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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frequency of each component to be estimated directly from the maximum lines of its modulus, or ridges, and its value on the ridges, called the skeleton. Thus, the modal information of each mode can be retrieved from the CWT of the whole signal by using ridge extraction techniques. The procedure for using the CWT is not detailed here, due to the fixed number of pages in the publication and also because it is described in detail in Carpine's thesis (2022), including, for example, the technique for obtaining a single ridge shared by all the channels, which is both more practical and more accurate than having a ridge for each of them. Moreover having a single ridge for all channels allows to smooth out the individual channels irregularities.

Fig. 3. General shape of the deflection on any point of the bridge. In the shaded areas, expressions of the deflection are unknown.

A significant change in the instantaneous frequency of a mode, or deviation of its amplitude from an exponential decay with time, reflects nonlinear stiffness or damping, cf. Vacca & al. (2018). Two important issues are, first, to find a parameter to characterise the time-frequency resolution of the wavelet transform and, second, to account for edge effects. For this purpose, the quality factor Q , which defines the quality of the mother wavelet filter, was introduced by Le & Argoul (2003), and the authors gave a procedure for choosing the optimal quality factor, see also Carpine (2022). The second difficulty is due to the finite time precision of the CWT, edge effects are present at the beginning and end of the signal. These effects are proportional to the time spread ∆t, which is inversely proportional to f due to the dilatation of ψ. The simplest solution used in this paper to control the shape and magnitude of the edge effect is to consider the signal to be zero outside the interval where it is known (zero padding). There are alternative solutions for controlling edge effects other than zero padding, see Le & al. (2004). In order to choose a mother wavelet, four in particular have been studied in detail in Carpine (2022): the Morlet, the Cauchy-Paul, the harmonic and the Littlewood-Paley. A complete analysis of the three first can also be found in Le & al. (2004). The Cauchy-Paul complex wavelet has good uncertainty rather to 1/2 for a high enough value of Q , with a null Fourier transform for negative frequencies. For these reasons, the Cauchy-Paul mother wavelet was preferred in this paper. The Morlet mother wavelet would also have been a good choice. 4. Results and Discussion The proposed methodology for measurements of railway bridges was implemented on existing data, gathered on a French high speed line in 2003 in Alvandi & al. (2005) and Cremona (2003). Modal parameters of the bridge were estimated back then, as it was the purpose of the experimental campaign. 4.1. Case study The studied structure is a railway bridge built in the early 1980s with two independent spans, 17.5m long and 4.84m wide decks, made of concrete with embedded steel beams (Fig.1). Due to a previous analysis of the dynamic parameters of the bridge, a strengthening procedure was carried out in 2003, coupled with the measurement campaign. Acceleration, displacement and temperature data was gathered before, during and after the installation of support rods.