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Über dieses Buch

Finite element model updating has emerged in the 1990s as a subject of immense importance to the design, construction and maintenance of mechanical systems and civil engineering structures. This book, the first on the subject, sets out to explain the principles of model updating, not only as a research text, but also as a guide for the practising engineer who wants to get acquainted with, or use, updating techniques. It covers all aspects of model preparation and data acquisition that are necessary for updating. The various methods for parameter selection, error localisation, sensitivity and parameter estimation are described in detail and illustrated with examples. The examples can be easily replicated and expanded in order to reinforce understanding. The book is aimed at researchers, postgraduate students and practising engineers.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
This book addresses the problem of updating a numerical model by using data acquired from a physical vibration test. Modern computers, which are capable of processing large matrix problems at high speed, have enabled the construction of large and sophisticated numerical models, and the rapid processing of digitised data obtained from analogue measurements. The most widespread approach for numerical modelling in engineering design is the finite element method. The Cooley-Tukey algorithm, and related techniques, for fast Fourier transformations have led to the computerisation of long established techniques, and the blossoming of new computer intensive methods, in experimental modal analysis. For various reasons, to be elaborated upon in the chapters that follow, the experimental results and numerical predictions often conspire to disagree. Thus, the scene is set to use the test results to improve the numerical model. It would be superficial to imagine that updating is straightforward or easy: it is beset with problems of imprecision and incompleteness in the measurements and inaccuracy in the finite element model. In model updating the improvement of an inaccurate model by using imprecise and incomplete measurements is attempted. But by what means can the proverb of two wrongs not making a right be defied?
M. I. Friswell, J. E. Mottershead

2. Finite Element Modelling

Abstract
In modern times the finite element method has become established as the universally accepted analysis method in structural design. The method leads to the construction of a discrete system of matrix equations to represent the mass and stiffness effects of a continuous structure. The matrices are usually banded and symmetric. No restriction is placed upon the geometrical complexity of the structure because the mass and stiffness matrices are assembled from the contributions of the individual finite elements with simple shapes. Thus, each finite element possesses a mathematical formula which is associated with a simple geometrical description, irrespective of the overall geometry of the structure. Accordingly, the structure is divided into discrete areas or volumes known as elements. Element boundaries are defined when nodal points are connected by a unique polynomial curve or surface. In the most popular (isoparametric, displacement type) elements, the same polynomial description is used to relate the internal, element displacements to the displacements of the nodes. This process is generally known as shape function interpolation. Since the boundary nodes are shared between neighbouring elements, the displacement field is usually continuous across the element boundaries. Figure 2.1 illustrates the geometric assembly of finite elements to form part of the mesh of a modelled structure.
M. I. Friswell, J. E. Mottershead

3. Vibration Testing

Abstract
Vibration testing is now a mature and widely used tool in the analysis of structural systems. It would be impossible and undesirable to review all the aspects of experimental technique in this chapter. A summary is presented which should be sufficient to enable the reader to appreciate the sources of errors in measured vibration data. The quality of this data is vitally important if the satisfactory updating of a finite element model is to be achieved. Although some errors in the data are inevitable they should be minimised by good experimental technique. Ewins (1984) gives a good introduction to modal testing. Allemang et al. (1987), Zaveri (1984) and Snoeys et al. (1987) gave more detail on aspects of vibration testing.
M. I. Friswell, J. E. Mottershead

4. Comparing Numerical Data with Test Results

Abstract
An important requirement in design is to be able to compare experimental results from prototype structures with predicted results from a corresponding finite element model. In finite element model updating the experimental and analytical databases should be compared to assess the improvement in the modelled response. In this comparison, major problems arise because of the large number of degrees of freedom in the analytical model; the limited number of transducers used to measure the response of the structure; and modelling inaccuracies, in particular the omission of damping in the numerical model. The various methods used to address these problems will be discussed in this chapter. Specific methods to produce global measures of correlation between measured and numerical mode shapes are considered.
M. I. Friswell, J. E. Mottershead

