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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4

Thus, the bridge’s normal modes are driven by uncoupled equations, that can be solved independently. Next, our attention will be focused on the right-hand term in Eq. 3. Our main hypothesis here is that the train is made up of a series of identical cars, that carry the same weight. That way, on an interval of the train’s length, its linear density distribution on the track λ is a periodic function, so regardless of the shape of ( ) , 〈 ( ), 〉 ( ) is periodic when the train is across the whole bridge (which happens only if it is longer than the bridge). Eq. 3 can be divided as follows:

(4)

where T 1 and T 2 are the moments the train starts entering and finishes leaving the bridge, ∆ T = L / c is the time it takes an axle to cross the bridge , ν T = c / d is the fundamental excitation frequency of the train on the bridge with d the spacial period of λ (distance between two cars), and A k ( p ) and α k ( p ) are Fourier coefficients. The term “-” in the previous equation indicates an indefinite form for the signal. An example of what this term looks like is given in Fig.2. Therefore, by Eq. 5, the shape of z k is also divided into five separate intervals. When the train is across the whole bridge (between T 1 + ∆ T and T 2 − ∆ T ), z k is a sum of harmonic responses to the fundamental frequency ν T = c / d of the right-hand term and its harmonics p ν T ( p, integer ), and a free response of the equivalent 1-DoF system. When the train is completely off the bridge, z k is also a free response. For both time intervals corresponding to train arrival and departure, the general shape of z k cannot be expressed in such a simple manner. In short, z k can be divided as follows:

(5)

where is

the harmonic response of mode k, are the system’s damped frequencies and is the frequency response function of the 1-DoF linear system associated with mode k , and B k , C k , β k and γ k are constants of integration.

Finally, using Eq.3, on any point x 0 of the bridge, the local deflection t → w ( x 0 , t ) will follow the shape presented in Fig. 3. The analysis of the local deflection can therefore be restricted to intervals I 1 =[ T 1 +∆ T , T 2 −∆ T ] and I 2 =[ T 2 , T f ],where T f is the end of the signal.

3. Use of the Continuous Wavelet Transform The CWT is a time-frequency analysis tool well suited to frequency and amplitude modulated signals. In the case of railway bridges crossed by trains, it allows a precise analysis of their modal parameters over time in the different parts of the signal. As discussed later in this paper, it can also be used to detect non-linear behaviour. For a signal made up of a sum of amplitude and phase modulated signals, the CWT allows the amplitude and instantaneous