## 6 Instability of Two Moving Masses
In Figure 13a, there are only starting crossings, implying the beginning of a larger unstable region with a higher degree of instability, but for lower \(\mu_{s}\), these values are quite close to \(\alpha_{cr_{1}}\). In Figure 13b,c, there are always two crossings, meaning that they delimitate an unstable interval, and after that, stability is recovered. This also means that in such cases, the rate of instability is generally low: lower than in the region with the starting branch. No irregular situation was found for \(\widetilde{d}=1\) or higher \(\mu_{s}\) than indicated in Figure 13. For simplicity, higher values than \(\kappa_{p}=300\) were not tested because the tendency for such high values is predictable. This is obvious from the fact that the points in graphs of Figure 13 are becoming closer as \(\kappa_{p}\) increases by constant steps. To conclude this analysis, the case with \(\mu_{s}=1\) and \(\kappa_{p}=300\), which was regular for one moving mass and now is irregular, is further examined for \(\widetilde{d}=1;1.25;1.5;1.75\) with \(\eta_{p}=\eta_{f}=0.05\). All instability branches are plotted until \(\alpha=10\) to cover also the region beyond \(\alpha_{cr_{2}}\). The results of this analysis are presented in Figures 14 and 15. In Figure 14a, a is limited by 1.1, which is more realistic. However, to show how complicated instability branches can be, how the number of asymptotes is affected by \(\widetilde{d}\) and the relation to \(\alpha_{cr_{j}}\) and \(\alpha_{per}\) are not preserved; also, the region with higher \(\alpha\) values is shown in Figure 14b,c for different scales of \(\eta_{M}\). These graphs should be accompanied by the associated real-valued \(q\), which are reported separately for each \(\widetilde{d}\) in Figure 15.  
It is worthwhile to compare Figure 14 with Figure 13. If the limit \(\eta_{M}=100\) is imposed, it can be concluded that for \(\widetilde{d}=1.5\) and \(\widetilde{d}=1.75\), there is an unstable interval in the subcritical velocity range. Crossings with discrete \(q\) values reported in Figure 13 do not indicate the minimum \(\eta_{M}\), but this value can be extrapolated. For \(\widetilde{d}=1\), there is no value lower than \(\eta_{M}=100\) in the subcritical velocity range, and for \(\widetilde{d}=1.25\), there is only the starting branch with values relatively close to \(\alpha_{cr_{1}}\). It should also be mentioned that by increasing the level of damping, the situation becomes generally worse [24; 25]. Figure 13: Identification of irregular cases for two moving masses: (**a**) \(\widetilde{d}=1.25\); (**b**) \(\widetilde{d}=1.5\); (**c**) \(\widetilde{d}=1.75\). The numbers in the legend indicate \(\mu_{s}\), starting \(\kappa_{p}\) and \(q\), respectively. The arrow indicates the direction of \(\kappa_{p}\) increase. For better clarity, single values are indicated by markers. An additional letter “S” indicates that this branch is starting without terminating the instability region. Figure 14: Instability lines for case characterized by \(\mu_{s}=1\), \(\kappa_{p}=300\), \(\eta_{N}=\eta_{s}=0\) and \(\eta_{p}=\eta_{f}=0.05\): (**a–c**) parts correspond to different scales of \(\alpha\) and \(\eta_{M}\), the dotted line 0.05 identifies the behavior of a single moving mass (CV2 = \(\alpha_{cr_{2}}\)). (In (b), CV1 = \(\alpha_{cr_{1}}\), FCV = \(\alpha_{fcr}\), CV2 = \(\alpha_{cr_{2}}\). Both CV1 and CV2 are represented by the same dashed line because they have the primary effect on the instability lines. This introduces no ambiguity because, by definition, CV1 < CV2.). Figure 15: _Cont._