5. Estimation Techniques

Abstract
In this chapter the formulation of least squares and related estimators is introduced in a general way, without specific reference to the model updating problem. The particular role of parameter estimation in model updating is elucidated in Chapters 8 and 9. Problems of noise contamination and rank deficiency are considered. Regularisation and the singular value decomposition are introduced. In writing this chapter the authors have drawn heavily on the excellent texts by Golub and Van Loan (1989) and Soderstrom and Stoica (1989), which can be consulted for a more detailed treatment.
M. I. Friswell, J. E. Mottershead

6. Parameters for Model Updating

Abstract
The choice of parameters is a crucial step in model updating. The measured data will contain a limited amount of information and to avoid possible problems of ill-conditioning the number of updating parameters should be kept small. Such parameters should be chosen with the aim of correcting recognised uncertainty in the model, and the data should be sensitive to them. This can rarely be achieved without the application of considerable physical insight.
M. I. Friswell, J. E. Mottershead

7. Direct Methods using Modal Data

Abstract
To understand the strengths and weaknesses of the direct methods one must understand what the methods do, where errors may occur and how they are propagated into the updated model. Of course, as the title implies, these methods have the great advantage of not requiring iteration and thus the possibilities of divergence and excessive computation are eliminated.
M. I. Friswell, J. E. Mottershead

8. Iterative Methods using Modal Data

Abstract
As with all model updating techniques, the objective of iterative methods using modal data is to improve the correlation between the measured data and the analytical model. The correlation is determined by a penalty function involving the mode shape and eigenvalue data: often the sum of squares of the difference between the measured and estimated eigenvalues is used. Because of the nature of this penalty function, the solution requires the problem to be linearised and optimised iteratively. These methods allow a wide choice of parameters to be updated and both the measured data and the initial analytical parameter estimates may be weighted. This ability to weight the different data sets gives the method its power and versatility, but requires engineering insight to provide the correct weights. The minimum variance algorithm is often used in commercial updating software.
M. I. Friswell, J. E. Mottershead

9. Methods using Frequency Domain Data

Abstract
The methods using measured FRF data optimise a penalty function involving the FRF data directly. Extracting natural frequencies and mode shapes for structures with close modes or a high modal density can be difficult. The FRF data may be used directly to update the finite element model without extracting the natural frequencies and mode shapes (Sestieri and D’Amrogio, 1989). Friswell and Penny (1992) discussed the use of a typical algorithm for structures with close modes. One problem with any method using the FRFs directly is that damping must be included in the finite element model. Methods using modal data, Chapters 7 and 8, are able to use undamped models because the measured natural frequencies and damping ratios may be separated. The inclusion of damping is vital to obtain a good correpondence between the measured and predicted FRFs. Since damping is so difficult to model accurately, proportional damping is often assumed (see Chapter 2). The rest of this chapter will assume a proportional viscous damping model, in which the constants of proportionality are unknown. Other damping models may be easily incorporated.
M. I. Friswell, J. E. Mottershead

10. Case Study: An Automobile Body

Abstract
This chapter considers the application of correlation and validation techniques to an automobile body, commonly called a body-in-white. The automobile was a 1991 GM Saturn four door Sedan (Brughmans et al., 1992). The finite element model of this car had 46830 degrees of freedom (half model). A multi-point experimental modal analysis (EMA) survey was performed with 360 response degrees of freedom. Classical techniques for correlation analysis such as the Modal Assurance Criterion (MAC, Chapter 4) were applied. Error localisation methods identified the regions of the finite element model that caused most of the discrepancies between the model and the measurements. The Bayesian, or minimum variance, updating method (Chapter 8) reduced the difference between the finite element model and measured modal model to acceptable limits.
Marc Brughmans, Jan Leuridan, Kevin Blauwkamp

11. Discussion and Recommendations

Abstract
The purpose of this final chapter is to briefly revisit some of the more important, and sometimes contentious, issues involved with model updating. The conditions under which particular methods can be recommended are outlined. Opportunities for new research and the further development of existing techniques are discussed.
M. I. Friswell, J. E. Mottershead

Backmatter

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