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
1st Edition
Published on 2. December 2010
XVIII, 470 pages
978-94-007-0289-9 (ISBN)
Description
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.
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Alexander K. Belyaev | Robin S. 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
Book
12/2010
Springer
€160.49
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Content
1 - Preface [Seite 6]
2 - Contents [Seite 8]
3 - Non-probabilistic and related approaches [Seite 20]
3.1 - Linear Bounds on an Uncertain Non-Linear Oscillator: An Info-Gap Approach [Seite 21]
3.1.1 - Introduction [Seite 21]
3.1.2 - Dynamics, Uncertainty and Robustness [Seite 22]
3.1.3 - Example: Uncertain Cubic Non-Linearity [Seite 25]
3.1.4 - Example: Multiple Uncertainties [Seite 28]
3.1.5 - Robustness as a Proxy for Probability [Seite 30]
3.1.6 - Conclusion [Seite 31]
3.1.7 - References [Seite 32]
3.2 - Quantification of uncertain and variable model parameters in non-deterministic analysis [Seite 33]
3.2.1 - Introduction [Seite 33]
3.2.2 - Numerical representation of parameter uncertainty and variability [Seite 34]
3.2.2.1 - Definitions [Seite 35]
3.2.2.2 - Discussion and extension of the definitions [Seite 36]
3.2.3 - Literature review on uncertain model and material data [Seite 37]
3.2.3.1 - Non-probabilistic models [Seite 37]
3.2.3.2 - Probabilistic models [Seite 38]
3.2.3.3 - Material data [Seite 40]
3.2.3.4 - Other model properties [Seite 43]
3.2.3.5 - Alternative approaches: non-parametric model concept and info-gap theory [Seite 43]
3.2.3.6 - Summary of observations [Seite 44]
3.2.4 - Conclusions [Seite 44]
3.2.5 - References [Seite 45]
3.3 - Vibrations of layered structures with fuzzy core stiffness/fuzzy interlayer slip [Seite 47]
3.3.1 - Introduction [Seite 47]
3.3.2 - Fuzzy sandwich beams [Seite 48]
3.3.2.1 - Three-layer beams [Seite 48]
3.3.2.2 - Modal analysis of the three-layer beam, hard-hinged support [Seite 51]
3.3.3 - Numerical results [Seite 53]
3.3.3.1 - Isosceles uncertainty [Seite 53]
3.3.3.2 - Constraints affected to the uncertain natural frequencies [Seite 56]
3.3.3.3 - Some effects of non-symmetric uncertainty [Seite 59]
3.3.4 - Conclusions [Seite 59]
3.3.5 - References [Seite 60]
3.4 - Vibration Analysis of Fluid-Filled Piping Systems with Epistemic Uncertainties [Seite 61]
3.4.1 - Introduction [Seite 62]
3.4.2 - Classification, Representation and Propagation of Uncertainty [Seite 62]
3.4.2.1 - Uncertainty Classification and Representation [Seite 62]
3.4.2.2 - Uncertainty Propagation Based on the Transformation Method [Seite 64]
3.4.3 - Fluid-Filled Piping System [Seite 66]
3.4.3.1 - Modeling Approach [Seite 66]
3.4.3.2 - Experimental Setup [Seite 69]
3.4.4 - Comprehensive Modeling and Simulation [Seite 69]
3.4.4.1 - Modeling of Epistemic Uncertainties [Seite 69]
3.4.4.2 - Simulation Results [Seite 71]
3.4.4.3 - Measures of Influence [Seite 72]
3.4.5 - Conclusions [Seite 73]
3.4.6 - References [Seite 73]
3.5 - Fuzzy vibration analysis and optimization of engineering structures: Application to Demeter satellite [Seite 75]
3.5.1 - Introduction [Seite 75]
3.5.2 - Aims of the study [Seite 76]
3.5.2.1 - Description of the study [Seite 76]
3.5.2.2 - Building of fuzzy optimization problem [Seite 77]
3.5.3 - Fuzzy vibration analysis [Seite 80]
3.5.3.1 - PAEM method [Seite 80]
3.5.3.2 - Numerical application [Seite 81]
3.5.4 - Fuzzy optimization [Seite 81]
3.5.4.1 - Design methodology [Seite 82]
3.5.4.2 - Improvement of the initial design [Seite 84]
3.5.5 - Conclusion [Seite 85]
3.5.6 - References [Seite 87]
3.6 - Numerical dynamic analysis of uncertain mechanical structures based on interval fields [Seite 88]
3.6.1 - Introduction [Seite 88]
3.6.2 - Interval finite element analysis [Seite 90]
3.6.3 - Interval fields [Seite 91]
3.6.3.1 - General concept [Seite 91]
3.6.3.2 - Interval fields as uncertain input parameters [Seite 92]
3.6.3.3 - Interval fields as uncertain analysis results [Seite 94]
3.6.4 - Application of interval fields for vibro-acoustic analysis [Seite 96]
3.6.4.1 - Vibro-acoustic analysis based on the ATV concept [Seite 96]
3.6.4.2 - Interval analysis based on structural FRF interval fields [Seite 97]
3.6.4.3 - Numerical example [Seite 98]
3.6.5 - Conclusions [Seite 99]
3.6.6 - References [Seite 100]
3.7 - From Interval Computations to Constraint-Related Set Computations: Towards Faster Estimation of Statistics and ODEs under Interval and p-Box Uncertainty [Seite 101]
3.7.1 - Formulation of the Problem [Seite 101]
3.7.2 - Interval Computations: Brief Reminder [Seite 104]
3.7.3 - Constraint-Based Set Computations [Seite 105]
3.7.4 - References [Seite 114]
3.8 - Dynamic Steady-State Analysis of Structures under Uncertain Harmonic Loads via Semidefinite Program [Seite 115]
3.8.1 - Introduction [Seite 115]
3.8.2 - Uncertain equations for steady state vibration [Seite 117]
3.8.2.1 - Governing equations [Seite 117]
3.8.2.2 - Uncertainty model [Seite 117]
3.8.2.3 - ULE in real variables [Seite 118]
3.8.3 - Bounds for complex amplitude [Seite 119]
3.8.3.1 - Upper bound for modulus of displacement amplitude [Seite 119]
3.8.3.2 - Lower bound for modulus of displacement amplitude [Seite 122]
3.8.3.3 - Bounds for phase angle [Seite 122]
