M I Friswell and J E Mottershead
Preface
Contents
Introduction
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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. The modern world is one in which the demand for improved performance of the products of engineering design must be achieved in the face of ever increasing energy and materials costs. The Japanese car companies have shown how attention to detail can lead to vast improvements in manufactured products: we have them to thank for the excellent reliability of modern motor cars. As designs become more and more refined, it is necessary that the search for improvement is involved with aspects of increasingly intricate detail. Analysts will appreciate the obvious analogy with the mathematical modelling of non-linear systems, where the inclusion of higher order (smaller) terms in the equations can reveal behavioural modes of whole systems which are not detected by less intricate mathematics. Computer based analysis techniques (especially the finite element method) have had a huge impact on engineering design and product development since the 1960s. In the case of many engineering products, we now stand at the point where more detailed finite element models are not capable of delivering the improvements in product performance that are demanded. Clearly, the approach of numerical predictions to the behaviour of a physical system is limited by the assumptions used in the development of the mathematical model. Model updating, at its most ambitious, is about correcting invalid assumptions by processing vibration test results.
Updating is a process fraught with numerical difficulties. These arise from inaccuracy in the model and imprecision and lack of information in the measurements. This book 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 analysis and estimation are described in detail. The book has been written in such a way that the level of mathematics required of the reader is little more than that covered at first degree level in engineering. It is interspersed with examples which are used to illustrate and highlight various points made in the text. The examples can be easily replicated (and in many cases expanded) by the novice in order to reinforce understanding.
We (M.I. Friswell and J.E. Mottershead) have benefited in our studies of model updating by working closely with our colleagues (especially Drs. J.E.T. Penny and R. Stanway and Professor A.W. Lees) and with many gifted Ph.D. students whom we had the good fortune to supervise. A draft manuscript was read by Professors G.M.L. Gladwell, D.J. Inman, A.W. Lees and M. Link and Dr. J.E.T. Penny: we would like to record our appreciation of the many helpful comments and suggestions made by them, which we feel have resulted in a much improved final version of the book. Finally, our wives and children are certainly not to be omitted from our acknowledgement: Wendy and Susan; Clare and Robert; and Stuart, James, Timothy and Elizabeth have been our staunchest supporters.
- Preface
- 1. Introduction
- Numerical Modelling
- Vibration Testing
- Estimation Methods
- Arrangement of Text
- Reference
- 2. Finite Element Modelling
- Shape Functions and Discretisation
- Finite Element Masses and Stiffnesses
- Multi-Degree of freedom Mass/Spring Systems, Normal Modes and Mass Normalisation
- Damping
- Eigenvalues, Eigenvectors and Frequency Response Functions
- The Calculation of Sensitivities
- Errors in Finite Element Modelling
- Assessment of Errors
- References
- 3. Vibration Testing
- Measurement Hardware and Methods
- Time, Frequency and Modal Domain Data
- Measurement Noise: Random and Systematic Errors
- Incomplete Data
- References
- 4. Comparing Numerical Data with Test Results
- The Modal Assurance Criterion
- Orthogonality Checks
- The Problem of Complex Modes
- Model Reduction
- Modal Expansion
- Optimising Transducer Locations
- References
- 5. Estimation Techniques
- Least Squares Estimators
- Problems of Bias
- Problems of Rank Defficiency, Ill-Conditioning and Under-Determination
- Singular Value Decomposition
- Regularisation
- References
- 6. Parameters for Model Updating
- Representational and Knowledge-Based Models
- Uniqueness, Identifiability and Physical Meaning
- Parameterisation Methods
- Error Localisation
- Selective Sensitivity and Adaptive Excitation
- References
- 7. Direct Methods using Modal Data
- Overview - Advantages and Disadvantages
- Lagrange Multiplier Methods
- Matrix Mixing
- Methods from Control Theory
- References
- 8. Iterative Methods using Modal Data
- Overview - Advantages and Disadvantages
- Penalty Function Methods
- Minimum Variance Methods
- Perturbed Boundary Condition Testing
- Discretisation Errors: A Two Level Gauss-Newton Method
- Assessing Model Quality
- References
- 9. Methods using Frequency Domain Data
- Equation and Output Error Formulations
- Equation Error Methods
- A Weighted Equation Error Method
- A Simulated Example using the Equation Error Methods
- Output Error Methods
- Frequency Domain Filters
- Combining Frequency and Modal Domain Data
- References
- 10. Case Study: An Automobile Body by M. Brughmans, J. Leuridan and K. Blauwkamp
- Updating Large Finite Element Models
- The Body-in-White
- Correlation Analysis
- Model Updating Approach
- Concluding Remarks
- References
- 11. Discussion and Recommendations
- Selection of Updating Parameters
- Updating Methods
- Index
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?
