Vibration in Continuous Media

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 hypotheses: equations with mixed variables 124

4.4.4. Love-Kirchhoff hypotheses: equations with displacement variables 127

4.4.5. Love-Kirchhoff hypotheses: equations with displacement variables obtained using Hamilton’s functional 129

4.4.6. Some comments on the formulations of transverse vibrations 130

4.5. Coupled movements 130

4.6. Equations with polar co-ordinates 133

4.6.1. Basic relations 133

4.6.2. Love-Kirchhoff equations of the transverse vibrations of plates 135

4.7. Conclusion 138

Chapter 5. Vibratory Phenomena Described by the Wave Equation 139

5.1. Introduction 139

5.2. Wave equation: presentation of the problem and uniqueness of the solution 140

5.2.1. The wave equation 140

5.2.2. Equation of energy and uniqueness of the solution 142

5.3. Resolution of the wave equation by the method of propagation (d’Alembert’s methodology) 145

5.3.1. General solution of the wave equation 145

5.3.2. Taking initial conditions into account 147

5.3.3. Taking into account boundary conditions: image source 151

5.4. Resolution of the wave equation by separation of variables 154

5.4.1. General solution of the wave equation in the form of separate variables 154

5.4.2. Taking boundary conditions into account 157

5.4.3. Taking initial conditions into account 163

5.4.4. Orthogonality of mode shapes 165

5.5. Applications 168

5.5.1. Longitudinal vibrations of a clamped-free beam 168

5.5.2. Torsion vibrations of a line of shafts with a reducer 172

5.6. Conclusion 178

Chapter 6. Free Bending Vibration of Beams 181

6.1. Introduction 181

6.2. The problem 182

6.3. Solution of the equation of the homogenous beam with a constant cross-section 184

6.3.1. Solution 184

6.3.2. Interpretation of the vibratory solution, traveling waves, vanishing waves 186

6.4. Propagation in infinite beams 189

6.4.1. Introduction 189

6.4.2. Propagation of a group of waves 191

6.5. Introduction of boundary conditions: vibration modes 197

6.5.1. Introduction 197

6.5.2. The case of the supported-supported beam 197

6.5.3. The case of the supported-clamped beam 201

6.5.4. The free-free beam 206

6.5.5. Summary table 209

6.6. Stress-displacement connection 210

6.7. Influence of secondary effects 211

6.7.1. Influence of rotational inertia 212

6.7.2. Influence of transverse shearing 215

6.7.3. Taking into account shearing and rotational inertia 221

6.8. Conclusion 227

Chapter 7. Bending Vibration of Plates 229

7.1. Introduction 229

7.2. Posing the problem: writing down boundary conditions 230

7.3. Solution of the equation of motion by separation of variables 234

7.3.1. Separation of the space and time variables 234

7.3.2. Solution of the equation of motion by separation of space variables 235

7.3.3. Solution of the equation of motion (second method) 237

7.4. Vibration modes of plates supported at two opposite edges 239

7.4.1. General case 239

7.4.2. Plate supported at its four edges 241

7.4.3. Physical interpretation of the vibration modes 244

7.4.4. The particular case of square plates 248

7.4.5. Second method of calculation 251

7.5. Vibration modes of rectangular plates: approximation by the edge effect method 254

7.5.1. General issues 254

7.5.2. Formulation of the method 255

7.5.3. The plate clamped at its four edges 259

7.5.4. Another type of boundary conditions 261

7.5.5. Approximation of the mode shapes 263

7.6. Calculation of the free vibratory response following the application of initial conditions 263

7.7. Circular plates 265

7.7.1. Equation of motion and solution by separation of variables 265

7.7.2. Vibration modes of the full circular plate clamped at the edge 272

7.7.3. Modal system of a ring-shaped plate 276

7.8. Conclusion 277

Chapter 8. Introduction to Damping: Example of the Wave Equation 279

8.1. Introduction 279

8.2. Wave equation with viscous damping 281

8.3. Damping by dissipative boundary conditions 287

8.3.1. Presentation of the problem 287

8.3.2. Solution of the problem 288

