## 6 Stable Travel Speeds of Road Vehicle Systems
\begin{array}{c}0\\ 0\\ 1\\ -2 Bv\end{array} \right] \tag{35}\]  
where \(\kappa\) is the reference frequency ratio given by \(\kappa=\omega_{2}/\omega_{1}\). The coefficients \(x_{c}\), \(x_{s}\), \(y_{c}\), \(y_{s}\) of the sinusoidal set-up are calculated by means of Equation (35) and inserted into  
\[A_{y}=\sqrt{y_{s}^{2}+y_{c}^{2}},\qquad\quad\text{and}\qquad\quad A_{x}=\sqrt {x_{s}^{2}+x_{c}^{2}},\]  
which are plotted in Figure 5b against the related speed of the vehicle. The amplitude \(A_{y}\) of the car body is marked by red and the wheel amplitude \(A_{x}\) by pink. In unstable speed ranges, both amplitudes are marked by thin black lines. The vertical vibration amplitudes of both masses are calculated for the reference frequencies \(\omega_{1}=1/s\), \(\omega_{2}=2/s\), \((\kappa=2)\), for the stiffness-ratio \(\gamma=0.5\) and for the damping values \(D=0.3\), \(B=0.2\), and \(\beta=0.5\). The applied road wave is chosen by \(z_{o}\Omega=1\). This is equivalent to the height-length ratio \(z_{o}/L=0.159\) when the wave frequency \(\Omega=2\pi/L\) is inserted into \(z_{o}\Omega=1\). The frequency responses in Figure 5b possess two resonance speeds: the first at \(v_{1}=0.27\) and the second at \(v_{1}=1.29\). Both eigenvalues are valid for vanishing damping. In order to decide the realizability of the stationary vehicle speeds applied in Equation (35), the new speed Equation (32) is investigated in the dimensionless form  
\[\frac{m+M}{m}\dot{v}/\omega_{2}+2B(z_{o}\Omega)^{  
where \(x_{s}\) and \(x_{c}\) are calculated by means of Equation (35). New numerical evaluations of Equations (35) and (37) are plotted in Figure 5b and marked by green for stable speeds when the slope of the speed driving force characteristic is positive and by black-thin lines when the slope is negative. In the latter case, the speed applied in Equation (35) is unstable and physically not realizable. Finally, it is noted that both instability speed ranges are vanishing for growing damping coefficients.  
The instability behavior is numerically investigated by means of simulations applied to a slightly modified quarter car model with two equal masses \(m\) and a third damper \(B\), as shown in Figure 6a. The car is rolling with velocity \(\mathrm{v}\) on a wavy road described by \(z(s)=z_{o}\,\cos\Omega s\) in dependence on the travel way coordinate \(s\). The speed frequency \(v\Omega\) is related to the reference frequency \(\omega_{1}\) given by \(\omega_{1}^{2}=c/m\). The vertical vibrations of this car model are described by the two nonlinear equations of motion  
\[\ddot{y}_{1}+2d_{1}\omega_{1}\big{(}\dot{y}_{1}-\dot{y}_{2}\big{)}+2D\omega_{ 1}\big{(}\dot{y}_{1}-\dot{z}\big{)}+\omega_{1}^{2}(y_{1}-y_{2})=0, \tag{38}\]  
\[\ddot{y}_{2}+2d_{2}\omega_{1}\big{(}\dot{y}_{2}-\dot{z}\big{)}-2d_{1}\omega_{ 1}\big{(}\dot{y}_{1}-\dot{y}_{2}\big{)}+\omega_{1}^{2}(2y_{2}-y_{1}-z)=0, \tag{39}\]  
where \(\dot{z}=v\Omega u\) couples both vertical vibrations to the horizontal velocity described by  
\[2\dot{\mathrm{v}}+\omega_{1}[\omega_{1}(z-y_{2})+2d_{2}(\dot{z}-\dot{y}_{2})+ 2D\big{(}\dot{z}-\dot{y}_{1}\big{)}]\Omega u=f/m. \tag{40}\]  
The damping coefficients are introduced by \(B/m=2D\omega_{1}\) and \(b_{i}/m=2d_{i}\omega_{i}\) for \(i=1,2\). For numerical integration, the dimensionless time \(\tau=\omega_{1}t\) and the velocity \(v=\mathrm{v}\Omega/\omega_{1}\), together with the related coordinates \((\overline{z},\ \overline{u})=(z,u)/z_{o}\) and \((\overline{y}_{i},\overline{x_{i}})=(y_{i},x_{i})\Omega\), are inserted into the velocity Equation (40) and vehicle Equations (38) and (39). Figure 6b shows the average vehicle speed for a given driving force \(f\Omega/c\), marked by a blue thick line. It is calculated for the road level \(z_{o}\Omega=1\) and the damping values \(D=d_{1}=d_{2}=0.1\). In the middle range of the related driving force (\(0.60<f\Omega/c<0.82\)), there are five branches of stationary solutions: two solution branches are unstable and three are stable.