## 6 Stable Travel Speeds of Road Vehicle Systems  
To extend the above investigations to higher order models, consider the quarter car model with two degrees of freedom shown in Figure 5a. The motions of the car body of mass \(M\) and the wheel mass \(m\) are described by the equations of motion  
\[\ddot{y}+2D\omega_{1}\big{(}\dot{y}-\dot{x}\big{)}+\omega_{1}^{2}(y-x)=0, \hskip 28.452756 ptz(s)=z_{0}\cos(\Omega s), \tag{30}\]  
\[\ddot{x}+2B\omega_{2}\big{[}(1+\beta)\dot{x}-\dot{z}-\beta\dot{y}\big{]}+ \omega_{2}^{2}[(1+\gamma)x-z-\gamma y]=0, \tag{31}\]  
where the system frequencies \(\omega_{1}\), \(\omega_{2}\) and the damping coefficients \(D\), \(B\) are introduced together with the stiffness and damping ratios \(\gamma\) and \(\beta\), and defined as follows:  
\[\omega_{1}^{2}=c/M,\hskip 56.905512 pt2D\omega_{1}=d/M,\hskip 56.905512 pt \gamma=c/k,\]  
\[\omega_{2}^{2}=k/m,\hskip 56.905512 pt2B\omega_{2}=b/m,\hskip 56.905512 pt \beta=d/b.\]  
The parameter \(\gamma\) denotes the stiffness ratio of the car and wheel spring \(c\) and \(k\), respectively. Correspondingly, \(\beta\) is the damping ratio of the car and wheel damper \(d\) and \(b\), respectively. The reference frequencies \(\omega_{2}\) and \(\omega_{1}\) describe the decoupled vibrations of the wheel and car body. In addition to Equations (30) and (31), the dynamic balance in horizontal direction gives a third equation of motion that determines the travel speed, as follows:  
\[(M+m)\ \dot{v}=f+\big{[}k(x-z)+b\big{(}\dot{x}-\dot{z}\big{)}\big{]}\tan\alpha\, \hskip 28.452756 pt\tan\alpha=dz/ds. \tag{32}\]  
Figure 5: **(a)** Quarter car model with \(2+1/2\) DOF rolling on wavy ground without losing road contact. **(b)** Vertical vibration amplitudes of car body (red) and wheel (pink). Speed driving force characteristic marked by green for stable speeds and by black lines when the travel speed is unstable. Note that Equation (32) is of first order with respect to the car speed \(\
u\), and \(s\) denotes the longitudinal coordinate of the travel path. Note that both masses are assumed to be concentrated in the contact point of road and vehicle so that only planar translations are considered. Rotations are excluded. It is appropriate to introduce the dimensionless vibration and road coordinates by means of \((y,x)=(\overline{y},\overline{x})/\Omega\) and \((z,u)=z_{o}(\overline{x},\overline{u})\), respectively. The insertion of these coordinates into Equations (30) and (31) leads to the dimensionless equations of motion  
\[\ddot{\overline{y}}+2D\omega_{1}\Big{(}\ddot{\overline{y}}-\dot{\overline{x}} \Big{)}+\omega_{1}^{2}(\overline{y}-\overline{x})=0,\qquad\qquad\qquad\qquad v= \
u\Omega/\omega_{1}, \tag{33}\]  
\[\ddot{\overline{x}}+2\beta\omega_{2}\Big{[}(1+\beta)\dot{\overline{x}}-\beta \dot{\overline{y}}\Big{]}+\omega_{2}^{2}[(1+\gamma)\overline{x}-\gamma \overline{y}]=z_{o}\Omega\omega_{2}^{2}(\overline{z}+2B\overline{w}\overline {u}), \tag{34}\]  
where \(v\) is the related speed of the vehicle rolling on road with level \(\overline{z}=\cos\Omega s\) and slope \(\overline{u}=-\sin\Omega s\). In order to derive a first approximation, it is assumed that the oscillating speed of the vehicle can be averaged by \(v(\tau)=v=\text{const}\). In this case, the travel path is \(s=vt\) and the equations of motion become linear. They are solved by the set-up  
\[\overline{y}(t)=y_{c}\cos(v\omega_{1}t)+y_{s}\sin(v\omega_{1}t),\qquad\qquad \overline{z}(t)=\cos(v\omega_{1}t),\]  
\[\overline{x}(t)=x_{c}\cos(v\omega_{1}t)+x_{s}\sin(v\omega_{1}t),\qquad\qquad \overline{u}(t)=-\sin(v\omega_{1}t)\]  
In the stationary case, the insertion of these set-ups into Equations (33) and (34) and the coefficient comparison leads to the linear matrix equation  
\[\left[ \begin{array}{cccc}1&2 Dv&v^{2}-1&-2 Dv\\ -2 Dv&1&2 Dv&v^{2}-1\\ 1+\gamma-v^{2}\kappa^{2}&2B(1+\beta)v\kappa&-\gamma&-2B\beta v\\ -2B(1+\beta)v\kappa&1+\gamma-v^{2}\kappa^{2}&2B\beta\kappa&-\gamma\end{array}  \right]\left[ \begin{array}{c}x_{c}\\ x_{s}\\ y_{c}\\ y_{s}\end{array} \right]=z_{o}\Omega\left[