###### Abstract  
In this paper, the authors present a mathematical and engineering model to optimally calculate the dynamic equation on the pantograph-catenary interaction when considering a rigid catenary with an overlapping span. The model starts from well-known methods adapted to the special features of rigid catenary. As a result, an algorithm for the integration of a dynamic equation based on explicit methods is provided. Moreover, from this algorithm, a reliable, efficient, and user-friendly software tool called RICATI is developed in order to approach the model to railway-based companies. The results show the usefulness of an application. such as RICATI, to check the behavior of the configuration initially established for a catenary, allowing solutions to be obtained for the problems encountered when simulating the passage of the pantograph (or pantographs), not only for the overlapping span but also for the entire catenary. That encourages us to continue future works. t 1 (2021) 11 August 2  
performance. However, this issue also has special relevance when the interaction occurs with a rigid catenary, as is the case in tunnels and underground roads; however, few scientific studies have been conducted for this problem. In the case of this power system, the element that transmits the electrical energy is not a wire but a rigid rail. Due to its weight, this rigid rail cannot be maintained parallel to the track using tension or by suspending it from another wire with an amount of deflection; rather, it is necessary to increase the number of supports on which it must be suspended to decrease the distance between them. For example, a typical span of a rigid catenary has a length of about 10-12 m, whereas the span for the flexible catenary is around 50-60 m.  
It is well known that, in the cases of rigid catenary, the speed of the units is restricted to maximum values of around 100-120 km/h. A small number of studies have been published for this type of line (see references [19; 20; 21]). Thus, it is an important goal of scientific research to raise this maximum speed limit. In addition, to achieve greater rigidity of the catenary, which may be a cause of a significant fluctuation of the contact force, the sudden variation of this force must be added when the pantograph circulates through the spans of transition between two series of spans. These variations are more frequent because the length of the spans is notably less than in the case of flexible catenaries. No publications in the scientific literature address this problem in a satisfactory and detailed manner. In the case of overlapping spans, it is also necessary to configure the beams with a special slope-shaped geometry that ensures that the flow of the pantograph is as smooth as possible between the sequences of spans. In this study, we addressed the problem of dynamic pantograph-rigid catenary interaction with an overlapping span. This problem has significant practical importance and, as previously indicated, has not been closely studied in the scientific literature. For this purpose, we present a methodology based on the combination of well-known mathematical models. Based on this model, a computer application for the study and simulation of the dynamic interaction of the pantograph with a rigid catenary is developed. This model is applied to two catenary series with an overlapping span and several pantographs in circulation. In the first section of this paper, the mathematical foundations of the model are explained and the algorithm of the developed computer program is presented. In the subsequent section, the computational aspects and results of the simulations are discussed. The main novelty of this work lies in the application of well-known methods to study the dynamic interaction between the pantograph and rigid catenary, and the establishment of a development methodology. Therefore, the methods used in this paper are not novel. However, the application of all of them together to solve the problem of pantograph-rigid catenary interaction is a novelty.