## 3 Results and Discussion
### Time Integration Results  
This section discusses the results obtained after performing numerical integration in the time domain of the dynamic equations that describe the system, as presented in Section 2.1.2, and for the test cases shown in Table 3. The physical quantities that will be in the focus of the analysis are the velocity and the displacement of the wheel-rail contact point, as well as the nonlinear friction force applied to the wheel because of the imposed relative velocity between the wheel and the rail; they will be studied both in the time domain and in the frequency domain. Figure 7 shows the evolution in time of the tangential velocity for all test cases: the first 12 cases correspond to the modal-derived initial conditions from Table 3, whereas the last 10 correspond to the physically-derived initial conditions. One can observe that velocity evolves differently in time depending on the initial condition used, which is the expected outcome of nonlinear time integration; however, inside both categories of test cases, some major features can already be outlined from these time domain results, according to the following aspects: final peak-to-peak amplitude, number of stationary regimes reached, overshoot, stabilization time and maximum amplitude. In general, the system seems to be able to reach two primary families of final states: a low-energy regime, with a moderate peak-to-peak amplitude of 0.1172 m/s, and a high-energy one, with an amplitude almost doubled, reaching 0.2042 m/s. Cases \"0L3\", \"0L7\" and \"0L8\" stand out because they display these two types of stationary regimes with a well-defined transition between them. However, a more detailed analysis that will be presented later reveals that this apparent intermediate limit-cycle is not yet stabilized. In other cases, for instance \"All_unstable_V_2.0\", the vibrations directly reach a final high-energy regime. The existence of a low-energy and a high-energy stationary regime, observation previously made on the time history of the velocity (cf. Figure 7), is more clearly visualized in Figures 8 and 9, which show the phase plot (velocity vs. displacement) of the simulations, for the physically-derived and for the modal-derived conditions, respectively. Furthermore, one can observe that the low-energy cycles display a closed, almost elliptical phase plot, which corresponds to a periodic regime with one fundamental frequency and some of its harmonics, whereas high-energy regimes display a rather complex phase plot that corresponds to a quasi-periodic regime where multiple fundamental frequencies and their harmonic interactions coexist. For test case \"All_unstable_V2.0\" (cf. Figure 9), in which the initial velocity (marked with an \"x\") is considerably higher than the maximum velocity amplitude reached during the simulation, the system is still attracted to one of the two observed stationary regimes, namely the high-energy one; the same is true for cases \"U_0.5\" and \"U_-0.5\", which have a non-zero initial displacement, but they are attracted to a low-energy limit cycle. An interesting remark, observed for all simulations, regardless of the initial condition used, is that the mean displacement increases gradually during the transient phase, i.e., the phase plot shifts slowly to the right. For instance, in both test cases U_0.5 and U_-0.5, which start at a non-zero displacement, the system quickly returns to the quasi-static position at \(u=0\), but then the mean value starts to increase until a stationary regime is reached. This is related to the inclusion of the wheelset modes (modes 1 to 3) in the wheel's modal base, which contribute to the quasi-static behavior of the structure, due to their low natural frequency. Figure 7: Evolution of the tangential velocity at the wheel–rail contact point for all test cases from Table 3.  
Figure 8: Phase plot of the wheel–rail contact point for all physically-derived initial conditions from Table 3. Grey: complete history, black: final limit cycle reached, ‘x’: initial condition.