AN INVESTIGATION INTO THE EFFECTS OF FREQUENCY RESPONSE FUNCTION ESTIMATORS ON MODEL UPDATING

Mechanical Systems and Signal Processing (1999) 13(2), 315—334 Article No. mssp.1998.1210 Available on line at http://www.idealibrary.com on

AN INVESTIGATION INTO THE EFFECTS OF FREQUENCY RESPONSE FUNCTION ESTIMATORS ON MODEL UPDATING M. J. RATCLIFFE

AND

N. A. J. LIEVEN

Department of Aerospace Engineering, University of Bristol, Bristol BS8 1TR, U.K.

Model updating is a very active research field, in which significant effort has been invested in recent years. Model updating methodologies are invariably successful when used on noise-free simulated data, but tend to be unpredictable when presented with real experimental data that are—unavoidably—corrupted with uncorrelated noise content. In the development and validation of model-updating strategies, a random zero-mean Gaussian variable is added to simulated test data to tax the updating routines more fully. This paper proposes a more sophisticated model for experimental measurement noise, and this is used in conjunction with several different frequency response function estimators, from the classical H and H to more refined estimators that purport to be unbiased. Finite-element   model case studies, in conjunction with a genuine experimental test, suggest that the proposed noise model is a more realistic representation of experimental noise phenomena. The choice of estimator is shown to have a significant influence on the viability of the FRF sensitivity method. These test cases find that the use of the H estimator for model updating  purposes is contraindicated, and that there is no advantage to be gained by using the sophisticated estimators over the classical H estimator.   1999 Academic Press

1. INTRODUCTION

Finite-element (FE) model updating has been a very active research field in recent years. An abundance of techniques has been proposed in order to correct errors in the analytical model by using experimental vibration test data or experimental model parameters. Most updating strategies work successfully when the (simulated) data used are noise-free but addition of random noise to the simulated data causes many of these routines to behave inconsistently. Arguably the most successful updating methodology has been the frequency response function (FRF) sensitivity method [1]. Recently, subtle improvements have been made to the algorithm [2] that have enabled successful updating performances to be achieved when simulated data—with added Gaussian noise—have been used. Despite the success of this method, meaningful updates of complicated real structures remain difficult to attain. This suggests that the simple zero-mean Gaussian noise model that is used to corrupt simulated data is not representative enough of real experimental noise. Real noise will be neither Gaussian nor have a zero mean, and the effects of this have not yet been investigated. This paper describes an enhanced noise model that amalgamates existing work from two different authors [3, 4]. The model characterises three possible sources of error in the measurement process, and represents the phenomenon of force drop-off at resonance that so hinders mechanical mobility measurement. 0888—3270/99/020315#20 $30.00/0

 1999 Academic Press

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FRF estimation is a topic that has also been researched enthusiastically over the past few years, with several new estimators having been proposed [5—9]. Estimators produce biased approximations to the genuine FRFs, and this is precisely the sort of discrepancy between genuine experiments and simulations that has not yet been studied. The noise model developed in this paper is used to investigate the effects of FRF estimators on the behaviour of the FRF sensitivity method. The structure of the present paper is as follows: initially, a review of FRF estimators is presented. The new noise model is then described in detail, followed by an exposition of the basics of updating using FRF sensitivity. The effects of different estimators are then investigated using simulated experimental data, before progressing to a genuine experimental test case.

2. FREQUENCY RESPONSE FUNCTION ESTIMATORS

In a modal testing utopia, all that would be needed to determine experimental FRFs would be to apply an input force and measure the response to the applied force and then take the ratio between these two spectra. However, all model test data are subject to the following limitations: E a limited analysis resolution that gives rise to a bias error. This is described in depth in [10]; and E noise corruption, due to a number of sources, such as non-linear effects, extraneous structural noise or electrical noise in the instrumentation. It is generally not acceptable merely to average a number, n, ratios of force and response, as shown in equation (1), because of the effect of noise: 1 ½(u) H(u)" . n X(u)

(1)

