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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 25 (2011) 2275–2296

0888-32

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

The sensitivity method in finite element model updating: A tutorial

John E. Mottershead a,n, Michael Link b, Michael I. Friswell c

a Centre for Engineering Dynamics, University of Liverpool, UKb Institute for Statics and Structural Dynamics, University of Kassel, Germanyc School of Engineering, Swansea University, Wales, UK

a r t i c l e i n f o

Article history:

Received 29 July 2010

Received in revised form

14 October 2010

Accepted 19 October 2010Available online 31 October 2010

Keywords:

Model Updating

Sensitivity Method

70/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ymssp.2010.10.012

esponding author.

ail address: [email protected] (J.E. Mo

a b s t r a c t

The sensitivity method is probably the most successful of the many approaches to the

problem of updating finite element models of engineering structures based on vibration

test data. It has been applied successfully to large-scale industrial problems and

proprietary codes are available based on the techniques explained in simple terms in

this article. A basic introduction to the most important procedures of computational

model updating is provided, including tutorial examples to reinforce the reader’s

understanding and a large scale model updating example of a helicopter airframe.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Modern and highly sophisticated finite-element (FE) procedures are available for structural analysis, yet practicalapplication often reveals considerable discrepancy between analytical prediction and test results. The way to reduce thisdiscrepancy is to modify the modelling assumptions and parameters until the correlation of analytical predictions andexperimental results satisfies practical requirements. Classically, this is achieved by a trial and error approach, which isgenerally time consuming and may not be feasible in some cases. Thus computational procedures have been developed toupdate the parameters of analytical models using test data. In particular, modal data (natural frequencies and mode shapes)extracted from measured frequency response data have found broad application as a target for model parameter adjustment.This procedure was described in detail by Natke [1] and Friswell and Mottershead [2,3] and in recent years has developed intoa mature technology applied successfully for the correction of industrial-scale FE models.

Model updating is essentially a process of adjusting certain parameters of the finite element model. The user should beaware of numerous sources of modelling error and make necessary adjustments paying particular attention to those aspectsof the model that cannot be corrected by changing the values of selected model-updating parameters. Examples of sucherrors, listed in categories (1) and (2) below, are related to the mathematical structure of the model and generally referred toas model-structure errors. The errors listed under category (3) are typical of those that can be corrected by model updating.

(1)
Idealisation errors resulting from the assumptions made to characterise the mechanical behaviour of the physicalstructure. Such errors typically arise from:� simplifications of the structure, for example, when a plate is treated like a beam, which might or might not be
erroneous depending on the length to width ratio of the plate and the frequency range to be covered,� inaccurate assignment of mass properties, for example, when distributed masses are modelled with too few lumped

masses or when an existing eccentricity of a lumped mass is disregarded,

ll rights reserved.

ttershead).

www.elsevier.com/locate/jnlabr/ymssp

dx.doi.org/10.1016/j.ymssp.2010.10.012

mailto:[email protected]

dx.doi.org/10.1016/j.ymssp.2010.10.012

(2)

(3)

J.E. Mottershead et al. / Mechanical Systems and Signal Processing 25 (2011) 2275–22962276

� when the finite element formulation neglects particular properties, for example, when the influence of transverseshear deformation or warping due to torsion in beam elements is neglected,� errors in the connectivity of the mesh i.e. some elements are not connected or are connected to a wrong node,� erroneous modelling of boundary conditions, for example, when an elastic foundation is assumed to be rigid,� erroneous modelling of joints, for example, when an elastic connection is assumed to be rigid (clamped) or when an

eccentricity of a beam or a plate connection is omitted from the model,� erroneous assumptions for the external loads,� erroneous geometrical shape assumptions,� a non-linear structure assumed to behave linearly.

Discretisation errors introduced by numerical methods such as those inherent in the finite element method, for example:� discretisation errors when the finite element mesh is too coarse so that the modal data in the frequency of interest is

not fully converged,� truncation errors in order reduction methods such as static condensation,� poor convergence and apparent stiffness increase due to element shape sensitivity.

Erroneous assumptions for model parameters, for example:� material parameters such as Young’s modulus or mass density,� cross section properties of beams such as area moments of inertia,� shell/plate thicknesses,� spring stiffnesses or� non-structural mass.

When the model includes idealisation and discretisation errors it may only be updated in the sense that the deviationsbetween test and analysis are minimised. The same happens when the selected correction parameters are not consistent withthe real source and the location of the error. The parameters in such cases may lose their physical meaning after updating.A typical result of updating such inconsistent models is that they may be capable of reproducing the test data but may not beuseful to predict the system behaviour beyond the frequency range used in the updating. Similarly, they may not be able topredict the effects of structural modifications or to serve as a substructure model to be assembled as part of a model of theoverall structure.

The aim of all structural analyses to predict the structural response can only be achieved if all three kinds of modellingerrors are minimised with respect to the given purpose of the structural analysis. Models which fulfil these requirements shallbe called validated models. Model quality must therefore be assessed in three steps:

Step 1: Assessment of idealisation and numerical method errors (model structure errors) prior to parameter updating.Step 2: Correlation of analytical model predictions and test results and selection of correction parameters.Step 3: Assessment of model quality after parameter updating. Since a unique solution cannot be expected, this

requirement must be related to the intended purpose for which the model is used, for example:

�
to predict the system behaviour to types of load or response other than those used in the test, � to predict the system behaviour beyond the frequency range and/or at degrees of freedom other than those used for
updating,
� to predict the effects of structural modifications, � to check if the model, when used as a substructure within an assembled complete structure, will improve the response of
the whole model.

This article provides a basic introduction to the most important procedures of computational model updating and includessimple tutorial examples to reinforce the reader’s understanding together with a large scale model updating example of ahelicopter airframe in Section 6. For a complete exposition of the sensitivity method in model updating the reader is referredto the forthcoming book by the present authors [4].

2. Sensitivity analysis

The sensitivity method is based upon linearization of the generally non-linear relationship between measurable outputs,such as natural frequencies, mode shapes or displacement responses and the parameters of the model in need of correction.Thus the sensitivity method is developed from a Taylor series expansion truncated after the linear term

ez ¼ zm�zðhÞ � ri�Giðh�hiÞ ð1Þ

The residual, ri, is defined at the ith iteration as

ri ¼ zm�zi ð2Þ

so that linearization is carried out at h=hi. The measured and analytically predicted outputs are denoted by zm and zi=z(hi),which typically may be eigenfrequencies, mode-shapes or complex frequency response functions. The sensitivity matrix Gi is

J.E. Mottershead et al. / Mechanical Systems and Signal Processing 25 (2011) 2275–2296 2277

given by

Gi ¼@zj

@yk

� �h ¼ hi

ð3Þ

where j=1, 2, y, q denotes the output data points and k=1, 2, y, p is the parameter index. The sensitivity matrix Gi is computed atthe current value of the complete vector of parametersh=hi. The error,ez, is assumed to be small for parametersh in the vicinity ofhi.

