# Assessment of Tire Features for Modeling Vehicle Stability in Case of Vertical Road Excitation
## 2 Investigation the Tire-Road Interaction
### Investigation of Tread Deformation
The results obtained for different tires are presented in Table 3. The modulus of elasticity of the whole tread was calculated using the equation in [44]:  
\[\frac{1}{E_{1}}=\left(\frac{\delta_{1a}}{E_{1a}}+\frac{\delta_{1b}}{E_{1b}} \right)\cdot\frac{1}{\delta_{1}} \tag{13}\]  
This model does not accurately reflect the operating conditions of the tread. Deformation of the tread part with a pattern may be restricted by the contact with the road or a continuous breaker. Moreover, the deformation of the tread part with a pattern does not fully correspond to the ideal compression conditions, as the deformation of the ends at \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Tire Type** & **Part with a Pattern \(E_{1a}\)**, & **Continuous Part \(E_{1b}\)**, & **Tread \(E_{1}\)**, \\ \hline A & 5.72 & 7.84 & 6.16 \\ \hline B & 5.88 & 7.84 & 6.22 \\ \hline C & 7.08 & 7.84 & 6.51 \\ \hline D & 5.33 & 7.84 & 5.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Tread modulus of elasticity obtained by assessment using the first model, MPa. Figure 7: Tire tread patterns: (**a**) for a passenger car; (**b**) for a light-duty truck; and (**c**) for a truck. compression leads to a change in the element shape. The course of deformation would be more accurately estimated using the model of a prism subject for compression by applying a rubber-shock-absorber calculation methodology. The deformations of the tread part with a pattern can be assessed more precisely using the calculation methods applicable to the rubber shock absorbers. Under this approach, the tread element is depicted as a rubber prism, deformed with more or less constrained end deformations depending on the conditions. If the deformations of the both ends of the prism-shaped rubber shock absorber are constrained, the relationship between the load and the deformation is expressed by the equation:  
\[R_{z}=\frac{1}{3}\beta EA\left(\frac{1}{\lambda^{2}}-\lambda\right),\ \lambda=1-\frac{\Delta\delta}{\delta},\ \beta=\frac{4}{3}\left(1+\eta^{2}-\eta\right)\left(1-\frac{2}{\alpha\delta}th \frac{\alpha\delta}{2}\right)^{-1} \tag{14}\]  
where \(A\)--prism base area; \(\delta\)--prism height; \(\Delta\delta\)--prism deformation; and \(\alpha\) and \(\eta\)--shape change parameters. Whereas \(\alpha\) and \(\eta\) are interrelated, a system of equations derived from the energy minimum condition is proposed in the study [45]:  
\[ \begin{array}{c}\alpha^{2}=\ \frac{1}{b^{2}}\frac{48(1+\eta^{2}-\eta)}{\eta^{2 }(\frac{a}{b})+(1-\eta)^{2}},\\ \frac{\alpha\delta-2 th\frac{\alpha\delta}{2}}{\alpha\delta(th^{2}\frac{a\delta }{2}-1)+2 th\frac{\alpha\delta}{2}}=\ \frac{1}{2}+\frac{1+\eta^{2}-\eta}{1-2\eta}\frac{\eta(\frac{a}{b})^{2}+ \eta-1}{\eta^{2}(\frac{a}{b})^{2}+(1-\eta)^{2}}\end{array}  \tag{15}\]  
where \(a\), \(b\)--dimensions of the prism base. Equations (14) and (15) were solved by approximation. This model of the tread part with a pattern was applied to two cases: with the end deformations not constrained and with the end deformations completely constrained. If the deformations of the shock absorber ends were not constrained, \(\beta\) = 1. In case the deformations were constrained, the stiffness of the individual bands of the tread part with a pattern would increase by up to 4.5 times (Table 4). Under actual conditions, the constraints of the end deformations of the tread part with a pattern were not absolute. In relation to the contact with the road, they are determined by the tire-road friction, while the other end is in contact with the tread part without a pattern. Deformations of the tread part without a pattern are restricted by the breaker layers that have the anisotropic properties (Table 2), and the modulus of elasticity of the layer is only 1.2-1.5 times higher than the modulus of elasticity of rubber in the direction transverse to the reinforcement. Under these conditions, the deformation conditions of the tread part with a pattern are closer to those of a shock absorber with freely deforming ends. This option was used for further calculations.