Frequency resolution and number of averages effects on the accuracy and dispersion of operational modal analysis results

Frequency resolution and number of averages effects on the accuracy and dispersion of operational modal analysis results A. Cattaneo, S. Manzoni, M. Vanali Politecnico di Milano, Department of Mechanics, Milan, Italy ABSTRACT: Structural Health Monitoring techniques based on the study of the evolution of modal parameters aim to detect a structural modification relying on the results of a modal parameter estimation algorithm. Basically, a change in modal parameter values may indicate a structure evolution and possibly an ongoing damage situation. Beside structure changes there are other reasons that can cause changes in modal parameter values. Environmental and operational conditions naturally cause a variation of modal parameters. In addition the results of identification are affected by a dispersion due to the signal processing method. This paper is entirely focused on this latter aspect. The study exploits a Monte Carlo method relying upon the modal model of the G. Meazza stadium grandstands in Milan, where a permanent monitoring system is working since several years. Thanks to the modal model the response of the structure to random excitation is simulated. Numerical data are analyzed with the same algorithm used to process real data coming from the sensors that measure the dynamic response of the structure. Since the algorithm takes as input the power spectra associated to the acceleration responses of the monitored structure, modal parameter estimations are heavily affected by sub-record length, that is to say frequency resolution and number of averages used to compute power spectra. The work specifically discussed in the paper aims to investigate the separate effects of these two parameters on accuracy and spread of the estimated eigenfrequencies and non-dimensional damping ratios. Identification process spread is a key factor in assessing the real potential of the monitoring technique since represents a threshold under which is not possible to recognize a modal parameter change linked to a structural modification. Outcomes are discussed taking into account the complexity of the modal analysis performed on the proposed test case structure whose dynamic behaviour is characterized by the presence of close and highly damped modes. 1 INTRODUCTION Among Structural Health Monitoring (SHM) techniques, the one considered in this paper belongs to the family of strategies relying on modal parameter identification (Maeck, Peeters, De Roeck (2001), Mohanty, Rixen (2004)). Since these strategies try to relate structure changes by finding out dynamic response changes, the effective chance to asses those changes are unusual and possibly caused by damages requires a deep knowledge of the spread associated to the estimated modal parameters value. In this context, it is worth noticing that modal parameter changes, especially those of frequency and damping ratios, are very low. This represents a significant drawback of the proposed strategies and deserves the maximum attention in order to comprehend if a certain modal value change can be recognised or not. The carried on study is part of a wider work aimed at developing a permanent monitoring system of the Giuseppe Meazza stadium in Milan (Cigada, Caprioli, Redaelli, Vanali (2008), Vanali, Cigada (2009), Cigada et al. (2010)). The approach followed is the so-named Operational Modal Analysis (OMA) in order to provide a continuous estimation of the modal parameters of

2 IOMAC'11 4 th International Operational Modal Analysis Conference the structure (Herlufsen, Mller, Brincker (2002)). Under the hypothesis that the ambient excitation, mainly represented by wind and traffic, can approximate a white random noise on the frequency range of interest (Mohanty, Rixen (2004)), OMA allows to investigate the dynamic behaviour of the structure only measuring its response in a set of conveniently chosen points. Data collected by sensors are finally processed by the Polyreference Least Square Frequency Domain (PolyMAX) algorithm that proves to be suitable to estimate modal parameters of a highly damped structure like the Meazza stadium (Peeters, Van der Auweraer, Guillaume, Leuridan (2004), Caprioli, Cigada, Gentile, Vanali, (2009)). Several factors are responsible for modal parameter changes. Environmental (i.e. temperature, humidity, etc.) changes (Cigada, Manzoni, Vanali (2008)), excitation characteristics (i.e. level and frequency content) and changes of loads which the structure is subjected to (Cigada, Caprioli, Redaelli, Vanali (2008), Reynolds, Pavic, Ibrahim, (2004)) can be mentioned as examples. All these factors represent a source of variability of the modal parameter estimation process. In addition, modal analysis results are affected by a certain dispersion only due to the identification algorithm. Thanks to Monte Carlo simulations, this aspect is specifically investigated in order to define a threshold under which is not possible to find out any modal parameter change and therefore any structure change. Section 2 discusses the peculiarities of the test case structure and the simulation approach that leads to the results discussed in the paper. An overview of the analyzed tests and a description of the pursued objectives are provided in section 3. Finally, section 4 explains the outcomes. Thanks to new tests and to the comparison with the former ones (Cattaneo, Manzoni, Vanali (2010), Cattaneo, Manzoni, Vanali (2011)), it is shown how and dispersion of eigenfrequencies and non-dimensional damping ratios change when OMA algorithm is performed using different frequency resolution and number of averages to estimate the power spectra associated to the acceleration responses of the monitored structure. Since power spectra represent the input of the signal processing algorithm, frequency resolution and number of averages heavily affect the entire identification process. 2 TEST CASE STRUCTURE AND SIMULATION APPROACH As mentioned in the previous section, the modal analysis technique adopted for the continuous structural health monitoring of the Giuseppe Meazza Stadium in Milan is the Operational Modal Analysis (OMA). It should be pointed out that the stadium is a very complex structure and the study described in this paper specifically concerns a single grandstand (Cigada, Caprioli, Redaelli, Vanali (2008)). The huge amount of data collected during the years, thanks to the monitoring system and several measurement tests performed on the field, has allowed to validate a modal model of the stadium grandstand considered in this paper. The structure considered as test case consists of a pretensioned reinforced concrete box beam that carries the cantilevered frames sustaining the stand (Fig. 1). According to the results obtained with experimental modal analyses (EMA) performed with a known loading and to the study of the considered stand dynamic response with a finite-element method (FEM), it has been noticed that measurements gathered on the lower side of the stand allow to satisfactorily identify its eigenfrequencies, non dimensional damping ratios and modal shapes. The optimized set-up adopted for the field measurements is obtained placing 11 accelerometers (Fig. 1).

3 Figure 1: placement on the examined grandstand (only vertical direction has been considered in simulations) Operational Modal Analysis (OMA) performed on the grandstand, as recommended when applying Natural Excitation Technique (NEXT), assumes that the structure is excited by white random noise in the interesting frequency range (Mohanty, Rixen (2004)). The structure response measured by accelerometers is then processed with an automatic procedure that uses the modal identification PolyMAX method. It should be underlined that the input to PolyMAX are the non-scaled impulse response functions (Silva, Maia (1997), Ewins (2000)) estimated with auto and cross-correlations of the structure vibration signals measured in several locations. The need to carry out the continuous monitoring of the modal parameter evolution has made necessary the automation of the process with which they are estimated. This requirement is fulfilled with an automated algorithm based on the interpretation of the stabilization diagrams produced by the PolyMAX identification method. As stated in the introduction, authors aim is to characterize the dispersion just caused by the signal processing algorithm. As a matter of fact, the mentioned result dispersion puts a limit to the possibility to identify modal parameter changes related to a structure change. Identification process relies upon the solution of a least-squares problem to estimate modal parameters and the use of stabilization diagrams to identify the true physical poles representative of the dynamic behavior of the mechanical system. Both these aspects and the problem non-linearity do not