IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties (eBook)

Preface 6
Contents 8
Non-probabilistic and related approaches 20
Linear Bounds on an Uncertain Non-Linear Oscillator: An Info-Gap Approach 21
Introduction 21
Dynamics, Uncertainty and Robustness 22
Example: Uncertain Cubic Non-Linearity 25
Example: Multiple Uncertainties 28
Robustness as a Proxy for Probability 30
Conclusion 31
References 32
Quantification of uncertain and variable model parameters in non-deterministic analysis 33
Introduction 33
Numerical representation of parameter uncertainty and variability 34
Definitions 35
Discussion and extension of the definitions 36
Literature review on uncertain model and material data 37
Non-probabilistic models 37
Probabilistic models 38
Material data 40
Other model properties 43
Alternative approaches: non-parametric model concept and info-gap theory 43
Summary of observations 44
Conclusions 44
References 45
Vibrations of layered structures with fuzzy core stiffness/fuzzy interlayer slip 47
Introduction 47
Fuzzy sandwich beams 48
Three-layer beams 48
Modal analysis of the three-layer beam, hard-hinged support 51
Numerical results 53
Isosceles uncertainty 53
Constraints affected to the uncertain natural frequencies 56
Some effects of non-symmetric uncertainty 59
Conclusions 59
References 60
Vibration Analysis of Fluid-Filled Piping Systems with Epistemic Uncertainties 61
Introduction 62
Classification, Representation and Propagation of Uncertainty 62
Uncertainty Classification and Representation 62
Uncertainty Propagation Based on the Transformation Method 64
Fluid-Filled Piping System 66
Modeling Approach 66
Experimental Setup 69
Comprehensive Modeling and Simulation 69
Modeling of Epistemic Uncertainties 69
Simulation Results 71
Measures of Influence 72
Conclusions 73
References 73
Fuzzy vibration analysis and optimization of engineering structures: Application to Demeter satellite 75
Introduction 75
Aims of the study 76
Description of the study 76
Building of fuzzy optimization problem 77
Fuzzy vibration analysis 80
PAEM method 80
Numerical application 81
Fuzzy optimization 81
Design methodology 82
Improvement of the initial design 84
Conclusion 85
References 87
Numerical dynamic analysis of uncertain mechanical structures based on interval fields 88
Introduction 88
Interval finite element analysis 90
Interval fields 91
General concept 91
Interval fields as uncertain input parameters 92
Interval fields as uncertain analysis results 94
Application of interval fields for vibro-acoustic analysis 96
Vibro-acoustic analysis based on the ATV concept 96
Interval analysis based on structural FRF interval fields 97
Numerical example 98
Conclusions 99
References 100
From Interval Computations to Constraint-Related Set Computations: Towards Faster Estimation of Statistics and ODEs under Interval and p-Box Uncertainty 101
Formulation of the Problem 101
Interval Computations: Brief Reminder 104
Constraint-Based Set Computations 105
References 114
Dynamic Steady-State Analysis of Structures under Uncertain Harmonic Loads via Semidefinite Program 115
Introduction 115
Uncertain equations for steady state vibration 117
Governing equations 117
Uncertainty model 117
ULE in real variables 118
Bounds for complex amplitude 119
Upper bound for modulus of displacement amplitude 119
Lower bound for modulus of displacement amplitude 122
Bounds for phase angle 122
Bounds for nodal oscillation 124
Numerical experiments 124
Conclusions 126
References 127
SEA related methods and wave propagation 129
Universal eigenvalue statistics and vibration response prediction 130
Introduction 130
Eigenvalue statistics 131
The joint probability density function of the eigenvalues 131
The modal density 133
Universality of the ``local'' eigenvalue statistics 134
Application to natural frequency statistics 136
Application to response statistics 137
Fundamental concepts 137
Built-up systems: SEA 138
Built-up systems: the Hybrid method 139
Conclusions 140
References 141
Statistical Energy Analysis and the second principle of thermodynamics 143
Introduction 143
First principle of thermodynamics in SEA 144
Vibrational entropy, vibrational temperature 147
Second principle of thermodynamics in SEA 148
Entropy balance in SEA 149
Conclusion 150
Discussion 151
References 153
Modeling noise and vibration transmission in complex systems 154
Introduction 154
Complexity 155
Uncertainty 156
How much information is needed for noise and vibration design? 156
Modeling methods and frequency ranges 157
Low, mid and high frequency ranges 157
Low and High frequency modelling methods 158
The Mid-Frequency problem 159
The Hybrid FE-SEA method 160
Statistical subsystem 160
The direct and reverberant fields of a statistical subsystem 161
