36 S. Gonen et al. 0 5 10 15 20 25 30 35 40 Frequency (Hz) 1.5 2 2.5 3 3.5 4 PSD (g 2 /Hz) 10 -10 0 10 20 30 40 50 Frequency (Hz) 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized PSD 10 minutes 30 minutes 100 minutes Duration (a) (b) Fig. 4.4 (a) PSD of the ambient vibration data and modal frequency peaks. (b) Comparison of the PSDs from 10, 30, and 100 minutes of ambient vibration data estimated. In this study, the Welch method, a Hanning window, and a 50% overlapping are used to estimate the PSD matrix, Gyy. The PSD matrix is then decomposed using singular value decomposition (SVD), as in Eq. 4.1. Gyy ωj =UjSjU H j (4.1) In the SVD of the PSD matrix, Uj is an orthonormal matrix (UjUH j = I) that contains the singular vectors of Gyy(ωj), and Sj is a diagonal matrix containing the scalar singular values. The decomposition of the spectral matrix into a set of auto-spectral density functions enables obtaining the modal properties of a single degree of freedom system. The system’s modal frequencies can be obtained by plotting the first singular values for each frequency and using simple peak-picking, whereas the mode shapes are obtained using the first singular vectors at the corresponding frequency. The damping ratio of the bridge is not investigated in this study. 4.4.2 Results and Discussion Using ambient vibration and free vibration signals is common for identifying the modal parameters of railway bridges. Therefore, the ambient vibration data is used first to obtain the benchmark dynamic properties of the Bridge. Figure 4.4a shows the auto-spectrum plot where the first three singular values are plotted together and the first four modal frequencies are marked. It should be noted that system identification methods, especially frequency domain methods, are affected by the amount and quality of the data. As there are no guidelines to determine the amount of data to be used, a comparative study in Fig. 4.4b showing the normalized PSDs obtained from 10, 30, and 100 minutes of ambient vibration data highlights the differences in the outcome. Using 10-minute-long ambient vibration data, only the first two vibration modes could be identified. Increasing the length of vibration data to 30 minutes led to the identification of one more vibration mode in addition to the two identified using 10 minutes of data. A total of five vibration modes could be identified using 100 minutes of ambient vibration data. Although not depicted in Fig. 4.4b, 60 minutes of data is also used for modal identification, which enabled us to identify four vibration modes. This discrepancy highlights the effect of the length of the data used in the modal identification using ambient vibration data. The first four vertical mode shapes obtained from ambient vibration are presented in Fig. 4.5. In the first two mode shapes, the modal displacements are maximum at the middle spans. In contrast, in the third and fourth mode shapes, the maximum modal displacements are observed at the cantilever spans at both ends of the bridge. Recalling that the maximum accelerations are observed at the two ends of the bridge (Fig. 4.3), it can be argued that the dynamic behavior of the bridge is significantly influenced by the third and fourth modes. Here, it should also be noted that neither of these two modes could be
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