3.8.4 - Bounds for nodal oscillation [Seite 124]
3.8.5 - Numerical experiments [Seite 124]
3.8.6 - Conclusions [Seite 126]
3.8.7 - References [Seite 127]
4 - SEA related methods and wave propagation [Seite 129]
4.1 - Universal eigenvalue statistics and vibration response prediction [Seite 130]
4.1.1 - Introduction [Seite 130]
4.1.2 - Eigenvalue statistics [Seite 131]
4.1.2.1 - The joint probability density function of the eigenvalues [Seite 131]
4.1.2.2 - The modal density [Seite 133]
4.1.2.3 - Universality of the ``local'' eigenvalue statistics [Seite 134]
4.1.2.4 - Application to natural frequency statistics [Seite 136]
4.1.3 - Application to response statistics [Seite 137]
4.1.3.1 - Fundamental concepts [Seite 137]
4.1.3.2 - Built-up systems: SEA [Seite 138]
4.1.3.3 - Built-up systems: the Hybrid method [Seite 139]
4.1.4 - Conclusions [Seite 140]
4.1.5 - References [Seite 141]
4.2 - Statistical Energy Analysis and the second principle of thermodynamics [Seite 143]
4.2.1 - Introduction [Seite 143]
4.2.2 - First principle of thermodynamics in SEA [Seite 144]
4.2.3 - Vibrational entropy, vibrational temperature [Seite 147]
4.2.4 - Second principle of thermodynamics in SEA [Seite 148]
4.2.5 - Entropy balance in SEA [Seite 149]
4.2.6 - Conclusion [Seite 150]
4.2.7 - Discussion [Seite 151]
4.2.8 - References [Seite 153]
4.3 - Modeling noise and vibration transmission in complex systems [Seite 154]
4.3.1 - Introduction [Seite 154]
4.3.1.1 - Complexity [Seite 155]
4.3.1.2 - Uncertainty [Seite 156]
4.3.1.3 - How much information is needed for noise and vibration design? [Seite 156]
4.3.2 - Modeling methods and frequency ranges [Seite 157]
4.3.2.1 - Low, mid and high frequency ranges [Seite 157]
4.3.2.2 - Low and High frequency modelling methods [Seite 158]
4.3.2.3 - The Mid-Frequency problem [Seite 159]
4.3.3 - The Hybrid FE-SEA method [Seite 160]
4.3.3.1 - Statistical subsystem [Seite 160]
4.3.3.2 - The direct and reverberant fields of a statistical subsystem [Seite 161]
4.3.3.3 - Ensemble average reverberant loading [Seite 162]
4.3.3.4 - Coupling a deterministic and statistical subsystem [Seite 162]
4.3.4 - Application examples [Seite 163]
4.3.4.1 - Monte Carlo simulations [Seite 164]
4.3.4.2 - Numerical applications [Seite 164]
4.3.4.3 - Industrial applications [Seite 165]
4.3.5 - Concluding remarks [Seite 166]
4.3.6 - References [Seite 167]
4.4 - A Power Absorbing Matrix for the Hybrid FEA-SEA Method [Seite 170]
4.4.1 - Introduction [Seite 170]
4.4.2 - Cylindrical Waves and Energy Sinks [Seite 171]
4.4.2.1 - The Governing Equations [Seite 171]
4.4.2.2 - The Cylindrical Waves [Seite 173]
4.4.3 - Constructing the Power Absorbing Matrix [Seite 175]
4.4.3.1 - Discretization of the Power Integral, and Matrix Assembly [Seite 175]
4.4.3.2 - Numerical Issues [Seite 177]
4.4.4 - Numerical Results [Seite 178]
4.4.4.1 - A Simple System [Seite 178]
4.4.4.2 - System Randomization and Subsystem Response Prediction [Seite 178]
4.4.4.3 - Results [Seite 179]
4.4.5 - Conclusions [Seite 180]
4.4.6 - References [Seite 182]
4.5 - The Energy Finite Element Method NoiseFEM [Seite 183]
4.5.1 - Introduction [Seite 183]
4.5.1.1 - Motivation [Seite 184]
4.5.1.2 - Literature Overview [Seite 185]
4.5.2 - Components of NoiseFEM [Seite 185]
4.5.3 - Power Flow Between Structural Elements [Seite 185]
4.5.3.1 - Transmission Coefficients [Seite 186]
4.5.3.2 - The Coupling Matrix [Seite 188]
4.5.4 - Diffusive Energy Transport [Seite 190]
4.5.4.1 - Homogeneous Structural Elements [Seite 190]
4.5.4.2 - Stiffened Subsystems [Seite 191]
4.5.5 - Combining transport and coupling equations [Seite 192]
4.5.6 - Discretization [Seite 193]
4.5.7 - Validation of NoiseFEM with test structures [Seite 194]
4.5.8 - Application of NoiseFEM [Seite 195]
4.5.9 - Conclusions [Seite 196]
4.5.10 - References [Seite 196]
4.6 - Wave transport in complex vibro-acoustic structures in the high-frequency limit [Seite 198]
4.6.1 - Introduction [Seite 198]
4.6.2 - Wave energy flow in terms of the Green function [Seite 200]
4.6.3 - Linear phase space operators and DEA [Seite 201]
4.6.4 - A numerical example: coupled two-domain systems [Seite 205]
4.6.4.1 - The hp-adaptive Discontinuous Galerkin Method [Seite 205]
4.6.4.2 - FEM compared to DEA and SEA --- results [Seite 208]
4.6.5 - Conclusions [Seite 209]
4.6.6 - References [Seite 210]
4.7 - Benchmark study of three approaches to propagation of harmonic waves in randomly heterogeneous elastic media [Seite 212]
4.7.1 - Introduction [Seite 212]
4.7.2 - Method of integral spectral decomposition [Seite 213]
4.7.3 - The Fokker-Planck-Kolmogorov equation [Seite 216]
4.7.4 - The Dyson integral equation [Seite 220]
4.7.5 - Concluding remarks [Seite 225]
4.7.6 - References [Seite 225]
4.8 - Minimum-variance-response and irreversible energy confinement [Seite 226]
4.8.1 - Average Impulse Response and the Single Case [Seite 226]
4.8.2 - MIVAR: Minimum-Variance-Response [Seite 228]
4.8.3 - Application of the theory [Seite 231]
4.8.4 - References [Seite 238]
4.9 - High-frequency vibrational power flows in randomly heterogeneous coupled structures [Seite 240]
4.9.1 - Introduction [Seite 240]
4.9.2 - Transport model [Seite 242]
4.9.2.1 - Radiative transfer in an open domain [Seite 242]