An understanding of the purpose of the updated model is necessary before an answer to the above question can be given. In some cases, the only requirement of the updated model is that it should replicate the physical test data. Consider the updating of a turbomachinery model. If measured natural frequencies and mode shapes were available, then an updated model which reproduced such data might be quite useful for comparison with data obtained at another time or from another machine. If the model had been improved, not only with the intention of mimicking the test results but also by improving the physical parameters (upon which depends the distribution of finite element masses and stiffnesses), then it might be possible to locate a fault in a bearing, or a crack in a rotor which is responsible for the observed disparity between measurements and predictions. This can possibly be achieved by using the machine run-down data, which are readily available from large turbo-generator sets, and would eliminate the need for special modal tests that might involve considerable down-time of the machine.
In the car industry, the capacity of finite element models to predict vibration modes of bodies is limited (at frequencies above 80Hz) by inadequate modelling of joints and pressings, variations in the thickness of sheet metal and other model errors which might be improved by updating. If the physical meaning of such models can be improved, then the updated model can be used to assess the effect of changes in construction, such as the introduction of an additional rib, on the dynamics of a body-in-white. Updating by improving the physical meaning of the model always requires the application of considerable physical insight in the choice of parameters to update and the arrangement of constraints, force inputs and response measurements in the vibration test. Model updating brings together the skills of the numerical analyst and the vibration test engineer, and requires the application of modern estimation techniques to produce the desired improvement.
1.1 Numerical Modelling
A finite element model which will be updated requires, in its preparation, the consideration of factors not normally taken into account in regular model construction. Of these, the choice of updating parameters is the most important. The analyst should attempt to assess the confidence which can be attributed to various features of the model. For example, the main span of a beam, away from the boundaries, might be considered to be modelled with a high level of confidence. Joints and constraints could be considered to be less accurately modelled, and therefore in greater need of updating. The parameterisation of the inaccurate parts of the model is important. The numerical predictions (e.g. natural frequencies and mode shapes) should be sensitive to small changes in the parameters. Experimental results show that natural frequencies are often significantly affected by small differences in the construction of joints in nominally identical test pieces. But it can be very difficult to find joint parameters to which the numerical predictions are sensitive. If the numerical data is insensitive to a chosen parameter, then updating will result in a change to the parameter of uncertain value, because the difference between predictions and results has been reconciled by changes to other (more sensitive) parameters that might be less in need of updating. The result, in that case, will be an updated model which replicates the measurements but lacks physical meaning.
1.2 Vibration Testing
The extent to which a numerical model can be improved by updating depends upon the richness of information on the test structure contained in measurements. In general, the measurements will be both imprecise and incomplete. The imprecision takes the form of random and systematic noise. Electronic noise from instruments can be largely eliminated by the use of high quality transducers, amplifiers and analogue to digital conversion hardware. Signal processing errors, such as aliasing and leakage, may be reduced by the correct choice of filters and excitation signals. Systematic errors can occur when, for example, the suspension system fails to replicate free-free conditions, or when the mass of a roving accelerometer causes changes in measured natural frequencies. Rigidly clamped boundary conditions are usually very difficult to obtain in a physical test. Extreme care is necessary to either eliminate systematic errors, or to obtain an assessment of them which can be used in subsequent processing.