allow to asses OMA result dispersions by mean of an uncertainty propagation method (GUM, 1995) that relies upon close formulas. Hence a Monte Carlo method has been implemented in order to produce estimates of modal parameters and the associated spreads. Monte Carlo simulations allow to process numerical time-histories representative of the response of the structure with the same algorithm employed to analyze data acquired with real accelerometers. As explained below the generation of numerical data is performed thanks to a modal model of the grandstand investigated. By applying OMA algorithm on each data set, consisting of all accelerometers time-histories, the identification process leads to an estimation of modal parameters (i.e. eigenfrequencies, non-dimensional damping ratio and modal participation factors). All the results, obtained by processing all data sets, form a statistical population for each modal parameters. These statistical populations represent the starting point for any subsequent assessment on the modal parameters dispersion. 2.1 The modal model and the analysis procedure Numerical data generation entirely relies upon a modal model of the investigated grandstand. As a matter of fact, the model allows to compute the unit impulse response of the structure in the same points where accelerometers are place in real measurements (only vertical directions has been considered in simulations). Equation 1 reconstructs the x ik response at i th point for a unit impulse applied at k th point:

4 IOMAC'11 4 th International Operational Modal Analysis Conference N ϕϕ ri rk ζω r rt d xik () t = e sin ( ωr t) (1) m ω d r= 1 r r where t = time, m r = modal mass associated to the r th mode, φ ri = i th component of the eigenvector associated to the r th mode, φ rk = k th component of the eigenvector associated to the r th mode, ω r = eigenfrequency associated to the r th mode, ω d r = damped eigenfrequency associated to the r th mode, ζ r = non-dimensional damping ratio associated to the r th mode, N = number of considered modes. In order to exhaustively discuss how the simulation approach has been practically implemented, some further details on signal processing algorithm are provided. As previously mentioned, OMA technique gives modal parameter estimations without using a known and measured input. In this scenario it is worth noticing that the entire identification process is fulfilled by exploiting the power spectra of the signals, whether real or simulated, of the acceleration response of the structure. In particular PolyMAX reconstructs the free response of a physical system relying upon the Fourier transform of the auto/cross-correlation computed between the signals of the physical system responses measured in some points and one or more of these signals assumed as a reference. Power spectra estimations are performed as follow: the whole random signal time-histories are cut in sub-records; auto/cross correlation are computed for each sub-records; free response of the physical system can be related to the auto/cross-correlations decay. As explained in Peeters, Van der Auweraer, Vanhollebeke, Guillaume (2007) and Ewins (2000), only the first part of the auto/cross-correlations can be used to estimate satisfactorily the free response of the physical system. Hence only the first points of auto/cross-correlations, weighted by a proper exponential function, are kept; the first part of auto/cross-correlation is used to compute a Fourier transform; by averaging the Fourier transforms of each sub-records, power/cross-spectra of the signal time-history is obtained. The points listed above show how sub-record time length directly determine the number of averages used to estimate power spectra whereas their frequency resolution is determined by the time base of the auto/cross-correlations effectively used to estimate the free response of the system. Monte Carlo simulations presented in this paper shows how modal parameters estimated by the automatic procedure are heavily affected by the auto/cross-correlation time-base and the number of averages used to compute power spectra. The way these parameters are investigated is made clear, point by point, thanks to the description of the analysis procedure provided below: white random noise records have been generated numerically on the 0 100 Hz frequency range. These records have been adopted as structure excitation and they have been built by analyzing the accelerometer