# Assessment of Tire Features for Modeling Vehicle Stability in Case of Vertical Road Excitation
## 3 Vehicle Excitation by Road Unevenness
### Suspension Models  
When considering road excitation, it was necessary to specify which of the car's vertical motion models best met the requirements for stability models. Three vertical dynamics models are known: (1) quarter-car, (2) flat (2D), and (3) spatial (3D), examined in the studies [46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57]. The 2D model best describes the longitudinal vehicle dynamics, i.e., braking, acceleration, vertical obstacle clearance, etc., and will not be used in future investigations. The simplest model used for road-vehicle interaction analysis is the quarter-car model. It consists of three masses: tire element \(m_{1}\), unsprung masses (suspensions) \(m_{2}\), and sprung masses (body) \(m_{3}\) (Figure 2). This model enables the calculation of the maximum and characteristic values of the vertical displacements of the wheel and to specify the frequency characteristic of the suspension. More accurate models were found to require a more accurate assessment of tire performance, such as damping. The dimensions of the tire tread element were determined according to Figure 3, while the length of the contact was according to Equation (1) expression. The tread stiffness coefficient was determined using Equation (2). Quarter-car model behavior under kinematic excitation by road unevenness \(q\) was described using the following system of equations:  
\[ \begin{array}{l}m_{1}\ddot{z}_{1}+k_{1}(\dot{z}_{1}-\dot{q})+c_{1}(z_{1}-q)=0,\\ m_{2}\ddot{z}_{2}+k_{2}(\dot{z}_{2}-\dot{z}_{1})+c_{2}(z_{2}-z_{1})=0,\\ m_{3}\ddot{z}_{3}+k_{3}(\dot{z}_{3}-\dot{z}_{2})+c_{3}(z_{3}-z_{2})=0.\end{array}  \tag{27}\]  
where \(m_{1}\), \(m_{2}\), \(m_{3}\)--tread part of the tire, unsprung, and sprung mass, respectively; \(z_{1}\), \(z_{2}\), \(z_{3}\)--tire, suspension and body displacements, respectively; \(k_{1}\), \(k_{2}\), \(k_{3}\)--tread, tire, and suspension damping factors, respectively; \(c_{1}\), \(c_{2}\), \(c_{3}\)--stiffness of the tread, tire, and suspension, respectively; and \(q\)--height of road profile unevenness. When developing the 3D model of the car, the following assumptions were made: the car body would be symmetrical with respect to the longitudinal and transverse axes, and its center of gravity would be on the car symmetry plane. For the first approach, it was assumed that the masses of the wheels would move only in the vertical direction, with the track of the front and rear axes being the same. The vertical oscillations of the 3D car model on the longitudinal plane were calculated using the system of Equation (27). The transverse tilt of the car was expressed by the system of equations:  
\[\left[M_{\beta}\right]\cdot\left\{\tilde{\beta}\right\}+\left[K_{\beta}\right] \cdot\left\{\hat{\beta}\right\}+\left[C_{\tilde{\beta}}\right]\cdot\left\{\beta \right\}=\left[M_{q}\right] \tag{28}\]  
where \(\left\{\beta\right\}\)--transverse displacement vector consisting of \(\beta_{1g}=\frac{1}{l_{2}}(z_{12k}-z_{11d})\), \(\beta_{1p}=\frac{1}{l_{2}}(z_{11k}-z_{11d})\); indexes \(1i\)--front, \(2i\)--rear, \(k\)--left, \(d\)--right, \(t_{1}\)--front track of the car, and \(l_{2}\)--rear track of the car. Mass matrix:  
\[\left[M_{\beta}\right]=\left[ \begin{array}{ccc}I_{x}&0&0\\ 0&2m_{11}\left(\frac{t_{1}}{2}\right)^{2}&0\\ 0&0&2m_{12}\left(\frac{t_{2}}{2}\right)^{2}\end{array} \right] \tag{29}\]  
where \(I_{x}\)--moment of inertia of the sprung mass of the vehicle around the longitudinal axis of the tilt and \(t\)--wheel track of the respective axis. Damping matrix:  
\[\left[K_{\beta}\right]=\left[ \begin{array}{ccc}\frac{1}{2}\left(k_{21}t_{1}^ {2}+k_{22}\right)\left(\frac{t}{2}\right)^{2}&-\frac{k_{21}t^{2}}{2}&-\frac{k_ {22}t^{2}}{2}\\ -\frac{k_{21}t^{2}}{2}&\frac{(k_{11}+k_{12})t_{1}^{2}}{2}&0\\ -\frac{k_{22}t^{2}}{2}&k_{11}+k_{21}&\frac{(k_{12}+k_{22})t_{2}^{2}}{2}\end{array} \right] \tag{30}\]  
Stiffness matrix:  
\[\left[C_{\beta}\right]=\left[