Stochastic Dynamics of Structures E-Book

Abdelkhalak El Hami

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2016

Seitenzahl: 315

Beschreibung

This book is dedicated to the general study of the dynamics of mechanical structures with consideration of uncertainties. The goal is to get the appropriate forms of a part in minimizing a given criterion. In all fields of structural mechanics, the impact of good design of a room is very important to its strength, its life and its use in service. The development of the engineer's art requires considerable effort to constantly improve structural design techniques.

Leseprobe

Table of Contents

Cover

Title

Copyright

Preface

1 Introduction to Structural Dynamics

1.1. Composition of problems relating to dynamic structures

1.2. Structural optimization

1.3. Structures with uncertain parameters

1.4. Conclusion

2 Decoupled Systems

2.1. Introduction

2.2. Problems with structural dynamics

2.3. Acoustic problems

2.4. Conclusion

3 Coupled Systems

3.1. Introduction

3.2. Mathematical formulation

3.3. Variational formulation

3.4. Estimation by finite elements

3.5. Vibro-acoustic problem

3.6. Hydro-elastic problem

3.7. Reduction of the model

3.8. Conclusion

4 Reliability and Meshless Methods in Mechanics

4.1. Introduction to non-networking methods

4.2. Moving least squares

4.3. Galerkin mesh-free method

4.4. Imposition of essential limiting conditions

4.5. Integration in the EFG method

4.6. Description of EFG method algorithms

5 Mechanical Systems with Uncertain Parameters

5.1. Introduction

5.2. Monte Carlo simulation

5.3. Disturbance methods

5.4. Projection onto polynomial chaos

5.5. Conclusion

6 Modal Synthesis Methods and Stochastic Finite Element Methods

6.1. Introduction

6.2. Linear dynamic problems

6.3. Modal synthesis methods

6.4. Stochastic finite element methods

6.5. Conclusion

7 Stochastic Modal Synthesis Methods

7.1. Introduction

7.2. Taylor series expansion of the modal equations of a stochastic structure

7.3. Muscolino perturbation method

7.4. Stochastic fixed interface method

7.5. Stochastic modal synthesis method

7.6. Conclusion

8 Dynamic Response of a Structure with Uncertain Variables to a Given Excitation

8.1. Introduction

8.2. Perturbation method

8.3. Stochastic modal synthesis method

8.4. Projection onto homogeneous chaos

8.5. Coupling modal synthesis methods with projection onto homogeneous chaos

8.6. Conclusion

9 Stochastic Frequency Response Function

9.1. Introduction

9.2. Calculation of the stochastic frequency response function

9.3. Calculation of the stochastic frequency response function with modal synthesis methods

9.4. Conclusion

10 Modal Synthesis Methods and Reliability Optimization Methods

10.1. Introduction

10.2. Combining modal synthesis and RBDO methods

10.3. Conclusion

11 Stochastic Model of Transmission in a Wind Turbine

11.1. Introduction

11.2. Modeling the dynamic behavior of the gearing system in a wind turbine

11.3. Dynamic response of a two-step gear system in a wind turbine with uncertain variables

11.4. Conclusion

Bibliography

Index

End User License Agreement

List of Tables

2 Decoupled Systems

Table 2.1. Number of degrees of freedom (DoF) for a few models

Table 2.2. Properties of the slab studied

Table 2.3. The first six frequencies of the slab (Hz) calculated by different methods

Table 2.4. CPU time needed to calculate responses

Table 2.5. Properties of the rigid tank and water

Table 2.6. The first five frequencies (Hz) for the movement of water

Table 2.7. The first six acoustic frequencies (Hz) of the fluid calculated by different methods

3 Coupled Systems

Table 3.1. Data for the problem

Table 3.2. The first five frequencies for the beam with errors relative to the analytical result

Table 3.3. Data for the problem

Table 3.4. The first six frequencies for the structure with errors relative to the analytical result

Table 3.5. The first six frequencies of the submerged structure with errors relative to direct calculation

Table 3.6. CPU time necessary for calculation of the FRF of the submerged structure

