## 5 Instability of One Moving Mass
\[W(p,\mathrm{i}q)=\frac{1}{D_{1}(p,q)+\frac{D_{2}(p,q)D_{3}(p,q)}{D_{2}(p,q)+D_ {3}(p,q)}}\left(\frac{2\eta_{P_{1}}}{\mathrm{i}q}+\eta_{M_{1}}q^{2}\widetilde {W}(0,\mathrm{i}q)\right) \tag{40}\]  
Now, one can perform the inverse Fourier transform to return to the Laplace image, and by assuming \(\xi=0\), one obtains the following:  
\[\widetilde{W}(0,\mathrm{i}q)=\frac{2\eta_{P_{1}}}{\mathrm{i}q}\frac{K(0,q)}{2 \pi-\eta_{M_{1}}q^{2}K(0,q)} \tag{41}\]  
where  
\[K(\xi,q)=\int\limits_{-\infty}^{\infty}\frac{\mathrm{e}^{\mathrm{i}p\xi} \mathrm{d}p}{D_{1}(p,q)+\frac{D_{2}(p,q)D_{3}(p,q)}{D_{2}(p,q)+D_{3}(p,q)}} \tag{42}\]  
and finally  
\[\widetilde{W}(\xi,q)=\frac{2\eta_{P_{1}}}{\mathrm{i}q}\frac{K(\xi,q)}{2\pi- \eta_{M_{1}}q^{2}K(0,q)} \tag{43}\]  
where without losing generality, \(\widetilde{W}(\xi,\mathrm{i}q)\) was switched to \(\widetilde{W}(\xi,q)\). The final response in the time domain is obtained by the inverse Laplace transform, which in most cases can be well approximated by the sum of residues as a result of contour integration. However, this paper is concerned with the identification of intervals of velocities where the mass movement is unstable, and more specifically, to the identification of the irregular cases. The case with \(\mu_{s}=1\), \(\kappa_{p}=300\), \(\eta_{N}=\eta_{s}=0\) and variable damping levels is selected as an example of the regular, that is, the expected behavior. According to the previous section, this case has three resonances, which are calculated as \(\alpha_{\sigma_{1}}=0.707\)\(\alpha_{fcr}=1.899\) and \(\alpha_{cr_{2}}=4.573\). Without involving the D-decomposition method, instability lines can be directly determined by tracing the real-valued-induced frequency. This simply means to find a real \(q\) that fulfills the characteristic equation  
\[2\pi-\eta_{M}q^{2}K(0,q)=0 \tag{44}\]  
where subscript 1 is removed for simplicity and \(K(0,q)\) is proportional to the equivalent flexibility of the model. Because \(\eta_{M}\) must be real, this implies finding a \(q\) for which \(K(0,q)\) is real, and thus, the full analysis can be kept within the real domain. It can be easily verified that the lowest branch of the instability lines never intersects \(\alpha_{cr_{1}}\) or \(\alpha_{pcr}\); therefore, instability always occurs in the supercritical velocity range. In the regular cases, instability lines are formed by two branches, as demonstrated in Figure 8. The first branch is delimited by \(\alpha_{cr_{1}}\) (or \(\alpha_{pcr}\)) and \(\alpha_{cr_{2}}\), and both ends tend to infinite \(\eta_{M}\). The second branch approaches \(\alpha_{cr_{2}}\) from the left where it tends to infinite \(\eta_{M}\), and the other end tends to zero \(\eta_{M}\) at infinite \(\alpha\). \(\alpha_{fcr}\) is ignored and has no influence on the instability lines. With decreasing damping, the instability lines become closer to \(\alpha_{cr_{2}}\). From Figures 8 and 9, it can be concluded that the left sides of both branches are already quite close to \(\alpha_{cr_{2}}\), and therefore, the decrease in damping does not have much influence. However, the right part of the first branch is highly affected by the damping level. The irregular cases are identified by the fact that \(\alpha_{cr_{2}}\) is intersected by one of the branches. Additionally, there can be a third branch, which in some cases can be closed. It was proven by parametric analysis within the ranges of allowable parameters that the irregular cases only occur for \(\kappa_{p}\) on or below the line plotted in Figure 10. The curve in Figure 10 does not look smooth because the step in \(\kappa_{p}\) was selected as 0.01 and in \(\mu_{s}\), it was selected as 0.1. The line would be smoother for finer steps. As an example, one case selected from the irregular region: \(\mu_{s}=1.1\), \(\kappa_{p}=0.6\) and \(\eta_{N}=\eta_{s}=0\) with variable damping levels is discussed. The instability lines are presented in Figure 11, where intersections with \(\alpha_{cr_{2}}\) are shown.