## 4 Stable and Unstable Oscillation Amplitudes of Vehicle Speeds
Note that the four-dimensional time system (13) and (14) is not ergodic since the polar angle \(\varphi\) is a state variable that grows infinitely by permanent rotation. This disadvantage is avoided by eliminating the time variable by means of the polar angle, which leads to the three-dimensional angle system (15) and (16) where the polar angle now represents the independent integration variable restricted to one-periodic interval. The solutions of this new time-free equation system are ergodic, and multiplicative ergodic theorems are applicable in order to calculate characteristic numbers of the dynamic system of interest. For the new angle Equations (15) and (16), new analytical solutions are derived by means of the introduction of the Fourier expansions  
\[\overline{y}(\varphi)=y_{\varepsilon}\cos\varphi+y_{s}\sin\varphi+\cdots, \hskip 56.905512 pt\overline{z}(\varphi)=\cos\varphi \tag{17}\]  
\[\overline{x}(\varphi)=x_{c}\cos\varphi+x_{s}\sin\varphi+\cdots,\hskip 56.905512 pt \overline{u}(\varphi)=-\sin\varphi, \tag{18}\]  
\[v(\tau)=v_{o}+v_{c}\cos 2\varphi+v_{s}\sin 2\varphi+\cdots, \tag{19}\]  
Figure 3: (**a**) Vibration amplitudes of the vertical velocity (pink) and displacement (red) of the vehicle in comparison with the force speed characteristic (blue) in the under- and supercritical speed range. (**b**) Travel speed oscillation amplitudes (green) and extended force speed characteristic (red-cyan). where the zeroth Fourier coefficient \(v_{o}\) of the speed expansion in Equation (19) takes the role of the averaged speed \(v\) in Section 3. The insertion of the expansions (17) and (18) into Equation (15) and the coefficient comparison of the sinusoidal terms of \(\cos\varphi\) and \(\sin\varphi\) leads to the same result, already noted in Equations (10) and (11). All other terms are cutoff and can only be taken into account when higher expansions are introduced. The insertion of the expansions (17), (18), and (19) into the speed Equation (16) and the coefficient comparison leads to  
\[v_{s}=\frac{D(z_{o}\Omega)^{2}v_{o}^{2}\Big{[}(z_{o}\Omega)^{2}\big{(}1+4D^{2} v_{o}^{2}-v_{o}^{4}\big{)}-4v_{o}^{4}\Big{]}}{2\big{[}(v_{o}^{2}-1)^{2}+(2 Dv_{o} )^{2}\big{]}\Big{[}4v_{o}^{2}+(z_{o}\Omega)^{4}D^{2}\Big{]}}\ \, \tag{20}\]  
\[v_{c}=\frac{(z_{o}\Omega)^{2}v_{o}^{3}\left\{1+v_{o}^{2}\Big{[}4D^{2}-1+D^{2}( z_{o}\Omega)^{2}\Big{]}\right\}}{\Big{[}(v_{o}^{2}-1)^{2}+(2 Dv_{o})^{2}\Big{]} \Big{[}4v_{o}^{2}+(z_{o}\Omega)^{4}D^{2}\Big{]}}\ \ \, \tag{21}\]  
which represent new results for the amplitudes of the travel speed oscillations. In Figure 3b, the resultant speed amplitude \(A_{v}\) is plotted against the mean speed \(v_{o}=\text{v}\Omega/\omega_{1}\) by  
\[A_{v}=\sqrt{v_{c}^{2}+v_{s}^{2}}\,\ \lim_{v_{o}\to\infty}A_{v}=D(z_{o}\Omega)^{2}[4 +(z_{o}\Omega)^{2}]/8 \tag{22}\]  
and marked by a thick green line. Obviously, the speed amplitude vanishes if the vehicle slows. It is increasing up to the resonance speed and decreases again for further growing speed, up to the asymptote given in Equation (22). In addition to the above coefficient comparison of terms with \(\cos 2\varphi\) and \(\sin 2\varphi\), all terms with \(\cos 0\) lead to the extended speed driving force characteristic  
\[f\Omega/c=\frac{D(z_{o}\Omega)^{2}v_{o}^{3}}{\left(v_{o}^{2}-1\right)^{2}+(2 Dv _{o})^{2}}\big{\{}v_{o}^{2}+(z_{o}\Omega)^{2}\frac{v_{o}^{2}\Big{[}1-(z_{o} \Omega)^{2}D^{2}-4D^{2}\Big{]}-1}{2\Big{[}4v_{o}^{2}+(z_{o}\Omega)^{4}D^{2} \Big{]}}\ \big{\}} \tag{23}\]  
where the first part coincides with Equation (12) and the second part gives a correction of second order. The extended speed driving force characteristic in Equation (23) is plotted in Figure 3b. The red line marks the first approximation noted in Equation (12). Its second order approximation, noted in Equation (23), is marked by a thick cyan line. It is close to the red line of the first approximation. Thin black lines represent speed driving force characteristics with negative slope where the calculated solutions are unstable and therefore not realizable.