If the signal-to-noise (S/N) ratio at the input is small, then the denominator of equation (1) can become very small, resulting in unstable—or even undefined—FRF values. To overcome these problems, two FRF estimators using the cross- and auto-spectra were developed [11, 7]: G (u) G (u) H (u)" 67 , H (u)" 77 . (2)   G (u) G (u) 76 66 These yield identical estimates if there is no noise on the measurements, but under noisy conditions they produce differing approximations to the true FRF. H assumes all uncor related noise to be present on the output, and will equal the true FRF if this is the case. It has been widely reported that as the impedance of a structure decreases at resonance, then the input force will decrease markedly [4]; this is discussed in Section 4. As a result of this the input measurement can become dominated by noise, and thus H produces a relatively  poor estimate of H at resonance. Consideration of Fig. 1 provides G H(u) 34 H (u)" " . (3)  G #G 1#G /G 33 ++ ++ 33 This is clearly an underestimate of the true FRF. Mitchell [7] pointed out that ‘One is looking for an answer to the question ‘‘How bad is this resonance’’?’ and that consequently the low estimates provided by H are particularly worrisome. Therefore, H was designed   deliberately to overestimate the resonance.

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Figure 1. Simple system showing uncorrelated content.

H assumes all the uncorrelated content to be present on the input. Therefore, it gives rise  to almost unbiased estimates of H at resonance, since the output at resonance is high, and dominates any noise content. Conversely, when the output is small—for example at antiresonance—the H estimator is shown to produce severe overestimates of H, and  can sometimes suggest false modes due to peaks at antiresonance. The equivalent of equation (3) is






G #G G ,,"H(u) 1# ,, . H (u)" 44 (4)  G G 43 44 Mitchell went further and suggested two further possibilities for measurement. Firstly, he suggested that if the complete data analysis system was fully operational, but operated without input to the structure, then estimates of G and G could be obtained. Equation ++ ,, (4) can then be rearranged to






G !G 77 ,, . (5) G 77 This requires twice as much data collection, and significant post-processing. Secondly, Mitchell proposed a third estimator, H R. This is defined as the arithmetic average of  H and H . The use of the arithmetic average means that it is commonly dominated by the   larger of the two estimates—remember that FRFs are normally viewed on a log scale—and this will not be considered further. Vold et al. [8, 12] described another FRF estimator, H , that attempts to minimise the 4 error in a tensorial way and involves minimising the trace of the error vector. For the case of a single input FRF, the detailed analysis in [8, 12] showed that H reduces to the geometric 4 mean of H and H , and is therefore similar in concept to H , but does not suffer from the    scaling problem. In addition, it is trivial to program H(u)"H (u) 

H "(H H . (6) 4   A more general model that assumes corruption to be present on both input and output signals was postulated by Wicks and Vold [9] in 1986. This estimator, H , requires that the 1 magnitudes of the input and output error terms be equal, and this is ensured by applying a scaling factor, s, to the input. This enables an error expression to be formulated, which is - Mitchell’s notation was confused by Hong and Yun [6], who termed Fabunmi and Tasker’s H [5] as H . In   addition, Hong and Yun named their own estimator H . The notation H is unique to the present paper, but  7 avoids ambiguity.

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then minimised to produce G !sG #((sG !G )#4sG G 66 66 77 76 67 . H " 77 1 2sG 76

(7)

The analysis also yields a coherence function, which was shown [9] to be always greater than the ordinary coherence values, sG c"1! ++ . 1 G 77

(8)

In the absence of any information regarding the noise ratio, s, it is assumed constant at all frequencies. As discussed above this is not optimal, since uncorrelated content from different sources dominates at different frequencies. For this reason, the H estimator is 1 rarely used, and will not be considered further in this paper. In addition, the fact that the formula for H cannot easily be expressed in terms of H , H and c means that significant 1   modification of the data-acquisition equipment and subsequent signal processing may be necessary to output the two autospectra and the cross-spectrum. Another estimator that seeks to minimise errors on both input and output signals is H  [5] which starts by jointly minimising errors in H and H . In common with H , it uses   1 a normalising factor to scale the magnitudes of input and output error terms. Unlike the H estimator, though, reference [5] provided a method for a sensible choice of weighting 1 function. H is given by  H (u)[(c (u)F(u)/"H (u)")#1] $6  H (u)"  . (9)  (F(u)/"H (u)")#1  This estimator is weighted so as to switch between H at antiresonance and H at   resonance, provided F(u) is chosen appropriately. Promising results were suggested with a constant F(u), and the function used was S

 "G67 (u)" du . S "G66 (u)" du

F(u)" S S

(10)



