194 E. Mola et al. Table 23.3 Percentage differences in frequencies for TO-MI bridge Mode Relative frequencies error (%) 1 1.57 2 1.47 3 2.65 4 2.92 5 3.59 7 7.81 23.4.4 Test Results and Comparison with Tuned Model The comparison carried out in the following is carried out by using the numerical mode shapes derived after the model tuning, which are also the ones represented in Table 23.1. In order to illustrate the similarity between the two sets of modes, the relative frequencies error is calculated in Table 23.3. It is clear that percentage differences in frequencies are very low since they are lower than 8% in the worst case. Modal Assurance Criterion (MAC) is used to assess the correlation between two sets of modes. Herein MAC is used to find the correlation between experimental mode shapes and numerical mode shapes. The formula of MAC is: MACij D ˇ ˇ T is jsˇ ˇ 2 T is is T js js where is, js are values of the ith order and jth order mode shape corresponding to the degree of freedom s respectively. The MAC plot between numerical modes and experimental modes for TO-MI way and MI-TO way are shown in Fig. 23.9. The right color-column indicates the MAC level for each color, where 1 indicates perfect correlation, whereas 0 indicates uncorrelated modes. From Fig. 23.9, the diagonal MAC values are all nearly close to 1, which means that numerical modes and experimental modes are pretty closely coincident. It needs to be noticed that high correlation values are also reported for non-corresponding modes. For instance, experimental modes 1, 3 and 5 of TO-MI way have a good matching with simulated modes 1, 2 and 3. From Fig. 23.8 it is clear that these modes have the same shapes for the tested spans, then this high correlation is explained. In order to see a difference among these modes, the number of the spans under test should be increased. 23.5 Conclusions In this paper the numerical model validation of the bridges MI-TO and TO-MI have been achieved. Considering both results from numerical computation and experimental test, a good agreement between simulation and reality was obtained. The correspondence among the frequency shows a difference lower than 8% for the higher modes and the MAC values among the simulated and experimental modes are close to 100%. Nevertheless, it is rather difficult to distinguish all the modes in the frequency range 1–3 Hz, because the natural frequencies are very close and the modes are highly damped. It must be also noticed that the number of spans used for the experimental tests was insufficient to describe completely the modes shapes. Since the two bridges have all the spans of the same length, the mode shapes are similar. The differences among the modes could have been seen only if all the spans would have been instrumented, but it was impossible due to budget limits.
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