## 4 Discussion  
The inclusion of the slope in an overlapping span may be necessary from a critical length onwards to achieve optimum values of force in the common area. However, in general, for the assumed speed, it is not necessary to introduce a slope height to obtain good results. This is because, in the initial static configuration, the section already presents a certain inclination as a consequence of the action of gravity. However, the length of the section should not exceed a certain value from which the results of the contact force significantly fluctuate. As the authors state at the beginning of this paper, the main contribution of this paper consists in a model for the pantograph-rigid catenary interaction considering an overlapping span. This model is a particular model for this kind of catenary and not an elastic catenary where the stiffness is increased. In order to build this model, the authors start from well-known methods that appear in Finite Element Method literature, such as [22; 23], but particularized for the elements of a rigid catenary and its underlying problems. In fact, according to the literature, the model uses an explicit method as an integration method. The RICATI tool allows the study of dynamic behavior in which new profile designs are considered and compared with existing designs. Therefore, RICATI allows multiple simulations to determine the optimal catenary structure for any configuration. For example, with the same characteristics as the previous simulations, it is possible to vary the speed of the unit to observe the behavior at different speeds. We observed that, at speeds of 80 and 120 km/h, the behavior of the examples is similar. However, at 200 km/h, Figure 15 shows variations in the behavior and indicates two optimal points in the length of the sectioning around 1.2 and 3.9 m.  
This example shows the usefulness of an application, such as RICATI, to check the behavior of the initially established configuration for a catenary. The tool also allows solutions to be obtained for the problems encountered when simulating the passage of the pantograph (or pantographs), not only for the section, but also for the entire catenary. The proposed methodology provides a tool to simulate the pantograph-catenary behavior in a rigid catenary considering the transition between two cantons of the catenary with overlapping spans. To advance the presented method, the following lines of work are proposed:  
* Study of the behavior for different configurations in the geometry of the sectioning span. * Development of the model in three dimensions. * Study of the behavior and simulation in the transition between elastic and rigid catenary. * Optimization of the algorithm, taking into account the data structures underlying the problem, in addition to the parallelization of the calculations, to obtain results in significantly shorter times. * Development of RICATI following the paradigm of Software-as-a-Service. This model of the application in the cloud would allow access by more companies, from any point at which there is connectivity, and at a lower cost. In addition, this solution would allow different tests to be carried out simultaneously by requesting resources from the provider of the cloud service, and significantly more quickly, to find better solutions. In addition, as an important future work, the authors will address the problem of the irregularities in the beam following similar ideas of those described in [30; 31; 32]. **Author Contributions:** [PERSON] contributed to the conceptualization, methodology, investigation, and writing--original draft. [PERSON] contributed to software, methodology, writing--original draft, and supervision. [PERSON] contributed to the software, methodology, validation, investigation, and writing--original draft. P.T. : software, validation, and investigation. [PERSON] contributed to the software, methodology, investigation, and writing--original draft. All authors have read and agreed to the published version of the manuscript.