Hong and Yun [6] claimed that the desired properties of the normalising function are sometimes not achieved, and that the estimator therefore produces undesirable results over some frequency ranges. They proposed another estimator, H , that is also 7 a weighted average of H and H , although the method for determining the weighting   functions is significantly different from the approach used by Fabunmi and Tasker. The estimator is H (u)"(1!¼(u))H (u)#¼(u)H (u) 7   and the weighting function is the exponential function





¼(u)"exp !



u/u !1   . a

(11)

(12)

The spread, a, of the weighting function is determined by minimising the resolution bias error. Schmidt’s work [10] enabled approximate calculations of bias error for H and 

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319

Figure 2. Parameter of exponential weighting function for H versus damping ratio—after Hong and Yun [6]. 7

H for rectangular and Hanning windows to be calculated and these expressions are  substituted into equation (14) to determine the bias error of H : 7 e["H (u)"]"(1!¼(u))e["H (u)"]#¼(u)e["H (u)"]. A  

(13)

The spread of the weighting function is then determined by integrating the square of the bias error [equation (14)] over the significant frequency range, which is chosen from zero to twice the resonant frequency. The optimum weighting parameter is found to be an almost linear function of the damping ratio, as shown in Fig. 2, which implies an obligation to use an iterative process, since the weighting function cannot be determined until the damping ratios have been estimated. The H estimator can be used to enable accurate estimates of  the damping ratios to be extracted, and the use of H is thereafter straightforward. 7 Goyder [13] suggested a three-channel estimator called H that makes use of a direct ! measurement of the source signal from the signal generator, S, and is thus suitable only for an excitation involving a model shaker: G H " 71 . ! G 61

(14)

This estimator is analysed in some considerable depth in [3, 14] where it is shown to be unbiased as the number of averages tends to infinity. Obviously, this desirable property comes at the inconvenience of having to measure the extra channel.

3. THE CHOICE OF NOISE MODEL

The most common method for modelling the corruption present on experimental data is to add proportional Gaussian random noise to FRFs where each datum is factored as a(u)"a(u)(1#a ) s)

(15)

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where a is a normally distributed variable, and s is a user-specified standard deviation. It is unlikely that genuine experimental noise will be either Gaussian or proportional. The choice of noise model used in this paper was guided by observation of the trends typical to ordinary coherence plots. Coherence is calculated from the two autospectra and the cross-spectrum of the input and output signals as follows: "G " c " 67 . 67 G G 66 77 There are several possible reasons for poor coherence values:

(16)

E uncorrelated noise on the output or input channels; E a non-linear relationship between the output and input; and E FFT leakage problems. Non-linearities and leakage effects are a potentially important source of poor coherence, and in particular, leakage tends to cause coherence to drop off at resonance. However, modelling these two sources of error would require a time-domain analysis; in order to determine errors due to FRF estimators in isolation, these effects were not modelled. The reader is referred to the work of To and Ewins [15] for a description of the effects of leakage on various FRF estimators. Uncorrelated noise causes drops in coherence, and these are most often noticeable at resonance and antiresonance; at resonance the force drops away and is therefore dominated by ambient noise. Similarly, at antiresonance, it is the response signal that is small, and that also tends to be dominated by ambient noise. The standard Gaussian proportional noise model is not adequate for an investigation into the effects of estimators, because it does not reproduce the expected trends in coherence. The problem with the Gaussian proportional model is its very proportionality. Proportional noise will not dominate the response at antiresonance as genuine noise would since its level is set to be proportional to a small number—i.e. the response at antiresonance. For this reason a more sophisticated noise model was adopted in the hope that it would be rather more representative than the proportional model, and thus exhibit realistic trends in ordinary coherence functions. Cobb [3] simulated the FRF measurement process as shown in Fig. 3. This figure is explained as follows: E the random input signal, S(u), is applied to the structure through an amplifier and shaker. The amplifier, shaker and attachment hardware are modelled as a single system gain FRF, C(u); E this input is then corrupted by noise, or other uncorrelated signals, and these are modelled by K(u), which is added to the transduced signal, S(u)C(u), to form the ‘true’ input force, º(u); E the input force is measured by a force transducer, and the errors prevalent in force measurement are modelled by adding M(u) to the ‘true’ force, to generate the measured force, X(u); E the ‘true’ system response, »(u), is then calculated from »(u)"º(u)H(u), and finally this is corrupted by N(u) to simulate the ‘true’ output, ½(u). Cobb’s model enables the FRF estimators discussed in Section 2 to be calculated as well as exhibiting the correct trends in ordinary coherence away from resonance. However, at resonance, the coherence drops away because of force drop-out, and this is not modelled in Cobb’s work.