Table 3.7. Data of the problem

Table 3.8. The first six frequencies for the slab when completely submerged

Table 3.9. CPU time required for FRF calculation of the wet slab

4 Reliability and Meshless Methods in Mechanics

Table 4.1. Base functions with one, two and three dimensions

Table 4.2. Geometric and mechanical properties of the slab

Table 4.3. Movements and constraints in the slab

5 Mechanical Systems with Uncertain Parameters

Table 5.1. Values of dynamic matrices of a mechanical system

Table 5.2. Moments of the first six frequencies of the slab

Table 5.3. Moments of the random parameters of the paired system

Table 5.4. Moments of the first six frequencies specific to the submerged slab

Table 5.5. Calculation time of moments of the FRF for the submerged slab

Table 5.6. Polynomial chaos of order 4 in the case of a dimension

Table 5.7. Polynomial chaos of order 4 in the case of two dimensions

Table 5.8. Moments of the parameters of the problem

Table 5.9. Moments of the first six frequencies specific to the paired problem

Table 5.10. Calculation time of moments of FRF for the paired problem

Table 5.11. Data for the problem

Table 5.12. First frequencies specific to the dry and submerged block

Table 5.13. Moments of random parameters for the problem

Table 5.14. Moments of the first five frequencies for the submerged block

Table 5.15. Result of reliable calculation

Table 5.16. Physical properties of the propeller

Table 5.17. The first six frequencies for the propeller

Table 5.18. Moments of physical parameters of the propeller

Table 5.19. Moments of the first six frequencies for the propeller

Table 5.20. Result of the reliability study for the first frequency of the propeller

Table 5.21. Data for the problem

Table 5.22. Frequencies (Hz) of the tank

Table 5.23. Moments of physical parameters of the paired problem

Table 5.24. Moments of the first six frequencies of the full tank

Table 5.25. Result of the reliability study for the first frequency of the tank

7 Stochastic Modal Synthesis Methods

Table 7.1. Comparison between the normal modes of the entire model, the model condensed by the Craig–Bampton method and the model condensed by the free interface method with a single fixed interface mode for each sub-structure

Table 7.2. Comparison between the normal modes of the whole model, the model condensed by the Craig–Bampton model and the free interface method with two fixed interface modes for each sub-structure

Table 7.3. Comparison between the first two moments of the normal modes of the whole model and the model condensed by the Craig–Bampton method with a single fixed interface mode for each sub-structure

Table 7.4. Comparison between the first two moments of the normal modes of the whole model and the model condensed by the Craig–Bampton method with two fixed interface modes for each sub-structure

Table 7.5. Comparison between the first two moments of the normal modes of the whole model and the model condensed by the free interface method with deterministic reduced bases

Table 7.6. Number of modes of each sub-structure, for the fixed interface and free interface methods

Table 7.7. Comparison between the normal modes of the full model and the model reduced in the deterministic case

Table 7.8. Comparison between the stochastic normal modes of the full model and the model condensed by the Craig–Bampton method, with random and deterministic reduced basis applying the classical perturbation method

Table 7.9. Comparison between the stochastic normal modes of the full model and the model condensed by the Craig–Bampton method, applying the new method

Table 7.10. Comparison between the stochastic normal modes of the full model and the model condensed by the fixed interface method, applying the new method and the classical perturbation method

Table 7.11. Number of modes retained from the band [0 2f

u

] for the fixed interface and free interface methods without reducing the interface degrees of freedom

Table 7.12. Number of modes retained from the band [0 2f

u

] for the fixed interface and free interface methods with reduction in the interface degrees of freedom

Table 7.13. Number of modes retained from the band [0 3f

u

] for fixed interface and free interface methods without reducing the interface degrees of freedom

Table 7.14. Number of modes retained from the band [0 3f

u

] for the fixed interface and free interface methods with reduced interface degrees of freedom

9 Stochastic Frequency Response Function

Table 9.1. Number of modes retained in the band [0 2F

u

] for the fixed interface and free interface methods, without reduction of the interface degrees of freedom