At each iteration, Eq. (1) is solved for

Dhi ¼ h�hi ð4Þ

and the model is then updated to give

hiþ1 ¼ hiþDhi ð5Þ

This procedure continues until consecutive estimates hi and hi+ 1 are sufficiently converged.In practice, the first step is the definition of a residual. In this section we consider the following residuals, representing

probably the most widely used techniques: real eigenvalues, real mode shapes and weighted input forces.A comprehensive selection of these and other residuals with special consideration of statistically based weighting and the

statistical properties of the parameter estimates is given by Natke et al. [5].

2.1. Undamped eigenvalue residual

The linearised undamped eigenvalue residuals are defined by the differences between the vector of measured eigenvalueskm and their analytical counterparts k(h), which being undamped are entirely real. The eigenvalues in this case are defined asthe squares of the system natural frequencies, lj ¼o2

j , j¼ 1,2, . . ., at the linearisation point, ‘i’. Thus the eigenvalue residualand sensitivity are given by Eqs. (1)–(3) when z(h)=k(h); zm=km. It is necessary to ensure that the analytical and measuredeigenvalues correspond to the same physical mode. This process, usually referred to a mode pairing, may be achieved bycarrying out a modal correlation using the modal assurance criterion (MAC) [3].

The terms in the sensitivity matrix may be determined analytically [6] by differentiation of the undamped eigenvalue equation

@lj

@yk¼uT

j �lj@M

@ykþ@K

@yk

� �uj ð6Þ

where M, KARN�N are the FE mass and stiffness matrices, respectively, and uj is the jth mode shape. It is seen that only the jtheigenvalue and eigenvector are needed to calculate all the jth-eigenvalue sensitivities. As well as the analytical approach tocalculating the sensitivity matrix terms, it is also possible to obtain numerical approximations by the simple procedure ofperturbing the parameters in turn by a suitably small quantity and determining numerically the change in the predictedeigenvalues and eigenvectors. Although efficient procedures exist in codes such as MSC-NASTRAN, there is a considerableadvantage in using analytically determined sensitivities when large-scale structures are to be updated.

2.2. Undamped mode-shape residual

The linearised undamped mode-shape residuals are the differences between the measured mode shapes at a restrictednumber of degrees of freedom corresponding to the location of sensors and the analytical modes shapes at the samecoordinates. The differences are determined at NmoN measured degrees of freedom. Thus the mode-shape residual andsensitivity are given by Eqs. (1)–(3) when z=u(h); zm=um, where the vectors u(h) and um may contain many concatenatedvectors of the different analytical and measured mode shapes respectively. The analytical and experimental mode shapesshould be normalized in the same way.

The determination of mode-shape sensitivities is a significant computational task and several approaches are available.In this article we consider only the method of Fox and Kapoor [6]. The method, based on expanding the gradients into aweighted sum of the eigenvectors, is widely used due to its simplicity of implementation

@uj

@yk¼XH

h ¼ 1

ajkhuh; HrN ð7Þ

which, after substitution of Eq. (7) into the derivative of the eigenvalue equation, produces the factor ajkh in the form

ajkh ¼uT

h �lj@M@ykþ @K

@yk

� �uj

ðlj�lhÞ; haj ð8Þ

and

ajkj ¼�1

2uT

j

@M

@yk

� �uj ð9Þ

This expansion is exact if H=N modes are used. For HoN the expansion represents an approximation depending on thenumber of modal terms. Corrections to this approach have been investigated by several authors. Eqs. (7) and (8) show that the


expansion contains large factors ajkh for neighbouring eigenvalues ljElh that can cause convergence problems. Nelson’smethod [3] has the advantage that only the mode shape of interest is required. The antiresonance residual may be used as analternative to the undamped mode-shape residual, as was considered by D’Ambrogio and Fregolent [7].

2.3. Input force residual

The input force residual, also called equation error residual, results from substituting the measured displacementfrequency response xm into the equation of motion. Since the number of measured degrees of freedom is generally muchsmaller than the number of analytical degrees of freedom it is necessary to either expand the measured vector to full modelsize or to condense the model of order N to the number of measured degrees of freedom NmoN. Substituting the measuredfrequency-domain displacement response xm into the equation of motion leads to

zðhÞ ¼ fðhÞ ¼ Zðio,hÞxm ð10Þ

Zðio,hÞ ¼ ½�o2MðhÞþ ioCðhÞþKðhÞ� ð11Þ

where CARN�N is the FE damping matrix, o is the excitation frequency and i¼ffiffiffiffiffiffiffi�1p

. The system matrices in the aboveequation must be condensed to the measured degrees of freedom using methods such as static condensation though this isnot stated explicitly in Eqs. (10) and (11) for reasons of simplicity of notation.

The kth column of Giis given by

@fðhÞ@yk

h ¼ hi

¼ �o2 @M

@ykþ io @C

@ykþ@K

@yk

� �h ¼ hi

xm ð12Þ

The sensitivity matrix is generally contaminated with random measurement noise and systematic errors due tocondensation of the system matrices. This generally leads to bias in the parameter estimates. One way of overcoming thisproblem is by weighting the force residual so that it mimics an output error [8,9]. This may be achieved by using the weightingmatrix H(of, h0), the frequency response function matrix of the initial model at selected frequencies of, not necessarily thesame as the test frequencies. The output error estimator is unbiased when the measurement noise is restricted to xm. Theweighted force (with units of displacement) may be written as,

zðhÞ � xðhÞ ¼Hðom,h0ÞfðhÞ ð13Þ

The kth column of Gi then becomes

@xðhÞ@yk

h ¼ hi

¼Hðof , h0Þ �o2 @M

@ykþ io @C

@ykþ@K

@yk

� �h ¼ hi

xm ð14Þ

where

Hðof ,h0Þ ¼Xp

j ¼ 1

ujuTj =ðo

2j �o

2f þ i2zfofojÞ h ¼ h0

ð15Þ

is the frequency response matrix at the measured degrees of freedom expressed by modal data from the initial model,denoted by the index ‘0’, when modal damping or proportional damping is assumed. This matrix can also be interpreted as adynamic filter having filter frequencyof which is either kept constant over the updating iterations or allowed to vary over thefrequency range. Applying the weighted force residual vector with the initial system vibrating at the arbitrary filter frequencyof controls the magnification of the error. This has the effect of reducing the bias on the estimated parameters.

The force residual and the pseudo response technique have been investigated by several authors [10–12] with respect tothe bias and ill-conditioning problems mentioned above and with respect to the optimal choice of the excitation frequencies,which generally should be close to the peaks of the FRF, but not at the peaks or in the dips. A special case of the weighted forceresidual is obtained by letting the weighting matrix change with each iteration step and with each excitation frequency. Theweighting matrix of the previous paragraph, H(of,h0), is then replaced by H(om,hi). The subscript m is applied to frequencyoto indicate that the analytical frequency response function is determined at the measured frequencies. The pseudo responseand the sensitivity matrix are then given by Eqs. (13) and (14) except that h0 is replaced by hi.