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Figure 3. Model for noise sources in an FRF measurement—after Cobb [3].

4. MODELLING FORCE DROP-OFF AT RESONANCE

A significant amount of research has been undertaken in recent years regarding force drop-off at resonance and attempts to ensure a uniform input spectrum [16—19]. Rao [16] reported seemingly disparate views regarding the best method of reducing drop off. Olsen [17] reported that drop-off is reduced by reducing the mass of the shaker armature, whereas the classical picture, documented by Unholtz [18], is that increasing the armature mass is the way towards eliminating drop-off. Rao reformulated the problem and concluded that a grounded structure needs a light armature to minimise drop-off, whereas, for a free—free structure, a heavy armature is optimal. The work directed at minimising the force drop-off is, in a sense, not relevant to the present work where the objective is to model the noise manifest on FRFs as representatively as possible. Tomlinson’s work [4, 19] enabled a detailed force spectrum to be derived, dependent on the properties of the structure being measured. The following section summarises a derivation described in [4]. The interested reader is referred to that work for the details of the analysis. If the shaker is modelled as an sdof system, then the resultant force applied by the shaker can be modelled as F "F eHSR>(!(m x¨ #c xR #k x).  C C C C The force applied by the shaker is given by




(17)



k l jk k $ F" ! k !m u#jc u# $ x. C R#ju¸ C C C R#ju¸

(18)

In order to quantify the interaction between the structures characteristics and the exciter, the receptance at a point q is considered for the rth mode: x L

P P O O O" . k !m u#jc u F P P  P P This can be combined with equation (18) and rearranged to yield

where

A F"  1#B L P P /C P O O k l $ A" (R#iu¸)

(19)

(20)

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Figure 4. Theoretical point FRFs, forcing function and coherence.

B"k !m u#jc u C C C C"k !m u#jc u. P P P This enables a detailed force spectrum to be synthesised provided estimates for the shaker parameters are available. The force spectrum is incorporated in Cobb’s noise model as C(u). A synthesised force spectrum is shown in Fig. 4 together with the point FRF from the structure under test—which is described in Section 6. Figure 5 shows the equivalent point FRF and input spectrum from a preliminary experimental test. The experimental and theoretical force spectra can be seen to be qualitatively similar, and most significantly both of the coherence plots exhibit the same trends. This suggests that the noise model developed in this section is more representative of the true situation than the simple Gaussian model.

5. UPDATING USING FRF SENSITIVITY

The FRF sensitivity method [1] is considered as one of the most successful updating methods available, and so is used as the study vehicle for this work. It is outlined briefly in this section.

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Figure 5. Experimental point FRF, force spectrum and coherence.

5.1. THE FRF SENSITIVITY METHOD If two single-input—multiple-output systems—one experimental and one representing the FE model—are subjected to a single input force, then it can be shown that [a (u)][DZ]+a (u),"+Da(u),  6

(21)

where +Da(u),"+a (u),!+a (u), and [DZ] is the error between analytical and experi 6 mental dynamic stiffness matrices. In order to proceed to a solution, a form for the errors must now be assumed, and this is done by selecting a set of design parameters to vary in order to rectify the differences between the experimental and FE model responses. The design parameters, which represent fractional changes in their original values, are known as p-values. The updated system is a function of the p-values, and can be written as a Taylor series expansion about the original FE model as *[Z] [Z]"[Z ]#[DZ]"[Z ]# p #O(p ). G   *p G G G

(22)

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Neglecting the higher-order terms and substituting back into equation (21) yields [a (u)] 



*[Z] *[Z] +a (u), 2 +a (u), 6 6 *p *o ,N 

which can be written





p  $ "+Da,. p ,N

[S] +p,"+Da,

(23)

(24)

where S is the sensitivity matrix which can be interpreted as the changes in response for a unit change in an updating parameter,