4.9.2.2 - Radiative transfer in a bounded domain [Seite 243]
4.9.2.3 - Radiative transfer with a sharp interface [Seite 244]
4.9.3 - Numerical examples [Seite 246]
4.9.3.1 - Coupled beams [Seite 247]
4.9.3.2 - Coupled shells [Seite 249]
4.9.4 - Conclusions [Seite 252]
4.9.5 - References [Seite 252]
4.10 - Uncertainty propagation in SEA using sensitivity analysis and Design of Experiments [Seite 254]
4.10.1 - Introduction [Seite 254]
4.10.2 - SEA equations [Seite 256]
4.10.3 - Uncertainty propagation in SEA [Seite 257]
4.10.3.1 - Approach using sensitivity [Seite 258]
4.10.3.2 - Approach using Design of Experiments [Seite 259]
4.10.4 - Results [Seite 261]
4.10.5 - Conclusions [Seite 265]
4.10.6 - References [Seite 265]
4.11 - Phase reconstruction for time-domain analysis of uncertain structures [Seite 266]
4.11.1 - Introduction [Seite 266]
4.11.2 - Explanation of minimum phase [Seite 267]
4.11.2.1 - Defining minimum phase [Seite 267]
4.11.2.2 - The Hilbert transform and analytic systems [Seite 267]
4.11.2.3 - The Hilbert Transform and minimum phase systems [Seite 268]
4.11.2.4 - Further interpretation of minimum phase [Seite 268]
4.11.3 - Using minimum phase reconstruction [Seite 269]
4.11.3.1 - Approximating the Hilbert Transform [Seite 269]
4.11.3.2 - Errors using MPR for non-minimum phase systems [Seite 272]
4.11.4 - Application: peak shock prediction in uncertain structures [Seite 274]
4.11.4.1 - Modelling an uncertain structure [Seite 275]
4.11.4.2 - Ensemble average results [Seite 276]
4.11.4.3 - Changing the correlation of modal amplitudes [Seite 277]
4.11.5 - Conclusions [Seite 278]
4.11.6 - References [Seite 278]
5 - Probabilistic Methods [Seite 279]
5.1 - Uncertain Linear Systems in Dynamics: Stochastic Approaches [Seite 280]
5.1.1 - Introductory Remarks [Seite 280]
5.1.2 - Overview of Available Methods [Seite 281]
5.1.3 - Response variability [Seite 282]
5.1.3.1 - Perturbation Method [Seite 282]
5.1.3.2 - Spectral methods [Seite 285]
5.1.3.3 - Direct Monte Carlo Simulation [Seite 288]
5.1.3.4 - Random matrix approach [Seite 291]
5.1.4 - Computational Efficiency [Seite 291]
5.1.5 - Summary [Seite 292]
5.1.6 - References [Seite 293]
5.2 - Time domain analysis of structures with stochastic material properties [Seite 296]
5.2.1 - Introduction [Seite 296]
5.2.2 - Preliminary concepts [Seite 297]
5.2.3 - Application of the perturbation approach [Seite 298]
5.2.4 - Moments of the uncertain structure response [Seite 299]
5.2.5 - Application [Seite 302]
5.2.6 - Conclusions [Seite 303]
5.2.7 - References [Seite 308]
5.3 - Vibration Analysis of an Ensemble of Structures using an Exact Theory of Stochastic Linear Systems [Seite 309]
5.3.1 - Introduction [Seite 309]
5.3.2 - Description of the Stochastic System [Seite 310]
5.3.3 - Expression of Mean, Variance, and Covariance [Seite 312]
5.3.3.1 - Parameterized Response [Seite 312]
5.3.3.2 - Mean Response [Seite 313]
5.3.3.3 - Variance and Covariance of the Responses [Seite 313]
5.3.3.4 - Multirank Disturbance [Seite 314]
5.3.3.5 - Discussion of the Theory [Seite 316]
5.3.4 - Stochastic Coefficients in the case of a Gaussian Probability Density Function [Seite 316]
5.3.5 - Application examples [Seite 317]
5.3.5.1 - Comparison to a Monte-Carlo Simulation [Seite 318]
5.3.5.2 - Transition from low to high modal density [Seite 319]
5.3.5.3 - Variance and covariance of responses at different frequencies [Seite 321]
5.3.6 - Conclusion [Seite 322]
5.3.7 - References [Seite 322]
5.4 - Structural Uncertainty Identification using Vibration Mode Shape Information [Seite 324]
5.4.1 - Introduction [Seite 324]
5.4.2 - Maximum Likelihood Estimation of Uncertain Structural Parameters [Seite 326]
5.4.2.1 - Uncertainty Estimation via the Perturbation Method [Seite 326]
5.4.3 - ML estimates of uncertain point-mass position statistics using natural frequency information on a cantilever beam structure [Seite 327]
5.4.4 - ML Estimation of uncertain point mass position on a plate structure using mode shape information [Seite 330]
5.4.5 - Discussion of Results [Seite 334]
5.4.6 - Conclusions [Seite 336]
5.4.7 - References [Seite 336]
5.5 - Extremely strong convergence of eigenvalue-density of linear stochastic dynamical systems [Seite 337]
5.5.1 - Introduction [Seite 338]
5.5.2 - Uncertainty quantification of dynamic response [Seite 339]
5.5.3 - Wishart random matrix model [Seite 340]
5.5.4 - Density of eigenvalues [Seite 342]
5.5.4.1 - Linear eigenvalue statistic [Seite 342]
5.5.4.2 - Self averaging property and the Marcenko-Pastur density [Seite 343]
5.5.5 - Numerical investigations [Seite 346]
5.5.5.1 - Plate with randomly inhomogeneous material properties: parametric uncertainty problem [Seite 348]
5.5.5.2 - Plate with randomly attached spring-mass oscillators: nonparametric uncertainty problem [Seite 349]
5.5.6 - Conclusions [Seite 349]
5.5.7 - References [Seite 350]
5.6 - Stochastic subspace projection schemes for dynamic analysis of uncertain systems [Seite 352]
5.6.1 - Introduction [Seite 352]
5.6.2 - Preliminaries [Seite 353]
5.6.3 - Frequency domain analysis of linear stochastic structural systems [Seite 354]
5.6.3.1 - Preconditioner [Seite 358]
5.6.3.2 - Postprocessing [Seite 359]
5.6.4 - The algebraic random eigenvalue problem [Seite 359]
