Belyaev / Langley IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties

Proceedings of the IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties held in St. Petersburg, Russia, July 5–9, 2009
1. Auflage 2010
ISBN: 978-94-007-0289-9
Verlag: Springer Netherland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 27, 470 Seiten, eBook

Reihe: IUTAM Bookseries

The Symposium was aimed at the theoretical and numerical problems involved in modelling the dynamic response of structures which have uncertain properties due to variability in the manufacturing and assembly process, with automotive and aerospace structures forming prime examples. It is well

known that the difficulty in predicting the response statistics of such structures is immense, due to the complexity of the structure, the large number of variables which might be uncertain, and the inevitable lack of data regarding

the statistical distribution of these variables.

The Symposium participants presented the latest thinking in this very active research area, and novel techniques were presented covering the full frequency spectrum of low, mid, and high frequency vibration problems. It was demonstrated that for high frequency vibrations the response statistics

can saturate and become independent of the detailed distribution of the uncertain system parameters. A number of presentations exploited this physical behaviour by using and extending methods originally developed in both

phenomenological thermodynamics and in the fields of quantum mechanics and random matrix theory.

For low frequency vibrations a number of presentations focussed on parametric uncertainty modelling (for example, probabilistic models, interval analysis, and fuzzy descriptions) and on methods of propagating this uncertainty through a large dynamic model in an effi cient way. At mid frequencies

the problem is mixed, and various hybrid schemes were proposed.

It is clear that a comprehensivesolution to the problem of predicting the vibration response of uncertain structures across the whole frequency range requires expertise across a wide range of areas (including probabilistic and non-probabilistic methods, interval and info-gap analysis, statistical energy analysis, statistical thermodynamics, random wave approaches, and large

scale computations) and this IUTAM symposium presented a unique opportunity to bring together outstanding international experts in these fields.

Zielgruppe

Research

Autoren/Hrsg.

Belyaev, Alexander K.

Langley, Robin S.

