Comparison between different operational modal analysis techniques for the identification of large civil structure modal parameters

Transcription

1 Comparison between different operational modal analysis techniques for the identification of large civil structure modal parameters Alessandro Caprioli, Alfredo Cigada, Marcello Vanali Politecnico di Milano, Dipartimento di Meccanica, via La Masa Milano Italy 1. Abstract A reliable modal analysis is fundamental in vibration-measurements-based monitoring of large civil structures, such as the grandstands of the Meazza stadium in Milan. In this paper different techniques to extract the modal parameters out of data gathered under normal operational conditions have been compared: these methods work both in time and frequency domain,. A large amount of data has been collected since summer 03 during different sort of events: concerts, football matches and windy days; in all events a lot of different measurement points have been acquired and this allowed to apply all the identification techniques with different kinds of excitation. The comparison also takes into account the capability to correctly identify the modal parameters while reducing the number of transducers put on the structure. This will allow for a reduction of the number of transducers being part of the permanent monitoring system that is going to equip the structure in a close future. 2. Introduction In the case of large civil structures open to the public and suitable to be used for different kind of events (e.g. theaters, stadia, arenas), structural monitoring is an important diagnosis technique to ensure safety of the structure itself and so of the people present on the stands. A vibration-measurements-based system is the natural choice because it allows the structure monitoring on a long time basis, through its modal parameters evolution; at the same time control of anomalous events such as the trespassing of the maximum allowed stress levels is allowed. The most reliable estimation of structural modal parameters can be obtained through traditional modal analysis, exciting the building with a known force and measuring the response in the greatest possible number of points. This solution is clearly not affordable as it is expensive and time consuming, especially for large structures as stadia and arenas. The proper choice can therefore be operational modal analysis; in this case the excitation comes from the in field excitation, for instance coming from the event taking place in the structure; in this case the input to the structure is not completely known nor measurable, while response is observed at some points. Clearly the identification of modal properties improves as the number of measured response points increases. Figure 1 Internal view of all the G. Meazza stadium of Milan, the 3rd ring extends around three of the four sides of the football field, on the background the 4 th side. The number of measurements points can be a problem in the case of very large structures, like The G. Meazza stadium in Milano, when a permanent monitoring system has to be built. Certainly it will not be

2 possible to place 8- accelerometers on each of the sub-structures composing the stadium as this will mean an enormous number of sensors, nearly 0 only for the 3 rd ring of stands (see par.3) (Figure 1). In the present study some of the techniques for the operational modal analysis are investigated with the final purpose of designing the permanent monitoring system of the G. Meazza stadium. Aim of this study is to investigate, in a practical case, the efficiency of various techniques suitable to identify the stadium modal parameters in a stable and reliable way. To this purpose a preliminary and fairly good knowledge of the structural behavior is needed, as a starting point, to fix a reference for the various method s comparison. In the case of the considered structure, a preliminary and deep description of the response function has been provided, as the considered sub-structure has been monitored for a long time, so a lot of data exist coming from football matches and concerts held in the stadium. In addition the same portion of the 3 rd ring has also undergone a complete traditional modal analysis, with proper and controlled excitation: this has given the reference data be used as a reference term for the next operational modal analysis techniques. At first a description of the stadium is given, with particular attention devoted to the part of the 3 rd ring being analyzed, then the choices related to data acquisition performed during the last two years will be described followed by the identification of the sub-structure analyzed in this paper. Then some parameter identification strategies will be selected and the results will be shown and commented. 3. The structure: Meazza s 3 ring The structure considered in the paper is one of the three main parts of the G. Meazza Stadium in Milano. This stadium was built in different years from 19 up to 1990 and is composed of three different parts mutually independent from each other. These parts are commonly referred as 1 st, 2 nd, and 3 rd ring. The 3 ring extends around three of the four sides of the football field, it is composed by 11 towers (about m high) sustaining prestressed reinforced concrete box beams (4 straight and 6 curved). Inside the girders, with a separation of m, are transverse diaphragms which allow to link prefabricated pretensioned cantilevers bearing the weight of the concrete floorboard. This structure has absolutely no links with the previously existing ones: the gap between the 2 and the 3 ring is covered by steel plates and there is no way to pass between the two. The 4 corner towers are higher than the others (1 m against m) and carry huge iron beams that support the shelter roof protecting the grandstands against rain or snow. All the analysis presented in this paper deal with the straight box beams between towers A4 and A (Figure 3). Different sorts of event have been monitored in the last 3 years, Figure 2 3d model of the box beam, complete of the stands, object of the analysis in particular the concerts of the 03, 04 and 0 summers, a lot of football matches (both Italian championship and champions league); also some windy days have been considered a good identification occasion. These events produce different dynamic loads to the structure: during concerts spectators jump and dance following the rhythm of the music, so the load is very well frequency-localized [1]; during football matches it is possible to identify different situations, the public entrance, the match, the half time break, the exit of people out of the stadium [1,2]. During football matches spectators remain seated and stand up only in particular occasions, goals, penalties and so on, giving a kind of excitation which is not suitable for operational modal analysis purposes during most of the time; during entrance and exit the presence of people on the stands varies, causing a variable dynamic response of the structure; during the mid-match break, the number of people on the stands is quite constant and they move in untidy way causing a dynamic forcing much similar to a random band limited one, that is the best one for operational modal analysis [3].