signals acquired at the stadium while it was empty, and thus excited just by wind and traffic. Three different time-lengths (1800 s, 3600 s and 7200 s) are adopted in simulations in order to separately study the effect of time-base and number of averages used to carry out OMA operations; the acceleration responses in seven grandstand locations (Fig. 1) have been reconstructed by applying the excitations generated at the previous point and exploiting the modal model described above. This has been made possible by convolving (D antona, Ferrero (2006)) the excitation signals with the structure unit impulse responses (Equation 1) obtained thanks to the modal model and inverse Fourier Transform operations. Each response in the seven points where accelerometers have been simulated has been obtained by the following relation (Equation 2): ( ) x ( t ) h ( t ) Ne y ( t ) = (2) j i = 1 i ij

where y j (t) is the time response in the j th point, x i (t) is the input at the i th point and h ij (t) is the response at the j th point to a unit impulse applied at the i th point. Finally, Ne is the number of points where excitations have been applied; each data set has been analyzed. A single set is composed by the seven acceleration responses, each with a length of 1800 s, 3600 s or 7200 s chosen in order to vary independently time-base and frequency resolution used to carry out OMA identification process. As result of this operation the grandstand modal parameters are estimated. A statistical population for each modal parameter is generated by applying OMA to all data sets available. The data set number depends on the Monte Carlo method convergence (an adaptive Monte Carlo simulation has been considered in order to lower the number of required iterations GUM,(1995) and GUM, Supplement1 (1995)).The obtained statistical populations allow to estimate the dispersion associated to each modal parameter estimation only due to the OMA algorithm. The modal parameter values adopted in the numerical model and estimated by means of data measured at the stadium are gathered in Tab. 1. It is worth noticing that modes taken into account allow to fully describe the dynamic behavior that the structure exhibits when excited by environmental sources (e.g. wind and traffic). Looking at eigenfrequency and non-dimensional damping ratio values it is clear how the structure investigated proves to be a very interesting case to test OMA algorithm both on isolated and well-defined modes (i.e. mode 7), and modes close in frequency with high damping ratios (i.e. modes 3, 4 and 5). Table1: Modal parameter values used to build up the grandstand modal model; the eigenvector components are referred to Fig. 1; eigenvectors are normalized at point 1/3 L of Fig. 1. Mode number 1 2 3 4 5 6 7 5 Eigenfrequency [Hz] 1.01 2.79 3.05 3.18 3.75 4.36 5.22 Non-dimensional damping ratio [%] 1.5 2.5 2.5 2.1 1.9 2.6 1.8 point 0 1.00 0.20 0.00 0.00 0.00 0.20 0.10 point 1/6 L 1.00 0.70 0.70 0.70 0.60 0.50 0.65 point 1/3 L 1 1 1 1 1 1 1 point 1/2L 1.00 1.50 1.30 1.30 0.00 1.20 0.00 point 2/3 L 1.00 1.00 1.00 1.00-1.00 1.00-1.00 point 5/6 L 1.00 0.70 0.70 0.70-0.60 0.50-0.65 point L 1.00 0.20 0.00 0.00 0.00 0.20-0.10

6 IOMAC'11 4 th International Operational Modal Analysis Conference Thanks to the information provided on the Operational Modal Analysis algorithm, on the structure investigated and on the simulation approach employed to assess the results dispersion associated to the modal parameters estimation process, the subject matter of the paper can be considered outlined. It has been highlighted that frequency resolution and number of averages heavily affects the modal parameter estimation. How the effect of these parameters is investigated is discussed in the following section. 3 SIMULATIONS TARGETS Monte Carlo simulations formerly carried out by the authors have allowed to deepen the problem inherent to the number of iterations required to have convergence for the simulation results (Cattaneo, Manzoni, Vanali, (2010)) and what are the combined effect of frequency resolution and number of averages (Cattaneo, Manzoni, Vanali (2011)) in the estimations of eigenfrequencies and non-dimensional damping ratios. According to considerations on the expected uncertainty value for the identified modal variables as stated in Cattaneo, Manzoni, Vanali, (2010), the number of significant digits chosen to check Monte Carlo method convergence continue to be the same adopted in the previous works (3 s.d. for the eigenfrequencies values, 2 s.d. for non-dimensional damping ratio values and eigenvector components and 1 s.d. for all the standard deviation values). In the attempt to comprehend the effects of frequency resolution and number of averages, in Cattaneo, Manzoni, Vanali (2011) were proposed the results obtained using three different sub-records length: 40, 80 and 150 s. As pointed out in Section 2, sub records time-length directly affects number of averages whereas auto/cross-correlations time-base directly determine the frequency resolution of the power spectra assumed as input of OMA algorithm. Time bases used have respectively a length of 25, 50 and 100 s. Tests carried out in a previous work (Cattaneo, Manzoni, Vanali (2011)) analyzed only timehistories of 3600 s length. Thus using different sub-record and auto/cross-correlations time-base length as described above, an improvement in frequency resolution inevitably leaded to lower the number of averages used to estimate the accelerometer power spectra. On the other hand, shorter sub-records and auto/cross-correlations time-base length meant worsening frequency resolution, though there was a benefit on estimation accuracy due to the increased number of averages with which power spectra was computed. This paper specifically deals with the influence of frequency resolution and number of averages on modal analysis results considering separately their effect. This is made possible analyzing the results of the tests listed in Tab. 2. Table 2: List of tests used to separately investigate the effect of frequency resolution and averages number on OMA algorithm results TEST #1 TEST #2 TEST #3 TEST #4 TEST #5 Time-histories length [s] 1800 s 3600 s 7200 s 3600 s 7200 s Sub-records length [s] (n of averages) 40 s (45) 80 s (45) 80 s (90) 150 s (24) 150 s (48) Auto/crosscorrelation time base [s] / frequency resolution [Hz] 25 s (0.04 Hz) 50 s (0.02 Hz) 50 s (0.02 Hz) 100 s (0.01 Hz) 100 s (0.01 Hz) Varying length of both time-histories and sub-records as well as of the time-base used for auto/cross-correlation estimation allows to assess the following aspects: result comparisons between test #2 versus test #3 and test #4 versus test #5 show what are the effects of doubling the number of averages while keeping the same frequency resolution;

result comparisons among test #1 versus test #2 and test #5 highlights the effects of changing frequency resolution while keeping the same number of averages. (In test #5 number of averages slightly increases, but this doesn t compromise the proposed analysis). Next section provides the detailed analysis of result dispersion and s of eigenfrequency and non-dimensional damping ratio estimations. 7 4 INFLUENCE OF FREQUENCY RESOLUTION AND NUMBER OF AVERAGES ON MODAL ANALYSIS RESULTS Simulation results are discussed focusing separately on estimated non-dimensional damping ratios and eigenfrequencies. In order to concisely show the outcomes obtained relying upon former and new simulations, Tabs. 3, 4 proposed in the following subparagraphs collect the and the ratio between standard deviation and mean for both the investigated modal parameters. Bias allows to quickly valuate the accuracy of the OMA algorithm and the ratio between standard deviation and mean provides an immediate feedback on the variability of results just due to the identification algorithm. 4.1 Effects on non-dimensional damping ratio estimation Before analyzing non-dimensional damping ratio results, a preliminary consideration has to be pointed out. As known from literature (Vanali, Cigada (2009)), non-dimensional damping ratio estimation is a critical process both in terms of accuracy and of dispersion of the obtained results. The results of Tab. 3 confirms this aspect: as a matter of fact, both and the ratio between standard deviation and mean assumes relatively large values. Table 3: Bias and std. dev. on ratio for non-dimensional damping ratio estimations. For test #3 and #4, stabilization diagrams obtained applying PolyMAX do not show stable poles for mode 2 TEST #1 0.5 h, t 40, xcorr 25 TEST #2 1 h, t 80, xcorr 50 TEST #3 2 h, t 80, xcorr 50 TEST #4 1 h, t 150, xcorr 100 TEST #5 2 h, t 150, xcorr 100 Mode 1 191% 18% 94% 17% 90% 13% 47% 18% 43% 15% Mode 2-15% 31% -29% 22% NA NA NA NA -36% 19% Mode 3 25% 21% 4% 21% 12% 16% -18% 23% -8% 18% Mode 4 36% 18% 19% 16% 18% 12% 9% 17% 10% 13% Mode 5 36% 7% 15% 6% 18% 5% 1% 7% 5% 5% Mode 6 35% 9% 23% 7% 23% 5% 15% 