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.

Auteur
Jean-Louis Guyader is Professor of Vibration and Acoustics within the Mechanical Engineering Department at INSA, Lyon, France and Director of the Vibration and Acoustics Laboratory. His research covers the acoustic radiation of structures in light or heavy fluids and the energy propagation in vibrating structures and acoustic media.

Contenu

Preface 13

Chapter 1. Vibrations of Continuous Elastic Solid Media 17

1.1. Objective of the chapter 17

1.2. Equations of motion and boundary conditions of continuous media 18

1.2.1. Description of the movement of continuous media 18

1.2.2. Law of conservation 21

1.2.3. Conservation of mass 23

1.2.4. Conservation of momentum 23

1.2.5. Conservation of energy 25

1.2.6. Boundary conditions 26

1.3. Study of the vibrations: small movements around a position of static, stable equilibrium 28

1.3.1. Linearization around a configuration of reference 28

1.3.2. Elastic solid continuous media 32

1.3.3. Summary of the problem of small movements of an elastic continuous medium in adiabatic mode 33

1.3.4. Position of static equilibrium of an elastic solid medium 34

1.3.5. Vibrations of elastic solid media 35

1.3.6. Boundary conditions 37

1.3.7. Vibrations equations 38

1.3.8. Notes on the initial conditions of the problem of vibrations 39

1.3.9. Formulation in displacement 40

1.3.10. Vibration of viscoelastic solid media 40

1.4. Conclusion 44

Chapter 2. Variational Formulation for Vibrations of Elastic Continuous Media 45

2.1. Objective of the chapter 45

2.2. Concept of the functional, bases of the variational method 46

2.2.1. The problem 46

2.2.2. Fundamental lemma 46

2.2.3. Basis of variational formulation 47

2.2.4. Directional derivative 50

2.2.5. Extremum of a functional calculus 55

2.3. Reissner's functional 56

2.3.1. Basic functional 56

2.3.2. Some particular cases of boundary conditions 59

2.3.3. Case of boundary conditions effects of rigidity and mass 60

2.4. Hamilton's functional 61

2.4.1. The basic functional 61

2.4.2. Some particular cases of boundary conditions 62

2.5. Approximate solutions 63

2.6. Euler equations associated to the extremum of a functional 64

2.6.1. Introduction and first example 64

2.6.2. Second example: vibrations of plates 68

2.6.3. Some results 72

2.7. Conclusion 75

Chapter 3. Equation of Motion for Beams 77

3.1. Objective of the chapter 77

3.2. Hypotheses of condensation of straight beams 78

3.3. Equations of longitudinal vibrations of straight beams 80

3.3.1. Basic equations with mixed variables 80

3.3.2. Equations with displacement variables 85

3.3.3. Equations with displacement variables obtained by Hamilton's functional 86

3.4. Equations of vibrations of torsion of straight beams 89

3.4.1. Basic equations with mixed variables 89

3.4.2. Equation with displacements 91

3.5. Equations of bending vibrations of straight beams 93

3.5.1. Basic equations with mixed variables: Timoshenko's beam 93

3.5.2. Equations with displacement variables: Timoshenko's beam 97

3.5.3. Basic equations with mixed variables: Euler-Bernoulli beam 101

3.5.4. Equations of the Euler-Bernoulli beam with displacement variable 102

3.6. Complex vibratory movements: sandwich beam with a flexible inside 104

3.7. Conclusion 109

Chapter 4. Equation of Vibration for Plates 111

4.1. Objective of the chapter 111

4.2. Thin plate hypotheses 112

4.2.1. General procedure 112

4.2.2. In plane vibrations 112

4.2.3. Transverse vibrations: Mindlin's hypotheses 113

4.2.4. Transverse vibrations: Love-Kirchhoff hypotheses 114

4.2.5. Plates which are non-homogenous in thickness 115

4.3. Equations of motion and boundary conditions of in plane vibrations 116

4.4. Equations of motion and boundary conditions of transverse vibrations 121

4.4.1. Mindlin's hypotheses: equations with mixed variables 121

4.4.2. Mindlin's hypotheses: equations with displacement variables 123 4.4.3. Love-Kirchhoff hypothes...