Table 9.2. Number of modes retained in the band [0 4F

u

] for the fixed interface and free interface methods with a reduction in the interface degrees of freedom

10 Modal Synthesis Methods and Reliability Optimization Methods

Table 10.1. FRBDO results for the full model

Table 10.2. Results of RBDO integrating the CB method

Table 10.3. Results of RBDO integrating the FrI method

Table 10.4. Results of RBDO integrating the RCB method

Table 10.5. Results of RBDO integrating the RFrI method

Table 10.6(a). Material characteristics

Table 10.6(b). Geometric characteristics

Table 10.7. Results of the full-system hybrid frequency RBDO method

Table 10.8. Results of the RBDO method with reduction in the degrees of freedom by the condensation method (1,500 active degrees of freedom)

Table 10.9. Results of the hybrid dynamic approach before and after reducing the model with random variables following a normal distribution

Table 10.10. Results of the hybrid dynamic RBDO approach before and after reducing the model with random variables following a log-normal distribution

11 Stochastic Model of Transmission in a Wind Turbine

Table 11.1. Numerical data for the studied model

Guide

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Mathematical and Mechanical Engineering Set

coordinated by

Abdelkhalak El Hami

Volume 2

Stochastic Dynamics of Structures

Abdelkhalak El Hami

Bouchaib Radi

First published 2016 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd

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London SW19 4EU

UK

www.iste.co.uk

John Wiley & Sons, Inc.

111 River Street

Hoboken, NJ 07030

USA

www.wiley.com

© ISTE Ltd 2016

The rights of Abdelkhalak El Hami and Bouchaib Radi to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2016950151

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-949-6

Preface

Over the past few years, engineers, scientists and authorities have shown a real interest in stochastic methods and the optimization of mechanical systems. These two areas have received increasing attention because of their theoretical complexities and industrial applications. At present, deterministic models do not take into account the variability of factors, often falsely identified, and show results that do not correspond to the reality of the specific problem.

This book includes the most recent ideas from research and industry in the domain of stochastic dynamics and optimization of mechanical structures. It contains 11 chapters that focus on different tools including uncertainties, stochastic methods, reliability and the optimization of systems. The chapters discuss the interaction between optimization and reliability of structural dynamics in order to consider the uncertainties in modeling and resolve the problems that are encountered. Each chapter clearly sets out the technological methods used and developed along with illustrative and relevant examples.

Chapter 1 explains the problems in mechanical structure dynamics. The works used include the exposition of uncertainties with regard to the problems of optimization. The last section is dedicated to describing the use of different tools to analyze structures that consist of uncertain factors. Chapter 2 presents decoupled mechanical structures. Issues regarding structure and liquid are studied separately. In each area, initial theories are used. Equations are expressed in the variational form and then discretized by finite element in order to display the matrix systems required to resolve the separate problem numerically. Chapter 3 presents the equations that show a fluid–structure coupled system and are put into a variational form. They are then discretized using the finite element method to obtain the matrix systems so they can be solved numerically. Finally, the application of modal reduction methods to coupled systems is considered. Modal reduction methods are then applied to vibro-acoustic and hydro-elastic issues.

Chapter 4 demonstrates the modeling methods of mesh-free structures and the formal theory of the element-free Galerkin method. Different mesh-free methods will be presented, together with the approximation of lower movable squares and the utilization of the weak form of elasticity equations in order to determine the unknown nodal values in the mesh-free Galerkin approximation. Chapter 5 aims to combine the modal reduction methods with non-frequentative stochastic methods and apply them to the study of (paired and unpaired) mechanical systems dynamics with uncertain parameters. It then evaluates the impact of modal reduction with regard to saving calculation time when reliably analyzing non-deterministic systems. Chapter 6 presents general dynamic equations by aiming to apply them to stochastic modal synthesis methods. It recalls the sub-structuring method, initially formulated for static problems. This involves considering a structure like a network of interconnected sub-structures. Synthesis methods differ in terms of the choice of method, in order to represent the dynamics of each sub-structure, and in assembly procedures. Then, a DDL junction reduction strategy will be proposed after assembly. This strategy is based on using calculations of modes of interface, which are obtained via Guyan condensation at the interfaces of the complete structure. Chapter 7 explores frequencies and appropriate modes for a dynamic conservative system, in which mass and rigid matrices are functions of random parameters. Two perturbation methods are used. The first method uses a second-order Taylor series expansion. The second is a method proposed by Muscolino, which uses first-order expansion. The objective is to highlight the advantages of methods of modal synthesis in predicting the dynamic behavior of stochastic structures. The traditional solution to the stochastic problem will be compared with that which uses sub-structuring methods.

