Operational modal Analysis of the Guglia Maggiore of the Duomo in Milano

Transcription

1 Operational modal Analysis of the Guglia Maggiore of the Duomo in Milano Busca Giorgio, Cappellini Anna, Cigada Alfredo, Vanali Marcello Politecnico di Milano, Dipartimento di Meccanica, Milan, Italy ABSTRACT: Every city has its own symbol, its very heart. In Milano this symbol, with its six centuries of history, is the Duomo. One of the most impressive buildings in Italy, this unique Gothic cathedral continues to awe citizens and travelers with its timeless beauty. A fascinating characteristic is the building process of the Duomo: to date the church is in fact technically still under construction. The conservation of the Duomo is an activity which will never end which requires a continuous upkeep. It involves various phases: the observation, the studying of the decay phenomena and their monitoring, the settlement and the realization of a solution which prevents the time from damaging the monument. There is then need to provide people in charge of the Duomo s Restoration with tools capable to describe the behavior of this structure as time goes by. Among a wide number of static parameter monitoring, another instrument is the use of modal analysis. This paper describes the dynamic analysis that has been performed inside the restoration plan of the Guglia Maggiore aimed at providing a description of the dynamic behavior of this work of art. However the features of the Duomo make it difficult to excite the structure under test with an imposed force. In this work the identification of the modal parameters of the Guglia Maggiore of the Duomo has been performed by means of Operational Modal Analysis, that is without knowing nor controlling the input excitation. Special attention has been paid to the influence of environmental conditions, i.e. wind, people coming and going (daytime and nighttime), on the response of the structure. 1 INTRODUCTION Modal identification from output-only systems has gained a lot of attention from researchers in the last few decades. One of the applications of this kind of analysis is the identification of modal parameters of civil engineering structures (Farrar (1997)). It has been shown that from the dynamic behavior hints can be taken to asses structural integrity or to foresee vibration related issues (Doebeling et al. (1996)). Modal parameters are among the most commonly used representation of structures dynamic behavior. There are basically two main possibilities to evaluate these parameters. These methods are known as EMA (Experimental Modal Analysis) and OMA (Operational Modal Analysis). The first method is generally more accurate but implies the excitation to be known (Ewins (2001)). OMA, on the other hand, assuming the excitation to be white noise, introduces errors in the identification procedure but often suits better civil applications since it does not require the forces acting on the structure to be known or measured (Peeters (2004), Mohanty (2004)). In fact, even if EMA is certainly more accurate than OMA, its application is not always possible or easy on a civil structure. For instance in this specific application, modal analysis of a slender historical structure, it is not feasible to excite the structure with a known force. The modes that mostly influence the structural response are in fact the flexional and torsional ones. Long-stroke actuators with a low frequency range, placed in the hori-

2 2 IOMAC'11 4 th International Operational Modal Analysis Conference zontal plane, should be used to excite the structure properly. The structure configuration makes this practically impossible. The aim of this work is then to provide a first description of the dynamic behavior of the Guglia Maggiore of the Duomo by means of OMA and to identify some of the critical aspects of using this method for the analysis of the structure. A specific attention is given to the environmental factors, that influence the dynamic response, and to the identification methods used. In particular in order to extract the modal parameters and mode shapes, two different identification methods were used. The first considered technique is the well known erence frequency domain method (Peeters et al. (2004)). The second is the Singular Values Decomposition (), proposed by Agneni et al. (2009), a technique specifically developed to identify the modal parameters of coupled mode structures. The Guglia Maggiore, due to its symmetry, is in fact an example of structure with closely spaced poles. In the first section the experimental set-up will be presented and then a comparison between the modal parameters obtained by applying these two methods with different environmental conditions will be proposed. 2 EXPERIMENTAL SETUP In order to evaluate the behavior of the structure by OMA it is necessary to measure accelerations or displacements in correspondence of defined points of interest. The employed sensors should be able to measure low vibration levels due to environmental excitation sources. Therefore, high sensitivity piezoelectric accelerometers, with high signal to noise ratios, were used. This allowed high performances in terms of measured signal quality and in terms of measurement reliability. Table1 shows the metrological characteristics of the used sensors. Table1: Sensors metrological characteristics Sensor Full scale Bandwidth Sensitivity Piezoelectric accelerometer PCB 393B g 0.1 Hz 500 Hz 10 V/g The accelerometers were located on 4 different levels of the Guglia Maggiore. Figure 1 shows the accelerometers location for each level, while Figure 2 shows the levels locations. The positions were chosen in order to have an accurate description of the system behavior and to avoid redundancies at the same time. Level 4 Level 1 2 Level 2 Level Figure 1: Measurement points The first level corresponds to the higher terrace, the second was located along the internal stairs, the third was located on the lower terrace, while the fourth was placed by the bearing wall that sustains the dome. In correspondence of the first and third levels two measurement points were placed. Each measurement point consists of two accelerometers along the x and y directions. The last measurement point was chosen to verify that the lower part could be considered a

