fem odal updating book

8/12/2019 Fem Odal Updating Book

1/5

Finite Element Model Updating in StructuralDynamics

M I Friswell and J E Mottershead

Kluwer Academic Publishers, 1995, 286 pp., ISBN 0-7923-3431-0.

PrefaceContentsIntroduction

If you want to read any more please buy the book!

Preface

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 inwhich the demand for improved performance of the products of engineering design must be achieved in the face ofever increasing energy and materials costs. The Japanese car companies have shown how attention to detail can leadto vast improvements in manufactured products: we have them to thank for the excellent reliability of modern motorcars. As designs become more and more refined, it is necessary that the search for improvement is involved withaspects of increasingly intricate detail. Analysts will appreciate the obvious analogy with the mathematical modellingof non-linear systems, where the inclusion of higher order (smaller) terms in the equations can reveal behaviouralmodes 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 elementmodels are not capable of delivering the improvements in product performance that are demanded. Clearly, theapproach of numerical predictions to the behaviour of a physical system is limited by the assumptions used in thedevelopment of the mathematical model. Model updating, at its most ambitious, is about correcting invalidassumptions by processing vibration test results.

Updating is a process fraught with numerical difficulties. These arise from inaccuracy in the model and imprecisionand lack of information in the measurements. This book sets out to explain the principles of model updating, not onlyas a research text, but also as a guide for the practising engineer who wants to get acquainted with, or use, updatingtechniques. It covers all aspects of model preparation and data acquisition that are necessary for updating. Thevarious 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 thatcovered at first degree level in engineering. It is interspersed with examples which are used to illustrate and highlightvarious points made in the text. The examples can be easily replicated (and in many cases expanded) by the novice inorder to reinforce understanding.

We (M.I. Friswell and J.E. Mottershead) have benefited in our studies of model updating by working closely with ourcolleagues (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 manyhelpful comments and suggestions made by them, which we feel have resulted in a much improved final version ofthe 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.

Top of Page


2/5

Contents

Preface1. Introduction

Numerical ModellingVibration TestingEstimation MethodsArrangement of TextReference

2. Finite Element ModellingShape Functions and DiscretisationFinite Element Masses and StiffnessesMulti-Degree of freedom Mass/Spring Systems, Normal Modes and Mass NormalisationDampingEigenvalues, Eigenvectors and Frequency Response FunctionsThe Calculation of SensitivitiesErrors in Finite Element ModellingAssessment of ErrorsReferences

3. Vibration Testing

Measurement Hardware and MethodsTime, Frequency and Modal Domain DataMeasurement Noise: Random and Systematic ErrorsIncomplete DataReferences

4. Comparing Numerical Data with Test ResultsThe Modal Assurance CriterionOrthogonality ChecksThe Problem of Complex ModesModel ReductionModal Expansion

Optimising Transducer LocationsReferences

5. Estimation TechniquesLeast Squares EstimatorsProblems of BiasProblems of Rank Defficiency, Ill-Conditioning and Under-DeterminationSingular Value DecompositionRegularisationReferences

6. Parameters for Model UpdatingRepresentational and Knowledge-Based Models

Uniqueness, Identifiability and Physical MeaningParameterisation MethodsError LocalisationSelective Sensitivity and Adaptive ExcitationReferences

7. Direct Methods using Modal DataOverview - Advantages and DisadvantagesLagrange Multiplier MethodsMatrix MixingMethods from Control TheoryReferences

8. Iterative Methods using Modal DataOverview - Advantages and DisadvantagesPenalty Function MethodsMinimum Variance MethodsPerturbed Boundary Condition TestingDiscretisation Errors: A Two Level Gauss-Newton Method


3/5

Assessing Model QualityReferences

9. Methods using Frequency Domain DataEquation and Output Error FormulationsEquation Error MethodsA Weighted Equation Error MethodA Simulated Example using the Equation Error MethodsOutput Error MethodsFrequency Domain Filters

Combining Frequency and Modal Domain DataReferences10. Case Study: An Automobile Body by M. Brughmans, J. Leuridan and K. Blauwkamp

Updating Large Finite Element ModelsThe Body-in-WhiteCorrelation AnalysisModel Updating ApproachConcluding RemarksReferences