5.6.4.1 - Stochastic Basis Vectors [Seite 360]
5.6.4.2 - Bubnov-Galerkin Projection [Seite 361]
5.6.4.3 - Postprocessing [Seite 361]
5.6.5 - Numerical Studies [Seite 362]
5.6.6 - Concluding Remarks [Seite 364]
5.6.7 - References [Seite 365]
6 - Probabilistic Methods, Applications [Seite 366]
6.1 - Reliability Assessment of Uncertain Linear Systems in Structural Dynamics [Seite 367]
6.1.1 - Introduction [Seite 367]
6.1.2 - Methods of Analysis [Seite 368]
6.1.2.1 - Representation of uncertain excitation [Seite 368]
6.1.2.2 - Uncertain structural systems [Seite 370]
6.1.2.3 - Stochastic conditional response [Seite 370]
6.1.2.4 - Conditional reliability [Seite 371]
6.1.2.5 - Design point for stochastic structural systems [Seite 372]
6.1.2.6 - First excursion probability for stochastic systems [Seite 374]
6.1.3 - Numerical example [Seite 376]
6.1.3.1 - General remarks [Seite 376]
6.1.3.2 - Structural system [Seite 376]
6.1.3.3 - Dynamic excitation [Seite 378]
6.1.3.4 - Critical response [Seite 379]
6.1.3.5 - Reliability of critical component [Seite 379]
6.1.4 - Conclusions [Seite 381]
6.1.5 - References [Seite 382]
6.2 - On semi-statistical method of numerical solution of integral equations and its applications [Seite 383]
6.2.1 - Introduction [Seite 383]
6.2.2 - Short scheme of semi-statistical method [Seite 384]
6.2.3 - Statement of the problem of blade cascade flow [Seite 385]
6.2.4 - Scheme of application of semi-statistical method to the problem of blade cascade flow [Seite 387]
6.2.4.1 - Main formulas [Seite 387]
6.2.4.2 - Computation algorithm and optimization [Seite 388]
6.2.5 - Results of simulations [Seite 389]
6.2.6 - Analysis of efficiency of the density adaptation [Seite 389]
6.2.7 - Conclusion [Seite 390]
6.2.8 - References [Seite 392]
6.3 - An efficient model of drill-string dynamics with localised non-linearities [Seite 393]
6.3.1 - Introduction [Seite 393]
6.3.2 - Theoretical Framework [Seite 395]
6.3.2.1 - Linear Model [Seite 395]
6.3.2.2 - Coupling to Non-Linearities [Seite 397]
6.3.2.3 - Coupling to Subsystems [Seite 399]
6.3.3 - Example Simulations [Seite 400]
6.3.3.1 - Linear Behaviour [Seite 400]
6.3.3.2 - Coupling to non-linear friction law [Seite 401]
6.3.3.3 - Coupling to lumped inertia [Seite 402]
6.3.3.4 - Uncertainty Analysis of Stick-Slip Oscillation [Seite 403]
6.3.4 - Conclusions [Seite 405]
6.3.5 - References [Seite 405]
6.4 - Equivalent thermo-mechanical parameters for perfect crystals [Seite 407]
6.4.1 - Introduction [Seite 407]
6.4.2 - Hypotheses [Seite 408]
6.4.3 - Kinematics [Seite 410]
6.4.4 - Equation of momentum balance [Seite 412]
6.4.5 - Equation of angular momentum balance [Seite 414]
6.4.6 - Equation of energy balance [Seite 415]
6.4.7 - Constitutive relations for stress tensor and heat flux [Seite 417]
6.4.8 - Concluding remarks [Seite 419]
6.4.9 - References [Seite 420]
6.5 - Analysis of offshore systems in random waves [Seite 421]
6.5.1 - Introduction [Seite 421]
6.5.2 - Modeling aspects [Seite 422]
6.5.2.1 - Modeling of environmental forces [Seite 423]
6.5.2.2 - Modeling of multibody systems [Seite 424]
6.5.3 - Analysis of deterministic systems [Seite 425]
6.5.4 - Analysis of random systems [Seite 426]
6.5.4.1 - Monte Carlo simulation [Seite 426]
6.5.4.2 - Stochastic linearization [Seite 427]
6.5.5 - Selected Results [Seite 427]
6.5.6 - Conclusions [Seite 431]
6.5.7 - References [Seite 431]
6.6 - Statistical Dynamics of the Rolling Mills [Seite 432]
6.6.1 - Introduction [Seite 432]
6.6.2 - Cold Rolling Mills Chatter Vibrations [Seite 434]
6.6.2.1 - Rolling stand design and its modal analysis [Seite 434]
6.6.2.2 - Strip Elasto-Plastic Deformation [Seite 436]
6.6.2.3 - Horizontal work rolls vibration [Seite 438]
6.6.2.4 - Contact friction force variation [Seite 439]
6.6.2.5 - Chatter detection and control [Seite 440]
6.6.3 - Hot Rolling Mills Torsional Vibrations [Seite 441]
6.6.3.1 - Torsional vibration control and backlashes diagnostics [Seite 442]
6.6.4 - Conclusions [Seite 443]
6.6.5 - References [Seite 443]
6.7 - The application of robust design strategies on managing the uncertainty and variability issues of the blade mistuning vibration problem [Seite 446]
6.7.1 - Introduction [Seite 447]
6.7.2 - Basic concepts of the blade mistuning problem [Seite 448]
6.7.2.1 - The Amplification Factor (its significance and range) [Seite 449]
6.7.3 - Casting blade mistuning as a robust design problem [Seite 450]
6.7.3.1 - The Taguchi method of robust design [Seite 451]
6.7.3.2 - The robust optimisation method [Seite 451]
6.7.3.3 - Application of robust design methods to the blade mistuning problem [Seite 452]
6.7.4 - Improving the robustness of bladed discs by parameter design [Seite 453]
6.7.5 - Improving the robustness of bladed discs by tolerance design [Seite 455]
6.7.5.1 - The Small Mistuning approach [Seite 456]
6.7.5.2 - The Intentional Mistuning approach [Seite 456]
6.7.6 - Conclusions [Seite 458]
6.7.7 - References [Seite 459]
6.8 - Localized modeling of uncertainty in the Arlequin framework [Seite 460]
6.8.1 - Introduction [Seite 460]
6.8.2 - The classical Arlequin method [Seite 462]
6.8.3 - The continuous stochastic-deterministic Arlequin formulation [Seite 464]
6.8.3.1 - The stochastic monomodel [Seite 465]