3 3 rigid constraint, in accordance with the mathematical model used to simulate the structure behavior. Data were acquired for a few days in order to perform a preliminary analysis and to validate a first dynamic model. Sampling frequency has been fixed to 256 Hz and 24 bits ADCs with built in anti aliasing filters were used. Figure 2: Measurement levels The next section reports the results of the measurements analysis under different environmental conditions and with different methods. 3 RESULTS The available data, as previously explained, were acquired for a few days with the aim of validating a first model of the structure under test. During these tests the only information available were the accelerations in terms of time-histories and the wind intensity. At first, the Power spectral densities (PSD) have been estimated to have a check on the robustness of the acquired data and a first idea of the resonant frequencies. Figure 3 shows an example of PSD obtained with the measured data of all the signals. The analysis shown here considered the first 4 peaks as these were the ones of interest to validate the numerical model that is being developed. Since the Guglia Maggiore is a quasisymmetric structure, two of these peaks, the first, around 1.57 Hz, and the fourth, around 3.76 Hz, turned out to be related to coupled modes. Therefore, 6 modes in total were analyzed, as it will be shown in the next paragraph. Specific sets of data were considered separately for

4 4 IOMAC'11 4 th International Operational Modal Analysis Conference the analysis. In particular, conditions of strong and weak wind and day-night were considered separately. All the sets of data were analyzed with the erence (Peeters et al. (2004)) and (Agneni et al. (2009)) methods. As it can be seen in Figure 3, the vibrations of level 4 are much lower than the others. Therefore this level was not considered in the following analysis. Figure 3: Example of PSD 3.1 Effect of environmental conditions on the analysis With an eye to a future Structural Health Monitoring based on an Operational approach it is important to understand how modal parameters are influenced by environmental conditions. Therefore a preliminary analysis of these dependencies was performed. First of all the dependence of the vibration amplitude on the wind intensity and day-night condition was analyzed. Figure 4 shows the trend of the RMS, referred to the signal measured with the sensor placed on level 1, in position 1 and along the x direction, on the wind RMS. The RMSs were computed on a window of 1000s. The two lines represent day and night conditions respectively. There seems to be a quadratic dependence of the amplitude of vibration on the wind intensity. Furthermore, there is a constant offset in the day vibration level, probably due to traffic, people and so on, moreover day data seem to have a higher spread around the interpolant curve. Figure 4: Dependence of vibration on wind intensity

5 5 Subsequently, the dependence of the modal parameters on the environmental conditions was analyzed. Figure 5 shows an example of PSD around the first peak (1.57 Hz) with 4 different conditions (day-night, strong-weak wind). It is possible to notice that the peak frequency changes much more between day and night than with the vibration amplitude. Even if the amplitudes of vibration strongly depend on the wind intensity, the modal parameters seem to be affected by the night-day more than by the wind intensity. This could be due to a temperature/humidity etc. dependence of the parameters. Furthermore, the PSDs determined with strong wind conditions are generally more clear to be interpreted, better signal to noise ratio. Figure 5: Dependence of natural frequency on day-night and vibration level conditions It is generally noticeable that there is an influence of the environmental conditions on the modal parameters. Therefore, in order to perform a correct Heath Monitoring on this structure it should be necessary to determine an accurate correlation among all these conditions and the structural response. 3.2 Identification of modal parameters with erence method and As previously said, the Guglia Maggiore is a coupled mode structure. Therefore, it is necessary to find out which method is suitable to correctly identify the behavior of this structure. Two methods, erence and were considered in this work, with the aim of comparing the results of a commonly used method with the results of another method, specifically developed to identify coupled modes. From Table 2 to Table 5 it is possible to see the results of the analysis, in terms of natural frequencies and damping ratios, identified using the erence and methods for the 4 peaks (1.57 Hz, 2.4 Hz, 2.55 Hz, 3.77 Hz) for 4 different environmental conditions (night, day, strong wind, weak wind). The erence method was used taking first channels in x and then channels in y as reference. The peaks around 1.57 Hz and 3.7 Hz represent coupled modes. Table 2: Peaks around 1.6Hz Night/day Wind RMS [m/s] Natural frequency(ies) [Hz] Damping ratio(s) %rc P ref x P ref y P ref x P ref y Night Day Day Night

