## 3 The Method of Research  
Analysis of car model solutions, performed for the range of velocity \(v\) and its constant value in each simulation, is the essential element in the authors' research method. To explain the wheelset oscillations (its hunting motion), the theory of self-exciting vibrations is used. The range of vehicle velocity \(v\) starts, e.g., at 10 m/s, while its end is determined by the maximum value at which stable solutions of the model still exist. Generally, the model parameter observed in studies is the leading wheelset's lateral displacement \(y_{luv}\) in time or distance. The term stable solution is adopted when describing the \(y_{luv}\) solutions for constant or periodic (of a limit cycle nature) values. The same criteria for a stable solution on STs and CCs hold in the analyses discussed in this paper. Different from the mentioned forms of solutions are those classified as unstable, although they may fulfil other stability criteria, e.g., technical stability [37]. Example stable solutions, typical for vehicle motion velocity lower than the critical value (\(v<v_{n}\)) and above than critical value (\(v>v_{n}\)), are shown in Figure 3a,b, respectively. The vehicle motion in this example takes place on a route composed of ST, TC, and CC sections. The CC's radius \(R=2000\) m. The wheelset undergoes a lateral shift while negotiating the TC; thus, \(y_{luv}\
eq 0\) on the regular CC. The track's superelevation influences it also. The superelevation values \(h\) applied to the CC tracks for the studied \(R\) values are shown in Table 1. The \(h\) values are constant during the calculations on the CCs for each radius \(R\). That means that at lower speeds, excess superelevation (cant) exists, while at higher speeds, insufficient superelevation (cant) exists. To be precise, the superelevation values provide an ideal balance between the lateral components in the track plane of centrifugal and gravity forces at \(v=34.29\), 41.23, 46.45, 44.90, and 44.77 m/s for \(R=1200\), 2000, 3000, 4000, and 6000 m, respectively. \begin{table}
\begin{tabular}{l c c c c c c} \hline CC radius; \(R\) in m & 1200 & 2000 & 3000 & 4000 & 6000 & \(\infty\) \\ \hline Superelevation; \(h\) in m & 0.150 & 0.130 & 0.110 & 0.077 & 0.051 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: The tested curve radii and corresponding track superelevations. Figure 3: Example lateral displacements \(y_{luv}\) of the first wheelset as a function of distance: (**a**) stable stationary solution (40 m/s \(<v_{n}\)); (**b**) stable limit cycle (periodic solution, 65 m/s \(>v_{n}\)). The studies in [6] and the present research demonstrate that the wheelset's oscillations are a limit cycle, described as that of hard excitation. In other words, initial conditions of a given minimum value must occur for vibrations to start. When studying the stability of the model on CCs, the negotiation of the TC forces a lateral shift in the wheelset on the track, which means it falls out of equilibrium. This way, non-zero initial conditions for solutions on CCs are imposed. For the stability analysis on an ST, a single lateral irregularity in the track serves as the input (equivalent to the initial conditions). Usually, it is located 100 m from the beginning of the test route. Thanks to the negotiation of the irregularity, lateral shifts in all the wheelsets take place one by one. The shifts adopted are typically between 0.005 and 0.006 m. After each simulation, the maximum value of the lateral displacement of the leading wheelset in the first bogie \(y_{lu}\)_max_ has to be read from the final part of the simulation results. \(y_{lu}\)_max_ can be positive or negative (as in Figure 3), depending on the coordinate system orientation and the curve direction. When the curve turns to the left, this corresponds to a negative value. Actually, the simulation solutions for left- and right-hand side curve turns are antisymmetric to each other when the same conditions of motion occur. This enables us to avoid the problem of the sign.