Comparative study of model updating methods using simulated experimental data

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Comparative study of model updating methods using simulated experimental data

S.V. Modak, T.K. Kundra *, B.C. Nakra

Department of Mechanical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India Received 29 June 2001; accepted 9 January 2002

Abstract

The inverse eigensensitivity method and the response function method of analytical model updating have become relatively more popular among other methods and have been successfully applied to the practice of analytical model improvement. This paper gives a detailed comparison of these two approaches of model updating on the basis of computer simulated experimental data with the objective of studying the convergence of the two methods and the accuracy with which they predict the corrections required in a finite element model. The effect of the amount of experimental data used in the process of model improvement on the quality of an updated model is also studied. The test cases of complete, incomplete and noisy experimental data are considered. The updated models are compared on the basis of some error indices constructed to quantify error in the predicted natural frequencies, mode shapes and response functions.

Keywords: Model updating; Finite element model correction; Inverse eigensensitivity method; Response function method

1. Introduction

Accurate mathematical models of engineering structures are needed in order to predict their dynamic characteristics. Mathematical models can be derived analytically such as by the finite element method [1] or experimentally by modal testing [2]. A mathematical model derived analytically, at times, has been found to be inaccurate especially in the case of complex structures due to difficulties in the modelling of joints, boundary conditions and damping. The experimental approach to extract a model is faced with the problems due to limited number of measured coordinates and limited frequency range. While on the one hand a finite element based analytical model has the advantage of being complete and precise, the experimental data on the other hand are generally considered more accurate given the availability of reliable data acquisition and measuring equipment.

This has led to the development of technology of model updating that aims at reducing the inaccuracies present in an analytical model in the light of measured dynamic test data while allowing to simultaneously retaining a more detailed representation provided by a finite element model. Model updating thus can be viewed as an attempt to combine the best aspects of the two approaches.

A number of model updating methods have been proposed in recent years as shown in the surveys by Imregun and Visser [3] and Mottershead and Friswell [4]. Model updating methods can be broadly classified into direct methods, which are essentially non-iterative ones, and the iterative methods. A significant number of methods, which were among the first to emerge, be- longed to the direct category. Of such methods the one proposed by Baruch and Bar-Itzhack [5] assumes that the mass matrix is correct while the measured eigenvectors are updated by minimizing the weighted Eu- clidean norm of the difference between the measured and the analytical eigenvectors subjected to the orthogonal- ity constraints. The updated eigenvectors are then used

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Nomenclature

[K]

[M]

[Z]

A

U\

[a]

fn m

stiffness matrix mass matrix

dynamic stiffness matrix identity matrix

change in the corresponding quantity receptance FRF matrix

jth column of receptance FRF matrix rth eigenvalue

rth eigenvector frequency in rad/s natural frequency in Hz.

number of measured coordinates number of modes

nu nf

IP)

{u}

M

number of updating parameters number of frequency points vector of physical parameters

vector of fractional correction factors at some iteration

cumulative correction factor Subscripts

X experimental A analytical Superscripts

+ pseudo-inverse

to update the stiffness matrix. There have been other publications, like by Berman and Nagy [6] and Baruch [7], in which one of the three quantities, namely the measured modal data, the analytical mass matrix or the analytical stiffness matrix, is taken as a reference and the remaining quantities are updated. These methods though yielding updated matrices that reproduce measured modal data exactly, suffer from the drawbacks that the structural connectivity is generally not main- tained and the suggested corrections are not physically meaningful. Kabe [8] proposed to include an additional constraint to ensure connectivity while Lim [9] used submatrix-scaling factors that automatically guarantee structural connectivity. The error matrix method proposed by Sidhu and Ewins [10] is an another direct technique that aims at estimating the error in mass and stiffness matrices.

Iterative methods are based on minimizing an objective function that is generally a non-linear function of selected updating parameters. Quite often eigenvalues, eigenvectors or response data are used to construct an objective function. The use of eigendata sensitivity for analytical model updating in an iterative framework was first proposed by Collins et al. [11]. Chen and Garba [12] used matrix perturbation technique for re- computation of eigensolution and evaluation of eigendata sensitivities. The effect of including second order sensitivities was studied by Kim et al. [13]. Recently Lin et al. [14] proposed to employ both the analytical and the experimental modal data for evaluating sensitivity coefficients with the objective of improving convergence and widening the applicability of the method to cases where there is a higher error magnitude. There have been attempts to use directly the measured frequency response function (FRF) data for identifying the sys- tem matrices as done in Fritzen [15]. A method named

response function method (RFM), has been devel- oped by Lin and Ewins [16] that uses measured FRF data to update an analytical model. Imregun et al.

