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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
- Veröffentlichungsjahr: 2017
This book deals with the various aspects of stochastic dynamics, the resolution of large mechanical systems, and inverse problems. It integrates the most recent ideas from research and industry in the field of stochastic dynamics and optimization in structural mechanics over 11 chapters. These chapters provide an update on the various tools for dealing with uncertainties, stochastic dynamics, reliability and optimization of systems. The optimization-reliability coupling in structures dynamics is approached in order to take into account the uncertainties in the modeling and the resolution of the problems encountered. Accompanied by detailed examples of uncertainties, optimization, reliability, and model reduction, this book presents the newest design tools. It is intended for students and engineers and is a valuable support for practicing engineers and teacher-researchers.
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Table of Contents
Cover
Title
Copyright
Preface
Acknowledgments
1 Introduction to Inverse Methods
1.1. Introduction
1.2. Identification methods
1.3. Identification of the strain hardening law
2 Linear Differential Equation Systems of the First Order with Constant Coefficients: Application in Mechanical Engineering
2.1. Introduction
2.2. Modeling dissipative systems
2.3. Autonomous system general solution
2.4. General solution of the complete equation
2.5. Applications to mechanical structures
2.6. Inverse problems: expressions of the M, B, K matrices according to the intrinsic solutions
3 Introduction to Linear Structure Dynamics
3.1. Introduction
3.2. Problems in structure dynamics
4 Introduction to Nonlinear Dynamic Analysis
4.1. Introduction
4.2. Linear systems
4.3. The nonlinear 1 DOF system
4.4. Nonlinear
N
DOF systems
5 Condensation Methods Applied to Eigen Value Problems
5.1. Introduction
5.2. Mathematical generality: matrix transformation
5.3. Dynamic condensation methods
5.4. Guyan condensation
5.5. Rayleigh–Ritz method
5.6. Case of a temporary problem
6 Linear Substructure Approach for Dynamic Analysis
6.1. Generalities
6.2. Different types of Ritz vectors
6.3. Synthesis of eigen solutions of the assembled structure: formulation by an energetic method (Lagrange with multiplicators)
6.4. Craig and Bampton sub structuration method
6.5. Mixed method
6.6. Methods with eigen vectors with free common contours
6.7. Method systematically introducing an intermediary connection structure
7 Nonlinear Substructure Approach for Dynamic Analysis
7.1. Introduction
7.2. Dynamic substructuration approaches
7.3. Nonlinear substructure approach
7.4. Proper orthogonal decomposition for flows
7.5. Numerical results
8 Direct and Inverse Sensitivity
8.1. Introduction
8.2. Direct sensitivity
8.3. Sensitivity of eigen solutions
8.4. First derivative of a particular solution
8.5. Grouping the sensitivity relations together
8.6. Inverse sensitivity
9 Parametric Identification and Model Adjustment in Linear Elastic Dynamics
9.1. Introduction
9.2. Study in the elastic dynamics of mechanical structures
9.3. Parametric identification – use of a test for constructing weaker calculation models
9.4. Some basic methods in parametric identification
9.5. Parametric correction of finite elements models in linear elastic dynamics based on the test results
9.6. M model adjustment: k∈
R
c,c
by minimizing the matrix norms by the correction matrices δm, δk
9.7. M model adjustment: k∈
R
c,c
by minimizing residue vectors made up based on local correction matrices ΔM
I
, ΔK
I
10 Inverse Problems in Dynamics: Robustness Function
10.1. Introduction
10.2. Convex models
10.3. Robustness function
10.4. Solution methods
10.5. Numerical calculations
10.6. Applications
10.7. Conclusion
11 Modal Synthesis and Reliability Optimization Methods
11.1. Introduction
11.2. Design reliability optimization in structural dynamics
11.3. The SP method
11.4. Modal synthesis and RBDO coupling methods
11.5. Discussion
Appendix
A.1. Introduction
A.2. Finite elements approach
Bibliography
Index
End User License Agreement
List of Tables
1 Introduction to Inverse Methods
Table 1.1. Properties of the material used
Table 1.2. Swift parameters of the different hardening evolutions
3 Introduction to Linear Structure Dynamics
Table 3.1. Geometrical dimensions of the beam
Table 3.2. The first intrinsic frequencies of the beam studied
max
of the 1 DOF system
4 Introduction to Nonlinear Dynamic Analysis
Table 4.1. Static response |μ|x
max
of the 1 DOF system
5 Condensation Methods Applied to Eigen Value Problems
Table 5.1. The structure’s first 10 eigen modes
Table 5.2. The first 10 eigen modes of the beam
u
] for the fixed interface methods and free interface modes without reducing the junction DOF
6 Linear Substructure Approach for Dynamic Analysis
Table 6.1. Number of modes retained in the band [0 2f
u
] for the fixed interface methods and free interface modes without reducing the junction DOF
u
] for the fixed and free interface modes with reduced junction DOF