3. Parameter estimation

Parameter updating techniques aim at fitting the parameters of a given initial analytical model in such a way that themodel behaviour corresponds as closely as possible to the measured behaviour. The resulting parameters representestimated values rather than true values since the test data are unavoidably polluted by unknown random and systematicerrors. Also the mathematical structure of the initial analysis is not unique depending on the idealisations made by theanalyst for the real structure. The residuals containing the test/analysis differences may be formed by force and responseequation errors, by eigenfrequency and mode shape errors and by frequency response errors.


The first step in parameter estimation is the definition of a residual containing the difference between analytical andmeasured structural behaviour given explicity in Eq. (1). The objective is to minimise

JðhÞ ¼ eTz Weez ð16Þ

where the symmetric weighting matrix We has been included to account for the importance of each individual term in theresidual vector. We is difficult to estimate, although at the very least this weighting should include scaling to equalise theeffect of amplitude and a reasonable choice is We=[diag(zm)]�2. In general the model response vector z(h) represents a non-linear function of the parameters resulting in a non-linear minimisation problem. One of the techniques to solve this non-linear optimisation problem is to expand the model vector into a Taylor series about the current parameter estimate, hi,truncated after the linear term according to Eqs. (1) and (2).

The case when fewer measurements than parameters are available in Eq. (1) (qop) leads to an underdetermined systemwhose solution is not unique. Indeed, if rank(Gi)=q and riArange(Gi), then the model is able to reproduce the measurements,i.e. ez=0. Even if a minimum norm or a minimum parameter change solution is selected the resulting parameters will ingeneral not retain their physical meaning. In parameter updating the number of measurements should always be made largerthan the number of parameters (q4p) which yields overdetermined equation systems. A very effective way of choosingsuitable parameters is by the subset selection described by Lallement and Piranda [13], and Friswell et al. [14]. The datashould of course be sensitive to the selected parameters, which must be justified by physical understanding of the structureand the test arrangement. For the overdetermined case we minimise J with respect to Dhi to give an improved parameterestimate as

Dhi ¼ ½GTi WeGi�

�1GTi Weri ð17Þ

Even in the overdetermined case the condition of the sensitivity matrix G plays an important role for the accuracy and theuniqueness of the solution. A fundamental requirement to obtain a solution is that rank½GT

i WeGi� ¼ p.

4. Regularisation

The treatment of ill-conditioned, noisy systems of equations is a problem central to finite element model updating [15–18].Such equations often arise in the correction of finite element models by using vibration measurements. The classical weightedleast squares method described above can be extended in cases where it difficult to obtain a convergent solution because of anill-conditioned sensitivity matrix. The objective function, Eq. (16), is extended by the requirement that the parameter changesDh should be minimised, to give

JðhÞ ¼ eTz Weezþl

2DhTi WyDhi ð18Þ

The parameter weighting matrix Wy should be chosen to reflect the uncertainty in the initial parameter estimates [19–21].This may be formally related to Bayesian methods, where the optimum matrices We and Wy are the inverse of the output andparameter variances respectively [22–23]. However, the variance of the initial parameter estimates are rarely known inpractice and an alternative given by Link [24] relates the choice of weighting matrix, Wy, to the inverse of the squaredsensitivity matrix according to

Wy ¼meanðdiagðCÞÞ

meanðdiagðC�1ÞÞ

C�1; C¼ diag GTi WeGi

h ið19Þ; ð20Þ

This definition allows the parameter changes to be constrained according to their sensitivity. In consequence theparameters remain unchanged if their sensitivity approaches zero. Wy=I represents the classical Tikonov regularisation [25]used to solve ill conditioned systems of equations.

Eq. (17) is easily extended to penalise differences between the updated parameters and the corresponding initialestimates, or to penalise differences between nominally identical parameters [14,15]. For example, in a frame structure anumber of ‘T’ joints may exist that are nominally identical. Due to manufacturing tolerances the parameters of these jointswill be slightly different, although these differences should be small. Therefore a side constraint is placed on the parameters,so that both the residual and the differences between nominally identical parameters are minimised.

Minimising J in Eq. (18) gives the solution

Dhi ¼ ½GTi WeGiþl

2Wy��1GT

i Weri ð21Þ

The question remains how to choose the regularisation parameter l, that provides a balance between the measurementresidual, Je(h)=eTWee, and the side constraint (or parameter change), Jy hð Þ ¼DhT

i WyDhi. Link [24] suggested that the factor l2

lies in the range between 0 (no regularisation) and 0.3. Highl values are used if there are many insensitive parameters and thematrix GT

i WeGi is strongly ill-conditioned. If the ill-conditioning is not too strong l2=0.05. The higher the value of l the higheris the necessary number of iteration steps to achieve convergence.

This regularisation approach is very closely related to the optimisation of multiple objective functions. From Eq. (18) it isclear that the residual and side constraint are functions of l: Je(l) and Jy(l). The way in which these two terms are balanceddepends on the size of the regularisation parameter l. If l is too small then the problem will be too close to the originalill-posed problem, but if l is too large then the problem solved will have little connection with the original problem. A useful


approach is to plot the norm of the side constraint,ffiffiffiffiffiffiffiffiffiffiJyðlÞ

p, against the norm of the residual,

ffiffiffiffiffiffiffiffiffiffiJeðlÞ

p, for different values of l. For

multi-objective function optimisation this is called the Pareto front and in regularisation is called the L curve. Hansen [26]showed that the norm of the side constraint is a monotonically decreasing function of the norm of the residual. He pointed outthat for a reasonable signal-to-noise ratio and the satisfaction of the Picard condition, the curve is approximately vertical forlolopt, and soon becomes a horizontal line when l4lopt, with a corner near the optimal regularisation parameter lopt. Thecurve is called the L-curve because of this behaviour. The optimum value of the regularisation parameter, lopt, corresponds tothe point with maximum curvature at the corner of the log–log plot of the L-curve. This point represents a balance betweenconfidence in the measurements and the analyst’s intuition.

One difficulty in model updating is that the relationship between the parameters and the measurements is non-linear, andthe estimation problem is solved by constructing the linearised model and iterating until convergence. At each iterationregularisation may be applied, although often the corner of the L-curve disappears as the iterations progress (Titurus andFriswell [17]), and thus the value of the regularisation parameter is difficult to determine. Hence, it is often convenient to setthe regularisation parameter at the first iteration and retain this value until convergence.

Example: Two degree of freedom statically loaded system.We now consider the two degrees of freedom static example shown in Fig. 1. In the initial model the spring constants are all

1 N/m, so that k1=k2=k3=1 N/m. The measured data are taken from a system with k1=1.2 N/m, k2=0.8 N/m and k3=1.0 N/m. Thepurpose of the example is to show how the stiffness correction may be determined from measured static displacements u1 andu2. Two cases will be considered, namely the under-determined and over-determined cases.

