# Assessment of Tire Features for Modeling Vehicle Stability in Case of Vertical Road Excitation
## 2 Investigation the Tire-Road Interaction
### Investigation of the Tire Properties
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Tire**} & \multicolumn{2}{c}{**Tire Element**} \\  & **Tirchness, mm** & **Fiber Angle** \\ \hline  & & \(\delta_{1a}=8.0\) & – \\  & 175/70R13 & \(\delta_{1b}=7.0\) & – \\ A & Breaker, 3 layers & \(\delta_{2}=4.1\) & \(\theta_{2}=20^{\circ}\) \\  & Cord, 1 layer & \(\delta_{3}=1.2\) & \(\theta_{3}=90^{\circ}\) \\  & & \(\delta_{4}=1.0\) & – \\ \hline  & & \(\delta_{1a}=8.1\) & – \\  & 195/50R15 & \(\delta_{1b}=7.8\) & – \\ B & Breaker, 2 layers & \(\delta_{2}=3.9\) & \(\theta_{2}=20^{\circ}\) \\  & Cord, 2 layers & \(\delta_{3}=1.7\) & \(\theta_{3}=90^{\circ}\) \\  & & \(\delta_{4}=1.3\) & – \\ \hline  & & \(\delta_{1a}=12.0\) & – \\  & 185/75R14C & \(\delta_{1b}=11.0\) & – \\ C & Breaker, 2 layers & \(\delta_{2}=3.2\) & \(\theta_{2}=20^{\circ}\) \\  & Cord, 2 layers & \(\delta_{3}=1.8\) & \(\theta_{3}=90^{\circ}\) \\  & & \(\delta_{4}=1.0\) & – \\ \hline  & & \(\delta_{1a}=17.0\) & \\  & 12.00R20 & \(\delta_{1b}=15.0\) & \\ D & Breaker, 4 layers & \(\delta_{2}=6.4\) & \(\theta_{2}=20^{\circ}\) \\  & Cord, 6 layers & \(\delta_{3}=7.2\) & \(\theta_{3}=90^{\circ}\) \\  & & \(\delta_{4}=1.0\) & – \\ \hline \end{tabular} Note: \(\theta\)—cord angle in relation to the rolling direction. The indexes of the measures have the following meaning: \(1a\)—tread part with a pattern, \(1b\)—continuous tread part, \(2\)—breaker, \(3\)—cord, \(4\)—inner sealing liner.  
\end{table} 
Table 1: Data of the tire components. Figure 3: Tire load scheme: (**a**) tire deformation not assessed; (**b**) tire deformation assessed. Where the impact of the cord threads is assessed, the radial deformation model is used, and the tire reaction is assessed using single pressure \(q_{z}=C_{z}w\). The sidewalls react only to the displacements in the radial direction, and the reaction per length unit equals \(q_{s}=C_{zs}w\). An elastic outer layer with the thickness equal to \(H\) before deformation deforms by \(\Delta H\) in the contact area. Inelastic forces are not subject to assessment. Tread reaction:  
\[q_{z}=C_{zs}\Delta H0 \tag{2}\]  
Arc length \(x\) (Figure 4b) is measured from the center of contact. The main measures sought are [38]:  
* model deflection \(h_{z}\) assuming that the band does not deform in the radial direction: \[h_{z}=(w+\Delta H)_{\backslash x=0}\] (3)
* contact zone length--\(2x_{0}\);
* contact pressure distribution--\(q_{z}(x)\);
* load \(F_{z}\): \[F_{z}=2\int\limits_{0}^{x_{0}}q_{z}(x)dx\] (4)  
Figure 4: Model of the deformable ring of the tire: (**a**) deformable ring; (**b**) deformable ring element; (**c**) cross-section scheme for assessing the impact of the cord; 1—elastic layer (tread); 2—breaker layers; 3—elastic base (side walls); 4—disk. As a result of integration and rearrangement of Equation (4), the following is obtained:  
\[F_{z}=2k_{zx}\alpha_{0}\left[a_{0}\left(1-\frac{thax_{0}}{\alpha x_{0}}\right)-a_ {1}x_{0}^{2}\left(\frac{1}{3}-\frac{thax_{0}}{\alpha x_{0}}\right)\right] \tag{5}\]  
The unknown measures \(h_{z}\) and \(x_{0}\) are determined using the following dependencies [38]:  
\[h_{z}=\frac{x_{0}^{2}}{2R}\left(1+m\frac{2}{\beta x_{0}}\right),\ \beta=\sqrt{k_{z}/N_{0}},\ m= \left(1-\frac{\beta^{2}}{\alpha^{2}}\right)\frac{1-(thax_{0}/\alpha x_{0})}{1+( \beta/a)thax_{0}} \tag{6}\]  
The described methodology validates the specifics of the tire ring load, namely, that the tire pressure in the contact area is not equal to the internal pressure of the tire due to the breaker and the cord layers. In order to refine the model, the deformation characteristics of the ring consisting of the cord and breaker layers and the tread as a separate deformable element are determined.