Preface

(4)

Vibrations of Continuous Elastic Solid Media

(28)

Objective of the chapter

(1)

Equations of motion and boundary conditions of continuous media

(10)

Description of the movement of continuous media

(3)

Law of conservation

(2)

Conservation of mass

(1)

Conservation of momentum

(2)

Conservation of energy

(1)

Boundary conditions

(2)

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

(16)

Linearization around a configuration of reference

(4)

Elastic solid continuous media

(1)

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

(1)

Position of static equilibrium of an elastic solid medium

(1)

Vibrations of elastic solid media

(2)

Boundary conditions

(1)

Vibrations equations

(1)

Notes on the initial conditions of the problem of vibrations

(1)

Formulation in displacement

(1)

Vibration of viscoelastic solid media

(4)

Conclusion

(1)

Variational Formulation for Vibrations of Elastic Continuous Media

(32)

Objective of the chapter

(1)

Concept of the functional, bases of the variational method

(10)

The problem

(1)

Fundamental lemma

(1)

Basis of variational formulation

(3)

Directional derivative

(5)

Extremum of a functional calculus

(1)

Reissner's functional

(5)

Basic functional

(3)

Some particular cases of boundary conditions

(1)

Case of boundary conditions effects of rigidity and mass

(1)

Hamilton's functional

(2)

The basic functional

(1)

Some particular cases of boundary conditions

(1)

Approximate solutions

(1)

Euler equations associated to the extremum of a functional

(11)

Introduction and first example

(4)

Second example: vibrations of plates

(4)

Some results

(3)

Conclusion

(2)

Equation of Motion for Beams

(34)

Objective of the chapter

(1)

Hypotheses of condensation of straight beams

(2)

Equations of longitudinal vibrations of straight beams

(9)

Basic equations with mixed variables

(5)

Equations with displacement variables

(1)

Equations with displacement variables obtained by Hamilton's functional

(3)

Equations of vibrations of torsion of straight beams

(4)

Basic equations with mixed variables

(2)

Equation with displacements

(2)

Equations of bending vibrations of straight beams

(11)

Basic equations with mixed variables: Timoshenko's beam

(4)

Equations with displacement variables: Timoshenko's beam

(4)

Basic equations with mixed variables: Euler-Bernoulli beam

101

(1)

Equations of the Euler-Bernoulli beam with displacement variable

102

(2)

Complex vibratory movements: sandwich beam with a flexible inside

104

(5)

Conclusion

109

(2)

Equation of Vibration for Plates

111

(28)

Objective of the chapter

111

(1)

Thin plate hypotheses

112

(4)

General procedure

112

(1)

In plane vibrations

112

(1)

Transverse vibrations: Mindlin's hypotheses

113

(1)

Transverse vibrations: Love-Kirchhoff hypotheses

114

(1)

Plates which are non-homogenous in thickness

115

(1)

Equations of motion and boundary conditions of in plane vibrations

116

(5)

Equations of motion and boundary conditions of transverse vibrations

121

(9)

Mindlin's hypotheses: equations with mixed variables

121

(2)

Mindlin's hypotheses: equations with displacement variables

123

(1)

Love-Kirchhoff hypotheses: equations with mixed variables

124

(3)

Love-Kirchhoff hypotheses: equations with displacement variables

127

(2)

Love-Kirchhoff hypotheses: equations with displacement variables obtained using Hamilton's functional

129

(1)

Some comments on the formulations of transverse vibrations

130

(1)

Coupled movements

130

(3)

Equations with polar co-ordinates

133

(5)

Basic relations

133

(2)

Love-Kirchhoff equations of the transverse vibrations of plates

135

(3)

Conclusion

138

(1)

Vibratory Phenomena Described by the Wave Equation

139

(42)

Introduction

139

(1)

Wave equation: presentation of the problem and uniqueness of the solution

140

(5)

The wave equation

140

(2)

Equation of energy and uniqueness of the solution

142

(3)

Resolution of the wave equation by the method of propagation (d'Alembert's methodology)

145

(9)

General solution of the wave equation

145

(2)

Taking initial conditions into account

147

(4)

Taking into account boundary conditions: image source

151

(3)

Resolution of the wave equation by separation of variables

154

(14)

General solution of the wave equation in the form of separate variables

154

(3)

Taking boundary conditions into account

157

(6)

Taking initial conditions into account

163

(2)

Orthogonality of mode shapes

165

(3)

Applications

168

(10)

Longitudinal vibrations of a clamped-free beam

168

(4)

Torsion vibrations of a line of shafts with a reducer

172

(6)

Conclusion

178

(3)

Free Bending Vibration of Beams

181

(48)

Introduction

181

(1)

The problem

182

(2)

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

184

(5)

Solution

184

(2)

Interpretation of the vibratory solution, traveling waves, vanishing waves

186

(3)

Propagation in infinite beams

189

(8)

Introduction

189

(2)

Propagation of a group of waves

191

(6)

Introduction of boundary conditions: vibration modes

197

(13)

Introduction

197

(1)

The case of the supported-supported beam

197

(4)

The case of the supported-clamped beam

201

(5)

The free-free beam

206

(3)