Ensemble average reverberant loading 162
Coupling a deterministic and statistical subsystem 162
Application examples 163
Monte Carlo simulations 164
Numerical applications 164
Industrial applications 165
Concluding remarks 166
References 167
A Power Absorbing Matrix for the Hybrid FEA-SEA Method 170
Introduction 170
Cylindrical Waves and Energy Sinks 171
The Governing Equations 171
The Cylindrical Waves 173
Constructing the Power Absorbing Matrix 175
Discretization of the Power Integral, and Matrix Assembly 175
Numerical Issues 177
Numerical Results 178
A Simple System 178
System Randomization and Subsystem Response Prediction 178
Results 179
Conclusions 180
References 182
The Energy Finite Element Method NoiseFEM 183
Introduction 183
Motivation 184
Literature Overview 185
Components of NoiseFEM 185
Power Flow Between Structural Elements 185
Transmission Coefficients 186
The Coupling Matrix 188
Diffusive Energy Transport 190
Homogeneous Structural Elements 190
Stiffened Subsystems 191
Combining transport and coupling equations 192
Discretization 193
Validation of NoiseFEM with test structures 194
Application of NoiseFEM 195
Conclusions 196
References 196
Wave transport in complex vibro-acoustic structures in the high-frequency limit 198
Introduction 198
Wave energy flow in terms of the Green function 200
Linear phase space operators and DEA 201
A numerical example: coupled two-domain systems 205
The hp-adaptive Discontinuous Galerkin Method 205
FEM compared to DEA and SEA --- results 208
Conclusions 209
References 210
Benchmark study of three approaches to propagation of harmonic waves in randomly heterogeneous elastic media 212
Introduction 212
Method of integral spectral decomposition 213
The Fokker-Planck-Kolmogorov equation 216
The Dyson integral equation 220
Concluding remarks 225
References 225
Minimum-variance-response and irreversible energy confinement 226
Average Impulse Response and the Single Case 226
MIVAR: Minimum-Variance-Response 228
Application of the theory 231
References 238
High-frequency vibrational power flows in randomly heterogeneous coupled structures 240
Introduction 240
Transport model 242
Radiative transfer in an open domain 242
Radiative transfer in a bounded domain 243
Radiative transfer with a sharp interface 244
Numerical examples 246
Coupled beams 247
Coupled shells 249
Conclusions 252
References 252
Uncertainty propagation in SEA using sensitivity analysis and Design of Experiments 254
Introduction 254
SEA equations 256
Uncertainty propagation in SEA 257
Approach using sensitivity 258
Approach using Design of Experiments 259
Results 261
Conclusions 265
References 265
Phase reconstruction for time-domain analysis of uncertain structures 266
Introduction 266
Explanation of minimum phase 267
Defining minimum phase 267
The Hilbert transform and analytic systems 267
The Hilbert Transform and minimum phase systems 268
Further interpretation of minimum phase 268
Using minimum phase reconstruction 269
Approximating the Hilbert Transform 269
Errors using MPR for non-minimum phase systems 272
Application: peak shock prediction in uncertain structures 274
Modelling an uncertain structure 275
Ensemble average results 276
Changing the correlation of modal amplitudes 277
Conclusions 278
References 278
Probabilistic Methods 279
Uncertain Linear Systems in Dynamics: Stochastic Approaches 280
Introductory Remarks 280
Overview of Available Methods 281
Response variability 282
Perturbation Method 282
Spectral methods 285
Direct Monte Carlo Simulation 288
Random matrix approach 291
Computational Efficiency 291
Summary 292
References 293
Time domain analysis of structures with stochastic material properties 296
Introduction 296
Preliminary concepts 297
Application of the perturbation approach 298
Moments of the uncertain structure response 299
Application 302
Conclusions 303
References 308
Vibration Analysis of an Ensemble of Structures using an Exact Theory of Stochastic Linear Systems 309
Introduction 309
Description of the Stochastic System 310
Expression of Mean, Variance, and Covariance 312
Parameterized Response 312
Mean Response 313
Variance and Covariance of the Responses 313
Multirank Disturbance 314
Discussion of the Theory 316
Stochastic Coefficients in the case of a Gaussian Probability Density Function 316
Application examples 317
Comparison to a Monte-Carlo Simulation 318
Transition from low to high modal density 319
Variance and covariance of responses at different frequencies 321
Conclusion 322
References 322
Structural Uncertainty Identification using Vibration Mode Shape Information 324
Introduction 324
Maximum Likelihood Estimation of Uncertain Structural Parameters 326
Uncertainty Estimation via the Perturbation Method 326