3 Figure 3 Plan view and cross section of the structures of the Meazza stadium. The analyzed structure is highlight in red Other data have been recorded during windy days: in this case the load can be considered similar to a random one, but the different conditions (no spectators on the stands) don t allow a direct comparison with the analysis carried out on football match data; moreover, windy days are a non controlled, unpredictable excitation and this makes it impossible use wind as a stable excitation source for permanent monitoring purposes. Finally, it has been decided to perform a traditional modal analysis on the same stand on which attention has been focused: to the purpose, an electro-mechanical actuator has been employed, working as an inertial device, being amplitude and frequency controlled through stepped sine tests. This has offered the possibility to perform a preliminary validation, of the proposed operational modal analysis techniques tested on the same structure. Apart from this particular testing, requiring all the available transducers to be used on the same structure, the long lasting monitoring activity, carried out with a reduced number of transducers, also consisted in changing both the number and position of the transducer on the structure. The aim was to get the greatest amount of information with the minimum expense. The next paragraph will deal with measurement strategies. 4. Measurement set-up In order to measure the accelerations produced by the crowd action, but also those induced by the wind, high sensitivity accelerometers had to be adopted. The chosen sensors are servo accelerometers with sensitivity V/g and very high s/n ratio. The transducer bandwidth was wider than needed to cover the 0 1 Hz range, which is considered of interest. All transducers were sampled with two parallel systems, PC with A/D board and a digital recorder, used as a backup device. The settings for the PC were: sampling frequency 60 Hz, with anti aliasing filters set at Hz; the digital recorder was set to have a granted bandwidth of 80 Hz. For each measurement point two sensors were used in order to measure the acceleration in the vertical direction and in the horizontal direction, perpendicular to the football field. Axial stand movements (parallel to the football ground) have been neglected at this stage, due to the structural symmetry. A good description of the modal shapes can be obtained using a large amount of accelerometers: as already pointed out, the limited availability of high sensibility sensors has determined a testing strategy in which some sensors have been moved from one event to the other, while some others have always been kept in fixed positions to ensure a reference, to transfer data even to not directly measured locations. In the end this resulted in having a rather dense mesh, giving an accurate identification of the structure main modes, of some local modes involving only parts of the stand, in the end also allowing for a selection of the best measuring points for the final monitoring system. Figure 4 shows the sensor positioning: 2 sections of the considered stand have been instrumented, (1/3 and 1/2 span L of a single box beam), the sensors, named A, B, C and D, have been kept fixed all through the monitoring campaign, while position E and F were used only in the modal testing performed with the shaker. Positions C and D are placed inside the box beam.

4 Figure 4 Placement of the measuring points at the 3 rd ring Data gathered by these sensors have been analyzed using three different modal identification techniques, leading to the results explained in the following.. Modal identification techniques: overview of the adopted methods Aim of this paper is to identify and discuss which is, the modal identification technique, for this particular application, leading to results the most reliable also offering an easy-to-interpret analysis of the dynamic stand response. A clear identification in the monitoring process means that the adopted technique should provide an easy way to distinguish between the real poles of the dynamic response function, that is stable poles, and the other poles introduced by the mathematics and used to solve the problem [3]. In this paper three different techniques for the modal analysis has been tested on the operational data, two work in the time domain, Ibrahim and the Least Square Complex Exponentials (LSCE, Prony s solution), [3,4,] the third one, the Poly Reference Least Square Frequency Domain (known as POLYMAX), works in the frequency domain. The starting point of all these methods is that the free response of the structure can be obtained from the autocorrelation of the measured time histories, this applies as true if the excitation is an uncorrelated random noise with white spectra features. The time-domain techniques, such as LSCE and Ibrahim methods, are based on the structure free response analysis; this one is obtained from the recorded time history autocorrelation, a function defined in the time domain. The methods tries to reconstruct the time histories as a sum of exponential functions. The PolyMAX is a Frequency-domain method that can be successfully used in Operational Modal Analysis [6], it works on the auto spectra and on the cross spectra of the free response time histories, taken at different measurements points. Among the set of considered histories, one or more of them are taken as a reference point and the others as response points. The method operates in the frequency domain in a least square sense [6]. For all the considered methods the same data management has been applied to the raw time data, to compute the auto/cross correlation and the auto/cross spectra to be then used in the next steps that are the identification techniques..1 Data pre-processing Data gathered during all the considered events are affected by a certain amount of electrical noise, moreover the excitation given by the crowd is not a perfect white noise, equally distributed in the frequency domain. These two facts leads to some problems in estimating the free response of the structure by means of the time-history auto-correlations. According to literature [3,4] leakage reduction is one of the critical aspects in operational modal analysis techniques. One of the common ways to minimize the problem consists in