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;Non-probabilistic and related approaches;20
3.1;Linear Bounds on an Uncertain Non-Linear Oscillator: An Info-Gap Approach;21
3.1.1;Introduction;21
3.1.2;Dynamics, Uncertainty and Robustness;22
3.1.3;Example: Uncertain Cubic Non-Linearity;25
3.1.4;Example: Multiple Uncertainties;28
3.1.5;Robustness as a Proxy for Probability;30
3.1.6;Conclusion;31
3.1.7;References;32
3.2;Quantification of uncertain and variable model parameters in non-deterministic analysis;33
3.2.1;Introduction;33
3.2.2;Numerical representation of parameter uncertainty and variability;34
3.2.2.1;Definitions;35
3.2.2.2;Discussion and extension of the definitions;36
3.2.3;Literature review on uncertain model and material data;37
3.2.3.1;Non-probabilistic models;37
3.2.3.2;Probabilistic models;38
3.2.3.3;Material data;40
3.2.3.4;Other model properties;43
3.2.3.5;Alternative approaches: non-parametric model concept and info-gap theory;43
3.2.3.6;Summary of observations;44
3.2.4;Conclusions;44
3.2.5;References;45
3.3;Vibrations of layered structures with fuzzy core stiffness/fuzzy interlayer slip;47
3.3.1;Introduction;47
3.3.2;Fuzzy sandwich beams;48
3.3.2.1;Three-layer beams;48
3.3.2.2;Modal analysis of the three-layer beam, hard-hinged support;51
3.3.3;Numerical results;53
3.3.3.1;Isosceles uncertainty;53
3.3.3.2;Constraints affected to the uncertain natural frequencies;56
3.3.3.3;Some effects of non-symmetric uncertainty;59
3.3.4;Conclusions;59
3.3.5;References;60
3.4;Vibration Analysis of Fluid-Filled Piping Systems with Epistemic Uncertainties;61
3.4.1;Introduction;62
3.4.2;Classification, Representation and Propagation of Uncertainty;62
3.4.2.1;Uncertainty Classification and Representation;62
3.4.2.2;Uncertainty Propagation Based on the Transformation Method;64
3.4.3;Fluid-Filled Piping System;66
3.4.3.1;Modeling Approach;66
3.4.3.2;Experimental Setup;69
3.4.4;Comprehensive Modeling and Simulation;69
3.4.4.1;Modeling of Epistemic Uncertainties;69
3.4.4.2;Simulation Results;71
3.4.4.3;Measures of Influence;72
3.4.5;Conclusions;73
3.4.6;References;73
3.5;Fuzzy vibration analysis and optimization of engineering structures: Application to Demeter satellite;75
3.5.1;Introduction;75
3.5.2;Aims of the study;76
3.5.2.1;Description of the study;76
3.5.2.2;Building of fuzzy optimization problem;77
3.5.3;Fuzzy vibration analysis;80
3.5.3.1;PAEM method;80
3.5.3.2;Numerical application;81
3.5.4;Fuzzy optimization;81
3.5.4.1;Design methodology;82
3.5.4.2;Improvement of the initial design;84
3.5.5;Conclusion;85
3.5.6;References;87
3.6;Numerical dynamic analysis of uncertain mechanical structures based on interval fields;88
3.6.1;Introduction;88
3.6.2;Interval finite element analysis;90
3.6.3;Interval fields;91
3.6.3.1;General concept;91
3.6.3.2;Interval fields as uncertain input parameters;92
3.6.3.3;Interval fields as uncertain analysis results;94
3.6.4;Application of interval fields for vibro-acoustic analysis;96
3.6.4.1;Vibro-acoustic analysis based on the ATV concept;96
3.6.4.2;Interval analysis based on structural FRF interval fields;97
3.6.4.3;Numerical example;98
3.6.5;Conclusions;99
3.6.6;References;100
3.7;From Interval Computations to Constraint-Related Set Computations: Towards Faster Estimation of Statistics and ODEs under Interval and p-Box Uncertainty;101
3.7.1;Formulation of the Problem;101
3.7.2;Interval Computations: Brief Reminder;104
3.7.3;Constraint-Based Set Computations;105
3.7.4;References;114
3.8;Dynamic Steady-State Analysis of Structures under Uncertain Harmonic Loads via Semidefinite Program;115
3.8.1;Introduction;115
3.8.2;Uncertain equations for steady state vibration;117
3.8.2.1;Governing equations;117
3.8.2.2;Uncertainty model;117
3.8.2.3;ULE in real variables;118
3.8.3;Bounds for complex amplitude;119
3.8.3.1;Upper bound for modulus of displacement amplitude;119
3.8.3.2;Lower bound for modulus of displacement amplitude;122
3.8.3.3;Bounds for phase angle;122
3.8.4;Bounds for nodal oscillation;124
3.8.5;Numerical experiments;124
3.8.6;Conclusions;126
3.8.7;References;127
4;SEA related methods and wave propagation;129
4.1;Universal eigenvalue statistics and vibration response prediction;130
4.1.1;Introduction;130