6.8.3.2 - The Arlequin formulation [Seite 465]
6.8.4 - The discretized stochastic-deterministic Arlequin formulation [Seite 466]
6.8.5 - Example of application [Seite 468]
6.8.6 - Conclusion [Seite 470]
6.8.7 - References [Seite 470]
2 - Contents [Seite 8]
3 - Non-probabilistic and related approaches [Seite 20]
3.1 - Linear Bounds on an Uncertain Non-Linear Oscillator: An Info-Gap Approach [Seite 21]
3.1.1 - Introduction [Seite 21]
3.1.2 - Dynamics, Uncertainty and Robustness [Seite 22]
3.1.3 - Example: Uncertain Cubic Non-Linearity [Seite 25]
3.1.4 - Example: Multiple Uncertainties [Seite 28]
3.1.5 - Robustness as a Proxy for Probability [Seite 30]
3.1.6 - Conclusion [Seite 31]
3.1.7 - References [Seite 32]
3.2 - Quantification of uncertain and variable model parameters in non-deterministic analysis [Seite 33]
3.2.1 - Introduction [Seite 33]
3.2.2 - Numerical representation of parameter uncertainty and variability [Seite 34]
3.2.2.1 - Definitions [Seite 35]
3.2.2.2 - Discussion and extension of the definitions [Seite 36]
3.2.3 - Literature review on uncertain model and material data [Seite 37]
3.2.3.1 - Non-probabilistic models [Seite 37]
3.2.3.2 - Probabilistic models [Seite 38]
3.2.3.3 - Material data [Seite 40]
3.2.3.4 - Other model properties [Seite 43]
3.2.3.5 - Alternative approaches: non-parametric model concept and info-gap theory [Seite 43]
3.2.3.6 - Summary of observations [Seite 44]
3.2.4 - Conclusions [Seite 44]
3.2.5 - References [Seite 45]
3.3 - Vibrations of layered structures with fuzzy core stiffness/fuzzy interlayer slip [Seite 47]
3.3.1 - Introduction [Seite 47]
3.3.2 - Fuzzy sandwich beams [Seite 48]
3.3.2.1 - Three-layer beams [Seite 48]
3.3.2.2 - Modal analysis of the three-layer beam, hard-hinged support [Seite 51]
3.3.3 - Numerical results [Seite 53]
3.3.3.1 - Isosceles uncertainty [Seite 53]
3.3.3.2 - Constraints affected to the uncertain natural frequencies [Seite 56]
3.3.3.3 - Some effects of non-symmetric uncertainty [Seite 59]
3.3.4 - Conclusions [Seite 59]
3.3.5 - References [Seite 60]
3.4 - Vibration Analysis of Fluid-Filled Piping Systems with Epistemic Uncertainties [Seite 61]
3.4.1 - Introduction [Seite 62]
3.4.2 - Classification, Representation and Propagation of Uncertainty [Seite 62]
3.4.2.1 - Uncertainty Classification and Representation [Seite 62]
3.4.2.2 - Uncertainty Propagation Based on the Transformation Method [Seite 64]
3.4.3 - Fluid-Filled Piping System [Seite 66]
3.4.3.1 - Modeling Approach [Seite 66]
3.4.3.2 - Experimental Setup [Seite 69]
3.4.4 - Comprehensive Modeling and Simulation [Seite 69]
3.4.4.1 - Modeling of Epistemic Uncertainties [Seite 69]
3.4.4.2 - Simulation Results [Seite 71]
3.4.4.3 - Measures of Influence [Seite 72]
3.4.5 - Conclusions [Seite 73]
3.4.6 - References [Seite 73]
3.5 - Fuzzy vibration analysis and optimization of engineering structures: Application to Demeter satellite [Seite 75]
3.5.1 - Introduction [Seite 75]
3.5.2 - Aims of the study [Seite 76]
3.5.2.1 - Description of the study [Seite 76]
3.5.2.2 - Building of fuzzy optimization problem [Seite 77]
3.5.3 - Fuzzy vibration analysis [Seite 80]
3.5.3.1 - PAEM method [Seite 80]
3.5.3.2 - Numerical application [Seite 81]
3.5.4 - Fuzzy optimization [Seite 81]
3.5.4.1 - Design methodology [Seite 82]
3.5.4.2 - Improvement of the initial design [Seite 84]
3.5.5 - Conclusion [Seite 85]
3.5.6 - References [Seite 87]
3.6 - Numerical dynamic analysis of uncertain mechanical structures based on interval fields [Seite 88]
3.6.1 - Introduction [Seite 88]
3.6.2 - Interval finite element analysis [Seite 90]
3.6.3 - Interval fields [Seite 91]
3.6.3.1 - General concept [Seite 91]
3.6.3.2 - Interval fields as uncertain input parameters [Seite 92]
3.6.3.3 - Interval fields as uncertain analysis results [Seite 94]
3.6.4 - Application of interval fields for vibro-acoustic analysis [Seite 96]
3.6.4.1 - Vibro-acoustic analysis based on the ATV concept [Seite 96]
3.6.4.2 - Interval analysis based on structural FRF interval fields [Seite 97]
3.6.4.3 - Numerical example [Seite 98]
3.6.5 - Conclusions [Seite 99]
3.6.6 - References [Seite 100]
3.7 - From Interval Computations to Constraint-Related Set Computations: Towards Faster Estimation of Statistics and ODEs under Interval and p-Box Uncertainty [Seite 101]
3.7.1 - Formulation of the Problem [Seite 101]
3.7.2 - Interval Computations: Brief Reminder [Seite 104]
3.7.3 - Constraint-Based Set Computations [Seite 105]
3.7.4 - References [Seite 114]
3.8 - Dynamic Steady-State Analysis of Structures under Uncertain Harmonic Loads via Semidefinite Program [Seite 115]
3.8.1 - Introduction [Seite 115]
3.8.2 - Uncertain equations for steady state vibration [Seite 117]
3.8.2.1 - Governing equations [Seite 117]
3.8.2.2 - Uncertainty model [Seite 117]
3.8.2.3 - ULE in real variables [Seite 118]
3.8.3 - Bounds for complex amplitude [Seite 119]
3.8.3.1 - Upper bound for modulus of displacement amplitude [Seite 119]
3.8.3.2 - Lower bound for modulus of displacement amplitude [Seite 122]
3.8.3.3 - Bounds for phase angle [Seite 122]
3.8.4 - Bounds for nodal oscillation [Seite 124]
3.8.5 - Numerical experiments [Seite 124]
3.8.6 - Conclusions [Seite 126]
3.8.7 - References [Seite 127]
4 - SEA related methods and wave propagation [Seite 129]