6 6 IOMAC'11 4 th International Operational Modal Analysis Conference In Table 2 it is possible to notice that the always identifies the coupled modes, whereas in this case the erence method does it only in the first case and if the y reference is considered. Figure 6 shows an example of the singular values obtained with the, where the coupled modes are clearly seeable. Figure 6: : singular values Both algorithms confirm that the natural frequencies of this coupled mode change with the environmental conditions. The frequencies identified are similar for the two methods, but there is a dispersion of the damping ratios estimation. Besides, the method usually underestimates the damping ratios if compared to the erence results. Table 3: Peak around 2.4Hz Night/day Wind RMS [m/s] Natural frequency(ies) [Hz] Damping ratio(s) %rc P ref x P ref y P ref x P ref y Night Day Day Night Table 3 reports the analysis results around 2.4 Hz. The frequency identified is almost the same for the two methods and with different environmental conditions. In this case as well the method underestimates the damping ratios. Table 4: Peak around 2.55Hz Night/day Wind RMS [m/s] Natural frequency(ies) [Hz] Damping ratio(s) %rc P ref x P ref y P ref x P ref y Night Day Day Night Table 4 reports the analysis results around 2.55 Hz. Also in this case the frequency identified are similar for the two methods but, in the condition corresponding to the lowest vibration level considered, erence does not identify this mode. Analogous considerations to the previous may be done for the damping ratios.

7 7 Table 5: Peaks around 3.7Hz Night/day Wind RMS [m/s] Natural frequency(ies) [Hz] Damping ratio(s) %rc P ref x P ref y P ref x P ref y Night Day Day Night Table 5 reports the analysis results around 3.77 Hz. As for the first mode, also in this case it is possible to notice that while the always identifies the coupled mode, the erence method does it only in the first case and when the y reference is considered. The method underestimates the damping ratios. In conclusion, while always identifies the coupled modes, the erence method does it only in one out of the five different conditions. Furthermore, in one condition (lower vibration levels) erence does not identify the 2.55 Hz mode. On the other hand it is possible to notice that while the natural frequencies are almost the same for the two methods, the damping ratios are different. The method generally underestimates the damping in respect to the results obtained with the erence method. The damping ratios identified with the erence method little change with the environmental conditions and their standard deviations, calculated in every single identification among the stable poles, are rarely over 1% of the damping ratio itself. On the contrary, the damping ratios identified with the method show a higher dispersion. Therefore, it seems that the damping ratio could be better identified with the erence method. 3.3 Mode shapes Figure 7 and Figure 8 show the mode shapes identified with the method (night and strong wind conditions). Figure 7: mode shapes: 1.54 Hz, 1.57 Hz, 2.4 Hz The Modal Assurance Criterion (MAC) was used to compare the modes identified with the different methods and with different environmental conditions. The modes were considered separately along the x and the y directions.

8 8 IOMAC'11 4 th International Operational Modal Analysis Conference Figure 8: mode shapes: 2.55 Hz, 3.76 Hz, 3.77 Hz Table 6 show the MACs obtained using as reference the modes identified with condition of strong wind and night considering the x vibration direction. Day, s.w. Day, w.w. Night, w.w. Night, s.w. Day, s.w. Day, w.w. Night, w.w. Table 6: modes along x-axis - MAC - reference: night, strong wind 1.54 Hz 1.57 Hz 2.4 Hz 2.55 Hz 3.76 Hz 3.77 Hz Along the x direction, the only case where the result is completely uncorrelated is the one identified with erence in correspondence of weak wind and night (lowest vibration level). It has to be said that the mode having problems has a very low vibration component in the x direction, therefore the signal to noise ratio is highly worsened by the mode shape itself. Table 7 shows the same results when the y direction is considered. The reference is always the identification in strong wind conditions.