4.1.2;Eigenvalue statistics;131
4.1.2.1;The joint probability density function of the eigenvalues;131
4.1.2.2;The modal density;133
4.1.2.3;Universality of the ``local'' eigenvalue statistics;134
4.1.2.4;Application to natural frequency statistics;136
4.1.3;Application to response statistics;137
4.1.3.1;Fundamental concepts;137
4.1.3.2;Built-up systems: SEA;138
4.1.3.3;Built-up systems: the Hybrid method;139
4.1.4;Conclusions;140
4.1.5;References;141
4.2;Statistical Energy Analysis and the second principle of thermodynamics;143
4.2.1;Introduction;143
4.2.2;First principle of thermodynamics in SEA;144
4.2.3;Vibrational entropy, vibrational temperature;147
4.2.4;Second principle of thermodynamics in SEA;148
4.2.5;Entropy balance in SEA;149
4.2.6;Conclusion;150
4.2.7;Discussion;151
4.2.8;References;153
4.3;Modeling noise and vibration transmission in complex systems;154
4.3.1;Introduction;154
4.3.1.1;Complexity;155
4.3.1.2;Uncertainty;156
4.3.1.3;How much information is needed for noise and vibration design?;156
4.3.2;Modeling methods and frequency ranges;157
4.3.2.1;Low, mid and high frequency ranges;157
4.3.2.2;Low and High frequency modelling methods;158
4.3.2.3;The Mid-Frequency problem;159
4.3.3;The Hybrid FE-SEA method;160
4.3.3.1;Statistical subsystem;160
4.3.3.2;The direct and reverberant fields of a statistical subsystem;161
4.3.3.3;Ensemble average reverberant loading;162
4.3.3.4;Coupling a deterministic and statistical subsystem;162
4.3.4;Application examples;163
4.3.4.1;Monte Carlo simulations;164
4.3.4.2;Numerical applications;164
4.3.4.3;Industrial applications;165
4.3.5;Concluding remarks;166
4.3.6;References;167
4.4;A Power Absorbing Matrix for the Hybrid FEA-SEA Method;170
4.4.1;Introduction;170
4.4.2;Cylindrical Waves and Energy Sinks;171
4.4.2.1;The Governing Equations;171
4.4.2.2;The Cylindrical Waves;173
4.4.3;Constructing the Power Absorbing Matrix;175
4.4.3.1;Discretization of the Power Integral, and Matrix Assembly;175
4.4.3.2;Numerical Issues;177
4.4.4;Numerical Results;178
4.4.4.1;A Simple System;178
4.4.4.2;System Randomization and Subsystem Response Prediction;178
4.4.4.3;Results;179
4.4.5;Conclusions;180
4.4.6;References;182
4.5;The Energy Finite Element Method NoiseFEM;183
4.5.1;Introduction;183
4.5.1.1;Motivation;184
4.5.1.2;Literature Overview;185
4.5.2;Components of NoiseFEM;185
4.5.3;Power Flow Between Structural Elements;185
4.5.3.1;Transmission Coefficients;186
4.5.3.2;The Coupling Matrix;188
4.5.4;Diffusive Energy Transport;190
4.5.4.1;Homogeneous Structural Elements;190
4.5.4.2;Stiffened Subsystems;191
4.5.5;Combining transport and coupling equations;192
4.5.6;Discretization;193
4.5.7;Validation of NoiseFEM with test structures;194
4.5.8;Application of NoiseFEM;195
4.5.9;Conclusions;196
4.5.10;References;196
4.6;Wave transport in complex vibro-acoustic structures in the high-frequency limit;198
4.6.1;Introduction;198
4.6.2;Wave energy flow in terms of the Green function;200
4.6.3;Linear phase space operators and DEA;201
4.6.4;A numerical example: coupled two-domain systems;205
4.6.4.1;The hp-adaptive Discontinuous Galerkin Method;205
4.6.4.2;FEM compared to DEA and SEA --- results;208
4.6.5;Conclusions;209
4.6.6;References;210
4.7;Benchmark study of three approaches to propagation of harmonic waves in randomly heterogeneous elastic media;212
4.7.1;Introduction;212
4.7.2;Method of integral spectral decomposition;213
4.7.3;The Fokker-Planck-Kolmogorov equation;216
4.7.4;The Dyson integral equation;220
4.7.5;Concluding remarks;225
4.7.6;References;225
4.8;Minimum-variance-response and irreversible energy confinement;226
4.8.1;Average Impulse Response and the Single Case;226
4.8.2;MIVAR: Minimum-Variance-Response;228
4.8.3;Application of the theory;231
4.8.4;References;238
4.9;High-frequency vibrational power flows in randomly heterogeneous coupled structures;240
4.9.1;Introduction;240
4.9.2;Transport model;242
4.9.2.1;Radiative transfer in an open domain;242
4.9.2.2;Radiative transfer in a bounded domain;243
4.9.2.3;Radiative transfer with a sharp interface;244
4.9.3;Numerical examples;246
4.9.3.1;Coupled beams;247
4.9.3.2;Coupled shells;249
4.9.4;Conclusions;252
4.9.5;References;252