4.1 - Universal eigenvalue statistics and vibration response prediction [Seite 130]
4.1.1 - Introduction [Seite 130]
4.1.2 - Eigenvalue statistics [Seite 131]
4.1.2.1 - The joint probability density function of the eigenvalues [Seite 131]
4.1.2.2 - The modal density [Seite 133]
4.1.2.3 - Universality of the ``local'' eigenvalue statistics [Seite 134]
4.1.2.4 - Application to natural frequency statistics [Seite 136]
4.1.3 - Application to response statistics [Seite 137]
4.1.3.1 - Fundamental concepts [Seite 137]
4.1.3.2 - Built-up systems: SEA [Seite 138]
4.1.3.3 - Built-up systems: the Hybrid method [Seite 139]
4.1.4 - Conclusions [Seite 140]
4.1.5 - References [Seite 141]
4.2 - Statistical Energy Analysis and the second principle of thermodynamics [Seite 143]
4.2.1 - Introduction [Seite 143]
4.2.2 - First principle of thermodynamics in SEA [Seite 144]
4.2.3 - Vibrational entropy, vibrational temperature [Seite 147]
4.2.4 - Second principle of thermodynamics in SEA [Seite 148]
4.2.5 - Entropy balance in SEA [Seite 149]
4.2.6 - Conclusion [Seite 150]
4.2.7 - Discussion [Seite 151]
4.2.8 - References [Seite 153]
4.3 - Modeling noise and vibration transmission in complex systems [Seite 154]
4.3.1 - Introduction [Seite 154]
4.3.1.1 - Complexity [Seite 155]
4.3.1.2 - Uncertainty [Seite 156]
4.3.1.3 - How much information is needed for noise and vibration design? [Seite 156]
4.3.2 - Modeling methods and frequency ranges [Seite 157]
4.3.2.1 - Low, mid and high frequency ranges [Seite 157]
4.3.2.2 - Low and High frequency modelling methods [Seite 158]
4.3.2.3 - The Mid-Frequency problem [Seite 159]
4.3.3 - The Hybrid FE-SEA method [Seite 160]
4.3.3.1 - Statistical subsystem [Seite 160]
4.3.3.2 - The direct and reverberant fields of a statistical subsystem [Seite 161]
4.3.3.3 - Ensemble average reverberant loading [Seite 162]
4.3.3.4 - Coupling a deterministic and statistical subsystem [Seite 162]
4.3.4 - Application examples [Seite 163]
4.3.4.1 - Monte Carlo simulations [Seite 164]
4.3.4.2 - Numerical applications [Seite 164]
4.3.4.3 - Industrial applications [Seite 165]
4.3.5 - Concluding remarks [Seite 166]
4.3.6 - References [Seite 167]
4.4 - A Power Absorbing Matrix for the Hybrid FEA-SEA Method [Seite 170]
4.4.1 - Introduction [Seite 170]
4.4.2 - Cylindrical Waves and Energy Sinks [Seite 171]
4.4.2.1 - The Governing Equations [Seite 171]
4.4.2.2 - The Cylindrical Waves [Seite 173]
4.4.3 - Constructing the Power Absorbing Matrix [Seite 175]
4.4.3.1 - Discretization of the Power Integral, and Matrix Assembly [Seite 175]
4.4.3.2 - Numerical Issues [Seite 177]
4.4.4 - Numerical Results [Seite 178]
4.4.4.1 - A Simple System [Seite 178]
4.4.4.2 - System Randomization and Subsystem Response Prediction [Seite 178]
4.4.4.3 - Results [Seite 179]
4.4.5 - Conclusions [Seite 180]
4.4.6 - References [Seite 182]
4.5 - The Energy Finite Element Method NoiseFEM [Seite 183]
4.5.1 - Introduction [Seite 183]
4.5.1.1 - Motivation [Seite 184]
4.5.1.2 - Literature Overview [Seite 185]
4.5.2 - Components of NoiseFEM [Seite 185]
4.5.3 - Power Flow Between Structural Elements [Seite 185]
4.5.3.1 - Transmission Coefficients [Seite 186]
4.5.3.2 - The Coupling Matrix [Seite 188]
4.5.4 - Diffusive Energy Transport [Seite 190]
4.5.4.1 - Homogeneous Structural Elements [Seite 190]
4.5.4.2 - Stiffened Subsystems [Seite 191]
4.5.5 - Combining transport and coupling equations [Seite 192]
4.5.6 - Discretization [Seite 193]
4.5.7 - Validation of NoiseFEM with test structures [Seite 194]
4.5.8 - Application of NoiseFEM [Seite 195]
4.5.9 - Conclusions [Seite 196]
4.5.10 - References [Seite 196]
4.6 - Wave transport in complex vibro-acoustic structures in the high-frequency limit [Seite 198]
4.6.1 - Introduction [Seite 198]
4.6.2 - Wave energy flow in terms of the Green function [Seite 200]
4.6.3 - Linear phase space operators and DEA [Seite 201]
4.6.4 - A numerical example: coupled two-domain systems [Seite 205]
4.6.4.1 - The hp-adaptive Discontinuous Galerkin Method [Seite 205]
4.6.4.2 - FEM compared to DEA and SEA --- results [Seite 208]
4.6.5 - Conclusions [Seite 209]
4.6.6 - References [Seite 210]
4.7 - Benchmark study of three approaches to propagation of harmonic waves in randomly heterogeneous elastic media [Seite 212]
4.7.1 - Introduction [Seite 212]
4.7.2 - Method of integral spectral decomposition [Seite 213]
4.7.3 - The Fokker-Planck-Kolmogorov equation [Seite 216]
4.7.4 - The Dyson integral equation [Seite 220]
4.7.5 - Concluding remarks [Seite 225]
4.7.6 - References [Seite 225]
4.8 - Minimum-variance-response and irreversible energy confinement [Seite 226]
4.8.1 - Average Impulse Response and the Single Case [Seite 226]
4.8.2 - MIVAR: Minimum-Variance-Response [Seite 228]
4.8.3 - Application of the theory [Seite 231]
4.8.4 - References [Seite 238]
4.9 - High-frequency vibrational power flows in randomly heterogeneous coupled structures [Seite 240]
4.9.1 - Introduction [Seite 240]
4.9.2 - Transport model [Seite 242]
4.9.2.1 - Radiative transfer in an open domain [Seite 242]
4.9.2.2 - Radiative transfer in a bounded domain [Seite 243]
4.9.2.3 - Radiative transfer with a sharp interface [Seite 244]
4.9.3 - Numerical examples [Seite 246]
4.9.3.1 - Coupled beams [Seite 247]
4.9.3.2 - Coupled shells [Seite 249]