The updating equation is formulated using a force residual

ez ¼f1

f2

!�

k1þk2 �k2

�k2 k2þk3

" #um1

um2

!¼ rziþGdk ð22Þ

where G¼

@f1

@k1

@f1

@k2

@f1

@k3

@f2

@k1

@f2

@k2

@f2

@k3

26664

37775¼

um1 um1�um2 0

0 �um1þum2 um2

" #is the sensitivity matrix and doesn’t depend on the current stiffness

value. Thus the updating equation is of the same form as Eq. (1).

Case 1. Under-determined system.

Suppose that the force applied to the springs is f1 ¼ ½1 0�T , and the measured static deflection is um1 ¼0:60811

0:27027

�. The

vector of stiffness changes that reproduces the measured displacement, for the minimum norm of the stiffness change, may

be obtained as dk1=0.1398, dk2=�0.0916 and dk3=0.1354. Fig. 2 shows the results obtained by including the side constraint

of minimum stiffness change, for various values of regularisation parameter, l. In this case the matrix G is not ill-conditioned,

and hence the side constraint yields a solution that is a weighted average of the minimum norm solution and the zero change

solution, as shown by the smooth variation in the spring stiffnesses with l in Fig. 2. This is also the reason the residual plot

does not have the L-curve shape. Note also that the simulated stiffness change is never recovered by the estimation procedure.

Case 2. Over-determined system.

Suppose that a second force, f2 ¼ 0 1� T

, is applied to the springs, giving the measured static deflection of

um2 ¼0:27027

0:67568

�. If this data is combined with the data for the first load case f1 ¼ 1 0

� T, then the estimation is

overdetermined. Since there is no noise or modelling errors the exact spring stiffnesses are obtained: dk1=0.2, dk2=�0.2

and dk3=0.0

Regularisation is most useful when the coefficient matrix is relatively ill-conditioned and noise is present. In this simple

example the ill-conditioning is obtained by considering measurements taken at two force levels, namely f1 ¼ 1 0� T

and

f3 ¼ 1:05 0� T

. The exact ‘measured’ displacement for f3 is um3 ¼0:63851

0:28378

�. Noise is added to the measurements:

�0:0002

0:0003

�to um1 and

0:0010

�0:0002

�to um3. Fig. 3 shows the results obtained by including the side constraint of minimum

Fig. 1. Two degree of freedom discrete example.

10−4

10−3

10−2

10−1

10−3

10−2

10−1

norm

(St

iffn

ess

Cha

nge)

norm (Residual)

L curve

10−4

10−2

100

102

10−4

10−3

10−2

10−1

norm

Stiffness ChangeResidual

10−4

10−2

100

102

−0.2

0

0.2

Regularization Parameter, λ2 St

iffn

ess

Cha

nge

k1 k2 k3

Fig. 2. The estimation results for the under-determined case. Dotted lines represent the simulated stiffness change.

10−4

10−3

10−2

10−1

10−3

10−2

10−1

100

norm

(St

iffn

ess

Cha

nge)

norm (Residual)

L curve

10−10

10−5

100

10−4

10−2

100

norm


10−10

10−5

100

−1

0

1

Regularization Parameter, λ2

Stif

fnes

s C

hang

e

k1 k2 k3

Fig. 3. The estimation results for the over-determined case with the minimum norm parameter change. Dotted lines represent the simulated stiffness

change. Circles denote the corner of the L curve and the associated values of residuals and stiffness changes.


stiffness change, for various values of regularisation parameter, l. The L-curve now has the classical shape, with a definedcorner. For low values of regularisation parameter the estimation is clearly ill-conditioned leading to large estimated stiffnesschanges. The condition number of G in this example is 880. At the corner of the L-curve the stiffness changes areapproximately those obtained from the minimum norm solution for the under-determined case. For a regularisationparameter slightly less than that at the corner of the L-curve the stiffness changes are very close to those simulated, althoughthis is impossible to determine solely from the estimated results. Fig. 4 shows the results when using the parameterweighting matrix given by Eqs. (19), (20). Although similar to the results of Fig. 3, the updated parameters at the corner of theL curve are now much closer to the simulated values, as shown in Table 1.

5. Parameterisation

The amount of information that can be obtained from vibration test data is limited and therefore taking moremeasurements in the same frequency range would not necessarily result in more information. Neither will the additional

10−4

10−3

10−2

10−1

10−3

10−2

10−1

100

norm

(St

iffn

ess

Cha

nge)

norm (Residual)

L curve

10−10

10−5

100

10−4

10−2

100

norm


10−10

10−5

100

−1

0

1

Regularization Parameter, λ2 St

iffn

ess

Cha

nge

k1 k2 k3

Fig. 4. The estimation results for the over-determined case with parameter weighting matrix given by Eq. (18). Dotted lines represent the simulated stiffness

change. Circles denote the corner of the L curve and the associated values of residuals and stiffness changes.

Table 1Updated parameter estimates for the over-determined two degree of freedom static example.

l k1 (N/m) k2 (N/m) k3 (N/m)

No regularisation 0 1.779 �0.243 �0.304

L curve corner, Fig. 3 0.0093 1.149 0.889 1.112

L curve corner, Fig. 4 0.0179 1.179 0.836 1.046

‘Exact’ 1.200 0.800 1.000


measurements allow more parameters to be estimated necessarily. The number of parameters should be considerablysmaller than the number of measurements. The objective should be that the model-updating problem will be over-determined. Often the resulting equations are ill-conditioned and it is then necessary to apply additional information in theform of a side constraint by regularisation as described previously. The parameters should be justified by physicalunderstanding of the structure under test and the test set-up. Ideally the chosen parameters should have a physical meaningdirectly, but this is not always possible in practice. Equivalent models and their parameters often lead to improved modelswhen ‘physical’ parameters cannot be found. The data should be sensitive to small changes in the parameters. Difficultfeatures, such as joints, may be made more or less sensitive by choosing different types of parameters. A study that includesseveral different parameterisations of the same joint in a space-frame structure is described by Mottershead et al. [27].

When choosing parameters it is always advisable to try to understand the behaviour of the structure globally, and locallyin those regions where local modelling inaccuracies might be responsible for discrepancies in predictions. For example, closestudy of finite element mode shapes is able to reveal the motion of joints at each of the measured natural frequencies.Parameters can then be chosen that influence this motion and their significance in model updating easily confirmed bysensitivity analysis and subset selection. In regions of high strain energy one would usually choose stiffness parameterswhereas mass parameters would be useful in regions of high kinetic energy. Stiffness is generally more difficult to model thanmass and it is therefore more likely that errors in stiffness modelling are responsible for inaccurate predictions than masserrors. Damping is in many respects a special case, whereas finite element mass and stiffness matrices may be readily derivedfrom variational or energy principles, similar derivations for damping are generally not available. Joints and boundaryconditions are particularly difficult to model closely. In principle it is possible to design tests that increase the sensitivity ofchosen parameters, but this is extremely difficult to achieve in practice.