4.10;Uncertainty propagation in SEA using sensitivity analysis and Design of Experiments;254
4.10.1;Introduction;254
4.10.2;SEA equations;256
4.10.3;Uncertainty propagation in SEA;257
4.10.3.1;Approach using sensitivity;258
4.10.3.2;Approach using Design of Experiments;259
4.10.4;Results;261
4.10.5;Conclusions;265
4.10.6;References;265
4.11;Phase reconstruction for time-domain analysis of uncertain structures;266
4.11.1;Introduction;266
4.11.2;Explanation of minimum phase;267
4.11.2.1;Defining minimum phase;267
4.11.2.2;The Hilbert transform and analytic systems;267
4.11.2.3;The Hilbert Transform and minimum phase systems;268
4.11.2.4;Further interpretation of minimum phase;268
4.11.3;Using minimum phase reconstruction;269
4.11.3.1;Approximating the Hilbert Transform;269
4.11.3.2;Errors using MPR for non-minimum phase systems;272
4.11.4;Application: peak shock prediction in uncertain structures;274
4.11.4.1;Modelling an uncertain structure;275
4.11.4.2;Ensemble average results;276
4.11.4.3;Changing the correlation of modal amplitudes;277
4.11.5;Conclusions;278
4.11.6;References;278
5;Probabilistic Methods;279
5.1;Uncertain Linear Systems in Dynamics: Stochastic Approaches;280
5.1.1;Introductory Remarks;280
5.1.2;Overview of Available Methods;281
5.1.3;Response variability;282
5.1.3.1;Perturbation Method;282
5.1.3.2;Spectral methods;285
5.1.3.3;Direct Monte Carlo Simulation;288
5.1.3.4;Random matrix approach;291
5.1.4;Computational Efficiency;291
5.1.5;Summary;292
5.1.6;References;293
5.2;Time domain analysis of structures with stochastic material properties;296
5.2.1;Introduction;296
5.2.2;Preliminary concepts;297
5.2.3;Application of the perturbation approach;298
5.2.4;Moments of the uncertain structure response;299
5.2.5;Application;302
5.2.6;Conclusions;303
5.2.7;References;308
5.3;Vibration Analysis of an Ensemble of Structures using an Exact Theory of Stochastic Linear Systems;309
5.3.1;Introduction;309
5.3.2;Description of the Stochastic System;310
5.3.3;Expression of Mean, Variance, and Covariance;312
5.3.3.1;Parameterized Response;312
5.3.3.2;Mean Response;313
5.3.3.3;Variance and Covariance of the Responses;313
5.3.3.4;Multirank Disturbance;314
5.3.3.5;Discussion of the Theory;316
5.3.4;Stochastic Coefficients in the case of a Gaussian Probability Density Function;316
5.3.5;Application examples;317
5.3.5.1;Comparison to a Monte-Carlo Simulation;318
5.3.5.2;Transition from low to high modal density;319
5.3.5.3;Variance and covariance of responses at different frequencies;321
5.3.6;Conclusion;322
5.3.7;References;322
5.4;Structural Uncertainty Identification using Vibration Mode Shape Information;324
5.4.1;Introduction;324
5.4.2;Maximum Likelihood Estimation of Uncertain Structural Parameters;326
5.4.2.1; Uncertainty Estimation via the Perturbation Method;326
5.4.3;ML estimates of uncertain point-mass position statistics using natural frequency information on a cantilever beam structure;327
5.4.4;ML Estimation of uncertain point mass position on a plate structure using mode shape information;330
5.4.5;Discussion of Results;334
5.4.6;Conclusions;336
5.4.7;References;336
5.5;Extremely strong convergence of eigenvalue-density of linear stochastic dynamical systems;337
5.5.1;Introduction;338
5.5.2;Uncertainty quantification of dynamic response;339
5.5.3;Wishart random matrix model;340
5.5.4;Density of eigenvalues;342
5.5.4.1;Linear eigenvalue statistic;342
5.5.4.2;Self averaging property and the Marcenko-Pastur density;343
5.5.5;Numerical investigations;346
5.5.5.1;Plate with randomly inhomogeneous material properties: parametric uncertainty problem;348
5.5.5.2;Plate with randomly attached spring-mass oscillators: nonparametric uncertainty problem;349
5.5.6;Conclusions;349
5.5.7;References;350
5.6;Stochastic subspace projection schemes for dynamic analysis of uncertain systems;352
5.6.1;Introduction;352
5.6.2;Preliminaries;353
5.6.3;Frequency domain analysis of linear stochastic structural systems;354
5.6.3.1;Preconditioner;358
5.6.3.2;Postprocessing;359
5.6.4;The algebraic random eigenvalue problem;359
5.6.4.1;Stochastic Basis Vectors;360
5.6.4.2;Bubnov-Galerkin Projection;361
5.6.4.3;Postprocessing;361
5.6.5;Numerical Studies;362
5.6.6;Concluding Remarks;364
5.6.7;References;365