4.9.4 - Conclusions [Seite 252]
4.9.5 - References [Seite 252]
4.10 - Uncertainty propagation in SEA using sensitivity analysis and Design of Experiments [Seite 254]
4.10.1 - Introduction [Seite 254]
4.10.2 - SEA equations [Seite 256]
4.10.3 - Uncertainty propagation in SEA [Seite 257]
4.10.3.1 - Approach using sensitivity [Seite 258]
4.10.3.2 - Approach using Design of Experiments [Seite 259]
4.10.4 - Results [Seite 261]
4.10.5 - Conclusions [Seite 265]
4.10.6 - References [Seite 265]
4.11 - Phase reconstruction for time-domain analysis of uncertain structures [Seite 266]
4.11.1 - Introduction [Seite 266]
4.11.2 - Explanation of minimum phase [Seite 267]
4.11.2.1 - Defining minimum phase [Seite 267]
4.11.2.2 - The Hilbert transform and analytic systems [Seite 267]
4.11.2.3 - The Hilbert Transform and minimum phase systems [Seite 268]
4.11.2.4 - Further interpretation of minimum phase [Seite 268]
4.11.3 - Using minimum phase reconstruction [Seite 269]
4.11.3.1 - Approximating the Hilbert Transform [Seite 269]
4.11.3.2 - Errors using MPR for non-minimum phase systems [Seite 272]
4.11.4 - Application: peak shock prediction in uncertain structures [Seite 274]
4.11.4.1 - Modelling an uncertain structure [Seite 275]
4.11.4.2 - Ensemble average results [Seite 276]
4.11.4.3 - Changing the correlation of modal amplitudes [Seite 277]
4.11.5 - Conclusions [Seite 278]
4.11.6 - References [Seite 278]
5 - Probabilistic Methods [Seite 279]
5.1 - Uncertain Linear Systems in Dynamics: Stochastic Approaches [Seite 280]
5.1.1 - Introductory Remarks [Seite 280]
5.1.2 - Overview of Available Methods [Seite 281]
5.1.3 - Response variability [Seite 282]
5.1.3.1 - Perturbation Method [Seite 282]
5.1.3.2 - Spectral methods [Seite 285]
5.1.3.3 - Direct Monte Carlo Simulation [Seite 288]
5.1.3.4 - Random matrix approach [Seite 291]
5.1.4 - Computational Efficiency [Seite 291]
5.1.5 - Summary [Seite 292]
5.1.6 - References [Seite 293]
5.2 - Time domain analysis of structures with stochastic material properties [Seite 296]
5.2.1 - Introduction [Seite 296]
5.2.2 - Preliminary concepts [Seite 297]
5.2.3 - Application of the perturbation approach [Seite 298]
5.2.4 - Moments of the uncertain structure response [Seite 299]
5.2.5 - Application [Seite 302]
5.2.6 - Conclusions [Seite 303]
5.2.7 - References [Seite 308]
5.3 - Vibration Analysis of an Ensemble of Structures using an Exact Theory of Stochastic Linear Systems [Seite 309]
5.3.1 - Introduction [Seite 309]
5.3.2 - Description of the Stochastic System [Seite 310]
5.3.3 - Expression of Mean, Variance, and Covariance [Seite 312]
5.3.3.1 - Parameterized Response [Seite 312]
5.3.3.2 - Mean Response [Seite 313]
5.3.3.3 - Variance and Covariance of the Responses [Seite 313]
5.3.3.4 - Multirank Disturbance [Seite 314]
5.3.3.5 - Discussion of the Theory [Seite 316]
5.3.4 - Stochastic Coefficients in the case of a Gaussian Probability Density Function [Seite 316]
5.3.5 - Application examples [Seite 317]
5.3.5.1 - Comparison to a Monte-Carlo Simulation [Seite 318]
5.3.5.2 - Transition from low to high modal density [Seite 319]
5.3.5.3 - Variance and covariance of responses at different frequencies [Seite 321]
5.3.6 - Conclusion [Seite 322]
5.3.7 - References [Seite 322]
5.4 - Structural Uncertainty Identification using Vibration Mode Shape Information [Seite 324]
5.4.1 - Introduction [Seite 324]
5.4.2 - Maximum Likelihood Estimation of Uncertain Structural Parameters [Seite 326]
5.4.2.1 - Uncertainty Estimation via the Perturbation Method [Seite 326]
5.4.3 - ML estimates of uncertain point-mass position statistics using natural frequency information on a cantilever beam structure [Seite 327]
5.4.4 - ML Estimation of uncertain point mass position on a plate structure using mode shape information [Seite 330]
5.4.5 - Discussion of Results [Seite 334]
5.4.6 - Conclusions [Seite 336]
5.4.7 - References [Seite 336]
5.5 - Extremely strong convergence of eigenvalue-density of linear stochastic dynamical systems [Seite 337]
5.5.1 - Introduction [Seite 338]
5.5.2 - Uncertainty quantification of dynamic response [Seite 339]
5.5.3 - Wishart random matrix model [Seite 340]
5.5.4 - Density of eigenvalues [Seite 342]
5.5.4.1 - Linear eigenvalue statistic [Seite 342]
5.5.4.2 - Self averaging property and the Marcenko-Pastur density [Seite 343]
5.5.5 - Numerical investigations [Seite 346]
5.5.5.1 - Plate with randomly inhomogeneous material properties: parametric uncertainty problem [Seite 348]
5.5.5.2 - Plate with randomly attached spring-mass oscillators: nonparametric uncertainty problem [Seite 349]
5.5.6 - Conclusions [Seite 349]
5.5.7 - References [Seite 350]
5.6 - Stochastic subspace projection schemes for dynamic analysis of uncertain systems [Seite 352]
5.6.1 - Introduction [Seite 352]
5.6.2 - Preliminaries [Seite 353]
5.6.3 - Frequency domain analysis of linear stochastic structural systems [Seite 354]
5.6.3.1 - Preconditioner [Seite 358]
5.6.3.2 - Postprocessing [Seite 359]
5.6.4 - The algebraic random eigenvalue problem [Seite 359]
5.6.4.1 - Stochastic Basis Vectors [Seite 360]
5.6.4.2 - Bubnov-Galerkin Projection [Seite 361]
5.6.4.3 - Postprocessing [Seite 361]
5.6.5 - Numerical Studies [Seite 362]
5.6.6 - Concluding Remarks [Seite 364]