5 applying a proper weighting to the free response, obtained from the correlation of two signals under white noise input hypothesis; to this purpose one of the classical time windows (Hanning etc..)is usually adopted. This operation modifies the original signal, smoothing the side ends of the considered time window and so limiting the leakage, but introducing an unknown bias error in the estimation of modal damping parameters. Peeters et al. [6] proposed to apply an exponential window to the correlation (free response) before computing the DFT (in the PolyMAX method): the advantage of this technique is that the exponential window introduces a known modification of the modal damping. In this way the pole estimation can be corrected, on the contrary this is not possible using other kinds of windowing. Other problems, related to the noncorrelated noise, have been reduced by splitting the global acquired time history in section lasting 80 s. Every section has been auto or cross correlated, obtaining the free response of the system, the last s have been truncated (since no useful information are there present) maintaining a time history of s in order to grant frequency resolution of 0,02 Hz in the spectral analysis. The so obtained free response time histories have then been weighted with the above mentioned exponential window and the last step consisted in averaging. 6. Results First data analysis relying on the use of all transducers is presented; the main structure resonances are evidenced and a first description of the corresponding modal shapes is hypothesized. From these results the best positions for the permanent monitoring system are chosen and analysis is repeated with the mentioned identification methods to find out which technique gives the best results with a reduced number of accelerometers. 6.1 Identification of the modal shapes and parameters with different methods using all the collected data The first analysis is aimed at defining the modal shapes and the corresponding resonant frequencies together with the related damping. A first comparison between different techniques is also presented. Data considered in this section have been recorded during the half-time break of an important match which took place in the last football season. It has been chosen to analyze data coming from a football match as this will be the kind of measurements mostly available for the permanent monitoring system to be installed. The first proposed comparison is about the two time domain methods, Ibrahim and Prony. Figure shows the stability diagrams obtained using these two methods, Ibrahim on the right and Prony on the left. The first technique has been worked using all the possible transducers at the same time, while the second was used just on the data recorded at the measuring point A ( at 1/3 L, see Figure 4). To clarity purposes the spectra of the analyzed data are superimposed on the diagram showing the identified poles at each increment of the model order used in the identification. The stability diagrams are made up by using of a series of 3 markers, depending on the stability type as summarized in Table 1: Marker Red Blue Black Stability Frequency and damping ratio are stable Only frequency is stable Unstable Table 1 Stability diagram markers The Prony diagram shows a robust identification of frequencies; on the other and the Ibrahim diagram show that, at this step of the research, this method does not guarantee adequate performances in the practical considered case. It has to be underlined that literature [] specifies that this method offers the best performances when the sampling frequency is about 0 times the frequency that has to be identified; in our case this would have meant to adopt a sampling frequency of about 800 Hz with all the related problems given by data size and management.