[17,18] conducted several studies using simulated and experimental data to gauge the effectiveness of this technique.

Over the recent years, among all the updating methods, the eigendata based and the FRF data based iterative methods have acquired increasing attention of the researchers due to the flexibility these methods offer in the choice of updating parameters. Imregun et al. [19]

have compared these two methods for the case of updating of an L-plate structure. The study is based on the measured test data and updated models are compared with the emphasis on closeness between predicted and measured response functions. From the foregoing study and other literature the present authors believe that these methods have a potential of wide usage. This warrants a detailed comparison to gain an insight in these methods with the intention of exploring if further improvements are possible. This work is directed to- wards a detailed comparison of these two approaches of model updating on the basis of computer simulated experimental data to update a finite element model of a beam structure. The objective of the comparison is to study the convergence of the two methods and the accuracy with which they predict the corrections required in a finite element model. The effect of the quantity of experimental data used in the process of model improvement on the quality of an updated model is also studied. The test cases of complete, incomplete and noisy experimental data are considered. The updated models are compared on the basis of some error indices constructed to quantify error in the predicted natural frequencies, mode shapes and response functions.

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2. Basic theory

A brief description of the two methods considered for the study is given in this section. Basic formulation of the methods is extended for fractional correction factors for the physical variables taken as the updating parameters.

2.1. Response function method

This method, proposed by Lin and Ewins [16], uses measured F R F data directly without requiring any modal extraction to be performed. A relationship in- volving measured and analytical F R F data and the dynamic stiffness correction matrix is being derived that essentially forms the basis for updating calculations. An alternative derivation giving the same final relationship is presented here.

The following identities relating the dynamic stiffness matrix [Z] and the receptance F R F matrix [a] can be written for the analytical model and for the actual structure respectively as,

[ZA][«A] = [I], (1)

(2) where subscripts A and X denote analytical or FE model and experimental model respectively. Expressing [ZX] in Eq. (2) as [ZA] + [AZ], where [DZ] represents the correction in the dynamic stiffness matrix of the analytical model, and then subtracting Eq. (1) from it, the following matrix equation is obtained,

[AZ][ax] — [ZA]([aA] — [ax]). (3) Pre-multiplying the above equation by [aA] and then making use of Eq. (1) gives,

[aA][AZ][ax] = [ aA] - [ ax] . (4) If only thejth column faXgj of the measured FRF matrix [aX], is available then Eq. (4) is reduced to, [aA][AZ]{ax}. — {aA}. — {ax} , (5) which is the basic relationship of RFM. A physical variable based updating parameter formulation is used in this paper. Let {p} — {pi,P2, • ; p nu g be a vector of physical variables associated with individual or group of finite elements with nu denoting the number of param- eters to be updated. Linearizing [DZ] with respect to fpg gives,

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the equations of motion for the free vibration are given by,

(7) For a harmonic displacement vector {x} — X0g sin xt the above equation can be simplified as,

([KA]-w2[MA]){X0} = (8)

The coefficient matrix ([KA] — co2[MA]) in the above equation is called dynamic stiffness matrix [Z]. If [DKA] and [DMA] represent the changes in the stiffness and mass matrices then the corresponding change in the dynamic stiffness matrix, [DZ], would be [AKA] — co2[AMA]. Substituting for [DZ] in Eq. (6), dividing and multiplying by pi and then writing ui in place of Dpi=pi, Eq. (6) becomes,

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Thus, {u} — u1;u2; • • ; u nu g is the unknown vector of fractional correction factors to be determined during updating. Eq. (5), after making substitution for [DZ]

from Eq. (9), can be written at various frequency points chosen within the frequency range of interest. The re- sulting equations can be expressed in the following matrix form,