5.6.7 - References [Seite 365]
6 - Probabilistic Methods, Applications [Seite 366]
6.1 - Reliability Assessment of Uncertain Linear Systems in Structural Dynamics [Seite 367]
6.1.1 - Introduction [Seite 367]
6.1.2 - Methods of Analysis [Seite 368]
6.1.2.1 - Representation of uncertain excitation [Seite 368]
6.1.2.2 - Uncertain structural systems [Seite 370]
6.1.2.3 - Stochastic conditional response [Seite 370]
6.1.2.4 - Conditional reliability [Seite 371]
6.1.2.5 - Design point for stochastic structural systems [Seite 372]
6.1.2.6 - First excursion probability for stochastic systems [Seite 374]
6.1.3 - Numerical example [Seite 376]
6.1.3.1 - General remarks [Seite 376]
6.1.3.2 - Structural system [Seite 376]
6.1.3.3 - Dynamic excitation [Seite 378]
6.1.3.4 - Critical response [Seite 379]
6.1.3.5 - Reliability of critical component [Seite 379]
6.1.4 - Conclusions [Seite 381]
6.1.5 - References [Seite 382]
6.2 - On semi-statistical method of numerical solution of integral equations and its applications [Seite 383]
6.2.1 - Introduction [Seite 383]
6.2.2 - Short scheme of semi-statistical method [Seite 384]
6.2.3 - Statement of the problem of blade cascade flow [Seite 385]
6.2.4 - Scheme of application of semi-statistical method to the problem of blade cascade flow [Seite 387]
6.2.4.1 - Main formulas [Seite 387]
6.2.4.2 - Computation algorithm and optimization [Seite 388]
6.2.5 - Results of simulations [Seite 389]
6.2.6 - Analysis of efficiency of the density adaptation [Seite 389]
6.2.7 - Conclusion [Seite 390]
6.2.8 - References [Seite 392]
6.3 - An efficient model of drill-string dynamics with localised non-linearities [Seite 393]
6.3.1 - Introduction [Seite 393]
6.3.2 - Theoretical Framework [Seite 395]
6.3.2.1 - Linear Model [Seite 395]
6.3.2.2 - Coupling to Non-Linearities [Seite 397]
6.3.2.3 - Coupling to Subsystems [Seite 399]
6.3.3 - Example Simulations [Seite 400]
6.3.3.1 - Linear Behaviour [Seite 400]
6.3.3.2 - Coupling to non-linear friction law [Seite 401]
6.3.3.3 - Coupling to lumped inertia [Seite 402]
6.3.3.4 - Uncertainty Analysis of Stick-Slip Oscillation [Seite 403]
6.3.4 - Conclusions [Seite 405]
6.3.5 - References [Seite 405]
6.4 - Equivalent thermo-mechanical parameters for perfect crystals [Seite 407]
6.4.1 - Introduction [Seite 407]
6.4.2 - Hypotheses [Seite 408]
6.4.3 - Kinematics [Seite 410]
6.4.4 - Equation of momentum balance [Seite 412]
6.4.5 - Equation of angular momentum balance [Seite 414]
6.4.6 - Equation of energy balance [Seite 415]
6.4.7 - Constitutive relations for stress tensor and heat flux [Seite 417]
6.4.8 - Concluding remarks [Seite 419]
6.4.9 - References [Seite 420]
6.5 - Analysis of offshore systems in random waves [Seite 421]
6.5.1 - Introduction [Seite 421]
6.5.2 - Modeling aspects [Seite 422]
6.5.2.1 - Modeling of environmental forces [Seite 423]
6.5.2.2 - Modeling of multibody systems [Seite 424]
6.5.3 - Analysis of deterministic systems [Seite 425]
6.5.4 - Analysis of random systems [Seite 426]
6.5.4.1 - Monte Carlo simulation [Seite 426]
6.5.4.2 - Stochastic linearization [Seite 427]
6.5.5 - Selected Results [Seite 427]
6.5.6 - Conclusions [Seite 431]
6.5.7 - References [Seite 431]
6.6 - Statistical Dynamics of the Rolling Mills [Seite 432]
6.6.1 - Introduction [Seite 432]
6.6.2 - Cold Rolling Mills Chatter Vibrations [Seite 434]
6.6.2.1 - Rolling stand design and its modal analysis [Seite 434]
6.6.2.2 - Strip Elasto-Plastic Deformation [Seite 436]
6.6.2.3 - Horizontal work rolls vibration [Seite 438]
6.6.2.4 - Contact friction force variation [Seite 439]
6.6.2.5 - Chatter detection and control [Seite 440]
6.6.3 - Hot Rolling Mills Torsional Vibrations [Seite 441]
6.6.3.1 - Torsional vibration control and backlashes diagnostics [Seite 442]
6.6.4 - Conclusions [Seite 443]
6.6.5 - References [Seite 443]
6.7 - The application of robust design strategies on managing the uncertainty and variability issues of the blade mistuning vibration problem [Seite 446]
6.7.1 - Introduction [Seite 447]
6.7.2 - Basic concepts of the blade mistuning problem [Seite 448]
6.7.2.1 - The Amplification Factor (its significance and range) [Seite 449]
6.7.3 - Casting blade mistuning as a robust design problem [Seite 450]
6.7.3.1 - The Taguchi method of robust design [Seite 451]
6.7.3.2 - The robust optimisation method [Seite 451]
6.7.3.3 - Application of robust design methods to the blade mistuning problem [Seite 452]
6.7.4 - Improving the robustness of bladed discs by parameter design [Seite 453]
6.7.5 - Improving the robustness of bladed discs by tolerance design [Seite 455]
6.7.5.1 - The Small Mistuning approach [Seite 456]
6.7.5.2 - The Intentional Mistuning approach [Seite 456]
6.7.6 - Conclusions [Seite 458]
6.7.7 - References [Seite 459]
6.8 - Localized modeling of uncertainty in the Arlequin framework [Seite 460]
6.8.1 - Introduction [Seite 460]
6.8.2 - The classical Arlequin method [Seite 462]
6.8.3 - The continuous stochastic-deterministic Arlequin formulation [Seite 464]
6.8.3.1 - The stochastic monomodel [Seite 465]
6.8.3.2 - The Arlequin formulation [Seite 465]
6.8.4 - The discretized stochastic-deterministic Arlequin formulation [Seite 466]
6.8.5 - Example of application [Seite 468]
6.8.6 - Conclusion [Seite 470]
6.8.7 - References [Seite 470]
