IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties

IUTAM Symposium on the Vibration Analysis

of Structures with Uncertainties

IUTAM BOOKSERIES

Volume 27

Series Editors

G.M.L. Gladwell, University of Waterloo, Waterloo, Ontario, Canada

R. Moreau, INPG, Grenoble, France

Editorial Board

J. Engelbrecht, Institute of Cybernetics, Tallinn, Estonia

L.B. Freund, Brown University, Providence, USA

A. Kluwick, Technische Universitt, Vienna, Austria

H.K. Moffatt, University of Cambridge, Cambridge, UK

N. Olhoff, Aalborg University, Aalborg, Denmark

K. Tsutomu, IIDS, Tokyo, Japan

D. van Campen, Technical University Eindhoven, Eindhoven,

The Netherlands

Z. Zheng, Chinese Academy of Sciences, Beijing, China

Aims and Scope of the Series

The IUTAM Bookseries publishes the proceedings of IUTAM symposia under the

auspices of the IUTAM Board.

For other titles published in this series, go to

www.springer.com/series/7695

Alexander K. Belyaev Robin S. Langley

Editors

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

Editors

Alexander K. Belyaev

Russian Academy of Sciences

Inst. Problems of Mechanical Engineering

Bolshoy Ave. V. O. 61

199178 St. Petersburg

Russia

[email protected]

Robin S. Langley

University of Cambridge

Dept. Engineering

Trumpington Street

CB2 1PZ Cambridge

UK

[email protected]

ISSN 1875-3507

ISBN 978-94-007-0288-2

e-ISSN 1875-3493

e-ISBN 978-94-007-0289-9

DOI 10.1007/978-94-007-0289-9

Springer Dordrecht Heidelberg London New York

© Springer Science+Business Media B.V. 2011

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Preface

The IUTAM Symposium on The Vibration Analysis of Structures with Uncertain￾ties was held in Saint-Petersburg, Russia, on July 5 – July 9 2009. The members

of the Scientific Committee were Alexander K. Belyaev (Chair) — Institute of

Problems in Mechanical Engineering of the Russian Academy of Sciences, Saint￾Petersburg, Russia; Robin Langley (Co-Chair) — University of Cambridge, UK;

Franz Ziegler (IUTAM Representative) — Vienna University of Technology, Aus￾tria; Antonio Carcaterra — University of Rome La Sapienza, Italy; Yakov Ben￾Haim — Technion-Israel Institute of Technology, Israel; Christian Soize — Univer￾site de Marne-la-Vallee, France; Dirk Vandepitte — Katholieke Universiteit Leuven,

Belgium and Richard Weaver — University of Illinois at Urbana-Champaign, USA.

The Symposium took place in Tsarskoe Selo in a palace designed by Prince

Kochubei. Tsarkoe Selo (a suburb of St. Petersburg, also known as Pushkin) was

a summer residence of the Tsar, and is well known for its many palaces, including

the Katherine Palace which houses the famous Amber Room.

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 complex￾ity 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 presenta￾tions 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 pre￾sentations focussed on parametric uncertainty modelling (for example, probabilistic

models, interval analysis, and fuzzy descriptions) and on methods of propagating

v

vi Preface

this uncertainty through a large dynamic model in an efficient way. At mid frequen￾cies the problem is mixed, and various hybrid schemes were proposed. It is clear

that a comprehensive solution 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, ran￾dom wave approaches, and large scale computations) and this IUTAM symposium

presented a unique opportunity to bring together outstanding international experts

in these fields.

The lectures were arranged such that 12 of the presentations were keynote

overviews and allocated 45 minutes, while the remaining 24 presentations were each

allocated 20 minutes. In addition to this, there was much discussion and fruitful in￾teraction, both during the technical sessions and over lunch and dinner. All of the

presented papers are collected together in the Proceedings.

The IUTAM grant and the financial support of the Russian Foundation for Basic

Research are gratefully acknowledged. Also, we would like to express our sincere

gratitude to Dr Dmitry Kiryan who took the trouble of preparing the camera ready

manuscript of the Proceedings. Robin Langley’s participation was funded through

the ITN Marie Curie project GA-214909 “MID-FREQUENCY – CAE Methodolo￾gies for Mid-Frequency Analysis in Vibration and Acoustics”.