7% 17% 5% Mode 7 32% 6% 17% 5% 17% 3% 8% 5% 9% 4% Looking at the results of Tab. 3, two comments can be drawn out: a better (lower) frequency resolution (comparison among test #1, #2 and #5) leads to lower and spread; a higher number of averages (comparison between test #2 versus #3 and #4 versus #5) decreases dispersion without an appreciable systematic effect on estimation accuracy. The general comments listed above deserve in some cases more detailed remarks taking into account all the peculiarities related to the investigated structure. First of all, as observed in a previous work (Cattaneo, Manzoni, Vanali (2011)), it is worth noticing that lower frequency resolution reduces the non-dimensional damping ratio value. This trend is common for all modes. For mode 2, in particular, non-dimensional damping ratio is always underestimated. Reducing the frequency resolution from a value of Δ f = 1/ 25 = 0.04Hz

8 IOMAC'11 4 th International Operational Modal Analysis Conference (test #1) to a value of Δ f = 1/ 50 = 0.02Hz (test #2) and Δ f = 1/100 = 0.01Hz (test #5), the underestimation worsens. Similarly for mode 3 a reduction of frequency resolution doesn t necessary imply to lower the (when frequency resolution changes from Δ f = 1/ 50 = 0.02Hz to Δ f = 1/100 = 0.01Hz, moves from a value of 4% to a value of -8%). Further comments on the results collected in Tab. 3 can be drawn examining the numerically simulated accelerometer PSDs obtained exploiting the modal model discussed in Section 2 (Fig. 2). Figure 2: Numerical accelerometers PSD obtained relying upon the modal model described in Section 2 Mode 2 (at 2.79 Hz) and mode 3 (at 3.05 Hz) distance is only 0.26 Hz and they have high non-dimensional damping ratios (Tab. 1). It follows that between these two modes there is a large modal overlap that makes more difficult the estimation process of non-dimensional damping ratio and could explain the lack of an improvement on when the smaller frequency resolution is used (test #5). Difficulties in estimating mode 2 occur especially in test #3 and test #4 where the stabilization diagrams produced by the PolyMAX algorithm do not show stable poles at the frequency of this mode. Finally, it has to be pointed out that mode 1, unlike the others, is mainly horizontal and, as consequence of this, its energy content scarcely contribute in determining the response of the structure in vertical direction. In other words, the very poor accuracy in estimating the non-dimensional damping ratio of mode 1 can be ascribed, as inferred looking Fig. 2, at the much lower PSDs values at first mode frequency if compared with those of the others. In order to complete the dissertation on the non-dimensional damping ratio results obtained with tests from #1 to #5, Figs. 3a,b provide a representation of the non-dimensional damping ratio value estimated for mode 3 and 4 and of their associated standard deviations. In both figures it can be noticed the lower non-dimensional damping ratio values obtained decreasing frequency resolution (test #1, #2 and #5). As previously described, for mode 3 the decreasing of frequency resolution does not necessarily imply an improvement in non-dimensional damping ratio estimation. More easily interpretable results are obtained for mode 4: smaller (better) frequency resolution (test #1, #2 and #5) improves damping estimation accuracy and decreases dispersion; higher number of averages (comparison between test #2 versus #3 and #4 versus #5) decreases dispersion without an appreciable effect on estimation accuracy.

9 (a) (b) Figure 3: non-dimensional damping ratio estimations for mode 3 (a) and mode 4 (b) varying frequency resolution (test #1, #2 and #5) and number of averages (test #2 vs. #3, and test #4 vs. #5); blue dots and triangles respectively indicate and plus/minus standard deviation. Discussed the results obtained for non-dimensional damping ratio, the study of the effect of frequency resolution and number of averages on operational modal analysis results continues in the following subparagraph with the analysis of frequency estimations. 