*[Z] *[Z] +a (u),2 +a (u), . (25) 6 6 *p *p  ,N Equation (24) provides N linear equation for N unknowns at a single frequency. Given N N different frequency points, then N sets of equations can be stacked to form N ;N D D D equation for N unknowns. This overdetermined problem is then solved by singular-value N decomposition [20]. It is convenient to use groups of the elemental matrices, or macro-elements, as updating parameters, because this maintains the connectivity of the original model. In addition, noisy numerical differentiation to calculate the derivatives in equation (24) is circumvented, as the derivatives are given by [S(u)]"[a (u)] 

*[Z] "!u[M ] , Q G *m G G

*[Z] "[K ] . Q G *k G G

(26)

5.2. FREQUENCY POINT SELECTION In addition to the computational saving afforded by excluding frequency points, it is well-known that the inclusion of certain frequency points can be detrimental to the solution of the updating problem. In particular, some frequency points may contaminate the solution more than others because of the following reasons: E the response at resonance is inaccurate if the H FRF estimator is used, and inaccurate at  antiresonance using the H estimator;  E some frequencies may be poorly excited; and E the validity of FRF expansion techniques is frequency-dependent. The following method of frequency point selection is suggested by Waters [2]. A preliminary iteration of the FRF sensitivity method is performed, and the residuals are calculated. Remembering that the updating procedure is seeking a solution to [S]+p,"+Da, then the residuals are given by #+r,#"#[S]+p,!+Da,#. (27)   The residual contributions are then sorted in order of magnitude, and this enables the analyst to make a refined selection of the number of frequency points to discard. Typically, a small number of frequencies contribute a very large proportion of the overall residual. The residual criterion is applied throughout the work presented in this paper.

6. SIMULATED CASE STUDIES

The test case model for the updating problem was chosen to be a free—free beam of constant thickness, made of mild steel which was 1 m long, 12 mm wide and 20 mm thick.

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TABLE 1 Modal correlation for test cases MAC Mode

Case 1

Case 2

1 2 3 4 5 6 7

0.9996 0.9983 0.9985 0.9975 0.9992 0.9980 0.9979

0.9566 0.9274 0.8743 0.8569 0.8175 0.6476 0.5760

This beam was modelled using 20 three-dimensional beam elements in ANSYS [21]. Flexible springs to ground were included in the FE model to ensure that the natural frequencies remained positive in the event of numerical rounding errors in the eigensolution. Two different simulated test cases were undertaken, with different amounts of perturbation applied to the model to generate the experimental FRFs; the perturbed model is shown in Fig. 6. Test case 1 was created by reducing the thickness of the fifth and sixth elements to 90% of their original thickness, and test case 2 was more strongly perturbed, with elements 5, 6 and 17—19 reduced to 50% of their original thickness. Since one of the objectives of this work is to model an experiment as accurately as possible, the undamped situation was not considered. Complex FRFs were produced by assuming proportional structural damping where [D]"0.01[K]. The sophisticated noise model developed in Sections 3 and 4 was used to corrupt the FRFs from the perturbed test cases. Estimates were made for the modal parameters of a shaker, and these were used to generate a forcing signal, C(u), using equation (20). The auto- and cross-spectra of the various noise sources described in Section 3 were calculated, and finally 200 averages were performed for every FRF with the forcing signal, S(u), set to a zero mean random vector of standard deviation unity. Figure 7 shows a zoom around two resonances for the second case study. It clearly shows the differences between the H , H and H estimators. The correct FRF is approached   4 accurately by the H estimator everywhere except at resonance, where H is more accu  rate. H is inaccurate at all spectral lines except when approaching resonance. H can be  4 seen to be inaccurate at both resonance and antiresonance. The more sophisticated estimators—H , H and H —overlaid the exact FRF very successfully; these are not ! 7  displayed graphically. 6.1. UPDATING TEST CASE 1 With a real experimental updating scenario, the ideal p-value allocation is never known. The problems associated with inadequate p-value selection are considered in a separate paper [22]; for the present work the correct p-value allocation is used, as shown in Fig. 6, in order to isolate errors from estimators alone. Figure 8 shows initial attempts to use the H estimator for updating using FRF  sensitivity and one hundred frequency points, equally spaced in the range 1—1600 Hz. It is clear that FRF sensitivity has failed to produce an adequate update, when using a typical number of frequency points. Acceptable results were eventually achieved using this

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Figure 6. Case study model.

estimator only by reducing the number of frequency points to 33. This reduction is so severe that too much information is rejected, and numerical effects due to the ill-conditioning of the problem begin to influence the solution. Divergence occurred when attempting to update the nine p-values shown in Fig. 6. Only when the number of p-values was reduced to seven—by replacing the three damping p-values with one that spanned the entire structure—was a convergent solution attained. The H estimator results in heavily corrupted data away from resonance. Off-resonant  response data is more important for model updating purposes than data near resonance since the residual component from out-of-range high-order modes is more accurately preserved. In addition, data near resonance yields updating equations that are nearly

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Figure 7. Overlaid FRFs from three basic estimators.

