Dynamics of Large Structures and Inverse Problems

This book deals with the various aspects of stochastic dynamics, the resolution of large mechanical systems, and inverse problems. It integrates the most recent ideas from research and industry in the field of stochastic dynamics and optimization in structural mechanics over 11 chapters. These chapters provide an update on the various tools for dealing with uncertainties, stochastic dynamics, reliability and optimization of systems. The optimization?reliability coupling in structures dynamics is approached in order to take into account the uncertainties in the modeling and the resolution of the problems encountered.

Accompanied by detailed examples of uncertainties, optimization, reliability, and model reduction, this book presents the newest design tools. It is intended for students and engineers and is a valuable support for practicing engineers and teacher-researchers.

Preface xi

Chapter 1 Introduction to Inverse Methods 1

1.1 Introduction 1

1.2 Identification methods 3

1.3 Identification of the strain hardening law 6

1.3.1 Example of an application 8

1.3.2 Validation test 9

1.3.3 Hydroforming a welded tube 11

Chapter 2 Linear Differential Equation Systems of the First Order with Constant Coefficients: Application in Mechanical Engineering 15

2.1 Introduction 15

2.2 Modeling dissipative systems 15

2.2.1 Intrinsic solutions of autonomous systems 17

2.2.2 Intrinsic solutions 17

2.2.3 Intrinsic solutions of the adjoining system 19

2.2.4 Relation between the intrinsic solutions of s and s* 19

2.2.5 Relation between modal matrices X and X* 20

2.3 Autonomous system general solution 21

2.3.1 Direct solution by using the exponential matrix 21

2.3.2 Indirect solution by modal transformation 23

2.4 General solution of the complete equation 24

2.4.1 Direct solution by the exponential matrix 24

2.4.2 Indirect solution by modal transformation 24

2.4.3 General solution in the particular case of harmonic excitation 26

2.5 Applications to mechanical structures 27

2.5.1 Discrete mechanical structure at n degrees of freedom, linear, regular and non-dissipative 27

2.5.2 Discrete mechanical structure at n DOF, linear, regular and dissipative 29

2.5.3 Intrinsic vector norm 32

2.5.4 Particular solution of the system with a harmonic force 34

2.6 Inverse problems: expressions of the M, B, K matrices according to the intrinsic solutions 36

Chapter 3 Introduction to Linear Structure Dynamics 41

3.1 Introduction 41

3.2 Problems in structure dynamics 41

3.2.1 Finite elements method 43

3.2.2 Modal superposition method 44

3.2.3 Direct integration 46

3.2.4 Newmark method 46

3.2.5 The θ Wilson method 47

3.2.6 Modal analysis of the sandwich beam 49

Chapter 4 Introduction to Nonlinear Dynamic Analysis 53

4.1 Introduction 53

4.2 Linear systems 54

4.2.1 Generalities 54

4.2.2 Simple examples of large displacements 56

4.2.3 Simple example of a variable 58

4.2.4 Simple example of dry friction 58

4.2.5 Material nonlinearities 59

4.3 The nonlinear 1 DOF system 60

4.3.1 Generalities 60

4.3.2 Movement without non-dampened excitation 61

4.3.3 Case of a stiffness in the form � (1 + �� 2) 62

4.3.4 Movement with non-dampened excitation 65

4.3.5 Movement with dampened excitation 68

4.4 Nonlinear N DOF systems 71

4.4.1 Generalities 71

4.4.2 Nonlinear connection with periodic movement 72

4.4.3 Direct integration of the equations 74

Chapter 5 Condensation Methods Applied to Eigen Value Problems 77

5.1 Introduction 77

Contents vii

5.2 Mathematical generality: matrix transformation 78

5.3 Dynamic condensation methods 80

5.4 Guyan condensation 84

5.5 Rayleigh–Ritz method 87

5.6 Case of a temporary problem 90

5.6.1 Simplification with a full modal basis 91

Chapter 6 Linear Substructure Approach for Dynamic Analysis 105

6.1 Generalities 105

6.2 Different types of Ritz vectors 107

6.2.1 Stress vectors of the j st substructure �� (j) 107

6.2.2 Attachment vectors of the j st substructure �� (j) 108

6.2.3 Displacement field type vectors in dynamic regimes 108

6.3 Synthesis of eigen solutions of the assembled structure: formulation by an energetic method (Lagrange with multiplicators) 111