6;Probabilistic Methods, Applications;366
6.1;Reliability Assessment of Uncertain Linear Systems in Structural Dynamics;367
6.1.1;Introduction;367
6.1.2;Methods of Analysis;368
6.1.2.1;Representation of uncertain excitation;368
6.1.2.2;Uncertain structural systems;370
6.1.2.3;Stochastic conditional response;370
6.1.2.4;Conditional reliability;371
6.1.2.5;Design point for stochastic structural systems;372
6.1.2.6;First excursion probability for stochastic systems;374
6.1.3;Numerical example;376
6.1.3.1;General remarks;376
6.1.3.2;Structural system;376
6.1.3.3;Dynamic excitation;378
6.1.3.4;Critical response;379
6.1.3.5;Reliability of critical component;379
6.1.4;Conclusions;381
6.1.5;References;382
6.2;On semi-statistical method of numerical solution of integral equations and its applications;383
6.2.1;Introduction;383
6.2.2;Short scheme of semi-statistical method;384
6.2.3;Statement of the problem of blade cascade flow;385
6.2.4;Scheme of application of semi-statistical method to the problem of blade cascade flow;387
6.2.4.1;Main formulas ;387
6.2.4.2;Computation algorithm and optimization;388
6.2.5;Results of simulations;389
6.2.6;Analysis of efficiency of the density adaptation;389
6.2.7;Conclusion;390
6.2.8;References;392
6.3;An efficient model of drill-string dynamics with localised non-linearities;393
6.3.1;Introduction;393
6.3.2;Theoretical Framework;395
6.3.2.1;Linear Model;395
6.3.2.2;Coupling to Non-Linearities;397
6.3.2.3;Coupling to Subsystems;399
6.3.3;Example Simulations;400
6.3.3.1;Linear Behaviour;400
6.3.3.2;Coupling to non-linear friction law;401
6.3.3.3;Coupling to lumped inertia;402
6.3.3.4;Uncertainty Analysis of Stick-Slip Oscillation;403
6.3.4;Conclusions;405
6.3.5;References;405
6.4;Equivalent thermo-mechanical parameters for perfect crystals;407
6.4.1;Introduction;407
6.4.2;Hypotheses;408
6.4.3;Kinematics;410
6.4.4;Equation of momentum balance;412
6.4.5;Equation of angular momentum balance;414
6.4.6;Equation of energy balance;415
6.4.7;Constitutive relations for stress tensor and heat flux;417
6.4.8;Concluding remarks;419
6.4.9;References;420
6.5;Analysis of offshore systems in random waves;421
6.5.1;Introduction;421
6.5.2;Modeling aspects;422
6.5.2.1;Modeling of environmental forces;423
6.5.2.2;Modeling of multibody systems;424
6.5.3;Analysis of deterministic systems;425
6.5.4;Analysis of random systems;426
6.5.4.1;Monte Carlo simulation;426
6.5.4.2;Stochastic linearization;427
6.5.5;Selected Results;427
6.5.6;Conclusions;431
6.5.7;References;431
6.6;Statistical Dynamics of the Rolling Mills;432
6.6.1;Introduction;432
6.6.2;Cold Rolling Mills Chatter Vibrations;434
6.6.2.1;Rolling stand design and its modal analysis;434
6.6.2.2;Strip Elasto-Plastic Deformation;436
6.6.2.3;Horizontal work rolls vibration;438
6.6.2.4;Contact friction force variation;439
6.6.2.5;Chatter detection and control;440
6.6.3;Hot Rolling Mills Torsional Vibrations;441
6.6.3.1;Torsional vibration control and backlashes diagnostics;442
6.6.4;Conclusions;443
6.6.5;References;443
6.7;The application of robust design strategies on managing the uncertainty and variability issues of the blade mistuning vibration problem;446
6.7.1;Introduction;447
6.7.2;Basic concepts of the blade mistuning problem;448
6.7.2.1;The Amplification Factor (its significance and range);449
6.7.3;Casting blade mistuning as a robust design problem;450
6.7.3.1;The Taguchi method of robust design;451
6.7.3.2;The robust optimisation method;451
6.7.3.3;Application of robust design methods to the blade mistuning problem;452
6.7.4;Improving the robustness of bladed discs by parameter design;453
6.7.5;Improving the robustness of bladed discs by tolerance design;455
6.7.5.1;The Small Mistuning approach;456
6.7.5.2;The Intentional Mistuning approach;456
6.7.6;Conclusions;458
6.7.7;References;459
6.8;Localized modeling of uncertainty in the Arlequin framework;460
6.8.1;Introduction;460
6.8.2;The classical Arlequin method;462
6.8.3;The continuous stochastic-deterministic Arlequin formulation;464
6.8.3.1;The stochastic monomodel;465
6.8.3.2;The Arlequin formulation;465
6.8.4;The discretized stochastic-deterministic Arlequin formulation;466
6.8.5;Example of application;468
6.8.6;Conclusion;470
6.8.7;References;470