## 4 Stable and Unstable Oscillation Amplitudes of Vehicle Speeds  
In Figure 3a, the vibration amplitudes of the vertical displacement (red) and velocity (pink) of the vehicle are plotted against the travel speed together with the speed driving force characteristic shown in blue. The displacement and velocity amplitude are  
\[A_{y}=\sqrt{y_{s}^{2}+y_{c}^{2}},\qquad\text{ and }\quad A_{x}=\sqrt{x_{s}^{2}+x_{c}^{2}},\]  
respectively, where the coefficients \(y_{s},\ y_{c}\) and \(x_{s},x_{c}\) are determined by Equations (10) and (11). The blue curve in Figure 3a represents the speed driving force characteristic dependent on speed. As already derived, stable travel speeds are only realizable in those speed regions where the slope of the characteristic is positive marked by thick blue lines in the under- and super-critical range. In the middle range, where the slope is negative, the blue curve is replaced by a thin black line, indicating that the calculated stationary speed is unstable and therefore not realizable. From this it follows that the calculated amplitudes \(A_{y}\) and \(A_{x}\) of the vertical vibrations of the vehicle are not realizable in the middle speed range, as well. In the super-critical speed range, however, all stationary vibration amplitudes are stabilized with growing speed and end into asymptotes marked by a dashed line, which are linearly growing, constant without increasing and decreasing for the force characteristic, the velocity, and the displacement amplitude, respectively. Obviously, the asymptote of the driving force corresponds to acceleration. Both quantities, force and acceleration, are linearly increasing with growing speed of the vehicle. In order to introduce efficient solution methods, the time increment \(d\tau\) in Equations (6) and (8) is eliminated by the angle increment \(d\varphi=-vd\tau\) that leads to the time-free equations  
\[v\overline{y}^{\circ}=-\overline{x},\hskip 56.905512 ptv\overline{x}^{\circ}=( \overline{y}+2D\overline{x})-z_{0}\Omega(\overline{x}+2 Dv\overline{u}), \tag{15}\]  
\[vv^{\circ}-2D(z_{0}\Omega)^{2}\overline{u}^{2}v=\left(z_{0}\Omega\right)^{2} \overline{z}\overline{u}-z_{0}\Omega(\overline{u}\overline{y}+2D\overline{u} \overline{x})-f\Omega/c, \tag{16}\]  
where prime in \((.)^{\prime}=d(. )/d\tau\) is replaced by a circle in \((.)^{\circ}=d(. )/d\varphi\) denoting differentiation with respect to the polar angle \(\varphi\). In Equations (15) and (16), the related road level and slope are sinusoidal and dependent on the polar angle \(\varphi\). Accordingly, the vertical vehicle vibrations and the longitudinal speed oscillations are calculable inside one period by means of shooting methods to obtain numerical solutions for \(v_{p}(\varphi),\overline{y}(\varphi)\) and \(\overline{x}(\varphi)\). The associated characteristic multipliers \(\rho_{i}\) are investigated by means of the new equation system  
\[v_{p}\left[ \begin{array}{c}\Delta\overline{y}\\ \Delta\overline{x}\\ \Delta v\end{array} \right]^{\circ}+\left[ \begin{array}{ccc}0&-1&\overline{ y}_{p}^{\circ}\\ -1&-2D&\overline{x}_{p}^{\circ}+z_{0}\Omega 2D\overline{u}\\ z_{0}\Omega\overline{u}&2 Dz_{0}\Omega\overline{u}&\varphi_{p}^{\circ}-2D(z_{ 0}\Omega\overline{u})^{2}\end{array} \right]\left[ \begin{array}{c}\Delta \overline{y}\\ \Delta\overline{x}\\ \Delta v\end{array} \right]=0\]  
which is obtained when the perturbations \(\overline{y}=\overline{y}_{p}+\Delta\overline{y}\), \(\overline{x}=\overline{x}_{p}+\Delta\overline{x}\), and \(v=v_{p}+\Delta v\) are inserted into Equations (15) and (16) and the perturbation equations are linearized in the \(\Delta\) quantities. According to the Floquet theory, the calculated periodic solutions are asymptotically stable with respect to \(\varphi\) if all multipliers \(\rho_{i}\) satisfy \(|\rho_{i}|<1\). For known velocity \(v_{p}(\varphi)\), the applied polar angle can be integrated back to time by which the sequence of associated time distances is determined.