## 6 Instability of Two Moving Masses  
When two masses are traversing the beam, then the only difference with respect to the previous analysis is the right-hand side of Equation (32), which reads as  
\[4\big{(}2\eta_{P_{1}}-\eta_{M_{1}}\widehat{w}_{,\tau\tau}\big{)}\delta(\xi)+4 \big{(}2\eta_{P_{2}}-\eta_{M_{2}}\widehat{w}_{,\tau\tau}\big{)}\delta\Big{(} \xi-\widehat{d}\Big{)} \tag{45}\]  
with additional parameters introduced as \(\eta_{P_{2}}=\frac{P_{2}}{P_{1}}\), \(\eta_{M_{2}}=\frac{M_{2}X}{m}\) and \(\widetilde{d}=d\chi\). The solution steps can follow the same pattern as before, which allows to conclude that the characteristic equation is now  
\[ \begin{array}{c}\big{(}\pi-2\eta_{M_{1}}q^{2}K(0,q)\big{)}\big{(}\pi-2\eta_ {M_{2}}q^{2}K(0,q)\big{)}\\ -4\eta_{M_{1}}\eta_{M_{2}}q^{4}K\Big{(}\widetilde{d},q\Big{)}K\Big{(}- \widetilde{d},q\Big{)}=0\end{array}  \tag{46}\]  
which for \(\widetilde{d}=0\) reduces to the characteristic equation of \(\eta_{M}=\eta_{M_{1}}+\eta_{M_{2}}\), as expected. To simplify the following analysis, it is assumed that \(\eta_{M_{1}}=\eta_{M_{2}}=\eta_{M}\) and \(\eta_{P_{1}}=\eta_{P_{2}}=\eta_{P}\). Equation (46) is not as straightforward for tracing the instability lines as Equation (44) was. Nevertheless, Equation (46) is quadratic for \(\eta_{M}\); therefore, it can be solved analytically for \(\eta_{M}\), and then two conditions for finding the real-valued induced frequency can be specified as  
\[\sqrt{K\Big{(}\widetilde{d},q\Big{)}K\Big{(}-\widetilde{d},q\Big{)}}-K(0,q) \text{ or }\sqrt{K\Big{(}\widetilde{d},q\Big{)}K\Big{(}-\widetilde{d},q\Big{)}}+K(0,q) \tag{47}\]  
Thus, if one of these functions has a real value for real \(q\), then the corresponding \(\eta_{M}\) can be computed from the associated root, and after that, the negative values can be discarded. After detailed analysis, it can be concluded that when two moving masses are considered, then \(\alpha_{cr_{j}}\) and \(\alpha_{prx}\) are not respected. Any combination of allowable parameters creates situations  
Figure 12: Real-valued mass induced frequency linked to the instability lines from Figure 11 (PCV = \(\alpha_{prx}\), CV = \(\alpha_{cr_{2}}\)). where \(\alpha_{cr_{1}}\) or \(\alpha_{pcr}\) are intersected, and thus, instability occurs in the subcritical velocity range. Additionally, \(\alpha_{cr_{2}}\) does not serve to indicate some position of any of the asymptotes. To identify the irregular cases, it is necessary to state when the instability occurs in the subcritical velocity range for some realistic \(\eta_{M}\). In railway applications, the mass and moving force are not linked by a mass-to-weight relation. Owing to the primary and the secondary suspension, it is a bit ambiguous what mass should be used in such simplified cases. It could be 10 t as a typical force transmitted by one wheel or just 880 kg, which is the wheel mass. Since according to Table 1, the value of \(\chi\) can vary between 0.3 and 2.7 m\({}^{-1}\), then the range for \(\eta_{M}\) is 50-497 for 10 t and 4.5-44 for 880 kg. Therefore, \(\eta_{M}=100\) was selected to be a limit value. To avoid tracing the instability lines for all possible cases, the analysis is performed by identifying crossings with a specific \(q\)-value. In fact, only values as low as 0.1-0.15 are worth studying. Thus, \(\mu_{s}\) was stepped by 0.1, \(\kappa_{p}\) was stepped by 2, real \(q\) was stepped by 0.01 and \(\widetilde{d}\) was stepped by 0.25 starting at 1. The level of damping was selected as \(\eta_{p}=\eta_{f}=0.05\). The tested region was specified by \(\alpha\) spanning from 0 to \(\alpha_{cr_{1}}\) or \(\alpha_{pcr}\). The obtained results are presented graphically in Figure 13.  
Figure 13: Cont. The graphs in Figure 13 are limited on the top by \(\eta_{M}=100\), as justified before, and on the left by \(\alpha_{cr_{1}}\) or \(\alpha_{per}\), which varies with \(\kappa_{p}\) and cannot therefore be directly indicated in these graphs.