Summary table

209

(1)

Stress-displacement connection

210

(1)

Influence of secondary effects

211

(16)

Influence of rotational inertia

212

(3)

Influence of transverse shearing

215

(6)

Taking into account shearing and rotational inertia

221

(6)

Conclusion

227

(2)

Bending Vibration of Plates

229

(50)

Introduction

229

(1)

Posing the problem: writing down boundary conditions

230

(4)

Solution of the equation of motion by separation of variables

234

(5)

Separation of the space and time variables

234

(1)

Solution of the equation of motion by separation of space variables

235

(2)

Solution of the equation of motion (second method)

237

(2)

Vibration modes of plates supported at two opposite edges

239

(15)

General case

239

(2)

Plate supported at its four edges

241

(3)

Physical interpretation of the vibration modes

244

(4)

The particular case of square plates

248

(3)

Second method of calculation

251

(3)

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

254

(9)

General issues

254

(1)

Formulation of the method

255

(4)

The plate clamped at its four edges

259

(2)

Another type of boundary conditions

261

(2)

Approximation of the mode shapes

263

(1)

Calculation of the free vibratory response following the application of initial conditions

263

(2)

Circular plates

265

(12)

Equation of motion and solution by separation of variables

265

(7)

Vibration modes of the full circular plate clamped at the edge

272

(4)

Modal system of a ring-shaped plate

276

(1)

Conclusion

277

(2)

Introduction to Damping: Example of the Wave Equation

279

(30)

Introduction

279

(2)

Wave equation with viscous damping

281

(6)

Damping by dissipative boundary conditions

287

(10)

Presentation of the problem

287

(1)

Solution of the problem

288

(6)

Calculation of the vibratory response

294

(3)

Viscoelastic beam

297

(6)

Properties of orthogonality of damped systems

303

(5)

Conclusion

308

(1)

Calculation of Forced Vibrations by Modal Expansion

309

(46)

Objective of the chapter

309

(1)

Stages of the calculation of response by modal decomposition

310

(12)

Reference example

310

(7)

Overview

317

(4)

Taking damping into account

321

(1)

Examples of calculation of generalized mass and stiffness

322

(2)

Homogenous, isotropic beam in pure bending

322

(1)

Isotropic homogenous beam in pure bending with a rotational inertia effect

323

(1)

Solution of the modal equation

324

(12)

Solution of the modal equation for a harmonic excitation

324

(6)

Solution of the modal equation for an impulse excitation

330

(2)

Unspecified excitation, solution in frequency domain

332

(1)

Unspecified excitation, solution in time domain

333

(3)

Example response calculation

336

(11)

Response of a bending beam excited by a harmonic force

336

(4)

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

340

(3)

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

343

(4)

Convergence of modal series

347

(6)

Convergence of modal series in the case of harmonic excitations

347

(3)

Acceleration of the convergence of modal series of forced harmonic responses

350

(3)

Conclusion

353

(2)

Calculation of Forced Vibrations by Forced Wave Decomposition

355

(32)

Introduction

355

(1)

Introduction to the method on the example of a beam in torsion

356

(9)

Example: homogenous beam in torsion

356

(2)

Forced waves

358

(1)

Calculation of the forced response

359

(2)

Heterogenous beam

361

(2)

Excitation by imposed displacement

363

(2)

Resolution of the problems of bending

365

(4)

Example of an excitation by force

365

(3)

Excitation by torque

368

(1)

Damped media (case of the longitudinal vibrations of beams)

369

(2)

Example

369

(2)

Generalization: distributed excitations and non-harmonic excitations

371

(8)

Distributed excitations

371

(4)

Non-harmonic excitations

375

(2)

Unspecified homogenous mono-dimensional medium

377

(2)

Forced vibrations of rectangular plates

379

(6)

Conclusion

385

(2)

The Rayleigh-Ritz Method based on Reissner's Functional

387

(22)

Introduction

387

(1)

Variational formulation of the vibrations of bending of beams

388

(3)

Generation of functional spaces

391

(1)

Approximation of the vibratory response

392

(1)

Formulation of the method

392

(5)

Application to the vibrations of a clamped-free beam

397

(9)

Construction of a polynomial base

397

(2)

Modeling with one degree of freedom

399

(3)

Model with two degrees of freedom

402

(2)

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

404

(2)

Conclusion

406

(3)

The Rayleigh-Ritz Method based on Hamilton's Functional

409

(26)

Introduction

409

(1)

Reference example: bending vibrations of beams

409

(6)

Hamilton's variational formulation

409

(2)

Formulation of the Rayleigh-Ritz method

411

(3)

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

414

(1)

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

415

(5)

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

420

(3)

Elastic boundary conditions

423

(3)

Introduction

423

(1)

The problem

423

(1)

Approximation with two terms

424

(2)

Convergence of the Rayleigh-Ritz method

426

(6)

Introduction

426

(1)

The Rayleigh quotient

426

(2)

Introduction to the modal system as an extremum of the Rayleigh quotient

428

(3)

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

431

(1)

Conclusion

432

(3)

Bibliography and Further Reading

435

(4)

Index

439

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Vibration in Continuous Media

9781905209279

1905209274

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