5.1. Mass, damping and stiffness matrix multipliers

Probably the simplest parameters for model updating are non-dimensional scalar multipliers applied at the element orsubstructure level. The updated model takes the form

M¼M0þa1M1þ . . .þarMrþ . . .þaRMR ð23Þ


C¼ C0þb1C1þ . . .þbsCsþ . . .þbSCS ð24Þ

K¼K0þg1K1þ . . .þguKuþ . . .þgUKU ð25Þ

where in this case U stiffness parameters, S damping parameters and R mass parameters are chosen for updating. Thesubscript ‘0’ denotes the analytical model before updating. Of course the parameter ar, bs or gu may be applied to more thanjust one element. In this way the parameters may be applied to substructures, when those elements sharing the sameupdating parameter are connected, or there may be elements dispersed through the mesh that for some physical reason are tobe updated in the same way. One reason why different parts of the structure might be updated using the same parameterwould be the sensitivity of the eigenvalues (and eigenvectors) in the frequency range of interest to small changes in theparameters is very similar. In that case separating the elements whose changes have a similar effect would be a bad choice,possibly leading to ill-conditioning of the sensitivity matrix.

It is good practice to scale the updating equations so that the parameters a1, y, ar, y, aR, b1, y, bs, y, bS andg1, y, gu, y, gU take similar numerical values. One way to do this is by dividing the parameter correction by the initialparameter values, which is done implicitly in Eqs. (23)–(25). Returning to the specific discussion of matrix-multiplierparameters, it is seen that the terms @M=@ar , @C=@bs, @K=@gu are simply the rth element mass matrix, Mr, sth elementdamping matrix, Cs and the uth element stiffness matrix, Ku, a fact which makes the computation of the various sensitivitiesespecially simple.

5.2. Material properties, thicknesses and sectional properties

The most common material property parameters are Young’s modulus and mass density, identical to the gu and ar abovesince the element stiffness and mass matrices are linear in E and r.

In the case of a beam having a rectangular cross section the element matrices are

Ke¼

EI

l3

12 6l �12 6l

6l 4l2 �6l 2l2

�12 �6l 12 �6l

6l 2l2 �6l 4l2

26664

37775, I¼

bt3

12ð26Þ

and

Me¼

rAl

420

156 22l 54 �13l

22l 4l2 13l �3l2

54 13l 156 �22l

�13l �3l2 �22l 4l2

26664

37775, A¼ bt ð27Þ

where l denotes the element length and the breadth and thickness of the cross section are denoted by t and b, respectively.Young’s modulus and mass density are represented by E and r.

The choice of parameters E and I independently would lead to redundancy and ill-conditioning since they lead to identicaleigenvalue sensitivities (except for a scaling factor). I and A independently would be a difficult choice to justify physically. But,both Ke and Me depend upon b and t, so that the choice of these two parameters would allow the correction of a beam cross-section meaningfully. Other useful parameters include the thicknesses and dimensions of thin-walled sections and platethicknesses.

The material parameters, thicknesses and cross-sectional dimensions tend to be powerful updating parameters becausethey often apply throughout a finite element mesh affecting a large number of elements. Thus a small change in theseparameters often affects the natural frequencies very considerably.

5.3. Offset nodes

Joints and boundary conditions are difficult to represent accurately and it is in these regions of the model that assumptionsare often made. Probably the most common assumption is that the connection made at a joint or boundary is rigid when infact there is flexibility. The problem of introducing flexibility into joints and boundaries assumed to be rigid can be tackled in anumber of different ways, all resulting in equivalent models with parameters that cannot be justified on physical grounds.However, the dynamic behaviour of the model and its physical usefulness will most definitely be improved by thisapproach.

One approach, useful in many applications, is to make use of offset finite element nodes and to use the offset dimensions tocorrect the model [28]. Lengthening or shortening an offset dimension usually corresponds to making the joint more flexibleor to stiffening it, and in this way it is possible to reconcile the modification with engineering understanding of the structure.

Example: Parameterisation of a ‘T’ joint.

Fig. 5. ‘T’ joint.


The ‘T’ joint under consideration is shown in Fig. 5. It consists of three element, two horizontal and one vertical element,each of which is inextensible so that x1=x4, x2=x5 and y3=y6.

The shaded region may be considered rigid in which case the degrees of freedom at nodes 1, 2 and 3 are referred to node 30

according to the connection matrix

y1

W1

y2

W2

x3

W3

y4

W4

y5

W5

x6

W6

8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;

¼

1 �b1

1 b1

1 a1

1

1

1

1

1

1

26666666666666666666666664

37777777777777777777777775

x3u

y3u

W3u

y4

W4

y5

W5

x6

W6

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;

ð28Þ

Construction of the overall stiffness matrix of the ‘T’ joint by the usual methods results in a matrix that contains the offsetsa and b. If the overall dimensions of the joint remain unchanged then increasing the offsets causes the joint to become stifferwhile reducing them makes the joint more flexible. Parameters such as a and b have been found to be extremely effective inthe correction of inaccurately modelled joints.

5.4. Generic elements

Generic elements offer perhaps the most sophisticated and most general way to parameterise a finite element model[29–31]. The basic idea is to change the element formulation within the limitations imposed by the number of nodes and thedegrees of freedom available. A number of different methods are available, based either on eigenvalue decomposition or theapplication of constraints (not discussed in the present article). Whichever of the different methods are chosen, genericelements are applied at the element (or substructure) level.

Quite often the mass matrix is considered to be accurate and only the stiffness matrix needs to be updated. Then aneigenvalue decomposition of the stiffness may be carried out

Ke¼W

0 0

0 P

� �WT

ð29Þ

where the matrix of eigenvectors is orthogonal, since Ke is a real symmetric matrix

WTW¼ I ð30Þ

The eigenvalues and eigenvectors of the stiffness matrix are not the same as those of the element generalised eigenvalueproblem, ðKe

�ljMeÞuj ¼ 0, j¼ 1,2,3, . . .. Each eigenvalue can be thought of as a spring coefficient for a deflection defined by

its eigenvector.A new matrix of eigenvectors may be introduced

W¼WS ð31Þ


so that a new stiffness matrix may be written as

Ke¼WjWT

ð32Þ

j¼ S0 0

0 P

� �ST

ð33Þ

and from Eqs. (30) and (31) S is an orthogonal matrix.It seems at first that the number of updating parameters might be quite large, but the number can be reduced very

considerably by applying engineering judgement. For example, the first two stiffness eigenvalues might be very sensitiveparameters for many of the lower vibration modes of the complete structure. In this case only the first two diagonal terms,p1, p2, would be chosen. Otherwise modifying the terms of j will lead to changes in both the stiffness eigenvalues andeigenvectors.

Example: Eigenvalue decomposition of a beam element.