Table 6.2. Number of the modes retained in the band [0 2f
u
] for the fixed and free interface modes with reduced junction DOF
7 Nonlinear Substructure Approach for Dynamic Analysis
Table 7.1. Geometric and physical parameters
Table 7.2. Characterization of the inertial force
Table 7.3. Analytic and numerical calculations of eigen frequencies for the immersed elastic ring. For a color version of this figure, see www.iste.co.uk/elhami/dynamics.zip
Table 7.4. Eigen frequencies of the circular cavity
Table 7.5. Eigen frequency of the elastic ring
Table 7.6. Eigen frequencies of the immersed ring
8 Direct and Inverse Sensitivity
Table 8.1. Sections of bars
Table 8.2. Displacements of the gate nodes
Table 8.3. Determination of the most influential sections for each DOF
9 Parametric Identification and Model Adjustment in Linear Elastic Dynamics
Table 9.1. Perturbation introduced into the initial model
Table 9.2. Perturbation 1 introduced into the initial model
Table 9.3. Petturbation 2 introduced into the initial model
10 Inverse Problems in Dynamics: Robustness Function
Table 10.1. Number of modes of each substructure for the fixed interface method and free interface methods, case 1
Table 10.2. Number of modes of each substructure for the fixed interface method and free interface methods, case 2
Table 10.3. Number of modes used in the band [0 2.f.] for the fixed interface and free interface methods without reduced junction DOF
Table 10.4. Number of modes used in the band [0 4.f.] for the fixed interface and free interface methods with reduced junction DOF
11 Modal Synthesis and Reliability Optimization Methods
Table 11.1. DDO and HRBDO results in the frequential interval
Table 11.2. Geometrical dimensions of the plane wing
Table 11.3. Results for a normal distribution
Table 11.4. Results for a log-normal distribution
Table 11.5. Results for a uniform distribution
Table 11.6. RBDO results for complete model
Table 11.7. RBDO results in CB integer
Table 11.8. RBDO results in IL integer
Table 11.9. RBDO results in CBR integer
Table 11.10. RBDO results in ILR integer
Guide
Cover
Table of Contents
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Mathematical and Mechanical Engineering Setcoordinated byAbdelkhalak El Hami
Volume 5
Dynamics of Large Structures and Inverse Problems
Abdelkhalak El Hami
Bouchaib Radi
First published 2017 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 Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com
© ISTE Ltd 2017The rights of Abdelkhalak El Hami and Bouchaib Radi to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2017937773
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-84821-952-6
Preface
In recent years, engineers, scientists and decision makers have expressed a strong interest in the dynamics of large structures and inverse problems. These two fields have attracted growing interest due to their industrial applications. The problems in structure dynamics are very important, notably the trend of building structures that are more and more supple and subject to frenzies that fluctuate more and more quickly in time. However, a dynamic analysis of large industrial structures is often based on model reduction techniques. With this aim, we will present some solution methods for large systems.
Within the sphere of calculating structures, the finite elements method enables one to determine a structure’s physical response to an applied force. This technique not only enables one to determine the stress states on a mechanical structure’s interior, but also to model the complete manufacturing processes, for example. Nowadays, the significantly reduced calculation time allows us to address so-called inverse problems. By repeating the calculations by finite elements while modifying the material’s parameters or the structure’s geometry, one comes to identify an optimal solution for the problem in question. The procedure, which couples optimization and calculations by finite elements, is of utmost importance for the manufacturing industry, for example, as this virtual development reduces the time and costs involved in developing new products. Those who understand the difference use the terminology of “inverse problem”, as opposed to that of “direct problem”, to refer to solving a differential equation based on the known parameters in order to calculate the system’s response. In the instance of an inverse problem, the system’s response is assumed to be known. Therefore, we aim to determine the physical or geometrical parameters which, when used in direct problems, allow one to find the prescribed system’s response. Inverse problems also involve an objective function, to be constructed according to the application, measuring a gap between the known response and the responses obtained from the sets of different parameters, by solving the direct problem. There are two large categories of techniques for solving an inverse problem:
1) Gradient-type techniques: They consist of identifying the minimum of the objective function as a point where the gradient of this function is cancelled.
