## 2 Materials and Methods
### Beam Model for Rigid Catenary  
In this work, the rigid catenary is modeled as a flexible beam. To derive a model for the catenary, the finite element method is used, as explained in references [22; 23]. As the flexible beam is a simpler structure, the stiffness matrix is a band matrix with fewer elements which has a positive impact, not only on the computational cost in terms of calculation, but also in terms of storage, as it is treated as a sparse matrix.  
The total length of the catenary is divided into a series of segments that behave as beams with flexural rigidity, according to the Euler-Bernoulli equation. In the discretization,  
Figure 1: Interaction of one pantograph with two catenaries.  
each node presents two coordinates corresponding to the displacement and rotation, resulting in matrices of 4 \(\times\) 4 for each element, according to Figure 2.  
A punctual mass matrix is assumed for the masses. Thus, the components of the coordinates and matrices for the beam element are:  
\[q_{c}=\left(\begin{array}{c}y_{i}\\ \theta_{i}\\ y_{j}\\ \theta_{j}\end{array}\right),\;\;M_{c}=\frac{\pi}{2}\left(\begin{array}{ cccc}1&0&0&0\\ 0&\frac{t^{2}}{2}&0&0\\ 0&0&1&0\\ 0&0&0&\frac{t^{2}}{2}\end{array}\right),\;\;K_{c}=\frac{Et}{L^{3}}\left( \begin{array}{cccc}12&6L&-12&6L\\ 6L&4L^{2}&-6L&-2L^{2}\\ -12&-6L&12&-6L\\ 6L&2L^{2}&-6L&4L^{2}\end{array}\right),\;\;R_{c}=\left(\begin{array}{c}- \frac{pL^{2}}{-\frac{pL^{2}}{2}}\\ -\frac{pL^{2}}{2}\\ -\frac{pL^{2}}{2}\\ \frac{pL^{2}}{2}\end{array}\right) \tag{3}\]  
where \(y_{i}\) and \(\theta_{i}\) represent the displacement and rotation in the node \(i\), respectively; \(L\) represents the length of the element; \(m\) represents its mass; \(p\) is the weight of the beam per unit of length; \(E\) is the elastic modulus of the beam material; and \(I\) is the inertia moment of the section. These matrices are repeated for the different elements of the two catenaries. It can be seen that the different terms of the mass and stiffness matrices, and the vector of external loads, are constant in this case. For the catenary damping matrix, Rayleigh damping is assumed according to references [22; 23], wherein the said matrix is a linear combination of the stiffness and mass matrix, per the equation:  
\[C_{c}=\alpha K_{c}+\beta M_{c} \tag{4}\]  
where \(\alpha\) and \(\beta\) are two constants that are determined from two assumed values of the critical damping for two different frequencies of interest. In our case, for 1 Hz, a critical damping percentage of 0.5% is assumed, and for 15 Hz, the assumed percentage is 1%. These values are obtained experimentally by trial and error. The experiment consists of the following. A video recording is made on two points along the line and the movement of the two points after passing the pantograph is recorded. Then, two points on the line coinciding with the real points are considered in the simulation program, and for two frequencies of interest (1 Hz and 15 Hz in this case) various damping percentages are assumed. The results of the simulation are compared with those obtained in the recording; it is then verified that the percentages given in the article are the ones that most closely match the real results.