A beam-element stiffness matrix with the properties, EI=1 and l=1, can be written as

Ke¼

12 6 �12 6

6 4 �6 2

�12 �6 12 �6

6 2 �6 4

26664

37775

and possesses eigenvalues 0, 0, 2, 30 (with units of stiffness), and eigenvector

W¼

1 1=2 0 2=ffiffiffiffiffiffi10p

0 �1 1=ffiffiffi2p

1=ffiffiffiffiffiffi10p

1 �1=2 0 �2=ffiffiffiffiffiffi10p

0 �1 �1=ffiffiffi2p

1=ffiffiffiffiffiffi10p

266664

377775

The rigid-body modes describe pure translation and pure rotation about the centre of mass of the beam. SinceKe¼S2

j ¼ 1wTjþ2pjwjþ2 it is seen that Ke

2 R4�4 is a rank 2 matrix, so that only the strain ‘modes’ contribute to the stiffness ofthe beam element. The rigid-body modes of (Ke

�pjI) are the same as the rigid-body modes of ðKe�ljM

eÞ so that the mass

matrix has no influence on the rigid-body modes, which span the null-space of the element stiffness matrix.Example: Generic element parameters for a pinned-pinned beam.

The eigenvectors of the stiffness matrix formed from two uniform beam elements with pinned ends are shown in Fig. 6.The first and third are symmetric whereas the second and fourth modes are anti-symmetric.

Fig. 6. Mode shapes of the pinned-pinned beam.


To select modifications to all the eigenvalues but only the symmetric eigenvectors, the matrix S is chosen so that

S¼

s11 s13

1

s31 s33

1

26664

37775

and

W¼ ST PS¼

ðs211p1þs2

31p3Þ 0 ðs11s13p1þs31s33p3Þ 0

0 p2 0 0

ðs13s11p1þs33s31p3Þ 0 ðs213p1þs2

33p3Þ 0

0 0 0 p4

266664

377775

The element stiffness matrix can be reconstructed as

Ke¼ w1 w2 w3 w4

h i k11 0 k13 0

0 k22 0 0

k31 0 k33 0

0 0 0 k44

266664

377775

wT1

wT2

wT3

wT4

2666664

3777775

and there are five updating parameters, k11, k13=k31, k22, k33 and k44.Example: Updating a system of 3 beams with offset central span.

We consider the system of three beams connected in-line but with the axis of the central beam offset from the axes of thetwo outer beams as shown in Fig. 7. The breadth of all three beams is 0.2 m and the material is steel (E=210 GN/m2,r=7860 kg/m3). The system is represented by a finite element model consisting of 10 beam elements as indicated in thefigure with rigidly fixed ends. Each node has three degrees of freedom, axial and transverse displacements and a rotationabout the third axis. The 4th and 7th elements have offset nodes at the left-hand and right-hand ends, respectively.

The connection matrix defining the offset node at the 4th element is given by

1 0 a

0 1 0

0 0 1

264

375

In the first case a finite element model with initial parameters E0=180 GN/m2, a0=0.015 m is updated using the correctchoice of parameters. Updating is carried out using the standard Moore–Penrose pseudo-inverse based on convergence of thefirst five natural frequencies. It is seen from Fig. 8 that exact convergence to the true solution is obtained in two iterations.

In the second case the finite element model is corrected using generic element parameters. This time the finite-element massmatrix is correct but in the stiffness matrix the offset is initially given by 0.8 of its true value. The elastic modulus is correct. Thegeneric element is formed from the group of elements, 3, 4, 7, 8, at the junctions between the thick and thin beam sections.The stiffness matrices from elements 3 and 4 are uncoupled from elements 7 and 8 but each pair of element has identicaleigenvalues. Therefore the eigenvalues of the generic element stiffness matrix occur in pairs. It is found that the first naturalfrequency of the beam is not very sensitive to the generic element parameters, which represent (shape) stiffnesses at thejunctions—well away from the most strained portion of the beam in the middle. The second and third natural frequencies arevery sensitive and the fourth mode is axial, and therefore insensitive. A single generic element parameter is not sufficient toproduce good results but in Figs. 9 and 10 results are shown from two updating parametersk11,k22. It is seen that the second andthird natural frequencies converge correctly after approximately 20 iterations when the parametersk11, k22 are fully converged.

Updating was achieved using the weighted (regularised) updating Eq. (21) using the first ten natural frequencies. Thesecond and third diagonal terms of We were given by values of 13,000 with the remaining diagonal terms set to unity

0.1

0.05

0.3 0.4 0.3

1 2 34 5 6 7

8 9 10

Fig. 7. System of 3-beams with offset central span (dimensions in metres).

Fig. 8. Parameter convergence—E and a.

Fig. 9. Parameter convergence—k11, k22.

Fig. 10. Convergence of natural frequencies—generic element parameters.


(off-diagonal terms set to zero), Wy ¼ diagð 250 50Þ. It is seen that the generic-element parameters are able to provide anupdated model that accurately reproduces the dynamic behaviour of the offset beam (in the frequency range considered)even though the updated model lacks physically meaning. Nevertheless, such a model may be meaningful within theupdating frequency range.

Fig. 11. Lynx helicopter in flight.

Fig. 12. XZ649 Baseline GVT configuration showing the sling assembly.


6. Model updating of a Lynx helicopter airframe

This section describes the updating of a baseline Lynx helicopter carried out by the Universities of Bristol, Liverpool and Kasseltogether with DLR, Eurocopter, NLR, ONERA and QinetiQ under the auspices of GARTEUR (the Group for Aeronautical Research and

Technology in Europe)—Action Group AG-14 [32,33]. Fig. 11 shows a photograph of the military Lynx helicopter in flight.The basic airframe structure consisting of the cabin, tail boom and tailplane was chosen as the ‘baseline’ configuration. The

skids were removed and there were no large concentrated masses anywhere on the structure. The mass was approximately800 kg with dimensions of 12 m in length, 2 m in width and 2.5 m in height.

To measure the elastic normal modes, the experimental test arrangement must represent the free-free flight boundaryconditions. This requires a very soft suspension system with frequencies significantly lower than the first elastic frequency.The most adequate suspension point is the rotor hub, which allows the same pre-load condition to be obtained as in flight.Fig. 12 shows a view of the ground vibration trials (GVT) performed at QinetiQ, Farnborough with the airframe in its baselinecondition. Note the baseline configuration does not include any gearboxes or engines and that the airframe is suspended viastrops between the yellow sling and a spring assembly at the top of the gantry. Detail of the suspension is shown in Fig. 13.

The location of 29 response points and 4 excitation attachments are shown in Fig. 14. Translational acceleration responses inthe x-, y- and z-axis directions were measured at each of the response points. As it was not possible to acquire all of this data in asingle test, a series of tests was performed and signal connections were changed for each test. The structure was not disturbedbetween tests which were conducted as soon as possible, one after another. Some responses were measured in all of the tests –notably the drive point responses – and this provided the data for repeatability checks between the individual tests.

