## 3 Results and Discussion
### Time Integration Results
The modal competition patterns in the transient phase are different for both cases as well: in the first case, mode 0L3 dominates from the beginning, while the other modes die out rapidly; then, modes that present a harmonic correlation, namely 0L3, 0L6, 0L7 and 0L8, suddenly revert their decaying tendency and present an exponential growth until they finally reach saturation around \(t=1\) s, where the vibrations stabilize and give a limit cycle whose signature is the one from Figure 7:3 In the second case, on the contrary, mode 0L2, which has the most negative equivalent damping according to the stability analysis, dominates the modal competition from the beginning and all other modes die out rapidly; this mode alone dominates the limit cycle. These results indicate that not only the global value of the initial condition, in this case the velocity at the contact point, has an impact on the nature of the final limit cycle obtained, but also the initial condition given to each particular mode. Figure 16 shows the evolution of the modal projections of the harmonically-correlated modes for the test cases in which only one of these instabilities is excited in the initial condition (test case \"0L2\" is included for reference). Here, the development of a quasi-periodic cycle is more clearly visualized: while for test case \"0L2\" the modal projections of non-excited modes cease to grow in amplitude and stabilize completely at a certain point, in the other three test cases, the non-excited but harmonically-correlated instabilities are growing exponentially in the background, while an apparent stationary regime dominated by the initially-excited mode is reached. This is why the existence of an intermediate limit cycle, as discussed at the beginning of this section and as shown in Figure 9, could be nuanced by considering that all modes are not yet completely stabilized, which results in an apparent limit cycle that is slowly evolving to the definitive, stabilized limit cycle. From test case \"All_unstable_V1.0\" (Figure 15) and test cases \"0L3\", \"0L7\" and \"0L8\" (Figure 16) it can be pointed out that the harmonic correlation phenomenon between modes in order to give birth to a quasi-periodic limit cycle occurs after saturation of the friction has taken place, so that the nonlinear domain of the friction law is fully established. This implies that in this stage, the modal growth rates of modes such as 0L6, 0L7, 0L8, no longer correspond to that predicted by the stability analysis, which is the case of the linear phase in test case \"0L1\" from Figure 7:4 the slopes of the curves differ. Figure 7:5 Comparison of the modal projections of the unstable 0 Ln modes for a modal-derived test case vs. a physically-derived test case (Y-scale is shared by both graphs). In summary, physically-derived initial conditions lead to only two scenarios: either a periodic limit cycle with a moderate peak-to-peak amplitude, or a quasi-periodic limit cycle with a higher peak-to-peak amplitude. The same two cycles (one periodic and the other quasi-periodic) were always obtained. In addition, the results suggest that the appearance of the quasi-periodic limit cycle is related to initial conditions quite far from the quasi-static equilibrium. On the other hand, modal-derived initial conditions lead to more scenarios: the same quasi-periodic cycle as that obtained with physically-derived initial conditions but several periodic limit cycles with different fundamental frequencies (related to the modal content of initial conditions). Thus, it is worth noting that some periodic limit cycles could only be obtained with modal initial conditions and not with physically-derived ones. Moreover, contrary to the case of physical-derived initial conditions, the quasi-periodic limit cycle can be obtained with initial conditions close to the quasi-static equilibrium but with a specific modal content.