Figure 8. Updating using the H estimator. 

,H ; 

, H with 33 fps; 

, Correct p-values.

identical and thus are highly ill-conditioned [2]. Therefore, the use of any estimator that results in the off-resonant data being inaccurate—such as H or H —is strongly  4 discouraged. Residual curves generated by equation (27) from updating with H and H are repro  duced in Fig. 9. There is clearly a marked difference between the two curves at around the 35th frequency point. The ranked residual for the H update begins to increase significantly 

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Figure 9. Residual curves for H and H estimators.  

Figure 10. Update of test case 1 with classical FRF sensitivity. , Correct.

, H ; 

, H ; !

, H ; 4

, H ; 

, H ; 7

before the equivalent curve for H . This is further evidence of the fact that more frequency  points are in error with H than for H .   Figures 10 and 11 show the results of updating case 1 using the estimators other than H .  Figure 10 shows an investigation into updating using classical FRF sensitivity, i.e. without the frequency point and dof selection proposed by Waters [2]. This was undertaken to see if

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Figure 11. Update of test case 1 with enhanced FRF sensitivity. , Correct.

,H ; 

, H ; !

, H ; 4

, H ; 

,H ; 7

the use of an essentially unbiased estimator, such as H , could resolve the problems caused ! by insensitive dofs and noisy data without rejecting potentially useful information. It is clear that the H estimator performs significantly worse than any other estimator 4 considered here, except H . This is not surprising, since taking the geometric mean of H   and H merely results in erroneous FRF data at both resonance and antiresonance. H   outperforms the more sophisticated estimators to a small degree. In general, the estimation of p-values is successful, with the least accurate p-values being the hysteretic damping parameter for the first macro-element. This relatively poor performance is because the response will be the least sensitive to this p-value. When frequency point and dof selection are incorporated into the FRF sensitivity method, the performance improves considerably, as shown in Fig. 11. The rejection of some frequency points has resolved the problems with the H estimator; the other estimators 4 perform comparably. 6.2. TEST CASE 2 The second test case was deliberately chosen to be more disparate from the original FE model, and thus present a more challenging problem for the updating routines. The final estimated p-values are shown in Figs 12 and 13, and once again very little difference is observed between the performance of the different estimators. H produces the worst final 4 p-values, but the problems are resolved when frequency point selection is used. Again the hysteretic p-values are estimated least successfully, because of the insensitivity of the response to the damping. H is consistently slightly more reliable than the other estimators. Therefore, ! this is further evidence of the increased accuracy available from using an unbiased estimator. 7. EXPERIMENTAL CASE STUDY

Despite the efforts made to model the noise phenomena in a representative way as possible, no simulated case can ever truly replace real experimental tests. Any real

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Figure 12. Update of test case 2 with classical FRF sensitivity. , Correct.

, H ; 

, H ; !

, H ; 4

, H ; 

, H ; 7

Figure 13. Update of test case 2 with enhanced FRF sensitivity. , Correct.

,H ; 

, H ; !

, H ; 4

, H ; 

,H ; 7

experiment will include sources of error not modelled in the simulation described in Sections 3 and 4. These include: E signal processing errors such as leakage due to use of the FFT; E imperfect boundary conditions; E discretisation errors due to the fact that the FE model is a discrete representation of a continuous structure; and E structure/shaker interaction. The unperturbed FE model described in Section 6 was constructed and suspended on elastic cords to ensure that support modes remained below the frequency range of interest.