Excitation of the structure was achieved using electromagnetic shakers driven by constant current power amplifiers. Forlateral or fore-aft excitation, a shaker was suspended on a chain hoist from a mobile crane. The suspension length variedaccording to the height of the support and the height of the excitation attachment. Reaction masses of 50 kg were bolted to therear of the shakers to lower the suspension frequencies and to provide a reaction force with very little movement of the shaker.For an excitation in which the line of action of a shaker was well away from the centre of gravity of the structure only one 50 kgmass was used. Otherwise, two reaction masses were bolted to the rear of the shaker. Vertical excitation was achieved with theshaker either bolted to a heavy support frame, or bolted to two reaction masses and rested on the laboratory floor.

Separate measurements were performed over two baseband frequency ranges, each using 2048 spectral lines; one to64 Hz (Df=0.03125 Hz) and another to 128 Hz (Df=0.0625 Hz). Four averages were taken. Twelve modes of vibration (up to75 Hz) were extracted from the baseline-structure test.

Fig. 13. Lifting beam and strop assembly.

Fig. 14. Location of response and excitation points for baseline tests.


The finite element model, in Fig. 15, was produced using MSC-NASTRAN by an assembly of shell and beam elements–massand inertia properties were introduced into the model using an array of concentrated mass elements.

Initially 109 different finite-element material properties were used to parameterise the baseline model. It was observed that tenmaterial properties, spread over different parts of the model, were most important. The model was partitioned to form smallergroups, thereby defining a set of parameters based on engineering judgement as well as topological and technical considerations.Then an eigenvalue sensitivity study for the whole helicopter model was carried out using the elastic moduli for 147 groups ofelements. The first 36 columns of the scaled sensitivity matrix1 (the most sensitive parameters) are shown in Fig. 16. In the horizonal

1 In general, when the sensitivity matrix is scaled in this way then Eq. (1) takes the form,

e1

z0;1

^ej

z0;j

^eq

z0;q

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA¼

r1

z0;1

^rj

z0;j

^rq

z0;q

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA�

@z1

@y1

y0,1

z0,1� � �

@z1

@yk

y0,k

z0,1� � �

@z1

@yq

y0,p

z0,1

^ ^ ^@zj

@y1

y0,1

z0,j� � �

@zj

@yk

y0,k

z0,j� � �

@zj

@yq

y0,p

z0,j

^ ^ ^@zq

@y1

y0,1

z0,q� � �

@zq

@yk

y0,k

z0,q� � �

@zq

@yp

y0,p

z0,q

266666666666664

377777777777775

dy1

y0,1

^dyk

y0,k

^dyp

y0,p

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA

which has the advantage that large numerical values of the sensitivity terms are moderated by the initial values of the outputs and parameters.

Fig. 15. GARTEUR baseline FEM of Mk7 Lynx fuselage, 2 views.

Fig. 16. Eigenvalue sensitivity to Young’s modulus: 36 most important groups.


plane of the figure the parameter indices are shown against the first eight natural frequencies. The height of the columns denoted thescaled sensitivity of each natural frequency to each of the 36 elastic moduli, the updating parameters. The sensitivities are all positivesince an increase in stiffness generally results in an increase in all the natural frequencies.

Fig. 17. Three groups of closely clustered parameters in the cabin region—numbers shown denote the columns of the sensitivity matrix.

Fig. 18. Correlation of the finite element model and test data.


In many cases small changes to the chosen parameters have a similar effect on the natural frequencies. This means that thecorresponding columns of the sensitivity matrix are almost identical. One way to test this is by the cosine distance betweentwo columns, expressed as

cosgab ¼gTagb

ðgTagaUgT

bgbÞ1=2

ð34Þ

where gab is the angle between two vectors ga and gb, being the ath and bth columns of the sensitivity matrix G, respectively.Fig. 17 shows three examples of so-called ‘clusters’ of parameters in the cabin region with close sensitivity columns within a51 angle of each other. Of course these parameters should be combined together for updating not only because they have asimilar effect, but also because making clusters in this way tends to improve the conditioning of the sensitivity matrix andreduces the number of parameters to be determined.

Fig. 18 shows a comparison between the finite-element predicted modes and the test modes. The pairing of FE and testmodes is seen to be good from the diagonal dominance of the MAC array up to the eighth mode. Table 2 shows that the naturalfrequencies predicted by the analytical model (bending modes) are higher than the test results and amongst them; thevertical ones are more in error than the lateral ones. The tail has a quasi-axi-symmetric structure and mostly dominates thelower frequency modes, which appear in pairs. The physical tailcone shows more flexibility in the test than does the model,due to its riveted multi-layer skin.

The tail spar was one of the most sensitive areas of the model and close inspection revealed that it had been incorrectlymodelled. After correction, by adding an offset, the FE-test correlation was significantly improved. Although the modificationreduced the error for mode 4 to 8% it was not successful in improving mode 3 as can be seen in columns 6 and 7 of Table 2.

Significant improvements were achieved by the clustering of parameters and the application of model updating. Since thesensitivity matrix was well-conditioned, regularisation was unnecessary. The average frequency error was reduced from7.09% to 2.19% and the MAC was improved too. The final results obtained after model updating, using eight natural

Table 2Predicted natural frequencies (Hz) and errors (%) between the finite element model (baseline) and test data: initial, modified and updated models.

Mode no. Mode description Test(Hz)

InitialFEM (Hz)

Error(%)

ModifiedFEM (Hz)

Error(%)

UpdatedFEM (Hz)

Error(%)

1 1st vertical bending 12.45 13.30 7.01 13.25 6.39 12.06 �3.12

2 1st lateral bending 13.29 14.00 5.13 14.07 5.93 13.14 �1.08

3 Tail plane fore/aft bending 18.52 19.60 6.06 20.03 8.15 18.41 �0.61

4 Tail plane vertical bending 19.85 22.80 15.10 21.57 8.66 19.94 0.47

5 2nd vertical Bending 32.67 35.30 7.97 36.62 12.11 34.15 4.55

6 2nd lateral Bending 36.91 38.50 4.38 38.08 3.18 36.83 �0.20

7 Fuselage torsion 50.34 47.10 �6.50 49.11 �2.44 48.25 �4.16

8 3rd vertical bending 52.57 56.30 7.19 57.74 9.85 54.33 3.36P9%Error9/8 7.42 7.09 2.19

Fig. 19. MAC table after offset correction and updating.

Table 3The selected parameters and correction factors for Young’s modulus.

Parameter Correction factor

1 E1312001 4.67�10�1

2 E1326203 6.42�10�1

3 E1326209 8.78�10�1

4 E1355010 8.44�10�1

5 E1250000 4.16�10�1

6 E1250001 8.43�10�1

7 E1201081 4.44�10�1

7 E2190001 4.44�10�1

7 E9170001 4.44�10�1


frequencies and seven parameters, are presented in columns 8 and 9 of Table 2 and in Fig. 19. The correlation of first sevenmodes are seen be very good, though the eighth mode (not shown in Fig. 19) was not improved. The chosen updatingparameters together with correction factors are shown in Table 3 and Fig. 20. Convergence was achieved in 5 iterations. Itshould be recalled that the chosen elastic moduli are equivalent parameters representing changes to the stiffness of thestructure not readily available for correction directly because of the extreme detail present in such a large finiteelement model.

