This book is dedicated to the general study of the dynamics of mechanical structures with consideration of uncertainties. The goal is to get the appropriate forms of a part in minimizing a given criterion. In all fields of structural mechanics, the impact of good design of a room is very important to its strength, its life and its use in service. The development of the engineer's art requires considerable effort to constantly improve structural design techniques.

Abdelkhalak El Hami is Professor at the Institut National des Sciences Appliquées, Rouen, France. He is the author of many articles and books on optimization and uncertainty.

Bouchaib Radi is Professor in the Faculty of Sciences and Technology at the University of Hassan Premier, Settat, Morocco. His research interests are in such areas as structural optimization, parallel computation, contact problem and metal forming. He is the author of many scientific articles and books.

Chapter 1. Introduction to Structural Dynamics  1

1.1. Composition of problems relating to dynamic structures 2

1.1.1. Finite element method  4

1.1.2. Modal superposition method 5

1.1.3. Direct integration 6

1.2. Structural optimization  8

1.2.1. Design optimization 9

1.2.2. Shape optimization  9

1.2.3. Topological optimization 10

1.2.4. Definitions and formulation of an optimization problem  12

1.3. Structures with uncertain parameters 13

1.3.1. Monte Carlo simulation 14

1.3.2. Analytic method  15

1.3.3. Stochastic finite element method  15

1.3.4. Fluid logic method  16

1.3.5. Reliability method  17

1.3.6. Reliability optimization 21

1.4. Conclusion 23

Chapter 2. Decoupled Systems 25

2.1. Introduction  25

2.2. Problems with structural dynamics  25

2.2.1. Movement equation  25

2.2.2. Hooke’s law  26

2.2.3. Variational formulation  27

2.2.4. Estimation by finite elements  27

2.2.5. Resolution in the frequency domain 28

2.2.6. Solution in the temporal domain 29

2.2.7. Reduction of the model  31

2.3. Acoustic problems 42

2.3.1. Wave equation: formulation pressure  42

2.3.2. Variational formulation  43

2.3.3. Estimation by finite elements  43

2.3.4. Solution in the frequency domain  44

2.3.5. Model fluid reduction 45

2.4. Conclusion 55

Chapter 3. Coupled Systems  57

3.1. Introduction  57

3.2. Mathematical formulation 57

3.2.1. Behavior equations  57

3.2.2. Conditions for fluid–structure coupling 58

3.3. Variational formulation  59

3.4. Estimation by finite elements 59

3.4.1. Estimation of unknown physical values 59

3.4.2. Integration of variational forms 60

3.5. Vibro-acoustic problem  60

3.6. Hydro-elastic problem 61

3.6.1. Calculation of the elementary matrix of the fluid–structure interaction  64

3.6.2. Dynamic study  65

3.7. Reduction of the model  67

3.7.1. Modal superposition method for the paired system 67

3.7.2. Direct calculation 71

3.7.3. Calculation with modal reduction  72

3.7.4. Modal synthesis method for paired systems  74

3.7.5. Direct numerical calculation 81

3.7.6. Numerical calculation with modal superposition  83

3.8. Conclusion 84

Chapter 4. Reliability and Meshless Methods in Mechanics  85

4.1. Introduction to non-networking methods  85

4.2. Moving least squares 88

4.2.1. Properties of MLS form function  94

4.2.2. Base functions 95

4.2.3. Weight functions 96

4.3. Galerkin mesh-free method  98

4.4. Imposition of essential limiting conditions 103

4.4.1. Variational principle modified with Lagrange multipliers 103

4.4.2. Variational principle modified without Lagrange multipliers 104

4.4.3. Variational principle with a charge 105

4.4.4. Connection with meshing of finite elements  106

4.5. Integration in the EFG method  107

4.6. Description of EFG method algorithms 109

Chapter 5. Mechanical Systems with Uncertain Parameters 115

5.1. Introduction  115

5.2. Monte Carlo simulation  116

5.3. Disturbance methods 116

5.3.1. Expansion into a second-order Taylor series  118

5.3.2. Muscolino distortion method  124

5.3.3. Disturbance methods and modal reduction methods  127

5.4. Projection onto polynomial chaos  131

5.4.1. Moments of the response function in frequency 134

5.4.2. Moments of dynamic response 135

5.4.3. Projection onto polynomial chaos with modal reduction  137

5.5. Conclusion 149

Chapter 6. Modal Synthesis Methods and Stochastic Finite Element Methods  151

6.1. Introduction  151

6.2. Linear dynamic problems 152

6.2.1. Equations of motion 152

6.2.2. Solutions in the transient regime 153

6.2.3. Solutions in the harmonic regime  154

6.3. Modal synthesis methods 155

6.3.1. Introduction  155

6.3.2. Sub-structure assembly technique  157

6.3.3. Fixed interface method  158

6.3.4. MacNeal’s free interface method  161

6.3.5. Free interface method 163

6.3.6. Hybrid method  166

6.3.7. Reduction in degrees of freedom of the interface 166

6.4. Stochastic finite element methods  168

6.4.1. Introduction  168

6.4.2. Discretization of random fields 169

6.4.3. Methods of moments 171

6.5. Conclusion 179

Chapter 7. Stochastic Modal Synthesis Methods  181

7.1. Introduction  181

7.2. Taylor series expansion of the modal equations of a stochastic structure 181

7.2.1. Expression of the mean values and covariances  184

7.3. Muscolino perturbation method 184

7.3.1. Expansion of the modal equations of a stochastic structure 185

7.4. Stochastic fixed interface method  186

7.4.1. Taylor series expansion  186

7.5. Stochastic modal synthesis method 191

7.5.1. Introduction  191

7.6. Conclusion 236

Chapter 8. Dynamic Response of a Structure with Uncertain Variables to a Given Excitation 237

8.1. Introduction  237

8.2. Perturbation method  238

8.2.1. Taylor series expansion of the equations of motion 238

8.2.2. Muscolino perturbation method 241

8.3. Stochastic modal synthesis method 242

8.4. Projection onto homogeneous chaos 245

8.5. Coupling modal synthesis methods with projection onto homogeneous chaos 248

8.6. Conclusion 264

Chapter 9. Stochastic Frequency Response Function  265

9.1. Introduction  265

9.2. Calculation of the stochastic frequency response function 266

9.3. Calculation of the stochastic frequency response function with modal synthesis methods  270

9.4. Conclusion 281

Chapter 10. Modal Synthesis Methods and Reliability Optimization Methods  283

10.1. Introduction  283

10.2. Combining modal synthesis and RBDO methods 283

10.3. Conclusion  294

Chapter 11. Stochastic Model of Transmission in a Wind Turbine  295

11.1. Introduction  295

11.2. Modeling the dynamic behavior of the gearing system in a wind turbine 295

11.3. Dynamic response of a two-step gear system in a wind turbine with uncertain variables  296

11.3.1. Dynamic model of a two-step wind turbine transmission  296

11.3.2. Study using the polynomial chaos method  299

11.3.3. Perturbation method study 309

11.3.4. Comparison of the different methods 315

11.4. Conclusion  317

Bibliography 319

Index 327