2) Stochastic methods [ELH 16].
This book includes the most recent ideas resulting from research and from the industry in the field of large structure dynamics and inverse problems. It consists of 11 chapters. These chapters take stock of the different tools used to handle condensation methods, linear and nonlinear model synthesis, identification, resetting, sensitivity, optimization, reliability and some inverse problems.
Each chapter has clear explanations of the techniques used and developed, and are accompanied by fully illustrated examples.
Chapter 1 introduces problems related to inverse problems.
Chapter 2 encompasses analyzing and solving first-order linear differential equations with constant coefficients. The chapter introduces a way in which it is applied to mechanical engineering for dynamic systems.
Chapter 3 presents an introduction to linear dynamics of structures. In each of the various industrial sectors (automobile, aeronautics, civil engineering, nuclear engineering, defense, aerospace, oceanic and marine engineering, etc.), it is important to determine the structure’s response to different applied forces for designing and dimensioning it. To assess this response (displacements, stresses, speed and acceleration) to a dynamic load (variable in time), there are two approaches: the determinist approach and the stochastic or non-determinist approach [ELH 16]. In this chapter, we present the general principles of linear determinist structure dynamics. This study enables one to establish the essential relations when calculating dynamic responses, when calculating frequencies, normal modes and response functions in frequencies. Finally, a few simple examples are introduced.
Chapter 4 introduces the dynamics of nonlinear structures. The objective of the chapter is to raise awareness about nonlinear specific characteristic in basic cases. Returning to the linear structure, a few basic avenues for analysis are presented, which may be sufficient for certain industrial applications.
In Chapter 5, some condensation methods are introduced. Currently, discrete models for forward calculations of structural behavior tend to be the finite elements type. Given the complexity of industrial structures, these knowledge models often involve a significant amount of degrees of freedom (DOFs). When making a dynamic analysis of such models, the size may exceed the capacity of the computers available. The discrete mechanical models considered are conservative linear models of the second order.
Chapter 6 is dedicated to introducing linear modal synthesis methods. The reader is reminded of the substructuration strategy, which was initially formulated for static problems. It consists of processing structures such as assembling substructures that are interconnected with each other. The modal synthesis methods differ in the choice of modes for representing the dynamics of each substructure and in assembly procedures. We then propose a strategy for reducing junction DOFs, after assembly. This strategy is based on the use of interface modes. These modes are obtained from condensation on the complete structure’s Guyan interfaces.
Chapter 7 introduces different reduction methods for models in nonlinear dynamics.
Chapter 8 is dedicated to analyzing a model’s sensitivity. It studies the variations of the output variations compared to the input parameters. It enables one to have improved understanding of the model’s behavior and to quantify the influence of different input parameters on the variability of the system’s output. We are often led to assess the dynamic behavioral variations due to given modifications of design variables. These are the direct problems. The design modifications variables leading to a given variation of dynamic behavior are the inverse problems. In this chapter, direct and inverse sensitivity methods are introduced.
Chapter 10 presents the robustness function in structure dynamics for inverse problems. In the probabilistic approach, the parameters are described by the probability densities and we aim to propagate this probabilistic characteristic through the mechanical model. The approach by convex models of uncertainty problems in mechanics has mainly been approached by Ben-Haim [BEN 90]. The “info-gap” convex models of uncertainty are defined as the gap between what is known, the nominal values of parameters, and what we want to determine, the uncertainties, to satisfy a given design criterion. We present two methods for solving inverse problems. The first one is based on interval arithmetic. The second one is a minimization problem under stress. Finally, we introduce some digital applications. A structure’s own pulse is chosen as a performance function. We use different model synthesis methods to calculate this function. We compare the results obtained with the complete model.
The objective of Chapter 11 is to introduce a methodology that couples modal synthesis techniques with optimizing the design’s reliability. We introduce an algorithm that enables modal synthesis methods to be integrated into the reliability optimization process. Finally, we assess this algorithm when used on different applications to show the effectiveness and robustness of the method presented.
Finally, this book constitutes an invaluable support for teachers and researchers. It is also aimed at engineering students, practicing engineers and master’s and PhD engineering students.
Acknowledgments
We would like to thank every person who has contributed in both big and small ways to the development of this publication, our families and particularly, the Rouen INSA PhD students for whom we have been responsible for the last few years.
Abdelkhalak EL HAMIHAMI Bouchaïb RADIMay 2017
