## 4 Critical Velocity of a Constant Moving Force  
To determine the critical velocity of a moving force, Equations (1)-(3) are first switched to moving coordinates and then changed to a dimensionless form. After removing the time-dependent terms not required for the steady-state regime, they read as  
\[\widetilde{w}_{\xi\xi\xi\xi}+4\kappa^{2}\widetilde{w}_{\xi\xi}+4\eta_{N} \widetilde{w}_{\xi\xi}+8\eta_{p}\alpha\big{(}-\widetilde{w}_{\xi\xi}+ \widetilde{u}_{\xi\xi}\big{)}+4\kappa_{p}\big{(}\widetilde{w}-\widetilde{u} \big{)}=8\eta_{P_{1}}\delta(\xi) \tag{12}\]  
\[\mu_{s}\alpha^{2}\widetilde{u}_{\xi\xi}-\eta_{s}\widetilde{u}_{\xi\xi}-2\eta_ {p}\alpha\big{(}-\widetilde{w}_{\xi\xi}+\widetilde{u}_{\xi\xi}\big{)}-\kappa_ {p}\big{(}\widetilde{w}-\widetilde{u}\big{)}-2\eta_{f}\alpha\widetilde{u}_{ \xi}+\widetilde{u}=0 \tag{13}\]  
where, besides the parameters specified in Equations (9)-(11),  
\[\widetilde{w}=\frac{w}{w_{st}},\ \widetilde{u}=\frac{u}{w_{st}}\ \text{with}\ w_{st}=\frac{P_{1}\chi}{2k_{f}},\ \text{and}\ \eta_{P_{1}}=\frac{P_{1}}{P_{1}} \tag{14}\]  
\[v_{ref}=\sqrt[4]{\frac{4k_{f}EI}{m^{2}}}=\frac{1}{\chi}\sqrt{\frac{k_{f}}{m}},\ \alpha=\frac{v}{v_{ref}} \tag{15}\]  
Next, the Fourier transform is applied. In the transferred space, it is more convenient to write the equations in the matrix form to facilitate the analytical solution  
\[ \begin{bmatrix}D_{1}+D_{2}&-D_{2}\\ -D_{2}&D_{2}+D_{3}\end{bmatrix}\begin{Bmatrix}W\\ U\end{Bmatrix} = \begin{Bmatrix}2\eta_{P_{1}}\\ 0\end{Bmatrix}  \tag{16}\]  
where  
\[D_{1}(p)=\ \frac{p^{4}}{4}-\alpha^{2}p^{2}-\eta_{N}p^{2} \tag{17}\]  
\[D_{2}(p)=-2\mathrm{i}\eta_{p}\alpha p+\kappa_{p} \tag{18}\]  
\[D_{3}(p)=-\mu_{s}\alpha^{2}p^{2}-2\mathrm{i}\eta_{f}\alpha p+\eta_{s}p^{2}+1 \tag{19}\]  
and \(W\), \(U\) and \(p\) represent \(\widetilde{w}\), \(\widetilde{u}\) and \(\xi\) in the Fourier space. Since the critical velocities should be determined without damping, expressions (18) and (19) can be further simplified. To analytically determine the critical velocity, it is necessary to find at least one real double \(p\)-root of the determinant of system (16). This means finding a real-valued solution to a system of two equations, where the first one is the determinant and the second one is its derivative with respect to \(p\). The determinant can be written as a cubic polynomial with real coefficients for \(\overline{p}=p^{2}\), and the second equation as a quadratic polynomial with real coefficients for \(\overline{p}\)  
\[ \begin{array}{l}\overline{p}^{3}\big{(}\mu_{s}\alpha^{2}-\eta_{s}\big{)}- \overline{p}^{2}\big{(}4\big{(}\mu_{s}\alpha^{2}-\eta_{s}\big{)}\big{(}\eta_{N }+\alpha^{2}\big{)}+\kappa_{p}+1\big{)}\\ +4\overline{p}\big{(}\alpha^{2}\big{(}\kappa_{p}\mu_{s}+\kappa_{p}+1\big{)}+ \eta_{N}\big{(}\kappa_{p}+1\big{)}-\eta_{s}\kappa_{p}\big{)}-4\kappa_{p}=0 \end{array}  \tag{20}\]  
The results of the parametric analyses of four selected cases are presented in Figure 3; in all of them, \(\eta_{N}=\eta_{s}=0\) and for better clarity, the scale does not start at \(\alpha=0\). As an example for the regular situation, \(\mu_{s}=1\), \(\kappa_{p}=0.36\) are selected, and for the irregular one: \(\mu_{s}=1.1\), \(\kappa_{p}=0.6\). In the regular case, as shown in Figure 3a, it can be seen that the analytically determined values for the resonances 0.681, 0.707 and 0.913 indicate the lower critical velocity (\(\alpha_{cr_{1}}\)), the false critical velocity (\(\alpha_{cr_{2}}\)) and the higher critical velocity (\(\alpha_{cr_{2}}\)), as predicted. In the irregular case, as shown in Figure 3b, it is observed that the only analytically determined value of 0.996 is \(\alpha_{cr_{2}}\) and \(\alpha_{cr_{1}}\) is replaced by the pseudocritical velocity (\(\alpha_{prc}\)) with a value of 0.692. Other cases concern the situations at the boundary of the domain shown in Figure 2, i.e., when one of the roots is double.