[C(W)](»xnf)x(nu){M}nuxl = {Aa(W)}(»xnf)xl. ð10Þ

For an undamped FE model, having its stiffness and mass matrices represented by [KA] and [MA] respectively,

where n represents the number of degrees of freedom at which measurements are available, while nf represents the number of frequency points used in setting the equations. The above matrix equation is solved for fug using a routine available in MATLAB [21] for finding the pseudo-inverse of a matrix [C] in the present case, the pseudo-inverse, calculated by the routine using sin- gular value decomposition of a matrix, is related to the least-squares problem. This is because the value of fug that minimizes ||[C]{M} — {Aa}||2 can be obtained as {u} — [C]+{Aa} [22]. The superscript þ denotes pseudo- inverse. The fug so found is used to update vector of physical variables {p} and then the updated version of an analytical finite element model is built using this new set of physical variables. This process is repeated in an iterative way until convergence is obtained. It should be noted that the fugj found in jth iteration represents the vector of fractional correction factors for the current {p}, i.e. fpgj, and does not represent a cumulative cor- rection with respect to the {p} existing before updating, the fpg0. At the end of jth iteration the ith cumulative fractional correction factor is given by,

ð11Þ

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2.2. Inverse eigensensitivity method

This method uses modal data, namely the natural frequencies, the mode shapes and the damping ratios, which are obtained in practice by modal analysis of measured FRFs. The updating parameters corresponding to an analytical model are corrected to bring the analytical modal data closer to the experimentally derived one. Most often updating equations are based on a linear approximation of the modal data that is generally a non-linear function of updating parameters. For rth eigenvalue, X (square of the rth natural frequency in rad/s), and rth eigenvector, fUgr (the mode shape), lin- earisation gives,

ð12Þ

ð13Þ

where the eigenvalue and the eigenvector sensitivities, represented by the derivatives, can be calculated from the following relationships derived by Fox and Kapoor [20],

d[K]- l \ ð14Þ

I f /

ð15Þ Using Eqs. (14) and (15), Eqs. (12) and (13) can be written for the chosen number m of modes. These equations together, after dividing and multiplying by pⁱ and then writing uⁱ in place of Dpi=pi, can be written in the following matrix form,

xl = {Ae}(»+ ð16Þ

where n represents the number of degrees of freedom at which measurements are available. The above matrix equation is solved in the same way as Eq. (10) as ex- plained in Section 2.1. The {u} so found is used to up- date the vector of physical variables {p} and then the updated version of analytical finite element model is built using this new set of physical variables. Due to the use of an approximate relationship, as in (12) and (13), the updating is required to be carried out in an iterative way until convergence is obtained. The cumulative or

final value of correction factors is evaluated using Eq. (11).

3. Description of the case study

The case study considered here for comparison is that of updating of an undamped finite element model of a simple but representative fixed-fixed beam structure, as shown in Fig. 1, using simulated experimental data. The dimensions of the beam are 9100 x 50 x 5 mm. The modulus of elasticity and density are assumed to be 2:0 x 1011 N/m2 and 7800 kg/m3 respectively. The beam is modeled using 32 noded beam elements with nodes at the ends fixed. The displacements in x and y directions and the rotation about the z-axis are taken as three degrees of freedom at each of the nodes.

The simulated modal and FRF data, which are treated as experimental data, are obtained by generating a finite element model by introducing certain known discrepancies in the thickness of all the finite elements with respect to an analytical model. The details of these introduced discrepancies are given in Table 1. The frequency range from 0 to 1 kHz covering seven modes is taken as the measured frequency range. The procedure for generating a finite element model and for updating it has been implemented in MATLAB. In the present study Eqs. (10) and (16) are solved with upper and lower bounds of þ0.5 and —0.5 respectively on {u} to avoid excessive variations in the unknown parameters in an iteration. It is to be noted that the imposed bounds do not put any restriction on the final cumulative value of the unknown parameters (as calculated from Eq. (11)) during the iterations. They simply limit the maximum/

minimum value that any unknown parameter can ac- quire in a particular iteration. The convergence criterion is based on the norm of the difference between two successive vectors of cumulative fractional correction factors. A value of 0.001 is used for the present study.

The modal data based and the FRF based error indices are also monitored so that the iterations could be stopped when no further improvement is seen from one iteration to the next.