11. Discussion and RecommendationsSelection of Updating Parameters

Updating MethodsIndex

Top of Page

1. Introduction

This book addresses the problem of updating a numerical model by using data acquired from a physical vibrationtest. 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 fromanalogue measurements. The most widespread approach for numerical modelling in engineering design is the finiteelement method. The Cooley-Tukey algorithm, and related techniques, for fast Fourier transformations have led tothe computerisation of long established techniques, and the blossoming of new computer intensive methods, inexperimental modal analysis. For various reasons, to be elaborated upon in the chapters that follow, the experimentalresults and numerical predictions often conspire to disagree. Thus, the scene is set to use the test results to improvethe numerical model. It would be superficial to imagine that updating is straightforward or easy: it is beset withproblems of imprecision and incompleteness in the measurements and inaccuracy in the finite element model. Inmodel updating the improvement of an inaccurate model by using imprecise and incomplete measurements isattempted. 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 begiven. 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 atanother time or from another machine. If the model had been improved, not only with the intention of mimicking thetest results but also by improving the physical parameters (upon which depends the distribution of finite elementmasses and stiffnesses), then it might be possible to locate a fault in a bearing, or a crack in a rotor which isresponsible for the observed disparity between measurements and predictions. This can possibly be achieved byusing the machine run-down data, which are readily available from large turbo-generator sets, and would eliminatethe 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 (atfrequencies above 80Hz) by inadequate modelling of joints and pressings, variations in the thickness of sheet metaland other model errors which might be improved by updating. If the physical meaning of such models can beimproved, then the updated model can be used to assess the effect of changes in construction, such as theintroduction of an additional rib, on the dynamics of a body-in-white. Updating by improving the physical meaningof the model always requires the application of considerable physical insight in the choice of parameters to update


4/5

and the arrangement of constraints, force inputs and response measurements in the vibration test. Model updatingbrings together the skills of the numerical analyst and the vibration test engineer, and requires the application ofmodern 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 normallytaken 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 levelof confidence. Joints and constraints could be considered to be less accurately modelled, and therefore in greaterneed 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. Experimentalresults show that natural frequencies are often significantly affected by small differences in the construction of jointsin nominally identical test pieces. But it can be very difficult to find joint parameters to which the numericalpredictions are sensitive. If the numerical data is insensitive to a chosen parameter, then updating will result in achange to the parameter of uncertain value, because the difference between predictions and results has beenreconciled by changes to other (more sensitive) parameters that might be less in need of updating. The result, in thatcase, 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 onthe 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 largelyeliminated by the use of high quality transducers, amplifiers and analogue to digital conversion hardware. Signalprocessing 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 whenthe mass of a roving accelerometer causes changes in measured natural frequencies. Rigidly clamped boundaryconditions are usually very difficult to obtain in a physical test. Extreme care is necessary to either eliminatesystematic 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 thesampling rate) will be much shorter than that of the numerical model which might typically contain tens or hundredsof 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 thenumber of measurement stations is generally very much smaller than the number of degrees of freedom in the finiteelement model. Rotational degrees of freedom are usually not measured and some degrees of freedom will beinaccessible. Spatial incompleteness often requires either the reduction of the model or the expansion of measuredeigenvectors.

1.3 Estimation Methods

The estimation methods used in model updating are closely related to those of system identification and parameterestimation which are regularly applied in other areas of science and engineering. System identification addresses theproblem of determining the order and structure of a mathematical model from measurement records. When the formof the structure has been decided upon, the coefficients are set by means of parameter estimation. In controlengineering the purpose of system identification (and parameter estimation) is usually the 'on-line' construction ofmodels which may be applied recursively in model-reference control schemes. In contrast, model updating instructural dynamics is usually performed 'off-line' using batch processing techniques. The aim is to generateimproved 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 notoccur in control system identification. The demand is that the mass, stiffness and damping terms should be based onphysically 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


5/5

more tests with modified boundary conditions or the addition of known masses. Regularisation techniques, which areoften related to the singular value decomposition (SVD), can be used to ensure that the updated parameters deviatefrom the finite element parameters by a minimal amount. A comprehensive survey (243 references) on modelupdating 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 elementtheory 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 anddiscretisation errors is especially relevant to model updating.

The elements of modern vibration testing are described in Chapter 3. The Fourier transformation of time domain dataand experimental modal analysis are introduced. The sections on measurement noise and incompleteness are theforerunners of more advanced discussion in the sequel.

Chapter 4 deals with the comparison of numerical predictions with test results. Important topics such as modelreduction, 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 updatingschemes. Problems of ill-conditioning and under-determination are described together with regularisation methodsand the singular value decomposition.

Chapter 6 addresses the problem of selecting the updating parameters. Methods of error localisation and thesensitising 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 capableof replicating the measured natural frequencies and mode shapes, but the changes to the mass and stiffness matricesbrought 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. Theseare powerful techniques which can result in updating parameters that have improved physical meaning. Detailedanalysis of over-determined and under-determined least squares approaches, and minimum variance methods isprovided.

The direct application of frequency response functions in updating has the potential of eliminating any errors whichmight have been introduced in the experimental modal analysis. Model updating by using frequency responsefunction sensitivities in equation error and output error formulations is described in Chapter 9. Eigenvaluesensitivities 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 theirapplication.

Reference

Mottershead, J.E. and Friswell, M.I. 1993. "Model Updating in Structural Dynamics: A Survey."Journal of Soundand Vibration,167(2), 347-375.

Top of Page

fem odal updating book

Documents