Structural Dynamics in Industry

Foreword xiii

Preface xv

Introduction xvii

Glossary xxiii

Chapter 1. General Introduction to Linear Analysis 1

1.1. Introduction 1

1.2. Motion types .2

1.2.1. Sine motion 2

1.2.1.1. Pure sine 2

1.2.1.2. Swept sine 4

1.2.1.3. Periodic motion 5

1.2.2. Transient motion 5

1.2.3. Random motion. 7

1.2.3.1. Random process 7

1.2.3.2. Time analysis 8

1.2.3.3. Statistical analysis 9

1.2.3.4. Power spectral densities 12

1.3. Time domain and frequency domain 14

1.3.1. Introduction 14

1.3.2. The time domain 15

1.3.3. The frequency domain 16

1.4. Frequency Response Functions 17

1.4.1. Introduction 17

1.4.2. Frequency Response Functions and responses 18

1.4.3. The nature of excitations and responses 19

1.4.4. The nature of Frequency Response Functions 21

1.5. Equations of motion and solution. 24

1.5.1. Equations of motion .24

1.5.2. Solution using the direct frequency approach 26

1.5.3. Solution using the modal approach 27

1.5.4. Modes and 1-DOF system 28

1.6. Analysis and tests 29

Chapter 2. The Single-Degree-of-Freedom System 33

2.1. Introduction 33

2.2. The equation of motion and the solution in the frequency domain 35

2.2.1. Equations of motion 35

2.2.2. Motion without excitation 35

2.2.2.1. The conservative system 35

2.2.2.2. Dissipative system 37

2.2.3. Solution in the frequency domain 39

2.2.4. Dynamic amplifications 43

2.2.5. Response to a random excitation 49

2.3. Time responses. 51

2.3.1. Response to unit impulse 51

2.3.2. Response to a general excitation 55

2.3.3. Response spectra 56

2.4. Representation of the damping 61

2.4.1. Viscous damping 61

2.4.2. Structural damping 62

2.4.3. Other representations 64

Chapter 3. Multiple-Degree-of-Freedom Systems 65

3.1. Introduction 65

3.2. Determining the structural matrices 66

3.2.1. Introduction 66

3.2.2. Local element matrices 67

3.2.3. Element matrices in global reference form 68

3.2.4. Assembly of element matrices 70

3.2.5. Linear constraints between DOF 72

3.2.5.1. Introduction 72

3.2.5.2. DOF elimination 73

3.2.5.3. DOF introduction. 77

3.2.6. Excitation forces 79

3.3. The finite element method 80

3.3.1. Introduction 80

3.3.2. The rod element 82

3.3.3. Beam finite element in bending 83

3.3.4. The complete beam finite element 86

3.3.5. Excitation forces 88

3.4. Industrial models 89

3.4.1. Introduction 89

3.4.2. The element types 89

3.4.3. Linear constraints 91

3.4.4. DOF management 91

3.4.5. Rules for modeling and verification of the model 93

3.4.6. Industrial examples 94

3.5. Solution by direct integration 95

3.5.1. Introduction 95

3.5.2. Example of explicit method 96

3.5.3. Example of implicit method 97

Chapter 4. The Modal Approach 99

4.1. Introduction 99

4.2. Normal modes 100

4.2.1. Introduction 100

4.2.2. Free structures 104

4.2.3. System static condensation 108

4.2.4. Eigenvalue problem solution 111

4.3. Mode superposition 115

4.3.1. Introduction 115

4.3.2. Equation of motion transformation 117

4.3.3. Problem caused by the damping 119

4.3.4. Frequency resolution 122

4.4. From the frequency approach to the modal approach 126

Chapter 5. Modal Effective Parameters  129

5.1. Introduction 129

5.2. Effective modal parameters and truncation 130

5.2.1. Definition of the effective modal parameters 130

5.2.2. Summation rules 133

5.2.2.1. Direct summation. 133

5.2.2.2. Flexibilities in the presence of rigid modes 134

5.2.2.3. Transmissibilities and effective masses by zones 137

5.2.2.4. Other summation rules 139

5.2.3. Correction of the truncation effects 140

5.3. Particular case of a statically determined structure 143

5.3.1. Introduction 143

5.3.2. Effective mass models 145

5.4. Modal effective parameters and dynamic responses 153

5.4.1. Frequency responses 153

5.4.2. Random responses 157

5.4.3. Time responses 159

5.4.4. Time response extrema 159

5.5. Industrial examples 161

Chapter 6. Continuous Systems 169

6.1. Introduction 169

6.2. The rod element 171

6.2.1. Introduction 171

6.2.2. Clamped-free rod 173

6.2.3. Free-free rod 178

6.2.4. Clamped-clamped rod 182

6.3. Bending beam element 184

6.3.1. Introduction 184

6.3.2. Clamped-free beam. 188

6.3.3. Free-free beam 193

6.3.4. Clamped-clamped beam 199

6.3.5. Shear and rotary inertia effects 204

6.4. Plate element 206

6.4.1.  Introduction  206

6.4.2. Some plate results in bending 207

6.4.3. Simply supported rectangular plate 208

6.5. Combined cases 210

6.5.1. Introduction 210

6.5.2. Combination rod + local mass or flexibility 213

6.5.3. Some typical results 215