Fig. 20. Selected updating parameters.


Parameter E1312001 represents the upper half of the tailcone, which is riveted, has many patches and open access pointsthat might lead to a stiffness reduction represented by a reduced Young’s modulus in the updated parameters. Anotherpossibility is that the added patches would lead to an increase in mass having the same effect as a stiffness reduction.

Regions of connectivity between the tapered section of the cabin and tail-cone, and the tapered section and the cabin itselfare represented by parameters E1326203 and E1326209. The results suggest that these physical transitions are less stiff thanin the model.

The region at the elbow of the tail-cone and at one end of the tie rods coincides with parameter E1335010. Again a stiffnessreduction is indicated.

Parameters E1250000 and E1250001 represent the tail spar and tail plane. The tail spar has been modelled using a singlebeam element. Over-stiffening of the joint in the FE model is the likely cause of the discrepancy.

The three groups E1201081, E2190001 and E9170001 were linked together to form a single updating parameter affectingmainly the third vertical bending mode. The boundary condition of the windshield with the cabin structure may beresponsible for the indicated stiffness reduction.

7. Closure

In this section we briefly introduce more advanced and recent developments for the interested reader without enteringinto an exhaustive review.

7.1. Model updating in damage detection

It is well known that low-frequency modal test data or static response data are not well suited for detecting and quantifyingsmall-sized local damage. Acceptable results can only be expected if high spatial resolution of the response data is available. Inparticular for bridge structures, static influence lines can be measured with high spatial resolution when static responses(deflections, deflection slopes, strains) are nearly continuously measured as a vehicle slowly crosses the bridge (Link et al.[34]). It is well known that time histories are sensitive to damage even if the damage is local [35]. Hernandez and Bernal [36]used time histories in conjunction with a state observer for model parameter updating. Lu and Law [37] used the sensitivitymethod with directly measured time histories due to impact, sine and random excitation. The analytical sensitivities and timehistories were calculated by classical numerical integration. Link and Weiland [38] investigated the potential of using timehistory data obtained from impact hammer excitation for estimating experimental frequency response functions. Thesensitivities and time histories were calculated by inverse Fourier transformation. Two techniques were applied to a beamstructure consisting of two thin facing sheets bonded together by an adhesive layer. Damage introduced locally by the adhesivelayer was identified by a multi-model updating method using time-domain and modal data separately.

7.2. Model updating using full-field vibration data

The increasing use of scanning laser velocimetry [39] and digital image processing [40,41] in modal analysis has made full-field measurement of vibration mode shapes available. A huge number of data points are available with highly redundantinformation which must be condensed down to the most significant parts. This can be achieve using image processing methodsgenerally based on orthogonal polynomials, typically the continuous Zernike polynomials [42] (ideal for the analysis of circularshapes), Fourier series or wavelets [43] (which have the advantage of being able to describe very localised modes-shape


features) or the discrete Tchebichef polynomial. The orthogonality property means that the shape information is retainedefficiently in the coefficients of these polynomials, known as shape features. Generally only a small number of shape features areneeded to reconstruct a measured full-field mode shape with excellent accuracy. The construction of multi-dimensional shapefeature vectors allows for a distance measure between measured and FE mode shapes. Also shape features have been usedsuccessfully in model updating [44].

7.3. Stochastic model updating

Beck and Katafygiotis [45] developed a Bayesian probabilistic frame-work for robust finite element model updating, which waslater employed by Beck and Au [46], to correct a two degree-of-freedom mass-spring system by using Markov chain Monte-CarloSimulation. They demonstrated that the method is capable of identifying multiple non-unique solutions. Soize et al. [47] presenteda methodology for robust model updating by using a non-parametric probabilistic approach [48]. The method leads to the solutionof a mono-objective optimisation problem with inequality probabilistic constraints. The statistics of the updated parameters areonly significant in that they represent the confidence in the estimate, for example in the presence of measurement noise. For moreor less noisy data one would expect to see greater or lesser variance in the estimated parameters. Also the use of more data willtend to reduce the parameter variances. Therefore, such distributions are not physically meaningful, for example in terms of thevariation in a parameter for multiple tests on different (but nominally identical) specimens.

There are generally two classes of uncertainty: epistemic and aleatory (irreducible) uncertainty. Epistemic uncertainty isdue to lack of knowledge, which is reducible by further information. For example random measurement noise on anaccelerometer signal is reducible by averaging. The methods discussed in the above paragraph are suitable for modelupdating in the presence of epistemic uncertainty. Systematic errors in the eigenvalues of FE code are reducible by improvedunderstanding if the constitutive equations and corresponding correction of the FE model. Vinot et al. [49] used an info-gaprobust-satisficing design approach for the base excitation of a vibration test. The method attempts to satisfy certainperformance criteria while maximising the robustness to epistemic uncertainties in a nominal model. Such uncertaintiesinclude, joint stiffnesses, geometric and material properties as well as possible unmodelled non-linear behaviour.

If a number of different structures, nominally identical to within manufacturing tolerances, are tested the results willgenerally be different and cannot be reduced by further testing on the same sample of structures. This is an example ofaleatoric uncertainty, which may be described with physical meaning using statistical measures (mean, variance, higher-order statistical moments), intervals or fuzzy analysis. Stochastic model updating, as opposed to deterministic modelupdating - and the statistical method described in the previous paragraphs - is capable of inferring not only the mean value ofthe updating parameters but also their variability (statistically or in terms of intervals) [50–54] from measured variability inoutputs such as natural frequencies and mode shapes.

7.4. Non-linear model updating

Hemez and Doebling [55] described a model updating programme using time domain data and a number of differenttestbeds covering a range of applications from non-linear vibrations to shock response with a single degree of freedom and athree dimensional multi-component assembly with non-linear material and contact interfaces. Several researchers use theharmonic balance method in non-linear model updating [56–58]. Meyer and Link [56] updated the linear and non-linearparameters of a beam with a cubic stiffness at one end. A mathematical model was developed in the frequency domain havingequivalent parameters identified using non-linear frequency response functions from stepped sine tests. Boswald and Link[57] assumed a model for bolted joints consisting of a bilinear stiffness and a quadratic non-linear damping function. Anapplication to the bolted flanges of aero-engine casings was carried out using a similar methodology [57]. Ahmadian and Jalali[58] developed a generic element formulation consisting of generic-stiffness and generic-damping matrices to model thenon-linear behaviour of bolted lap joints.

Acknowledgements

The authors wish to thank GARTEUR Ltd. for permission to reproduce material from Ref. [32] and QinetiQ Ltd. for the finiteelement model and test results from the Lynx helicopter.

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