Three test cases using simulated data of increasing complexity have been considered. In the first case, it is assumed that the measurements are available at all the degrees of freedom of the finite element model. This case

9100 All dimensions in mm

Fig. 1. Beam structure.

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Table 1

Discrepancies between Element no.

Percentage deviation

the finite element and 3 in thickness þ20

the experimental 5

þ40

model 11 þ25

16 þ40

25 þ30

29 þ30

All other elements + 10

has been referred to as the case of complete data. Thus, in this case one complete column of the F R F matrix and all the eigenvalues and eigenvectors falling in the measurement frequency range will be known.

In practice, it is not realistic to assume that all the degrees of freedom specified in a FE model have been measured, either due to physical inaccessibility or due to the difficulties faced in the measurement like that for rotational degrees of freedom. The second case considered, referred to as the case of incomplete data, therefore is the one where it is assumed that the measurements are not available at all the degrees of freedom of the finite element model. Two levels of incompleteness are considered by assuming that measurements are available only in the y-direction at all the 29 nodes and then at only 15 nodes (the node numbers being 3, 5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27 and 29). These have been referred to as cases of 33.33% and 17.24% incomplete data respectively, the percentages representing the pro- portion of the total degrees of freedom of the FE model that are taken as measured. In the case of RFM the coordinate incompleteness has been dealt by using analytically generated FRFs. This is done by replacing the responses of unmeasured coordinates by their analytical counterparts. In the inverse eigensensitivity method (IESM) the coordinate incompleteness does not require any treatment as updating equations can be framed using only the measured coordinates.

In practical measurements the measured FRFs will be contaminated by measurement noise and consequently the extracted modal parameters will also be affected. In the third case to compare the performance of the chosen methods under such conditions, the FRFs, the natural frequencies and the mode shapes corresponding to the simulated experimental data are polluted with random errors. While the incomplete FRFs and the mode shapes are polluted by 2% random noise, the natural frequencies are polluted by 0.2% noise under the assumption that the measurement error in natural frequencies can be taken around 1/10th of that in mode shapes.

For each test case the effect of the quantity of the measured data used in updating on the quality of an updated model is also studied in terms of the effect on the error levels after updating and on the convergence and stability of iterations of the two methods. This is done by varying the updating frequency range to cover 1-5 and then 6 modes. For all the cases reported here the thickness of all the individual finite elements were taken as the parameters to be updated giving 30 unknowns.

To assess the progress of iterations and to compare different updated models certain model quality indices have been constructed. Percentage average error in natural frequencies (AENF), percentage average error in mode shape (AEMS) and percentage average error in FRFs (AEFRF) are calculated using following expres- sions:

AENF = — V a b s m

tT

ffA-fx\

v fX i ;

AEMS = 100 m x n _{j=\ i=\}

ð17Þ

ð18Þ

AEFRF = 100

nf x n *-, abs ^]^a^)x

ð19Þ

where fA and fX are the natural frequencies, f/gA and f/gX a r e the mode shapes while [a]A and [a]x are the receptance F R F matrices corresponding to the analytical and the experimental model respectively.

Percentage average error in fractional correction factors (AECF) is calculated as error in the predicted correction factors as a percentage of the known exact correction factors,

T absðu^EXACT - UPREDICTEDÞ^I

1UU- ð20Þ

4. Results and discussion

For the case of complete data updating iterations by both the RFM and the IESM converged in about 4-5 iterations and predicted the unknown fractional correction factors to the elemental thickness exactly. This is obtained irrespective of the number of modes covered by the updating frequency range. Fig. 2 shows a comparison of the convergence of cumulative fractional correction factors, as evaluated by Eq. (11), plotted against iteration number when the updating range covers four modes. Due to the completeness of simulated measured data, the convergence is quite rapid and stable, that is,

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Table 2

RFM

2 3 Iteration Number

IESM

2 3 Iteration Number

Fig. 2. Convergence of fractional correction factors for the case of complete data.