Chapter 7. Complex Modes 219

7.1. Introduction 219

7.2. Dissipative systems 220

7.2.1. Complex modes. 220

7.2.2. Mode superposition 224

7.2.3. Modal effective parameters and dynamic amplifications 226

7.2.4. Simple example 229

7.3. Gyroscopic effects 232

7.3.1. Introduction 232

7.3.2. Mode superposition 234

7.4. A more general case 236

7.4.1. Introduction 236

7.4.2. Complex modes 237

7.4.3. Mode superposition 240

7.4.4. Modal effective parameters and dynamic amplifications 242

7.5. Applications 245

7.5.1. Simple example 245

7.5.2. Industrial case 248

Chapter 8. Modal Synthesis 249

8.1. Introduction 249

8.2. General approach 251

8.2.1. Analysis of substructures 251

8.2.2. Coupling of substructures 253

8.2.3. Recovery 255

8.3. Choice of mode 256

8.3.1. Introduction 256

8.3.2. Boundary conditions 258

8.3.3. Normal modes 259

8.3.4. Static flexibilities 260

8.3.5. Junction modes 262

8.3.6. Illustration 263

8.3.7. Possible combinations 265

8.4. Some methods 266

8.4.1. Craig-Bampton method 266

8.4.2. Craig-Chang method 271

8.4.3. Benfield-Hruda method 276

8.4.4. Effective mass models 281

8.4.5. Reduced models 283

8.5. Case study 287

8.5.1. Benfield-Hruda truss 287

8.5.2. Industrial cases 290

Chapter 9. Frequency Response Synthesis 295

9.1. Introduction 295

9.2. Frequency Response Functions 296

9.2.1. FRF and other dynamic characteristics 296

9.2.2. Transformation of the FRF 298

9.2.3. Simple examples 299

9.3. Coupling by FRF 301

9.3.1. FRF necessary for coupling 301

9.3.2. Solution of the coupling 303

9.3.3. Recovery 304

9.3.4. Summary 305

9.4. The basic cases 306

9.4.1. Introduction 306

9.4.2. Free substructures at the connections 306

9.4.3. Substructures constrained at the connections 308

9.4.4. Mixed conditions at the connections 309

9.5. Generalization 310

9.5.1. Introduction 310

9.5.2. Stiffness approach 311

9.5.3. Flexibility approach 312

9.5.4. Comparison of the two approaches 314

9.5.5. Particular cases 317

9.6. Comparison with other substructuring techniques 318

9.6.1. The matrix level 318

9.6.2. The modal level 319

9.6.3. The frequency response level 320

9.6.4. Conclusion 321

Chapter 10. Introduction to Non-linear Analysis 323

10.1. Introduction 323

10.2. Non-linear systems 324

10.2.1. Introduction 324

10.2.2. Simple examples of large displacements 326

10.2.3. Simple example of variable link 328

10.2.4. Simple example of dry friction 328

10.2.5. Material non-linearities 329

10.3. Non-linear 1-DOF system 329

10.3.1. Introduction 329

10.3.2. Undamped motion without excitation 331

10.3.3. Case of a stiffness of form k (1 x 2 ) 332

10.3.4. Undamped motion with excitation 336

10.3.5. Damped motion with excitation 340

10.4. Non-linear N-DOF systems 343

10.4.1. Introduction 343

10.4.2. Non-linear link with periodic motion 344

10.4.3. Direct integration of equations 346

Chapter 11. Testing Techniques 349

11.1. Introduction 349

11.2. Dynamic tests 350

11.2.1. Development plan of a structure 350

11.2.2. Types of tests 352

11.2.3. Test hardware 353

11.3. The identification tests 358

11.3.1. Introduction 358

11.3.2. Modal parameters to be identified 359

11.3.3. Phase resonance modal tests 362

11.3.4. Phase separation modal tests 364

11.3.5. Extraction of modal parameters 366

11.3.6. Single DOF (SDOF) methods 368

11.3.7. Multi-DOF (MDOF) methods 370

11.4. Simulation tests 372

11.4.1. Introduction 372

11.4.2. Tests with shakers 373

11.4.3. Shock device tests 375

11.4.4. The tests in a reverberant acoustic chamber 376

11.4.5. Elaboration of specifications 377

11.4.6. Impact of a structure on its environment 379

Chapter 12. Model Updating and Optimization 385

12.1. Introduction 385

12.2. Sensitivity analysis 387

12.2.1. Introduction 387

12.2.2. Sensitivity of the natural frequencies 388

12.2.3. Sensitivity of the eigenvectors 388

12.2.4. Sensitivity of the modal effective parameters 389

12.2.5. Simple example 390

12.3. Ritz reanalysis 392

12.3.1. Introduction 392

12.3.2. Utilization of the normal modes 392

12.3.3. Utilization of additional modes 393

12.3.4. Simple example 393

12.4. Model updating 395

12.4.1. Physical parameters 395

12.4.2. Test/analysis correlation 398

12.4.3. Updating procedure 400

12.5. Optimization processes 401

12.5.1. Introduction 401

12.5.2. Non-linear optimization methods 402

12.5.3. Non-linear simplex method 403

12.6. Applications 404

12.6.1. Optimization of a simple system 404

12.6.2. Updating a simple system 405

12.6.3. Industrial case 407

Bibliography 411

Index 417