## 5 Stability in Mean and Robustness with Regard to Disturbances  
Note that the velocity Equation (13) is determined by the products of road level \(z\) and slope \(\mathbf{u}\) multiplied by the two vibration coordinates \(y\) and \(x=y^{\prime}\). These products are called covariances [2]. They determine the dependency of the response of the system from the road excitation. The application of the product rule of differentiation to the road Equation (7) and the vehicle Equation (8) leads to the covariance equation system  
\[\left[ \begin{array}{c}\overline{zy}\\ \overline{u}y\\ \overline{z}\overline{x}\\ \overline{u}x\end{array} \right]^{\prime}+\left[ \begin{array}{cccc}0&-v&-1&0 \\ v&0&0&-1\\ 1&0&2D&-v\\ 0&1&v&2D\end{array} \right]\left[ \begin{array}{c}\overline{z}\overline{y}\\ \overline{u}\overline{y}\\ \overline{z}\overline{x}\\ \overline{u}\overline{x}\end{array} \right]=z_{0}\Omega\left[ \begin{array}{c} 0\\ 0\\ \overline{z}^{2}+2 Dv\overline{u}\\ \overline{z}\overline{u}+2 Dv\overline{u}^{2}\end{array} \right], \tag{24}\]  
where the related speed \(v\) of the vehicle is determined by Equation (13) as  
\[v^{\prime}+2D(z_{0}\Omega)^{2}\overline{u}^{2}v=z_{0}\Omega(\overline{u} \overline{y}+2D\overline{u}\overline{x})-\left(z_{0}\Omega\right)^{2}\overline {z}\overline{u}+f\Omega/c. \tag{25}\]Level and slope of the road are \(\overline{z}(\tau)=\cos(\upsilon\tau)\) and \(\overline{u}(\tau)=-\sin(\upsilon\tau)\), respectively. Again, this is a DA equation system where transcendentals take the role of algebraic terms. For numerical integrations, the sinusoidal terms in Equation (3) are generated by means of the road oscillator, noted in Equation (7). The oscillator is transformed by means of the polar coordinates  
\[\overline{z}(\varphi)=\cos\varphi,\hskip 28.452756 pt\overline{u}(\varphi)=-\sin \varphi,\hskip 28.452756 pt\varphi^{\prime}=-v, \tag{26}\]  
which are inserted into Equations (24) and (25). The remaining equation \(\varphi^{\prime}=-v\) determines the derivative of the polar angle and couples the polar angle back to the speed of the vehicle. In order to derive analytical approximations, the averaging method is applied by taking the time mean values \(<\overline{z}^{2}>\ =\ <\overline{u}^{2}>\ =1/2\) and \(<\overline{zu}>\ =\ 0\) in Equations (24) and (25) that leads to  
\[\left(1-v^{2}\right)\overline{u}\overline{y}+2D\overline{u}\overline{x}=z_{ \circ}\Omega vD\]  
\[-2 Dv^{2}\ \overline{u}\overline{y}+\left(1-v^{2}\right)\ \overline{u}\overline{x}=\ z_{ \circ}\Omega v/2,\]  
when in Equation (24) for \((. )^{\prime}=0\) the first two rows are applied to eliminate \(\overline{zy}\) and \(\overline{x}\overline{x}\). Subsequently, the covariances \(\overline{u}\overline{y}\) and \(\overline{u}\overline{x}\) are calculated and inserted into Equation (25)  
\[D(z_{0}\Omega)^{2}v=z_{0}\Omega(\overline{u}\overline{y}+2D\overline{u} \overline{x})+f\Omega/c,\]  
which finally leads to the same speed driving force characteristic  
\[f\Omega/c=(z_{0}\Omega)^{2}Dv^{5}/[\left(v^{2}-1\right)^{2}+(2 Dv)^{2}]\,\]  
already noted in Equation (12). Note that the above applied time mean values are exact when the speed of the vehicle is constant. For oscillating speeds, the time mean values are approximations. Following [2], the averaged Equation (24) are applied to investigate the stability in mean of the averaged speed by means of the perturbation equation system  
\[\left[ \begin{array}{c}\Delta v\\ \Delta\overline{x}\overline{v}\\ \Delta\overline{u}\overline{y}\\ \Delta\overline{x}\overline{x}\\ \Delta\overline{u}\overline{x}\end{array} \right]+\left[ \begin{array}{cccc}D (z_{0}\Omega)^{2}&0&-z_{0}\Omega&0&-2 Dz_{0}\Omega\\ -\overline{u}\overline{y}_{0}&0&-v_{0}&-1&0\\ \overline{z}\overline{y}_{0}&v_{0}&0&0&-1\\ -\overline{u}\overline{x}_{0}&1&0&2D&-v_{0}\\ \overline{z}\overline{x}_{0}-z_{0}\Omega D&0&1&v_{0}&2D\end{array} \right] \left[ \begin{array}{c}\Delta v\\ \Delta\overline{z}\overline{v}\\ \Delta\overline{u}\overline{y}\\ \Delta\overline{x}\\ \Delta\overline{u}\overline{x}\end{array}