## 2 The Approach to Stability and the Corresponding Model  
The stability studies discussed in the present paper, originating from the project specified in the Acknowledgements section, represent, according to a general view, the bifurcation approach to stability. Within it, a simulation method for studying nonlinear lateral stability is applied. Such a method is beneficial when considering rail vehicle systems (vehicles) and their entire dimensions, when one prefers to consider the entiresystem's degrees of freedom (DoFs) rather than considering one or two degrees of freedom for a single wheelset. The latter may be an alternative for some authors. Its advantage is the use of analytical methods. These can be applied to small systems only. At the same time, numerical simulations are helpful no matter what the dimensions of a given system are. On the other hand, authors have used an original approach with the method of studying the lateral stability on CCs, as elaborated in [5,6]. With this method, an ST is taken as a CC with an infinite curve radius of \(R=\infty\). Vehicle velocity \(v\) is typically the bifurcation parameter, as it is here. The authors' original bifurcation diagrams are utilised, providing a convenient form for presenting the results. These diagrams are bipartite, where the maximum absolute value of the leading wheelset's lateral displacements \(\,|y_{lm}\,|\,max\) versus velocity and the peak-to-peak value of \(y_{lm}\) versus velocity [5, 6, 7] are represented correspondingly. When jointly presenting these coupled values for all the radii values \(R\), one obtains two complex bifurcation diagrams. The authors call such a pair of bifurcation diagrams a stability map [5,6]. The first stability map in this paper is Figure 5, which can serve as an example. Following the bifurcation approach and their method, the authors needed simulation models for a whole vehicle with many DoFs for their studies. Consequently, a rail vehicle model with Y25 bogies was also created and tested. As mentioned before, its first wheelset's lateral displacements \(y_{lm}\) were chosen for observation. In fact, a model of the whole vehicle-track system was created. The authors used the VI-Rail engineering software for this task. In it, type I Lagrange equations are used to build equations of the system dynamics. All the inertia terms for motion on a curved track are taken into account. The main nonlinear elements within the models are those arising from the nonlinear kinematics in the curved tracks, the nonlinear contact geometry, and the nonlinear contact forces in the wheel-rail pair. The suspension parameters were assumed to be linear. The VI-Rail software is a well-established commercial product of high credibility. Nevertheless, the authors verified it when the project mentioned in the Acknowledgement section was realised. This was a kind of benchmarking approach. The simulation results from the commercial software were contrasted with the results from software built by the authors, according to which complete insight into the program code was made possible. The results of these comparisons appeared satisfactory. The model built is a discrete model of a cargo gondola car type 412W Eaos (see Figure 2). Its bogie models represent a Y25 construction (alternatively named 25 TN in Poland). A total of 15 rigid bodies make up the model of the 412W gondola. These are a loading space (a gondola), two bogie frames, four wheelsets, and eight axle boxes. The connections between the rigid bodies are flexible elements, with linear and bi-linear characteristics for stiffness and dumping. The suspension structure of the Y52 bogie comprises dry friction dampers. However, in order to avoid the non-smooth problem in modelling dry friction (stick-slip effect in the suspension), viscous dampers are applied in the model. Furthermore, each dry friction damper in the Y25 bogie acts vertically and laterally. Thus, to describe the damping forces, a two-dimensional model of dry friction should be considered [30, 31, 32].