St. Petersburg, June 2010 Alexander K. Belyaev

Robin S. Langley

Symposium Co-Chairs

Contents

Part I Non-probabilistic and related approaches

Linear Bounds on an Uncertain Non-Linear Oscillator: An Info-Gap

Approach ............................. ......................... 3

Yakov Ben-Haim and Scott Cogan

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Dynamics, Uncertainty and Robustness . . . . . . . . . . . . . . . . . . . . . . . 4

3 Example: Uncertain Cubic Non-Linearity . . . . . . . . . . . . . . . . . . . . . 7

4 Example: Multiple Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Robustness as a Proxy for Probability . . . . . . . . . . . . . . . . . . . . . . . . 12

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Quantification of uncertain and variable model parameters in

non-deterministic analysis ........................................ 15

Dirk Vandepitte, David Moens

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Numerical representation of parameter uncertainty and variability . 16

2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Discussion and extension of the definitions . . . . . . . . . . . . 18

3 Literature review on uncertain model and material data . . . . . . . . . . 19

3.1 Non-probabilistic models . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Probabilistic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Material data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Other model properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Alternative approaches: non-parametric model concept

and info-gap theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Summary of observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

vii

viii Contents

Vibrations of layered structures with fuzzy core stiffness/fuzzy

interlayer slip ................................................... 29

Rudolf Heuer and Franz Ziegler

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Fuzzy sandwich beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.1 Three-layer beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Modal analysis of the three-layer beam, hard-hinged

support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 Isosceles uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Constraints affected to the uncertain natural frequencies . 38

3.3 Some effects of non-symmetric uncertainty . . . . . . . . . . . . 41

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Vibration Analysis of Fluid-Filled Piping Systems with Epistemic

Uncertainties ................................................... 43

M. Hanss, J. Herrmann and T. Haag

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2 Classification, Representation and Propagation of Uncertainty . . . . 44

2.1 Uncertainty Classification and Representation . . . . . . . . . . 44

2.2 Uncertainty Propagation Based on the Transformation

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Fluid-Filled Piping System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1 Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4 Comprehensive Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . 51

4.1 Modeling of Epistemic Uncertainties . . . . . . . . . . . . . . . . . 51

4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Measures of Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Fuzzy vibration analysis and optimization of engineering structures:

Application to Demeter satellite ................................... 57

F. Massa, A. Leroux, B. Lallemand, T. Tison, F. Buffe, and S. Mary

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2 Aims of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.1 Description of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.2 Building of fuzzy optimization problem . . . . . . . . . . . . . . . 59

3 Fuzzy vibration analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.1 PAEM method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Numerical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Fuzzy optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1 Design methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Improvement of the initial design . . . . . . . . . . . . . . . . . . . . 66

Contents ix

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Numerical dynamic analysis of uncertain mechanical structures based

on interval fields ................................................ 71

David Moens, Maarten De Munck, Wim Desmet, Dirk Vandepitte

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2 Interval finite element analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 Interval fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.1 General concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2 Interval fields as uncertain input parameters . . . . . . . . . . . 75

3.3 Interval fields as uncertain analysis results . . . . . . . . . . . . . 77

4 Application of interval fields for vibro-acoustic analysis . . . . . . . . . 79

4.1 Vibro-acoustic analysis based on the ATV concept . . . . . . 79

4.2 Interval analysis based on structural FRF interval fields . . 80

4.3 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

From Interval Computations to Constraint-Related Set Computations:

Towards Faster Estimation of Statistics and ODEs under Interval and

p-Box Uncertainty ............................................... 85

Vladik Kreinovich

1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2 Interval Computations: Brief Reminder . . . . . . . . . . . . . . . . . . . . . . . 88

3 Constraint-Based Set Computations . . . . . . . . . . . . . . . . . . . . . . . . . . 89

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Dynamic Steady-State Analysis of Structures under Uncertain

Harmonic Loads via Semidefinite Program .......................... 99

Yoshihiro Kanno and Izuru Takewaki

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2 Uncertain equations for steady state vibration . . . . . . . . . . . . . . . . . . 101

2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2.2 Uncertainty model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2.3 ULE in real variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3 Bounds for complex amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.1 Upper bound for modulus of displacement amplitude . . . . 103

