## 5 Instability of One Moving Mass  
To solve the problem of instability of one moving mass, the previously omitted time-dependent terms must be included back; thus, Equations (12) and (13) are now given by  
\[ \begin{array}{c}\widetilde{w}_{\zeta,\zeta\zeta\zeta\zeta\zeta}+4\eta_{N} \widetilde{w}_{\zeta,\zeta\zeta}+4a^{2}\widetilde{w}_{\zeta,\zeta\zeta}+4 \widetilde{w}_{\zeta,\zeta\zeta}+4\widetilde{w}_{\zeta,\zeta\zeta}+8\eta_{p} \big{(}\widetilde{w}_{\zeta,-}-\alpha\widetilde{w}_{\zeta,\zeta}-\widetilde{ u}_{\zeta,+}+\alpha\widetilde{u}_{\zeta,\zeta}\big{)}\\ +4\kappa_{p}\big{(}\widetilde{w}-\widetilde{u}\big{)}=\big{(}8\eta_{P_{1}}-4 \eta_{M_{1}}\widetilde{w}_{\zeta,\tau\tau}\big{)}\delta(\xi)\\ \end{array}  \tag{32}\]  
\[ \begin{array}{c}\mu_{s}\big{(}\widetilde{u}_{\zeta,\tau\tau}-2\widetilde{a} \widetilde{u}_{\zeta,\tau\tau}+\alpha^{2}\widetilde{u}_{\zeta,\zeta\zeta}^{2} \big{)}-2\eta_{p}\big{(}\widetilde{w}_{\zeta,-}-\alpha\widetilde{u}_{\zeta, \zeta}-\widetilde{u}_{\tau,+}+\widetilde{a}\widetilde{u}_{\zeta,\zeta}\big{)} \\ -\kappa_{p}(\widetilde{w}-\widetilde{u})-\eta_{s}\widetilde{u}_{\zeta,\zeta \zeta}+\widetilde{u}+2\eta_{f}\big{(}\widetilde{u}_{\zeta,\tau}-\alpha \widetilde{u}_{\zeta,\zeta}\big{)}=0\\ \end{array}  \tag{33}\]  
Figure 7: Graphs of resonances as a function of \(\kappa_{p}\) in logarithmic scale for discrete values of \(\mu_{s}\) from 1 to 5.5 by 0.5 (ten curves) and \(\eta_{N}=0.2\), \(\eta_{s}=0\): (**a**) \(\alpha_{cr_{1}}\) for \(\kappa_{p}>100\) (Formula (30) is not plotted because it perfectly matches the semi-analytical value; no irregular cases are detected); (**b**) \(\alpha_{cr_{1}}\) (orange), \(\alpha_{f\sigma}\) (blue) and \(\alpha_{cr_{2}}\) (yellow) for \(\kappa_{p}\in\langle 0.03;100\rangle\) (region of irregular cases is clearly identified). where  
\[\tau=\chi v_{ref}t,\;\eta_{M_{1}}=\frac{M_{1}\chi}{m} \tag{34}\]  
In the solution method, the Laplace transform must be applied first and then the Fourier one second. Initial conditions are assumed to be homogeneous. The following definition is used for the Laplace transform  
\[\widetilde{F}(\xi,\overline{q})=\int\limits_{0}^{\infty}f(\xi,\tau)\mathrm{e} ^{-\overline{q}\tau}d\tau \tag{35}\]  
to keep its formalism, \(\overline{q}=\mathrm{i}q\) will be used in the following with \(q\) designating the frequency. In the transformed space, the equations can be again written in the matrix--formally the same as Equation (16) except for the right-hand side  
\[ \begin{bmatrix}D_{1}+D_{2}&-D_{2}\\ -D_{2}&D_{2}+D_{3}\end{bmatrix}\begin{Bmatrix}W\\ U\end{Bmatrix} = \begin{Bmatrix}\frac{2\eta_{P_{1}}}{\mathrm{i}q}+\eta_{M_{1}}q^ {2}\widetilde{W}(0,\mathrm{i}q)\\ 0\end{Bmatrix}  \tag{36}\]  
and also  
\[D_{1}(p,q)=\;\frac{p^{4}}{4}-(q-\alpha p)^{2}-\eta_{N}p^{2} \tag{37}\]  
\[D_{2}(p,q)=\;2\mathrm{i}\eta_{p}(q-\alpha p)+\kappa_{p} \tag{38}\]  
\[D_{3}(p,q)=-\mu_{s}(q-\alpha p)^{2}+2\mathrm{i}\eta_{f}(q-\alpha p)+\eta_{s}p^ {2}+1 \tag{39}\]  
which shows that whenever there was \(-\alpha p\) in Equations (17)-(19) is now \(q-\alpha p\). In Equation (36), there is still \(\widetilde{W}(0,\mathrm{i}q)\), which must be solved for. For this, first, Equation (36) is solved for \(W(p,\mathrm{i}q)\).