6 Ibrahim: Identified Poles 3 - Modes Milan - Inter Prony: Identified Poles 3 - Modes Milan - Inter 3 3 Modal Order 2 Modal Order Figure Comparison of the stability diagrams regarding Prony (left) and Ibrahim (right) methods; analysis of data from a football match Due to the above explained reasons, attention at this stage will be focused on the Prony method only, if time domain identification techniques are considered, and the PolyMAX method in the frequency domain; it is Mean-Square Error Prony Milan-Inter x Mean-Square Error Result considered in the analysis 2 0 Modal Order Figure 6 Mean-Square Error of the reconstructed free response time history and the real signal planned to further investigate the behavior of the Ibrahim method, changing the sampling frequency according to the values suggested in literature ore relying on over-sampling techniques, well developed in telecomm research. Analyzing the data acquired during all of the 04/0 football season 3 main resonant frequencies appeared in all events, for the considered structure. The identified frequencies for these resonances were about 1.1, 3.1 and.1 Hz; these values remained more or less constant during all the season. Some researchers [7] have pointed out as external factors (i.e. temperature) influence the natural frequency values: this has a fallout on the frequency resolution to be chosen and on the threshold to be fixed by SHM techniques to assess damage. On the contrary the estimated damping ratios present a wider spread (in the worst case about 1%) depending on the different monitored event: it has been often discussed [2,8] as the crowd on the stands has an important but not controllable role in the damping definition; to reduce this uncertainty the method comparison will be about a single football match: Milan-Inter held on 0/04/06. This match has been chosen due to its relevance, Milan and Inter are both teams from the city of Milan, and so all the stands were filled with people. The analysis has been performed for other matches and the evidenced behavior has a high repeatability (although it has been observed that reproducibility, in the metrology meaning, is not so good; it has to be remembered that this latter term addresses to a change in the test conditions, i.e. the match, while repeatability refers to the same testing conditions). The analysis performed with the above mentioned methods on the first 3 natural frequencies is shown in Table 2: the identification of the natural frequencies and the related damping ratios is presented; every parameter is expressed in terms of mean and standard deviation (spread) of the results, considering the poles identified as stable ones, while increasing the modal order. For every considered model order a couple of parameters (frequency and damping ratio) have been calculated, not all of them have been considered in computing the mean and standard deviation: the choice of the kept values has been performed by looking at the mean-square error between the measured function and the reconstructed one: only when the error trend reaches a minimum and becomes constant the correspondent parameters are considered, in the case related to Figure 6 only the parameter obtained by the analysis performed with a modal order greater then are considered. Natural Damping ratio frequency Prony LSCF Prony LSCF (Milan Inter) Mean Dev std Mean Dev std Mean Dev std Mean Dev std I x x % 2.4 x % 2.2 x -1

7 II x x % 3.9 x % 2.7 x -1 III. 2.0 x x % 2.6 x % 2.7 x -1 Table 2 Modal parameters of the 3 ring computer by different parameter estimation methods using all the data available for the football match Milan Inter Model order It has to be underlined that the PolyMAX method allow to identify all the resonances performing a single analysis (Figure 7) that involves all the measured points, on the other hand, identification performed by means of the Prony method brings to different results, depending on the channel chosen for the analysis. The analyzed time history is obtained by an auto/cross correlation of 2 signals: a generic mode identification fails if at least 1 of the 2 accelerometers is located at a nodal point for that mode. To explain this, in Figure 8 an analysis of the same event is presented, considering two different cross correlations. It can be seen, in the left side of the picture, that the identification of the.1 Hz resonance fails. On the contrary, the first natural frequency (1.1 Hz) is correctly identified when the modal order is high enough even if it is not visible in the cross-correlation. The above mentioned consideration put into evidence that, when using the Prony method, a careful evaluation of the chosen points has to be performed for a proper identification. This can be done either by knowing the modal shapes to be identified a priori, or through an extensive analysis of all the measured signals PolyMAX Frequency Hz] Figure 7 Stability diagram performed with PolyMAX technique, all the measuring points have been considered Prony Prony 3 3 Model order 2 Model order Figure 8 Prony analysis performed with different transducers, on the left horizontal A and C (1/3 L) are considered, on the right horizontal and vertical A To the aim of reducing the number of transducers without missing any important information, an accurate description of the modal shapes is therefore necessary to understand which one of the measuring point is suitable to identify all the important resonances. For this reason a traditional modal analysis of the stands has been performed using an electro mechanical shaker giving inertial excitation. The number of available measuring points, even if not dense enough to perform a full modal shape description, allowed to give a first description of the three main identified resonances.