3.2 Lower bound for modulus of displacement amplitude . . . 106

3.3 Bounds for phase angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4 Bounds for nodal oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

x Contents

Part II SEA related methods and wave propagation

Universal eigenvalue statistics and vibration response prediction ........ 115

R.S. Langley

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

2 Eigenvalue statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.1 The joint probability density function of the eigenvalues . 116

2.2 The modal density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

2.3 Universality of the “local” eigenvalue statistics . . . . . . . . . 119

2.4 Application to natural frequency statistics . . . . . . . . . . . . . 121

3 Application to response statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.1 Fundamental concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.2 Built-up systems: SEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3.3 Built-up systems: the Hybrid method . . . . . . . . . . . . . . . . . 124

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Statistical Energy Analysis and the second principle of thermodynamics . 129

Alain Le Bot

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

2 First principle of thermodynamics in SEA . . . . . . . . . . . . . . . . . . . . . 130

3 Vibrational entropy, vibrational temperature . . . . . . . . . . . . . . . . . . . 133

4 Second principle of thermodynamics in SEA . . . . . . . . . . . . . . . . . . 134

5 Entropy balance in SEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Modeling noise and vibration transmission in complex systems ......... 141

Philip J. Shorter

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

1.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

1.2 Uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

1.3 How much information is needed for noise and

vibration design? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

2 Modeling methods and frequency ranges . . . . . . . . . . . . . . . . . . . . . . 144

2.1 Low, mid and high frequency ranges . . . . . . . . . . . . . . . . . . 144

2.2 Low and High frequency modelling methods. . . . . . . . . . . 145

2.3 The Mid-Frequency problem . . . . . . . . . . . . . . . . . . . . . . . . 146

3 The Hybrid FE-SEA method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

3.1 Statistical subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

3.2 The direct and reverberant fields of a statistical subsystem148

3.3 Ensemble average reverberant loading . . . . . . . . . . . . . . . . 149

3.4 Coupling a deterministic and statistical subsystem . . . . . . 149

4 Application examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.1 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Contents xi

4.2 Numerical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

4.3 Industrial applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

A Power Absorbing Matrix for the Hybrid FEA-SEA Method .......... 157

R.H. Lande and R.S. Langley

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

2 Cylindrical Waves and Energy Sinks . . . . . . . . . . . . . . . . . . . . . . . . . 158

2.1 The Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

2.2 The Cylindrical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

3 Constructing the Power Absorbing Matrix. . . . . . . . . . . . . . . . . . . . . 162

3.1 Discretization of the Power Integral, and Matrix

Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

3.2 Numerical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4.1 A Simple System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4.2 System Randomization and Subsystem Response

Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

The Energy Finite Element Method NoiseFEM....................... 171

Christian Cabos and Hermann G. Matthies

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

1.2 Literature Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

2 Components of NoiseFEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

3 Power Flow Between Structural Elements . . . . . . . . . . . . . . . . . . . . . 173

3.1 Transmission Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 174

3.2 The Coupling Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4 Diffusive Energy Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4.1 Homogeneous Structural Elements . . . . . . . . . . . . . . . . . . . 178

4.2 Stiffened Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

5 Combining transport and coupling equations . . . . . . . . . . . . . . . . . . 180

6 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7 Validation of NoiseFEM with test structures . . . . . . . . . . . . . . . . . . . 182

8 Application of NoiseFEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

xii Contents

Wave transport in complex vibro-acoustic structures in the

high-frequency limit ............................................. 187

Gregor Tanner and Stefano Giani

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

2 Wave energy flow in terms of the Green function . . . . . . . . . . . . . . . 189

3 Linear phase space operators and DEA . . . . . . . . . . . . . . . . . . . . . . . 190

4 A numerical example: coupled two-domain systems . . . . . . . . . . . . 194

4.1 The hp-adaptive Discontinuous Galerkin Method . . . . . . . 194

4.2 FEM compared to DEA and SEA — results . . . . . . . . . . . 197

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Benchmark study of three approaches to propagation of harmonic

waves in randomly heterogeneous elastic media ...................... 201

Alexander K. Belyaev

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

2 Method of integral spectral decomposition . . . . . . . . . . . . . . . . . . . . 202

3 The Fokker-Planck-Kolmogorov equation . . . . . . . . . . . . . . . . . . . . . 205

4 The Dyson integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Minimum-variance-response and irreversible energy confinement....... 215