4.2 Effects on frequency estimation As a general comment it can be noticed that and spread for frequency estimations are much lower than those for non-dimensional damping ratio. An evidence of this aspect is provided in Tab. 4. Table 4: Bias and std. dev. on ratio for frequency estimations. For test #3 and #4, stabilization diagrams obtained applying PolyMAX do not show stable poles for mode 2 TEST #1 0.5 h, t 40, xcorr 25 TEST #2 1 h, t 80, xcorr 50 TEST #3 2 h, t 80, xcorr 50 TEST #4 1 h, t 150, xcorr 100 TEST #5 2 h, t 150, xcorr 100 Mode 1 0.6% 0.6% 0.7% 0.4% 0.5% 0.4% 1.0% 0.3% 0.8% 0.3% Mode 2 2.3% 0.9% 0.9% 0.5% NA NA NA NA 0.5% 0.4% Mode 3 0.8% 0.7% 0.5% 0.6% 0.4% 0.5% 0.6% 0.6% 0.4% 0.5% Mode 4 0.3% 0.5% 0.2% 0.4% 0.2% 0.3% 0.2% 0.4% 0.2% 0.3% Mode 5 0.0% 0.2% 0.0% 0.2% 0.0% 0.1% 0.1% 0.2% 0.1% 0.1% Mode 6 0.1% 0.3% 0.0% 0.2% 0.0% 0.1% 0.0% 0.1% 0.0% 0.1% Mode 7 0.1% 0.1% 0.0% 0.1% 0.0% 0.1% 0.1% 0.1% 0.0% 0.1% Tab. 4 shows how eigenfrequency s always remain under the threshold of 1%. Mode 2 appears again the most critical: it is significant to notice that both test #3 and test #4 does not allow to identify mode 2 and that the largest on frequency occurs when OMA algorithm tries to identify mode 2 with the lowest number of averages and coarse frequency resolution (that is to say test #1). Both decreasing the frequency resolution with the same number of averages (comparison among test #1, #2 and #5) and increasing the number of averages keeping the same frequency resolution (comparison between test #2 versus test #3 and test #4 versus test #5) lead to lower spreads of frequency estimations.

10 IOMAC'11 4 th International Operational Modal Analysis Conference (a) (b) Figure 4: Frequency estimations for mode 3 (a) and mode 4 (b) varying frequency resolution (test #1, #2 and #5) and number of averages (test #2 vs. #3, and test #4 vs. #5); blue dots and triangles respectively indicate and plus/minus standard deviation. A detailed representation of frequency estimations for mode 3 and 4 are respectively proposed in Figs.4a,b. Is interesting to observe that frequency of mode 3 is always underestimated, while for mode 4 OMA algorithm leads always to an overestimation. Bias trend of modes 3 and 4 deserves special attention: when the underestimation of the former decreases (or increases) the same happens for the overestimation of the latter. For example, when frequency resolution is decreased (comparison between test #1 and test #2 results) both the underestimation of mode 3 and the overestimation of mode 4 decrease. Referring again to Fig. 2, this behaviour can be ascribed to the fact that mode 3 and 4 are very close in frequency and high damped. 5 CONCLUSIONS This paper represents the prosecution of works previously carried on by authors aiming at deepen the topic of modal parameter estimation dispersion only caused by operational modal analysis algorithm. The study has to be considered as part of a wider project finalized at developing a permanent monitoring system of the G. Meazza Stadium in Milan. Thanks to the huge amount of data gathered on the field, it has been possible to develop a modal model of a grandstand. Relying on this model and performing Monte Carlo simulations, the operational modal analysis algorithm has been tested in order to valuate modal parameter estimation dispersion just caused by it. The analysis this paper specifically deals with is focused on the effect of frequency resolution and number of averages selected to evaluate the response power spectra used as identification inputs. The comparison between former and new test results has allowed to separately study the influence of these two aspects. From a general point of view, the conclusion on non-dimensional damping ratio estimation is that a decreased frequency resolution leads to lower and spread, while the increase in number of averages lower dispersion without an appreciable systematic effect on estimation accuracy. Frequency estimations, on the other hand, show low and spread for all the combinations of frequency resolution and number of averages tested. In particular frequency estimations spread and are lowered both by decreasing the frequency resolution and increasing the number of averages. It has been pointed out that the specificity of the structure analyzed has some important implications on the results obtained. For high damped and very close in frequency modes, modal overlap between close modes heavily affects non-dimensional damping ratio estimation. In this scenario a lack of accuracy in non-dimensional damping ratio estimation cannot be improved acting on frequency resolution and number of averages. On the other hand, for modes very close in frequency, it has been observed that the on frequency estimations could be reduced choosing appropriately frequency resolution and number of averages.

