PSI - Issue 64

Claude Rospars et al. / Procedia Structural Integrity 64 (2024) 716–723 Rospars & al./ Structural Integrity Procedia 00 (2019) 000–000

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interaction between the two lanes. Such interaction between statically decoupled railway bridge spans is also found experimentally in Liu & al. (2009). The difference between the natural frequencies, damping ratios and mode shapes of the first mode of the bridge calculated from the intervals I 1 and I 2 can be explained by two factors. Firstly, although it was neglected in our theoretical analysis, the train inertia has some influence on the modal parameters of the bridge. This assumption is coherent with these results, as the train’s inertia should lower the whole system’s natural frequencies, and its suspensions should increase the damping ratios. Secondly, because the amplitude of Mode 1 is very different between intervals I 1 and I 2 , non-linear effects could play a significant role in these discrepancies. This second explanation accounts for most of the natural frequency differences, as can be seen in Figure 4. 4.2. Visible nonlinear behavior of the bridge As mentioned earlier, the instantaneous modal frequencies calculated from the CWT analysis show significant variations over time. For a linear system, they should be constant and equal to their natural frequencies, so the bridge exhibits non-linear behaviour. Moreover, since these variations are present at the interval I 2 when the train is no longer on the bridge, they cannot be attributed to any interaction effect that could have been neglected in our model. In order to characterise these non-linearities, the instantaneous frequencies of the first three modes, calculated from the ridges extracted during the CWT procedure, are plotted as functions of their amplitude A in Fig. 5.

Fig. 5. The nonlinear behavior of the bridge: Amplitude-frequency graphs, (a) for mode 1 and (b) for mode 2, for each passing train.

We suspect that these nonlinearities (Fig.5) may be due to a contact problem, either between the steel girders and the concrete, or between the bridge and the abutment. Due to the age of the data, which is over twenty years old, the causes of this non-linear behaviour are not investigated further in this paper. However, a similar pattern of amplitude dependent natural frequencies was observed on several railway bridges in Rebelo & al. (2008) . 5. Conclusions This paper presents a detailed procedure, based on the continuous wavelet transform, for analysing the dynamic response of railway bridges when crossed by trains with periodically spaced axles. It is shown that a good understanding of the theoretical shape of the signal is crucial for a valid modal analysis of the structure, for two reasons. Firstly, the signal must be correctly partitioned as it is composed of different time intervals which contain specific modal information. Incorrect or omitted partitioning can lead to unexpected edge effects in the case of CWT analysis and excessive leakage in the case of Fourier transform. Secondly, the frequencies resulting from train excitation must be clearly identified and separated from the natural frequencies of the system. Otherwise, spurious modes may be detected. The correct use of CWT for this type of signal is also discussed in this paper from Carpine's PhD thesis (2022). The mother wavelet must be correctly selected and tuned to obtain accurate results. A brief comparison is made