# Assessment of Tire Features for Modeling Vehicle Stability in Case of Vertical Road Excitation
## 3 Vehicle Excitation by Road Unevenness
### Excitation during Movement on the Asphalt Pavement  
The influence of nonlinear characteristics of the elastic elements of suspension on the vertical displacements of the body and wheels caused by the excitation frequency was investigated during modeling of the stability of the compact car by combining different quality road profile and car speed, as well as assessing the influence of the tire tread. The results were compared by assessing the dependence of the coefficient of dynamism (ratio of sprung and unsprung mass displacements to the height of unevenness) on the road  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Quarter-Car Model**} & \multicolumn{2}{c}{**3D Model**} \\ \cline{2-3}  & & **Front** & **Rear** \\ \hline Sprung mass \(m_{3}\), kg & 246 & 246 & 205 \\ \hline Unsprung mass \(m_{2}\), kg & 27.5 & 27.5 & 32 \\ \hline Mass of the tire element \(m_{1}\), kg & 0.3 & 0.3 & 0.3 \\ \hline Suspension spring stiffness \(c_{3}\), kN/m & 18 & 18 & 24 \\ \hline Tire stiffness \(c_{2}\), kN/m & 157 & 157 & 157 \\ \hline Tread stiffness \(c_{1}\), kN/m & 15,000 & 15,000 & 15,000 \\ \hline Shock absorber damping \(k_{3}\), Ns/m & 1835 & 1835 & 1835 \\ \hline Tire damping \(k_{2}\), Ns/m & 475 & 475 & 475 \\ \hline Tread damping \(k_{1}\), Ns/m & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Quarter-car and 3D model element characteristics.  
Figure 19: Displacements of unsprung (**a**,**b**) and sprung (**c**,**d**) masses (front) of the compact car, calculated using the quarter-car and 3D models: 1—quarter-car model; 2—3D model; (**a**,**c**)—\(v\) = 40 km/h; and (**b**,**d**)—\(v\) = 80 km/h.  
excitation. Figure 20a,b shows the influence of nonlinear damping of shock absorbers and nonlinear stiffness of suspension on the sprung and unsprung mass displacements.  
With nonlinear elements, the frequency dependences shifted towards higher frequency because the damping of the shock absorbers due to nonlinearity increased suddenly, and the displacement (coefficient of dynamism) increased as well. The damping of the vibrations of the unsprung masses was fairly effective, but at a certain excitation frequency, the second peak would occur, causing certain damping deficiencies.  
Figure 21a,b shows the influence of road unevenness on the tread displacements of the sprung and unsprung masses and the tire tread. As the unevenness increased, the coefficient of dynamism could be observed to increase as well. As long as the amplitude was small, the suspension and shock absorber would operate in a linear section. As the amplitude increased, the influence of the nonlinearity of the elements would be observed (Figure 21b).  
Figure 20: Influence of nonlinear damping of shock absorbers and nonlinear stiffness of suspension on the sprung (**a**) and unsprung (**b**) mass displacements: 1—linear shock absorber and linear suspension, tread not assessed; 2—tread assessed; 3—nonlinear shock absorber; and 4—nonlinear suspension.  
Figure 22a-c shows the influence of the car speed on the displacements of the sprung (body), unsprung (tire) masses, and tread displacements.  
Figure 21: The influence of road uneveness on the sprung (**a**) and unsprung (**b**) mass displacements: 1—amplitude of irregularities 10.5 mm; 2—21 mm; 3—31.5 mm; and 4—42 mm.  
Figure 22: Influence of vehicle driving speed on the displacements of sprung mass (**a**), unsprung mass (**b**), and tire tread (**c**): 1—vehicle speed 10 m/s; 2—20 m/s; 3—30 m/s; and 4—40 m/s.  
Damping of the sprung mass was effective at different speeds. The displacement reached its peak at low frequencies and afterward would decrease considerably. Tire deformations were the most sensitive to the effect of speed.