9 9 Day, s.w. Day, w.w. Night, w.w. Night, s.w. Day, s.w. Day, w.w. Night, w.w. Table 7: modes along y-axis - MAC - reference: night, strong wind 1.54 Hz 1.57 Hz 2.4 Hz 2.55 Hz 3.76 Hz 3.77 Hz Along the y direction the results are always consistent except for the last mode. In this case the identification completely changes with the identification method and, in case of the method, changes with the vibration level (strong wind and weak wind). The identified mode shape is not clear, see Figure 8, and probably problems related to the S/N ratio are experienced. 4 CONCLUSIONS An identification of the modal parameters of the Guglia Maggiore based on operational modal analysis approach was proposed in this paper. The first 6 modes of this structure were identified using the erence and the methods. It was seen that there is an influence of the environmental conditions and of the identification method on these modal parameters. In particular, the natural frequencies are influenced by the day-night conditions, in some cases the mode shapes are influenced by the identification method and by the vibration level,and the values of damping ratios identified with the method are always lower than the ones identified with the erence method. Other analysis on a wider range of data should be carried out in order to determine the environmental conditions influence on the accuracy of the identification method. Furthermore, a single identification method seems not enough to provide a correct model of the system. Two or more methods should be combined in order to obtain a better approximation of the actual dynamic behavior. Further measurements should be carried out in order to have a wider range of conditions and, hence, a better description of the system behavior. REFERENCES Agnegni, A. Et Al Output-only analysis of structures with closely spaced poles. Mechanical Systems and Signal Processing, 24(2010) Caprioli A., and Reynolds, P. and Vanali, M Evaluation of serviceability assessment measures for different stadia structures and different live concert events. IMAC XXV, Orlando, Florida Caprioli A. and Vanali M Comparison of different serviceability assessment measures for different events held in the G. Meazza Stadium in Milano. IMAC-XXVII. 09/2/ /2/2009, Orlando, FL, USA, 1-8 Doebling, W., Farrar, C. R., Prime, M. B., and Shevitz, D. W Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: A literature review. Tech. Rep. Los Alamos National Laboratory, Los Alamos, NM, USA, LA MS. Ewins, D. J Modal testing: Theory and practice. Research Studies Press Ltd., Hertfordshire, U.K. Farrar, C. R System identification from ambient vibration measurements on a bridge. J. of Sound and Vibration, 205(1), Mohanty, P., and Rixen, D. J Operational modal analysis in the presence of harmonic excitation. J. of Sound and Vibration, 270(1-2), Peeters, B. and De Roeck, G One-year monitoring of the Z24-Bridge: environmental effects versus damage events. Earthquake engineering structural dynamics, 30(2),

10 10 IOMAC'11 4 th International Operational Modal Analysis Conference Peeters, B., Van der Auweraer, H., Guillaume, P. and Leurida, J The PolyMAX frequency-domain method: a new standard for modal parameter estimation?. IOS Press Shock and Vibration, Publisher IOS Press, ISSN (Print) (Online), Issue Volume 11, Numbers 3-4/2004 Peeters, B., Vanhollebeke, F. and Van der Auweraer, H Operational PolyMAX for estimating the dynamic properties of a stadium structure during a football game. Proceedings of XXIII international modal analysis conference, Orlando, Fl, USA, Rebelo, C. et al, Cable tensioning and modal identification of a circular cable-stayed footbridge. Experimental Techniques Vol 34, No 4. Canadian Commission on Building and Fire Codes, User s Guide - NBC 2005: Structural Commentaries (Part 4 of Division B), National Research Council of Canada; Institute for Research in Construction, Ottawa, Eurocode EN1990, Basis of structural design. ISO 10137, Bases for design of structures, Serviceability of buildings and walkways against vibration ISO ,2 Evaluation of human exposure to whole body vibrations IStructE/DCLG/DCMS Joint Working Group, Dynamic performance requirements for permanent grandstands: recommendations for management design and assessment, Institution of Structural Engineers, London, 2008.