8.3.3. Calculation of the vibratory response 294

8.4. Viscoelastic beam 297

8.5. Properties of orthogonality of damped systems 303

8.6. Conclusion 308

Chapter 9. Calculation of Forced Vibrations by Modal Expansion 309

9.1. Objective of the chapter 309

9.2. Stages of the calculation of response by modal decomposition 310

9.2.1. Reference example 310

9.2.2. Overview 317

9.2.3. Taking damping into account 321

9.3. Examples of calculation of generalized mass and stiffness 322

9.3.1. Homogenous, isotropic beam in pure bending 322

9.3.2. Isotropic homogenous beam in pure bending with a rotational inertia effect 323

9.4. Solution of the modal equation 324

9.4.1. Solution of the modal equation for a harmonic excitation 324

9.4.2. Solution of the modal equation for an impulse excitation 330

9.4.3. Unspecified excitation, solution in frequency domain 332

9.4.4. Unspecified excitation, solution in time domain 333

9.5. Example response calculation 336

9.5.1. Response of a bending beam excited by a harmonic force 336

9.5.2. Response of a beam in longitudinal vibration excited by an impulse force (time domain calculation) 340

9.5.3. Response of a beam in longitudinal vibrations subjected to an impulse force (frequency domain calculation) 343

9.6. Convergence of modal series 347

9.6.1. Convergence of modal series in the case of harmonic excitations 347

9.6.2. Acceleration of the convergence of modal series of forced harmonic responses 350

9.7. Conclusion 353

Chapter 10. Calculation of Forced Vibrations by Forced Wave Decomposition 355

10.1. Introduction 355

10.2. Introduction to the method on the example of a beam in torsion 356

10.2.1. Example: homogenous beam in torsion 356

10.2.2. Forced waves 358

10.2.3. Calculation of the forced response 359

10.2.4. Heterogenous beam 361

10.2.5. Excitation by imposed displacement 363

10.3. Resolution of the problems of bending 365

10.3.1. Example of an excitation by force 365

10.3.2. Excitation by torque 368

10.4. Damped media (case of the longitudinal vibrations of beams) 369

10.4.1. Example 369

10.5. Generalization: distributed excitations and non-harmonic excitations 371

10.5.1. Distributed excitations 371

10.5.2. Non-harmonic excitations 375

10.5.3. Unspecified homogenous mono-dimensional medium 377

10.6 Forced vibrations of rectangular plates 379

10.7. Conclusion 385

Chapter 11. The Rayleigh-Ritz Method based on Reissner’s Functional 387

11.1. Introduction 387

11.2. Variational formulation of the vibrations of bending of beams 388

11.3. Generation of functional spaces 391

11.4. Approximation of the vibratory response 392

11.5. Formulation of the method 392

11.6. Application to the vibrations of a clamped-free beam 397

11.6.1. Construction of a polynomial base 397

11.6.2. Modeling with one degree of freedom 399

11.6.3. Model with two degrees of freedom 402

11.6.4. Model with one degree of freedom verifying the displacement and stress boundary conditions 404

11.7. Conclusion 406

Chapter 12. The Rayleigh-Ritz Method based on Hamilton’s Functional 409

12.1. Introduction 409

12.2. Reference example: bending vibrations of beams 409

12.2.1 Hamilton’s variational formulation 409

12.2.2. Formulation of the Rayleigh-Ritz method 411

12.2.3. Application: use of a polynomial base for the clamped-free beam 414

12.3. Functional base of the finite elements type: application to longitudinal vibrations of beams 415

12.4. Functional base of the modal type: application to plates equipped with heterogenities 420

12.5. Elastic boundary conditions 423

12.5.1. Introduction 423

12.5.2. The problem 423

12.5.3. Approximation with two terms 424

12.6. Convergence of the Rayleigh-Ritz method 426

12.6.1. Introduction 426

12.6.2. The Rayleigh quotient 426

12.6.3. Introduction to the modal system as an extremum of the Rayleigh quotient 428

12.6.4. Approximation of the normal angular frequencies by the Rayleigh quotient or the Rayleigh-Ritz method 431

12.7. Conclusion 432

Bibliography and Further Reading 435

Index 439