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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in this paper, as will be explained later. The next approach would be to consider the inertia of the vehicle, either as a mass following the path of the bridge, or mounted on suspensions. All of these methods are discussed in Frýba (1972), and implemented in Lin & al. (2005) and then in Cantero & al. (2013). More sophisticated models that take into account the various moving parts of the train or the interactions between the track and the bridge can be used. However, due to their complexity, very few analytical results can be obtained from them and they are typically used for simulations combined with statistical investigations in Cantero & al. (2014). In this paper, the bridge will be modeled as a continuous, one dimensional, damped linear system. Damping is supposed to be low, thus the proportional damping approximation will be used cf. Géradin and Rixen (2015). Its motion equation can be expressed as in Fryba (1972) : 2 ( , )+ ( , )+ ( , )= ( , ), (1) where ( , ) is the deflection of the bridge, , and are the mass, stiffness and damping operators respectively, and ( , ) is the train weight force field. These field and operators are specified on the interval [0 , ] where is the total length of the bridge, and ( , ) satisifes to some linear boundary conditions at 0 and . As mentioned above, the inertia of the train is neglected. Its linear density is much smaller than that of the bridge (by about an order of magnitude), and the train's suspensions are effective enough to prevent any influence of the train's inertia. In fact, as explained later, the lowest frequency encountered in the experiment is about 4 Hz, while the typical high-speed train suspensions have a cut-off frequency of about 0.9 Hz cf. Zeng et al. (2018). In this way, the influence of the train on the bridge can be written as : ( , )= − ( − ), where is the gravitational acceleration, the linear density distribution of the train on the track at =0 , and its constant speed. For example, for a train with equally spaced axles, would be a sum of equally spaced Dirac delta distributions. 2.2. Projection onto the mode shapes of the bridge For linear systems, it is common to use modal decomposition theory and project the above partial differential equation in Eq. 1 onto the eigenmodes ( ) of the bridge. When the assumption of proportional damping is made, these modes are real, and by setting ( ) the generalized displacement of the k -th mode, this leads to: ( , )= ∑ ( ) ( ), (2) Introducing: , and respectively the modal mass, the natural frequency and the modal damping ratio for the k th mode such that: 〈 ( ), ( ) 〉 = , 〈 ( ), ( ) 〉 = (2 ) 2 and 〈 ( ), ( ) 〉 =4 , Eq.1 becomes:

(3)

Fig. 2. Example of the projection of the weight distribution of a train on the first two modes of a bridge. The model taken here is a 10m long bridge with uniform cross-section, which has sinusoidal mode shapes, and a train with 20 equally spaced (3 m) axles, same weight.