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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The first deck was fitted with eight vertical-axis accelerometers, among other sensors. Acceleration data was gathered using a 16-bit numeric converter with a 4096 Hz sampling frequency, combined with a preliminary analog low-pass filter with a 50 Hz cut-off frequency. For more details see Alvandi & al. (2005). During the measurements, the first deck was crossed by high speed trains traveling at about 300 . ℎ −1 , alone or in pairs. Each train had a total length of 193.15m, with 18.7m long cars and 17m long locomotives at their front and back ends – these lengths are given in terms of axles distribution, since it is what matters in regard to their weight influence on the track.

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Fig. 4. CWT analysis of the first train crossing data: (a) raw acceleration from the eight channels, Illustration of the absolute value of the CWT log scale (b) with Q = 18 for t ∈ I 1 = [ T 1 +∆ T , T 2 - ∆ T ] and (c) with Q = 7 for t ∈ I 2 = [ T 2 , T f ]. A modal analysis of the first three modes of the bridge was carried out using the CWT procedure described in Carpine (2022), for each of the nine samples corresponding to the train crossings. It should be noted that in this case the T f is greater than 55 s, so the edge effect can be neglected. The free response of the first three modes, as high intensity, exponentially decaying horizontal lines in the time-frequency plane, at frequencies around 6.5 Hz, 7.5 Hz and 14 Hz. The first mode, like the second, does not appear to be excited at all, and the third mode interferes with the second super-harmonic of the train excitation frequency 3 ν T . Its fundamental ν T and first super-harmonic 2 ν T also appear as high-intensity, constant-amplitude horizontal lines in the time-frequency plane at about 4:5 Hz and 9 Hz. Each ridge f r , which should be constant for linear systems, shows here a significant variation over time. As the train is already gone on interval I 2 , it can only be attributed to a nonlinearity in the bridge behavior (Fig.4). Amplitude does not follow a perfect exponential decay either, but that is much less uncommon, especially for a signal that spans more than two orders of magnitude. This is discussed further in part 4.3. Modal parameters, i.e. natural frequencies, damping ratios and mode shapes of the bridge can be plot easily. For the 7th train on interval I 2 , the ridge extraction for ν 3 was impossible because the mode did not seem to be excited at all. The similarities between the first two modal natural frequencies and mode shapes in Carpine (2022), that seem unexpected, can be explained by a coupling of the two decks - as can be seen in these mode shapes, the bridge’s abutment is not completely motionless, allowing for an