No. of modes inside updating frequency

1 2 3 4 5 6

Error calculated over entire measurement frequency range AENF

Before updating 10.74 10.74 10.74 10.74 10.74 10.74

After updating RFM 0 0 0 0 0 0

IESM 0 0 0 0 0 0

AEMS Before updating 8.6 8.6 8.6 8.6 8.6 8.6

After updating RFM 0 0 0 0 0 0

IESM 0 0 0 0 0 0

AEFRF Before updating 1350.80 1350.80 1350.80 1350.80 1350.80 1350.80

After RFM 0 0 0 0 0 0

updating IESM 0.07 0 0 0 0 0

without any excessive or oscillatory variation during the iterations.

For the case of 33.33% incomplete experimental data the three error indices defined in the last section calculated before and after updating by both the methods over entire measurement range are shown in Table 2. Both methods have eliminated the error com- pletely in all the cases of modes covered inside the updating range. It should be noted that the error indicated in this and other tables is up to two digits after the decimal place and anything less than this is being indicated as zero. The final values of the fractional correction factors obtained by the two methods when four modes were included inside the updating range are shown in Fig. 3. It is observed that the discrepancies have been correctly predicted by both methods.

With an increase in the level of incompleteness due to the reduction in number of measured coordinates as in

the case of 17.24% incomplete experimental data there is some increase in the error levels after updating as is seen from Table 3. This is particularly experienced for the case of IESM, while the error levels realized using the RFM seem to have been little affected. Symbol V is shown to indicate that the corresponding case was not solved since an over-determined set of equations could not be formed. When only two modes were included convergence is still obtained using the RFM but not with the IESM (letter 'N' has been put in the table to indicate case of no convergence). This probably points, for the IESM, to the insufficient over-determinacy of equations. This puts a constraint on the number of updating parameters in the case of IESM. A prior localization of error regions before updating may then be probably required to keep the number of updating parameters small. The presence of contribution of the out of range modes seems to be helping the convergence in the case of RFM even when the updat-

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RFM

•§ 0.4

ro

rection F

8 0.2

Fractional CD

1

i

1

1 1

1

1 1

I I I I I I I I I I H M I M I I M M M I 10 15 20 25 30 Correction Factor Number

IESM

•§ 0.4

ro

rection I CD

8 0.2

-ractional

1 |

1

•

III

1

1 5 10 15 20 25 30 Correction Factor Number

Fig. 3. Comparison of final values of fractional correction factors for the case of 33.33% incomplete data.

Table 3

Error indices for the case of 17.24% incomplete experimental data No. of modes

inside updating frequency range

1 2 3 4 5 6

Error calculated over AENF

Before updating 10.74 10.74 10.74 10.74 10.74 10.74

the entire

After updating RFM 0.04 0 0 0 0 0

IESM

* N 0.04 0.02 0 0

measurement frequency AEMS

Before updating 8.33 8.33 8.33 8.33 8.33 8.33

After RFM 0.41 0 0 0 0 0

range

updating IESM

* N 0.42 0.41 0.32 0.23

AEFRF Before updating 1226.26 1226.26 1226.26 1226.26 1226.26 1226.26

After RFM 3.8 0 0.07 0 0 0

updating IESM

* N 2.61 3.47 2.57 2 22

Table 4

Error indices for the case of 33.33% incomplete experimental data with 2% noise No. of modes

inside updating frequency range

3 4 5 6

Error calculated over AENF

Before updating 10.74 10.74 10.74 10.74

the entire

After updating RFM 0.26 0.09 0.07 0.05

IESM N 0.29 0.05 0.02

measurement AEMS Before updating 8.6 8.6 8.6 8.6

frequency

After

g RFM

3.59 2.31 1.59 1.1

range

updating IESM N

1.67 1.40 1.10

AEFRF Before updating 1352.03 1352.03 1352.03 1352.03

After RFM 40.40 19.54 10.69 12.49

updating IESM

N 37.59 12.56 3.82

ing frequency range is compressed to cover just two modes.

Now considering the case of 33.33% incomplete data with 2% noise, it is seen that the error levels as shown in Table 4 obtained by the two methods after updating are

not very different from each other. The error levels have generally gone down in case of both methods as updating range is extended to encompass a greater number of modes. The final values of the fractional correction factors obtained by the two methods when four modes

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0.4

0.3

o 0.2 o

ro o 0.1

RFM

10 15 20 25 30 Correction Factor Number

IESM

5 10 15 20 25 30 Correction Factor Number

Fig. 4. Comparison of final values of fractional correction factors for the case of 33.33% incomplete data with 2% noise.

