## 2 Coupled Vertical and Longitudinal Vehicle Road Dynamics  
Figure 1a shows the applied quarter car model rolling on a wavy road with vertical displacement \(z\) and derivative \(u\) taken along the travel way \(s\). In the following, the quantities \(z\) and \(u\) are called road level and slope, respectively. They generate vertical car vibration displacement \(y\) and velocity \(\dot{y}\), which are coupled by the vehicle speed \(v=\dot{s}\) and described by the two equations of motion  
\[\dot{v}=\left[\omega_{1}^{2}(y-z)+2D\omega_{1}\big{(}\dot{y}-\dot{z}\big{)} \right]\tan\alpha+f/m\, \tag{1}\]  
\[\ddot{y}+2D\omega_{1}\big{(}\dot{y}-\dot{z}\big{)}+\omega_{1}^{2}(y-z)=0,\ \dot{y}= \omega_{1}x \tag{2}\]  
where \(s\) is the travel displacement of the vehicle and \(x=\dot{y}/\omega_{1}\) denotes the coordinate of the vertical vibration velocity. In Equations (1) and (2), dots denote derivatives with respect to time \(t\). The parameter \(\omega_{1}^{2}=c/m\) determines the natural frequency \(\omega_{1}\) of the vertical vehicle vibrations, \(2D\omega_{1}=b/m\) denotes the damping, and \(f\) is the driving force which is constant or slightly decreasing with growing speed. In Figure 1b, both force characteristics are plotted in yellow-black. In the following, constant driving force is applied only. The nonlinear term in Equation (1) represents the damper and spring force multiplied by \(\tan\alpha\) that takes the horizontal component of the contact force \(N\) by means of \(\tan\alpha=dz/ds\) One finds Equations (1) and (2) already in the literature in [13; 14] and in [2; 3; 4; 5; 6; 7]. In first investigations, the road level and slope are assumed to be sinusoidal as  
\[z(s)=z_{0}\cos(\Omega s)\quad\text{and}\quad u(s)=-z_{0}\sin(\Omega s), \tag{3}\]  
Figure 1: (**a**) Quarter car model rolling on sinusoidal road surface driven by the constant force f. (**b**) Driving force speed characteristic for two parameters \(z_{0}\Omega\) of the road unevenness marked by red and cyan color, respectively, and two driving force characteristics marked by yellow-black lines. are applicable and inserted into Equation (8) in order to obtain the matrix equation  
\[\left[ \begin{array}{cccc}0&-v&-1&0\\ v&0&0&-1\\ 1&0&2D&-v\\ 0&1&v&2D\end{array} \right]\left[ \begin{array}{c}y_{s}\\ y_{c}\\ x_{s}\\ x_{c}\end{array} \right]=z_{0}\Omega\left[ \begin{array}{c}0\\ 0\\ 2 Dv\\ 1\end{array} \right]\]  
applying the coefficient comparison. The first two rows of this matrix equation yield \(x_{c}=vy_{s}\) and \(y_{c}=-x_{s}/v\). Both relations are inserted into the last two rows, obtaining  
\[\left(v^{2}-1\right)\,x_{s}+2 Dv^{2}y_{s}=z_{o}\Omega v,\]  
\[-2 Dx_{s}+\left(v^{2}-1\right)y_{s}=z_{o}\Omega\,2 Dv.\]  
This reduced equation system possesses the determinant \(\Delta=\left(v^{2}-1\right)^{2}+\left(2 Dv\right)^{2}\). It is solved by Cramer's rule in order to obtain first \(x_{s}\) and \(y_{s}\) and then \(x_{c}\) and \(y_{c}\), as follows:  
\[x_{s}=z_{o}\Omega v\Big{[}v^{2}-1-\left(2 Dv\right)^{2}\Big{]}/\Delta,\qquad \qquad x_{c}=z_{o}\Omega\Omega ZDv^{4}/\Delta, \tag{10}\]  
\[y_{s}=z_{o}\Omega 2 Dv^{3}/\Delta,\qquad\qquad\qquad y_{c}=z_{o}\Omega[1+v^{2} \Big{(}4D^{2}-1\Big{)}]/\Delta. \tag{11}\]  
These results coincide with the linear theory of a constant vehicle speed. They are inserted into the velocity Equation (6) to calculate the time dependent driving force  
\[f\Omega/c=2D(z_{0}\Omega)^{2}v\sin^{2}v\tau+z_{0}\Omega[(\overline{y}+2D \overline{x})-z_{0}\Omega\cos(v\tau)\mid\sin(v\tau)\]  
which is needed to keep the speed \(v\) constant. Its mean value is calculated as  
\[f\Omega/c\,=(z_{0}\Omega)^{2}Dv^{5}/[\left(v^{2}-1\right)^{2}+\left(2 Dv\right) ^{2}]\,\qquad\qquad\text{with}\ v=v\Omega/\omega_{1}. \tag{12}\]  
The approximated force speed characteristic in Equation (12) is plotted in Figure 1b for the two road surface parameters \(z_{0}\Omega=1.4\) and \(0.8\), marked by red and cyan lines, respectively.