Chapter 7 Application of recursive SSI algorithms in damage detection and
7.1.2 El Centro 100 gal
Consider the application of RSSI-COV algorithm to track recursively modal frequencies when the structure is no longer excited by a white noise but by an earthquake. Although the white noise assumption is violated, it can be overcome by increasing the moving window length and the block row number as proved before, besides, is it known that the structure vibrates under its own natural frequency when it is subjected to earthquake ground motion.
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Firstly, the recursive identification is applied to the cases when there is no sudden release of the stiffener but the frame is subjected to El Centro earthquake excitation.
The same window length of 2000 points and 100 block rows are used for both cases with brace and without brace. Two different choices of system order are considered:
firstly, 3 pairs of singular values is selected and later, this is increased to take into account more modes. The outcome is shown in Figure 7-7.
With the system order defined as 6, only the first three dominant modes can be extracted. From the plot shown in Figure 7-7 a), the 3rd X-translational modal frequency (5.2736 Hz) has shifted to the 2nd torsion mode (7.3282 Hz), the reason is that the signal power of the 3rd X-translational mode is lower than that of the 2nd torsion mode as that shown in the spectrogram of Figure 7-8 a), constructed with Short-Time Fourier Transform (STFT). Once the system order is increased to 14, both the three translational and torsion modes are identified, however, spurious modes also appear in the diagram. On the other hand, three X-direction translational modes dominates in NB case throughout the time history, this is also verified by the spectrogram shown in Figure 7-8 b). The torsion modes are not well excited in this NB case as that shown in Figure 7-7 d).
Consider now the cases when the brace is suddenly removed at 14.75 and 29.41 seconds. As the brace is removed at 14.75 seconds, the moving window length should be less than 2000 points (10 seconds) to leave a sufficient time length before the release of stiffener. Although a larger moving window length can enhance the tracking stability, it also implies that the algorithm will takes more time to detect system change. To make possible a faster detection of the instantaneous stiffness reduction, a moving window length of 1000 points (5 seconds) is adopted. The number of block rows is 100.
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Figure 7-9 shows the result for both RSSI-COV and rSSA-SSI-COV. The moving window length for rSSA is set to be 500 points (2.5 s) with the same number of block rows than RSSI-COV, and 6 principal components are extracted using rSSA. With a system order of 6, the RSSI-COV result for the brace removed at 14.75 seconds is shown in Figure 7-9 a), the change in the three translational modal frequencies in X direction was correctly traced, the unique drawback is that, even using a quite short moving window length, the sudden drop in frequency is delayed about 3 to 4 seconds to be reflected completely in the modal frequency trace. Moreover, the 1st modal frequency lost its stability after 40 seconds. From the spectrogram shown in Figure 7-10 a), the amplitude of the 1st translational mode was decreased after about 39 seconds and hence the 1st torsion mode takes place instead. The application of rSSA algorithm before RSSI-COV has enhanced the tracking stability as shown in Figure 7-9 b), the orthogonal projection performed in rSSA as a signal filter helps to RSSI-COV to extract signal component consistent with those modes has been traced.
From Figure 7-9 c) and d), when a higher system order is used, all excited modes are also revealed. There is a high frequency mode (about 16 Hz) appearing just after 14 seconds, which is the stiffener mode comparing with the off-line identification result.
The sudden release of the stiffener changes not also the translational modal frequencies in X direction, but torsion modes and coupling modes are also excited by this event although their contribution is much smaller than the dominant modes. This can be understood by comparing the singular spectrum between different time instants as shown in Figure 7-11: a) at the beginning there are only three modes clearly excited; b) once the stiffener is release, not only translational modes but torsion modes and coupling modes appear. It is also interesting to observe what is occurring in the segment from 35 to 45 seconds in Figure 7-9 d), the 3rd mode cannot be traced even with a
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higher system order because it is no longer excited as that shown in Figure 7-10 b).
This is very clear by comparing Figure 7-11 c) with a), there is only one pair above 102 in c), meanwhile there are three pairs in a) above the quantity. On the other side, Figure 7-9 e) and f) shows a comparison between RSSI-COV and rSSA-SSI-COV. Although rSSA was able to enhance the tracking stability, the coupled mode at 8.1 Hz appears instead of the 3rd X-translational mode after approximately 40 seconds.
The three translational mode shapes identified from the case where the brace is removed at 14.75 seconds are shown in Figure 7-12. These mode shapes are extracted from data point of 10 and 30 seconds, the first with brace and the second without brace.
For the another case where the brace is removed at 29.41 seconds, examples of mode shapes are taken from 40 seconds and the comparison with their corresponding mode shapes obtained by offline identification are shown in Figure 7-13. The 1st translational mode in X-direction is the same as that identified previously by offline analysis. The coupled X-Y translational mode has some phase difference, also the amplitude in Y-direction is also larger comparing with the offline reference. Torsion modes are similar with the reference, but the 3rd Y-translational mode (8.0540 Hz) obtained by offline identification is now coupled with the 2nd torsion mode, and the frequency is slightly higher: 8.2017 Hz.
As conclusions obtained from El Centro earthquake, although the RSSI-COV is able to track modal parameters and the pre-processing with rSSA can enhance the tracking stability, there are several challenges to face in use of recursive subspace algorithms. Besides the time delay to show up the system change due to the use of a moving window, it is hard to determine the system order, thus, there is an uncertainty about the total number of modes can be excited over time. To leave nothing out, usually
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the system order is defined in a way that even for modes with insignificant contribution can be considered, such as coupled modes. The fact leads to several consequences, first is that, as that shown in this case of El Centro earthquake, some modes are excited in some time periods and disappears in another periods, but once the mode disappears the trace goes to another mode or is just converted into a spurious mode, making quite confusing the time-frequency plot.
The second is, simpler models are required for online damage detection. As in this case of the 3-story steel frame with instantaneous stiffness reduction in X-direction, usually the sensors will be placed only in X-direction since the frame is symmetric and its structural dynamics are dominated by the X-direction translational modes. However, when the coupled modes engaged in due to the selection of a higher system order, it is possible to identify more than one modal frequencies for each mode as that occurs in Figure 7-9 d), i.e., coupling modes also appear. Since there are only sensors in X-direction, there is no way to distinguish coupled modes from true translational modes.