ML estimates of uncertain point-mass position statistics using natural frequency information on a cantilever beam structure 327
ML Estimation of uncertain point mass position on a plate structure using mode shape information 330
Discussion of Results 334
Conclusions 336
References 336
Extremely strong convergence of eigenvalue-density of linear stochastic dynamical systems 337
Introduction 338
Uncertainty quantification of dynamic response 339
Wishart random matrix model 340
Density of eigenvalues 342
Linear eigenvalue statistic 342
Self averaging property and the Marcenko-Pastur density 343
Numerical investigations 346
Plate with randomly inhomogeneous material properties: parametric uncertainty problem 348
Plate with randomly attached spring-mass oscillators: nonparametric uncertainty problem 349
Conclusions 349
References 350
Stochastic subspace projection schemes for dynamic analysis of uncertain systems 352
Introduction 352
Preliminaries 353
Frequency domain analysis of linear stochastic structural systems 354
Preconditioner 358
Postprocessing 359
The algebraic random eigenvalue problem 359
Stochastic Basis Vectors 360
Bubnov-Galerkin Projection 361
Postprocessing 361
Numerical Studies 362
Concluding Remarks 364
References 365
Probabilistic Methods, Applications 366
Reliability Assessment of Uncertain Linear Systems in Structural Dynamics 367
Introduction 367
Methods of Analysis 368
Representation of uncertain excitation 368
Uncertain structural systems 370
Stochastic conditional response 370
Conditional reliability 371
Design point for stochastic structural systems 372
First excursion probability for stochastic systems 374
Numerical example 376
General remarks 376
Structural system 376
Dynamic excitation 378
Critical response 379
Reliability of critical component 379
Conclusions 381
References 382
On semi-statistical method of numerical solution of integral equations and its applications 383
Introduction 383
Short scheme of semi-statistical method 384
Statement of the problem of blade cascade flow 385
Scheme of application of semi-statistical method to the problem of blade cascade flow 387
Main formulas 387
Computation algorithm and optimization 388
Results of simulations 389
Analysis of efficiency of the density adaptation 389
Conclusion 390
References 392
An efficient model of drill-string dynamics with localised non-linearities 393
Introduction 393
Theoretical Framework 395
Linear Model 395
Coupling to Non-Linearities 397
Coupling to Subsystems 399
Example Simulations 400
Linear Behaviour 400
Coupling to non-linear friction law 401
Coupling to lumped inertia 402
Uncertainty Analysis of Stick-Slip Oscillation 403
Conclusions 405
References 405
Equivalent thermo-mechanical parameters for perfect crystals 407
Introduction 407
Hypotheses 408
Kinematics 410
Equation of momentum balance 412
Equation of angular momentum balance 414
Equation of energy balance 415
Constitutive relations for stress tensor and heat flux 417
Concluding remarks 419
References 420
Analysis of offshore systems in random waves 421
Introduction 421
Modeling aspects 422
Modeling of environmental forces 423
Modeling of multibody systems 424
Analysis of deterministic systems 425
Analysis of random systems 426
Monte Carlo simulation 426
Stochastic linearization 427
Selected Results 427
Conclusions 431
References 431
Statistical Dynamics of the Rolling Mills 432
Introduction 432
Cold Rolling Mills Chatter Vibrations 434
Rolling stand design and its modal analysis 434
Strip Elasto-Plastic Deformation 436
Horizontal work rolls vibration 438
Contact friction force variation 439
Chatter detection and control 440
Hot Rolling Mills Torsional Vibrations 441
Torsional vibration control and backlashes diagnostics 442
Conclusions 443
References 443
The application of robust design strategies on managing the uncertainty and variability issues of the blade mistuning vibration problem 446
Introduction 447
Basic concepts of the blade mistuning problem 448
The Amplification Factor (its significance and range) 449
Casting blade mistuning as a robust design problem 450
The Taguchi method of robust design 451
The robust optimisation method 451
Application of robust design methods to the blade mistuning problem 452
Improving the robustness of bladed discs by parameter design 453
Improving the robustness of bladed discs by tolerance design 455
The Small Mistuning approach 456
The Intentional Mistuning approach 456
Conclusions 458
References 459
Localized modeling of uncertainty in the Arlequin framework 460
Introduction 460
The classical Arlequin method 462
The continuous stochastic-deterministic Arlequin formulation 464
The stochastic monomodel 465
The Arlequin formulation 465
The discretized stochastic-deterministic Arlequin formulation 466
Example of application 468
Conclusion 470
References 470