A. Carcaterra

1 Average Impulse Response and the Single Case . . . . . . . . . . . . . . . . 215

2 MIVAR: Minimum-Variance-Response . . . . . . . . . . . . . . . . . . . . . . . 217

3 Application of the theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

High-frequency vibrational power flows in randomly heterogeneous

coupled structures ............................................... 229

Eric Savin ´

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

2 Transport model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

2.1 Radiative transfer in an open domain . . . . . . . . . . . . . . . . . 231

2.2 Radiative transfer in a bounded domain . . . . . . . . . . . . . . . 232

2.3 Radiative transfer with a sharp interface . . . . . . . . . . . . . . . 233

3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

3.1 Coupled beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

3.2 Coupled shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Contents xiii

Uncertainty propagation in SEA using sensitivity analysis and Design of

Experiments.................................................... 243

Antonio Culla, Walter D’Ambrogio and Annalisa Fregolent

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

2 SEA equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

3 Uncertainty propagation in SEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

3.1 Approach using sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 247

3.2 Approach using Design of Experiments . . . . . . . . . . . . . . . 248

4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

Phase reconstruction for time-domain analysis of uncertain structures ... 255

L H Humphry

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

2 Explanation of minimum phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

2.1 Defining minimum phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

2.2 The Hilbert transform and analytic systems . . . . . . . . . . . . 256

2.3 The Hilbert Transform and minimum phase systems . . . . 257

2.4 Further interpretation of minimum phase . . . . . . . . . . . . . . 257

3 Using minimum phase reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 258

3.1 Approximating the Hilbert Transform. . . . . . . . . . . . . . . . . 258

3.2 Errors using MPR for non-minimum phase systems . . . . . 261

4 Application: peak shock prediction in uncertain structures . . . . . . . 263

4.1 Modelling an uncertain structure . . . . . . . . . . . . . . . . . . . . . 264

4.2 Ensemble average results . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

4.3 Changing the correlation of modal amplitudes. . . . . . . . . . 266

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Part III Probabilistic Methods

Uncertain Linear Systems in Dynamics: Stochastic Approaches......... 271

G.I. Schueller and H.J. Pradlwarter ¨

1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

2 Overview of Available Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

3 Response variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

3.1 Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

3.2 Spectral methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

3.3 Direct Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . 279

3.4 Random matrix approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

4 Computational Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

xiv Contents

Time domain analysis of structures with stochastic material properties . . 287

Giovanni Falsone and Dario Settineri

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

2 Preliminary concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

3 Application of the perturbation approach . . . . . . . . . . . . . . . . . . . . . . 289

4 Moments of the uncertain structure response. . . . . . . . . . . . . . . . . . . 290

5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Vibration Analysis of an Ensemble of Structures using an Exact Theory

of Stochastic Linear Systems ...................................... 301

Christophe Lecomte

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

2 Description of the Stochastic System . . . . . . . . . . . . . . . . . . . . . . . . . 302

3 Expression of Mean, Variance, and Covariance. . . . . . . . . . . . . . . . . 304

3.1 Parameterized Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

3.2 Mean Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

3.3 Variance and Covariance of the Responses . . . . . . . . . . . . . 305

3.4 Multirank Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

3.5 Discussion of the Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

4 Stochastic Coefficients in the case of a Gaussian Probability

Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

5 Application examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

5.1 Comparison to a Monte-Carlo Simulation . . . . . . . . . . . . . 310

5.2 Transition from low to high modal density . . . . . . . . . . . . . 311

5.3 Variance and covariance of responses at different

frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Structural Uncertainty Identification using Vibration Mode Shape

Information .................................................... 317

J. F. Dunne and S. Riefelyna

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

2 Maximum Likelihood Estimation of Uncertain Structural

Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

2.1 Uncertainty Estimation via the Perturbation Method . . . . 319

3 ML estimates of uncertain point-mass position statistics using

natural frequency information on a cantilever beam structure . . . . . 320

4 ML Estimation of uncertain point mass position on a plate

structure using mode shape information . . . . . . . . . . . . . . . . . . . . . . . 323

5 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329