The measurements will be incomplete in the sense that the measurement frequency range (determined by the sampling rate) will be much shorter than that of the numerical model which might typically contain tens or hundreds of thousands of degrees of freedom. An extreme case of incompleteness occurs when the inputs, or response sensors, are located at, or close to, vibration nodes so that the effect of one or more modes is obscured by measurement noise.
In addition to modal incompleteness, the measurements will also be spatially incomplete. This arises because the number of measurement stations is generally very much smaller than the number of degrees of freedom in the finite element model. Rotational degrees of freedom are usually not measured and some degrees of freedom will be inaccessible. Spatial incompleteness often requires either the reduction of the model or the expansion of measured eigenvectors.
1.3 Estimation Methods
The estimation methods used in model updating are closely related to those of system identification and parameter estimation which are regularly applied in other areas of science and engineering. System identification addresses the problem of determining the order and structure of a mathematical model from measurement records. When the form of the structure has been decided upon, the coefficients are set by means of parameter estimation. In control engineering the purpose of system identification (and parameter estimation) is usually the 'on-line' construction of models which may be applied recursively in model-reference control schemes. In contrast, model updating in structural dynamics is usually performed 'off-line' using batch processing techniques. The aim is to generate improved numerical models which may be applied in order to obtain predictions for alternative loading arrangements and modified structural configurations. This aim places a demand upon model updating techniques which does not occur in control system identification. The demand is that the mass, stiffness and damping terms should be based on physically meaningful parameters.
The incompleteness of the measured data usually leads to problems of rank deficiency in the sensitivity matrix, which may be masked by measurement noise. As a counter-measure further data can be acquired by carrying out more tests with modified boundary conditions or the addition of known masses. Regularisation techniques, which are often related to the singular value decomposition (SVD), can be used to ensure that the updated parameters deviate from the finite element parameters by a minimal amount. A comprehensive survey (243 references) on model updating has been conducted by Mottershead and Friswell (1993).
1.4 Arrangement of the Text
Chapter 2 gives an overview of finite element modelling with updating in mind. A treatment of the finite element theory and multi-degree of freedom dynamics is given in sufficient detail to provide the foundation for more advanced topics which appear in the later chapters. The discussion of sensitivity calculations, joint modelling and discretisation errors is especially relevant to model updating.
The elements of modern vibration testing are described in Chapter 3. The Fourier transformation of time domain data and experimental modal analysis are introduced. The sections on measurement noise and incompleteness are the forerunners of more advanced discussion in the sequel.
Chapter 4 deals with the comparison of numerical predictions with test results. Important topics such as model reduction, eigenvector expansion and the modal assurance criterion are described in detail.
The formulation of least squares and minimum variance estimators are described in a general way in Chapter 5. These topics are returned to in Chapters 8 and 9 where they are discussed in the context of particular updating schemes. Problems of ill-conditioning and under-determination are described together with regularisation methods and the singular value decomposition.
Chapter 6 addresses the problem of selecting the updating parameters. Methods of error localisation and the sensitising of measurements to selected parameters are considered in detail.
The description of the so-called 'direct' methods of model updating is given in Chapter 7. These methods are capable of replicating the measured natural frequencies and mode shapes, but the changes to the mass and stiffness matrices brought about by updating are seldom physically meaningful. The measured mode shapes need to be expanded.
Chapter 8 is devoted to the explanation of updating methods based on eigenvalue and eigenvector sensitivities. These are powerful techniques which can result in updating parameters that have improved physical meaning. Detailed analysis of over-determined and under-determined least squares approaches, and minimum variance methods is provided.
The direct application of frequency response functions in updating has the potential of eliminating any errors which might have been introduced in the experimental modal analysis. Model updating by using frequency response function sensitivities in equation error and output error formulations is described in Chapter 9. Eigenvalue sensitivities are generally more powerful than frequency response sensitivities.
Chapter 10 consists of a case study, namely the updating of a finite element model of the 1991 GM Saturn four door sedan with 46830 degrees of freedom.
Chapter 11 provides a final review of the various updating methods. Recommendations are made regarding their application.
Reference
Mottershead, J.E. and Friswell, M.I. 1993. "Model Updating in Structural Dynamics: A Survey." Journal of Sound and Vibration, 167(2), 347-375.