Chapter 8 is devoted to the dynamic response of a structure with an uncertain variable to a given agitation. Two stochastic methods are presented. The first is the perturbation method. The second involves projecting a solution onto polynomial chaos. Both methods are used to calculate the first two moments of the response (mean value and variance) using knowledge of the laws of probability in relation to the distribution of structural parameters. The use of modal synthesis methods will allow us to reduce the dimensions of the model before integrating the equation of motion. Then, the extension of modal synthesis methods will be presented to evaluate the stochastic response of a dynamic system to a given agitation using the perturbation method. The end of this chapter attempts to show the interaction between dynamic sub-structuring methods and the method for predicting homogeneous chaos in order to evaluate the variability of the response with respect to the variability of parameters of a large-scale model. Chapter 9 presents the development of the homogeneous chaos projection method to determine the function of the stochastic response in terms of frequency. Two methods are presented. In the first method, the calculation is performed directly. The second method is based on the use of modes specific to the structure. This is followed by the extension of modal synthesis methods, which is put forward to reduce the size of the mechanical model, allowing us to calculate the function of the response in terms of the frequency of a large-scale structure that has uncertain parameters. In general, the size of the model is reduced using transformation matrices that are constructed from the modes of each sub-structure, which can be normal modes of vibration, static modes, connection modes or rigid body modes. These modes contain the uncertain parameters of each sub-structure. Finally, numerical applications will be explored to show the efficiency and accuracy of using a homogeneous chaos with a mechanical model reduced by modal synthesis methods. Chapter 10 presents a methodology combining modal synthesis techniques with reliability optimization in design. This chapter presents an algorithm that allows us to incorporate modal synthesis methods in a reliability optimization process. Finally, it evaluates this algorithm using different applications to show the effectiveness and robustness of the proposed method. Chapter 11 aims to study the dynamic behavior of gear transmission in a wind turbine with uncertain parameters.

Finally, this book constitutes a useful source of information for teachers and researchers. It may also be informative for engineering students, engineers and students who are pursuing Master’s degree.

Acknowledgments

We would like to thank all who have contributed to the creation of this book, our families, and in particular the PhD students at INSA de Rouen who have helped us over the past few years.

Abdelkhalak EL HAMI

Bouchaib RADI

September 2016

1Introduction to Structural Dynamics

The aim of this chapter is to convey a non-exhaustive image of all areas considered, from near or far, in this work.

Section 1.1 is dedicated to the general study of structural dynamics. This study intends to attach the essential evaluations to the calculations of dynamic responses, frequencies, appropriate methods and their response functions. All of these aspects are consequently tackled using practical applications.

The dynamic balance equation system of a structure can be solved by using one of the traditional strategies [MOH 05]. The most frequent resolution strategy in dynamics is modal superposition, which is suited to linear structures whose first methods are only the ones that are agitated. In contrast, direct resolution methods incorporate movement equations in order to handle nonlinear structures. These structures can also be applied when the frequency contents of the disturbance cover a large number of methods of the mechanical structure studied.

In section 1.2, a non-exhaustive bibliographic study is put forward regarding the optimization of structures. The objective is to obtain suitable forms from an article by minimizing a given criterion. In every area of structural mechanics, knowing the impact of effective object design is very important in determining its resistance, lifetime and operation. This is one of the challenges faced by industries daily. The development of engineering requires considerable effort to constantly improve the techniques for designing structures. Optimization plays an important role in increasing performance and significantly reducing aerospace and motoring engineering equipment, while simultaneously substantially saving energy.