8 -1 Amplitude ratio Phase [deg] Figure 9 Hypothesis of the foremost modal shapes Figure Example of an Frf from the traditional modal analysis performed (point A at 1/3 L) The first modal shape (1.08 Hz) is quite certain: a rigid spin of whole structure around a center placed a few meters below the supporting girder; the other two shapes came out to be harder to understand, considering the relatively small number of measuring points. The first natural frequency has not been analyzed in the traditional modal analysis, due to the difficulty to excite such a low frequency even under resonance conditions, with the adopted shaker; all the same this resonance was actually the easiest to be identified, even working under operational conditions. At higher frequencies the shaker allowed to obtain good results, producing a better-defined exciting force component: an hypothesis of the modal shapes relative to the resonances at 3.1 Hz and.1 Hz is presented in Figure 9 (considering results from the data collected in the measurement points, indicated by bold arrows, and also the mirror symmetry of the stand): the most sensitive measuring position is located at 1/3 of the of the box girder span, at the lowest end of the grandstand, in this location all of the three natural frequencies are present with an important vibration amplitude. It has to be underlined that no further important resonances have been revealed by the traditional modal analysis (Figure ) confirming the operational results. Now all the measuring data points have been exploited, however in the definite monitoring system it is not planned to employ more than 1 measuring station (two-axis accelerometer) on each substructure ( couples of transducers are needed for the 3 rd ring only). So, in the next subsection results performed by means of the same operational modal analysis techniques (both in time and frequency domain) will be exposed considering data coming from 1 measurement point only. A comparison of the two situations is fundamental for a proper design of the monitoring system and of data management. 6.2 Identification of the modal shapes and parameters with different methods using only one measuring point data Only data acquired at point A (1/3 L, Figure 4) have been considered at this stage; analysis has been performed with both the Prony and PolyMAX methods. Results are expressed in terms of mean frequencies, mean damping coefficients and dispersion (see Table 3). Natural Damping ratio frequency Prony LSCF Prony LSCF (Milan Inter) Mean Dev std Mean Dev std Mean Dev std Mean Dev std I x x -3.% 1. x % 3.2 x -1 II x x % 3.6 x % 1.7 x -1 III. 1.3 x x % 1. x -1.1% 3.2 x -1 Table 3 Modal parameters of the 3 ring computer by different parameter estimation methods using only the data acquired in one measuring point during the football match Milan Inter In this case the two methods offer similar performances in terms of data spreading and frequency identification. Anyway, looking at the produced stability diagrams (Figure 11) Prony s method seems to guarantee a clearer and most stable pole identification as the model order increases; on the other hand the PolyMAX method increases the pole dispersion, while increasing the modal order.

9 PolyMAX Prony 3 3 Model order 2 Model order Figure 11 Comparison between Prony and PolyMAX method performed with the signal of 2 accelerometers The same data have been submitted to the Ibrahim method, giving the stabilization diagram shown in Figure 12. The only clearly identified resonance is that at 1.1 Hz; this is a confirmation of the poor effectiveness provided by this method, if applied to this particular case, unless some further care is adopted; but this means a waste of high sampling performances. 3 Ibrahim: Milan - Inter Modal Order Figure 12 Stabilizitation diagram given by Ibrahim method for one single measurement point 7. Concluding Remarks1 Three methods have been considered to identify the stadium modal parameters using operational measurement data. Only the LSCE and the Polyreference Least Square Frequency Domain (Polymax) seem to be reliable under all conditions and offer easy to interpret results. The Ibrahim method produces unclear stabilization diagrams, and seems to confirm the need for an higher sampling frequency to be as efficient as stated in literature. The Polymax method is well suited to be used when many measuring points are available making it possible to exploit all the measured data in one analysis. In this case the LSCE is very sensitive to the chosen measurement location on the structure: in addition data analysis is timeconsuming. On the other hand, concerning the G. Meazza vibration data once chosen a proper measurement point, it seems that the LSCE method offers a clearer way to identify the structural modes, producing stabilization diagrams that are easier to understand. 8. References 1. A. Caprioli, A. Castellani, A. Cigada, M. Vanali, VIBRATION MONITORING OF THE G. MEAZZA STADIUM IN MILANO DURING CONCERTS AND FOOTBALL MATCHES; IMAC XXIII, 0 2. P. Reynolds, A. Pavic, Z. Ibrahim, CHANGES OF MODAL PROPERTIES OF A STADIUM STRUCTURE OCCUPIED BY A CROWD; IMAC XXII, 04

10 3. N. M. M. Maia, J. M. M. Silva, THEORETICAL AND EXPERIMENTAL MODAL ANALYSIS; D. J. Ewins, MODAL TESTING theory, practice and application; 00. P. P. Sarkar, NEW IDENTIFICATION METHODS APPLIED TO THE RESPONSE OF FLEXIBLE BRIDGES TO WIND; PhD thesis, B. Peeters, H. Van der Auweraer, P. Guillaume, J. Leuridan, THE PolyMAX FREQUENCY-DOMAIN METHOD: A NEW STANDARD FOR MODAL PARAMETER ESTIMATION?; Shock and Vibration 11, 39-9, B. Peeters, J. Maeck, et al Dynamic Monitoring Of The Z24-Bridge: Separating Temperature Effects From Damage (00) (Make Corrections) 8. P. Mohanty, P. Reynolds, A. Pavic, STATISTICAL ANALYSIS OF ONLINE RESPONSE DATA OF A STADIUM STRUCTURE; IMAC XXIII, 0