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MODEL UPDATING

The response was measured at the 21 nodes using a laser Doppler velocimeter which has the advantage of affording FRFs of exceptional purity, and the FRFs were processed using a Bru¨el and Kjaer 2032 analyser. This analyser measures two channels simultaneously, not three, and this precluded the use of the H estimator in the experimental study. ! Close inspection of Fig. 5 shows ‘blips’ around 104, 920 and 1400 Hz and these are caused by the bending modes in the y-direction. Although the laser vibrometer used should exhibit no cross-axis sensitivity, the modes shown on the point FRF are excited and measured due to imperfections in the experimental set-up where the beam and shaker system tilted away from the ideal situation. The spectral lines adjacent to these resonances were discarded manually from the updating procedure. Residual-based frequency point selection was used after these points were discarded. The unperturbed FE model and the experimental model exhibited excellent correlation, and the FE model would not usually be considered in need of updating. For this reason, and so that a p-value assignment that was likely to be ‘correct’ could be made, the data from the unperturbed experimental beam was used to update an inaccurate FE model. This model was similar to the perturbed model used for the second simulated test case, in that elements 5, 6 and 17—19 were reduced to three-quarters of their original thickness. The p-value assignment used in the second simulated test case was again used. Naturally, we no longer know the correct p-values after the update, and so the most appropriate way of assessing the efficacy of the update is to consider the observed modal properties. The modal correlation is best seen by considering the following two measures: (i) the error in natural frequencies, f !f ; and 6  (ii) 1!MAC+ ,,+ ,), which is zero for perfect correlation [23]. 6  The resonant frequencies and mode shapes were extracted using the global rational fraction polynomial method [24]. Figures 14 and 15 display the results for the experimental updates. Figure 14 does not show the natural frequency errors between the original FE model and the experimental model because of the scaling of the figure; these discrepancies are so large that they obscure

Figure 14. Differences between experimental and updated natural frequencies. , H . 7

, H ; 

, H ; 

, H ; 4

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M. J. RATCLIFFE AND N. A. J. LIEVEN

Figure 15. Mode shape disparities.

,H ; 

, H ; 

, H ; 4

, H

7

, H . $#

the comparison between estimators. It suffices to say that the location of all the modes is improved considerably by the updates using all the estimators. One important difference between the experimental updates and the simulated cases is the successful updates attained using the H estimator experimentally. The success here is  because the beam exhibits much lower damping than that assumed for the simulated cases. Therefore, the mobility drop-off at antiresonance is much sharper than for the simulated cases, and so fewer spectral lines are affected by the inaccuracy of the H estimator here.  The mode shape disparities are shown in Fig. 15. Every estimator produces a significant improvement in modal correlation. In particular, H is seen to be very effective, and in fact  produces slightly better results than the more sophisticated estimators. Interestingly, the updated modes 3—7 are correlated better than the first two modes. This is a surprising result since low-order modes generally show discrepancies less well because they are relatively less sensitive to mass and stiffness errors. This result is explained by the fact that use of the residual method caused the lower frequencies to be discarded, and therefore higher frequencies—corresponding to the higher-order modes—were used.

8. CONCLUDING REMARKS

A new model for the noise corruption manifest in modal testing has been proposed, and examination of coherence plots has suggested that the new noise model is significantly more representative of genuine experimental noise than traditional methods of corrupting simulated experimental test cases. This model has been used in conjunction with the FRF sensitivity method to analyse the effects of FRF estimators on updating inaccurate FE models. Two simulated test cases and one experimental have suggested that the traditional H estimator—despite being biased  low, particularly at resonance—is perfectly adequate for updating.

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Careful selection of frequency points and dofs for updating has been shown to be more significant than the choice of FRF estimator. However, these investigations have also suggested that use of the H estimator, and the H estimator—which is the geometric  4 average of H and H —should be avoided. The use of more exotic estimators such as H ,   ! H and H has not been shown to be essential, despite the fact that these estimators produce  7 FRFs that approximate the true responses more closely. However, since the additional effort required to evaluate these estimators is small—particularly in the case of H and  H —the effects of these estimators on the updating of more complex structures should be 7 a subject for further research.