11 REFERENCES Bendat, J.S. and Piersol, A.G. 2000. Random Data: Analysis and Measurement Procedures, 3rd edition, Wiley-Blackwell. Caprioli, A., Cigada, A., Gentile, C., Vanali, M. 2009. Comparison of two different OMA techniques applied to vibration data measured on a Stadium grandstand, Proceedings of IOMAC'09 3rd International Operational Modal Analysis Conference, Ancona, Italy. Cattaneo, A., Manzoni, S., Vanali, M. 2010. Measurement uncertainty in operational modal analysis of a civil structure, Proceedings of the International Conference on Uncertainty in Structural Dynamics USD, p. 5103-5116, Leuven (Belgium). Cattaneo, A., Manzoni, S., Vanali, M. 2011. Numerical Investigations on the Measurement Uncertainty in Operational Modal Analysis of a Civil Structure, Proceedings of International Modal Analysis Conference (IMAC) XXIX, Jacksonville, Fl (USA). Cigada, A., Manzoni, S., Vanali, M. 2008. Vibro-acoustic characterization of railway wheels, Applied Acoustics, Vol. 69, No. 6, p. 530-545 Cigada, A., Caprioli, A., Redaelli M., Vanali, M. 2008. Vibration testing at Meazza stadium: Reliability of operational modal analysis to health monitoring purposes, Journal of Performance of Constructed Facilities, v 22, n 4, p. 228-237, 2008. Cigada, A. et al. 2010. The measurement network of the San Siro Meazza Stadium in Milan: origin and implementation of a new data acquisition strategy for structural health monitoring, Experimental Techniques, Vol 34, No 1, p. 70-81. D'Antona, G. and Ferrero, A. 2006. Digital signal processing for measurement systems: theory and applications (information technology: transmission, processing and storage), Springer. Ewins, D.J. 2000. Modal Testing: Theory, Practice and Applictions SECOND EDITION, Research Studies Press LTD. Herlufsen, H., Mller, N., Brincker, R. 2002. Modal parameters from a wind turbine wing by operational modal analysis, Proceedings of SPIE - The International Society for Optical Engineering, v 4753 II, p. 1426-1432 ISO/IEC Guide 98-3:2008, Uncertainty of measurement Part 3: Guide to the expression of uncertainty in measurement (GUM:1995). ISO/IEC GUIDE 98-3/Suppl.1:2008(E), Uncertainty of measurement, Part 3: Guide to the expression of uncertainty in measurement (GUM:1995), Supplement 1: Propagation of distributions using a Monte Carlo method. Júlio M.M. Silva, Nuno M.M. Maia 1997. Theoretical and Experimental Modal Analysis, Research Studies Press LTD.. Maeck, J., Peeters, B., De Roeck, G. 2001. Damage identification on the Z24 bridge using vibration monitoring. Smart Materials and Structures, Vol. 10, p. 512-517. Mohanty, P., Rixen, D.J. 2004. Operational modal analysis in the presence of harmonic excitation. Journal of Sound and Vibration. Vol. 270, No. 1-2, p. 93 109. Peeters, B., Van der Auweraer, H., Guillaume, P., Leuridan, J. 2004. The PolyMAX frequency-domain method: a new standard for modal parameter estimation?, Shock and Vibration, v. 11, n. 3-4, p. 395-409. Peeters B., Van der Auweraer, H., Vanhollebeke, F., Guillaume, P. 2007. Operational modal analysis for estimating the dynamic properties of a stadium structure during a football game, Shock and Vibration, v. 14, n. 4, p. 283-303. Reynolds, P., Pavic, A., Ibrahim, Z., 2004. Changes of modal properties of a stadium structure occupied by a crowd, Proceedings of International Modal Analysis Conference (IMAC) Orlando, Fl (USA). Vanali, M., Cigada, A. 2009. Long term operational modal analysis of a stadium grandstand to structural health monitoring purposes. IEEE Workshop on Environmental, Energy, and Structural Monitoring Systems.