6.3.1 Equilibrium equation of the k st isolated substructure �� (k) 111

6.3.2 Ritz basis for the k the substructure �� (k) 112

6.3.3 Compatibilities between substructure �� (1) and �� (2) 113

6.3.4 Lagrangian L of the assembled structure 113

6.4 Craig and Bampton substructuration method 116

6.4.1 Formulation of base relations in the case of two substructures 117

6.4.2 Assembly of two substructures 119

6.4.3 Restoring physical DOF 120

6.4.4 Comments 121

6.5 Mixed method 121

6.5.1 Formation in the case of a single secondary SS 122

6.5.2 Reconstructing the assembled structure 122

6.5.3 Comments 123

6.6 Methods with eigen vectors with free common contours 124

6.6.1 Stiffness method of coupling 124

6.6.2 Solution to [6.39] Ritz transformation 127

6.6.3 Formulation based on the dynamic flexibility matrices: search for the assembled structure’s eigen solutions 129

6.6.4 Formulation in the case of two �� (k) , k = 1,2, etc 130

6.7 Method systematically introducing an intermediary connection structure 133

6.7.1 Formation 133

6.7.2 Introducing Ritz vectors 136

6.7.3 Introducing fitting conditions 137

6.7.4 Equilibrium equations of the assembled structure 139

6.7.5 Normalization of the assembled structure’s eigen vectors 140

6.7.6 Critique of the method 141

Chapter 7 Nonlinear Substructure Approach for Dynamic Analysis 145

7.1 Introduction 145

7.2 Dynamic substructuration approaches 147

7.2.1 Linear case 148

7.2.2 Nonlinear case 149

7.3 Nonlinear substructure approach 151

7.3.1 Vibration equations of a substructure 152

7.3.2 Fixed interface problem 153

7.3.3 Static raising problem 155

7.3.4 Representation of the system in Craig-Bampton’s linear base 155

7.3.5 Model reduction with the Shaw and Pierre approach 157

7.3.6 Assembling substructures 159

7.4 Proper orthogonal decomposition for flows 160

7.4.1 Properties of the POD modes 161

7.4.2 POD snapshot 162

7.4.3 Script of low-order dynamic systems 163

7.5 Numerical results 168

7.5.1 Modal analysis 171

7.5.2 Decomposition of the circular acoustic cavity 173

7.5.3 Decomposition of the elastic ring 174

Chapter 8 Direct and Inverse Sensitivity 177

8.1 Introduction 177

8.2 Direct sensitivity 180

8.2.1 Definition of the state’s sensitivity matrix x(t) 180

8.2.2 Sensitivity equations 180

8.2.3 Simple direct applications 182

8.3 Sensitivity of eigen solutions 183

8.3.1 Direct numerical method 183

8.3.2 Derivatives of the eigen vectors according to the modal bases 184

8.3.3 Derivatives of eigen vectors based on the exact expressions 187

8.4 First derivative of a particular solution 190

8.4.1 Scalar case (primarily didactic) 190

8.4.2 General case 190

8.5 Grouping the sensitivity relations together 191

8.5.1 Variations 191

8.5.2 Grouping the eigen values and eigen vectors together 192

8.6 Inverse sensitivity 195

8.6.1 Overdetermined case: 2a > m 196

8.6.2 Unique solution: 2a = m 197

8.6.3 Underdetermined case: 2a < m 198

Chapter 9 Parametric Identification and Model Adjustment in Linear Elastic Dynamics 205

9.1 Introduction 205

9.2 Study in the elastic dynamics of mechanical structures 206

9.2.1 Provisional calculations of behavior based on mathematical models 207

9.2.2 Identification 207

9.3 Parametric identification – use of a test for constructing weaker calculation models 208

9.3.1 Introduction 208

9.3.2 Error minimization in the behavioral equation 209

9.3.3 Error minimization on the outputs 210

9.3.4 Combined estimation of the state and the parameters 211

9.4 Some basic methods in parametric identification 211

9.4.1 Linear dependency with respect to the parameters and estimation in the sense of the least squares 211

9.4.2 Estimation of parameters in the sense of maximum likelihood 212

9.4.3 Estimation of the vector p by the Gauss–Newton method Bayes formulation Vector z(p) nonlinear function of p 214

9.4.4 Non-random least squares method 218

9.4.5 Quasi-linearization method 220

9.5 Parametric correction of finite elements models in linear elastic dynamics based on the test results 221

9.5.1 Highlighting a few difficulties 222

9.6 M model adjustment: k∈ � c, c by minimizing the matrix norms by the correction matrices δm, δk 223

9.6.1 Principle of Baruch and Bar-Itzhack method 224

9.6.2 Kabe, Smith and Beattie methods 226

9.7 M model adjustment: k∈ � c, c by minimizing residue vectors made up based on local correction matrices ΔM I , ΔK I 227

9.7.1 Minimization of formed residue based on the behavior equation 228

9.7.2 Minimization of formed reside based on outputs 228

Chapter 10 Inverse Problems in Dynamics: Robustness Function 235

10.1 Introduction 235

10.2 Convex models 236

10.2.1 Definitions 236

10.2.2 Direct problem 237

10.2.3 Inverse problem 237

10.3 Robustness function 238

10.3.1 Monocriterion response 238

10.3.2 Multicriteria response 238

10.4 Solution methods 239

10.4.1 Interval arithmetic 239

10.4.2 Optimization method 240

10.5 Numerical calculations 244

10.6 Applications 245

10.6.1 Dual-recessed beam 245

10.6.2 Square 251

10.7 Conclusion 256

Chapter 11 Modal Synthesis and Reliability Optimization Methods 259

11.1 Introduction 259

11.2 Design reliability optimization in structural dynamics 260

11.2.1 Frequential hybrid method 260

11.2.2 Optimization condition of the hybrid problem 266

11.3 The SP method 270

11.3.1 Formulation of the problem 271

11.3.2 Implementation of the SP approach 273

11.4 Modal synthesis and RBDO coupling methods 281

11.5 Discussion 286

Appendix 289

Bibliography 299

Index 307

Bouchaib Radi, Setta Hassan First University, Morocco.