REFERENCES 1. W. J. VISSER and M. IMREGUN 1991 IMAC IX, 462—468. A technique to update finite element models using frequency response data. 2. T. P. WATERS 1995 Ph.D. thesis, Department of Aerospace Engineering, University of Bristol, UK. Finite element model updating using frequency response functions. 3. R. E. COBB 1988 Ph.D. thesis, Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, VA, USA. Confidence bands, measurement noise, and multiple input—multiple output measurements using the three-channel frequency response function estimator. 4. G. R. TOMLINSON IMAC V, 1479—1486. A simple theoretical and experimental study of the force characteristics from electrodynamic exciters on linear and nonlinear systems. 5. J. A. FABUNMI and F. A. TASKER 1988 ASME Journal of »ibration, Acoustics, Stress, and Reliability in Design 110, 345—349. Advanced techniques for measuring structural mobilities. 6. K.-S. HONG and C.-B. YUN 1993 Engineering structures 15(3), 179—188. Improved method for frequency domain identifications of structures. 7. L. D. MITCHELL 1982 ASME Journal of Mechanical Design 104, 277—279. Improved methods for the fast Fourier transform (FFT) calculation of the frequency response function. 8. H. VOLD, J. CROWLEY and G. T. ROCKLIN 1984 Sound and »ibration 18(11), 34—38. New ways of estimating frequency response functions. 9. A. WICKS and H. VOLD IMAC IV, 897—899. The H frequency response function estimator. 1 10. H. SCHMIDT 1985 Journal of Sound and »ibration 101(3), 347—427. Resolution bias errors in spectral density, frequency response and coherence function measurements, I—VI. 11. J. S. BENDAT and A. G. PIERSOL 1980 Engineering Applications of Correlation and Spectral Analysis. New York: Wiley-Interscience. 12. G. T. ROCKLIN, J. CROWLEY and H. VOLD, IMAC III, 272—278. A comparison of H , H and   H frequency response functions. 4 13. H. G. D. GOYDER 1984 2nd International Conference on Recent Advances in Structural Dynamics, II, 437—446. Foolproof methods for frequency response measurements. 14. L. D. MITCHELL, R. E. COBB, J. C. DEEL and Y. C. LUK, IMAC V, 364—373. An unbiased frequency response function estimator. 15. W. M. TO and D. J. EWINS 1990 IMAC VIII, 1101—1107. The characteristics of frequency response function estimators using random excitation. 16. D. K. RAO, IMAC V, 1142—1150. Electrodynamic interaction between a resonating structure and an exciter. 17. N. L. OLSEN, IMAC IV, 1160—1167. Using and understanding electrodynamic shakers in modal applications. 18. K. UNHOLTZ 1961 pp. 25.1—25.74. Shock and »ibration Handbook, C. M. Harris and C. E. Crede (eds). New York: McGraw-Hill. Vibration testing machines. 19. G. R. TOMLINSON 1979 Journal of Sound and »ibration 63(3), 337—350. Force distortion in resonance testing of structures with electrodynamic vibration exciters. 20. N. MAIA 1991 IMAC IX, 1515—1521. Fundamentals of singular value decomposition. 21. ANS½S Finite Element Software. Swanson Analysis Systems, Inc. 22. M. J. RATCLIFFE and N. A. J. LIEVEN 1997 ¹he Aeronautical Journal Submitted. An improved method for parameter selection in finite element model updating. 23. R. J. ALLEMANG and D. L. BROWN 1983 IMAC I, 110—116. A correlation coefficient for modal vector analysis.

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24. M. H. RICHARDSON and D. L. FORMENTI, IMAC I, 167—181. Parameter estimation from frequency response measurements using rational fraction polynomials.

APPENDIX A: NOMENCLATURE a k k k $ m r s D F  F C G ++ G ,, G 11 G 16 G 33 G 34 G 44 G 66 G 67 H H 2  K K(u) ¸ M M(u) N D N N N(u) R S S(u) a(u) C(u)

Normally distributed variable stiffness p-value back EMF constant for exciter force/current constant for exciter mass P-value residual standard deviation hysteretic damping matrix force applied to structure by electrodynamic exciter electromagnetic force induced in exciter coil autospectrum of uncorrelated content in input force X autospectrum of uncorrelated content in response ½ autospectrum of source signal S cross-spectrum between source signal S and input force X autospectrum of true input force º cross-spectrum between true input force º, and true output » autospectrum of true output » autospectrum of measured input force X cross-spectrum between measured input force X and measured response ½ frequency response function frequency response function estimator stiffness matrix uncorrelated content in source signal amplification and transduction electrodynamic exciter coil inductance mass matrix force measurement errors number of frequency points number of p-values output measurement noise resistance of electrodynamic exciter coil and amplifier sensitivity matrix random input signal to structure receptance datum amplifier and shaker attachment gain

Subscripts/Superscripts A analytical data X experimental data 1, 2, 2 type of FRF estimator