of Motion and Boundary Conditions of Plane Vibrations. 4.4 Equations of Motion and Boundary Conditions of Transverse Vibrations. 4.5 Coupled Movements. 4.6 Equations with Polar Co-ordinates. 4.7 Conclusion. Chapter 5. Vibratory Phenomena Described by the Wave Equation. 5.1 Introduction. 5.2 Wave Equation: Presentation of the Problem and Uniqueness of the Solution. 5.3 Resolution of the Wave Equation by the Method of Propagation. 5.4 Resolution of the Wave Equation by Separation of variables. 5.5 Applications. Chapter 6. Free Bending Vibration of Beams. 6.1 Introduction. 6.2 The Problem. 6.3 Solution of the Equation of the homogeneous Beam with a Constant Cross-section. 6.4 Propagation in Infinite Beams. 6.5 Introduction of Boundary Conditions: Vibration Modes. 6.6 Stress-displacement Connection. 6.7 Influence of Secondary Effects. 6.8 Conclusion. Chapter 7. Bending Vibration of Plates. 7.1 Introduction. 7.2 Posing the Problem: Writing Down Boundary Conditions. 7.3 Solution of the Equation of Motion. 7.4 Vibration Modes of Plates supported at Two Opposite Edges. 7.5 Vibration Modes of Rectangular Plates: Approximation by the Edge Effect Method. 7.6 Calculation of the Free Vibratory Response Following the Application of Initial Conditions. 7.7 Circular Plates. 7.8 Conclusion. Chapter 8. Introduction to Damping: Example of the Wave Equation. 8.1 Introduction. 8.2 Wave Equation with Viscous Damping. 8.3 Damping by Dissipstive Boundary Conditions. 8.4 Viscoelastic Beam. 8.5 Properties of Orthogonality of Damped Systems. 8.6 Conclusion. Chapter 9. Calculation of forced vibrations by modal expansion. 9.1 Objective of the Chapter. 9.2 Stages of Calculation of Response by Modal Decomposition. 9.3 Examples of Claculation of Generalized Mass and Stiffness. 9.4 Solution of the Modal Equation. 9.5 Example Response Calculation. 9.6 Convergence of Modal Series. 9.7 Conclusion. Chapter 10 Calculation of forced vibrations by forced wave decomposition. 10.1 Introduction. 10.2 Introduction to the Method on the Example of a Beam in Torsion. 10.3 Resolution of the Problems of Bending. 10.4 Damped Media. 10.5 Generalization: Distributed Excitations and Non-harmonic Excitations. 10.6 Forced Vibrations of Rectangular Plates. 10.7 Conclusion. Chapter 11. The Rayleigh-Ritz method based on Reissner's Functional. 11.1 Introduction. 11.2 Variational Formulation of the Vibrations of Bending of Beams. 11.3 Generation of Functional Spaces. 11.4 Approximation of the Vibratory Response. 11.5 Formulation of the Method. 11.6 Application to the Vibrations of a Clamped-free Beam. 11.7 Conclusion. Chapter 12. The Rayleigh-Ritz method based on Hamilton's Functional. 12.1 Introduction. 12.2 Reference Example: Bending Vibrations of Beams. 12.3 Functional Base of the Finite Elements Type: Application to. 12.4 Functional Base of the Modal Type: Application to Plates Equipped with Heterogenities. 12.5 Elastic Boundary Conditions. 12.6 Convergence of the Rayleigh-Ritz Method. 12.7 Conclusion. Bibliography and Further Reading. Index.