Table 5

Percentage error in the predicted fractional correction factors for the case of 33.33% incomplete experimental data with 2%

noise

No. of modes Percentage error in fractional correction factors

3 4 5 6

RFM 77.6 54.6 41.7 41.4

IESM N 27.3 24.1 12.8

were included inside updating range are shown in Fig. 4.

It is observed that in the presence of noise the discrepancies predicted by the IESM are found closer to the correct values as compared to those predicted by the RFM. This is also evident from Table 5 that shows the percentage error in the predicted fractional correction factors after updating for the cases of different number of modes covered inside the updating range.

The IESM gave no convergence for the case when just three modes were used in updating. For the present case of noisy data, it is also experienced that the RFM required several runs to be made with different sets of frequency points in terms of their number and location to find an acceptable solution.

Coming to the final case of 17.24% incomplete data with 2% noise, it is seen that the convergence process becomes unstable due to reduction in the number of measured coordinates combined with the presence of noise. This is evident from Fig. 5 that shows a comparison of the convergence of cumulative fractional

correction factors plotted against iteration number for cases when updating range spans to cover 3-5 and then 6 modes. It is also seen that the stability improves with inclusion of greater number of modes in the updating range. Again the error levels, as shown in Table 6, are not much different for the two methods though the RFM seems to be faring better in cases where updating range is limited to include fewer number of modes.

Error levels, though not reported here, were also calculated, using Eqs. (17)—(19), over the portion of measurement range that falls outside the updating frequency range. It was observed that lesser the extent of updating range more is the error left outside this range after updating. Fig. 6 shows a comparison of overlay of analytical and simulated experimental FRF after updating for the case of 17.24% incomplete data with 2%

noise when four modes are included in the updating range. It is seen that the analytical and the simulated experimental FRF have a poorer fit outside the updating range.

5. Conclusion

On the basis of above results the following conclu- sions are drawn.

Both methods predicted the errors in the finite element model in the case of complete and incomplete data with a good accuracy. In the presence of noise, the convergence for both methods is adversely affected making the iterations relatively unstable.

In the present study, the RFM seems to have worked better than the IESM for the case of incomplete exper-

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No. of

Modes RFM

0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1

•0.20 5 10 15 20 25 30 35 Iteration Number

5 10 15 20 25 30 35 40 Iteration Number

o.4

•g 0.3

o 0.2 0.1

5 10 15 20 Iteration Number

25 30

IESM

5 10 15 20 25 Iteration Number

Fig. 5. Convergence of fractional correction factors for the case of 17.24% incomplete experimental data with 2% noise when 3-6 number of modes are included inside updating frequency range.

imental data. But in the presence of noise, the IESM is found to be predicting the discrepancies more accurately specially when the updating range covers a greater number of modes. With the reduction in the number of modes included inside the updating range the convergence could still be obtained in the case of RFM. This seems to be because of use of the FRFs that account for the effect of out of range modes and due to the

availability of response data at many frequency points allowing setting equations that are sufficiently over- determined.

The selection of frequency points in terms of their location and number in the case of RFM seems to be very important for its convergence and this issue is yet to be resolved. The constraint on the number of updating parameters seems to be relatively more stringent

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Table 6

Error indices for the case of 17.24% incomplete experimental data with 2% noise No. of modes inside

updating frequency range

3 4 5 6

Error calculated over the entire measurement AENF

Before updating

10.74 10.74 10.74 10.74

IESM N 1.82 0.26 0.06

AEMS Before updating 8.31 8.31 8.31 8.31

frequency range

IESM N 3.62 1.34 0.99

AEFRF Before updating 1226.94 1226.94 1226.94 1226.94

After RFM 91.38 29 29 16.19 7.36

updating IESM N

138.0 27.32 12.03

RFM

200 400 600 800 1000 Frequency in Hz.

IESM

200 400 600 800 1000 Frequency in Hz.

Fig. 6. Comparison of overlay of analytical and experimental F R F (14y14y) after updating for the case of 17.24% incomplete data with 2% noise (' * *': analytical; '—': experimental).

in the case of IESM and a priori error localization may be necessary especially when amount of measured data is small.

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