## 3 Results and Discussion
### Time Integration Results
It is also observed that the most unstable mode predicted by the stability analysis (mode 0L2 with fundamental frequency of 386.4 Hz) does not have the most important contribution on the pseudo-periodic limit cycle. This reflects the nonlinear behavior of the force and interactions at the frictional interface, which the stability analysis is not able to represent. It is also interesting to highlight the frequency shift that can be observed in the spectrum from Figure 13, in which the frequency resolution is 0.8 Hz: while the stability analysis, a linear approach, predicted unstable frequencies 386.4, 1028.8, 4762.2 and 5794.7 Hz, nonlinear time integration gave a limit cycle with frequency components at 384.6, 1029.6, 4763.2 and 5792.8 Hz; these frequencies are slightly shifted in order to interact harmonically with each other. Again, this is a feature that the stability analysis alone is not able to represent. Figure 12: FFT of the nonlinear friction force of the periodic limit cycles obtained among all test cases (red: “0L4”, magenta: “0L5”, green: “0L6”, black (curves superpose): all other test cases from Table 3). Figure 14 shows the evolution of the modal projections of two modal-derived test cases, \"0L1\" and \"0L2\". Although both cases lead to a periodic limit cycle dominated by mode 0L2, the modal competition patterns are different in both cases. For instance, in test case \"0L2\", one can observe that, as only the unstable mode 0L2 is excited at the beginning of the simulation, the other unstable modes remain almost at a constant level, they do not interact, giving a rather smooth evolution of the vibrations (cf. phase plot of the contact point, Figure 9). On the contrary, in test case \"0L1\", where a stable mode, 0L1, is excited at the beginning of the simulation, two different stages of the time domain simulation can be clearly identified: first, a transient phase with exponential modal growth, where the slope of the curves (i.e., the modal growth rates) decreases with the order \(n\) of the unstable 0 Ln mode. This behavior is in good coherence with the results from the stability analysis (cf. Figure 3), as the behavior in this phase remains linear-like. Moreover, one can observe that there is a richer interaction between the different modes in this phase, hence giving a more complex phase plot (cf. Figure 9). In the end of the transient phase, saturation is reached and all modes other than 0L2 die out, introducing the second stage, where a stabilized regime is maintained. Figure 15 shows the modal competition of one modal-derived case (\"All_unstable_V1.0\") vs. one physically-derived case (\"V_1.0\"). In both cases, the initial velocity \(\vec{u}_{0}\) at the contact  
Figure 7:0 Comparison of the modal projections of the unstable 0 Ln modes for two modal-derived test cases (\(\gamma\)-scale is shared by both graphs). Figure 7:1 FFT of the velocity of a pseudo-periodic limit cycle (test case “All_unstable_V_2”). point is equal to the sliding velocity between the wheel and the rail, \(\alpha V_{x}\), but the way in which it is distributed into the different modes, in particular the unstable modes differs: in case \"All_unstable_V1.0\", all unstable modes have the same starting point, whereas for the case \"V_1.0\", they start at different levels, as illustrated in Figure 7:2 Although the initial velocity at the contact point is the same, in the first case, a quasi-periodic limit cycle is obtained, whereas a periodic one is reached for the second case.