# Assessment of Tire Features for Modeling Vehicle Stability in Case of Vertical Road Excitation
## 2 Investigation the Tire-Road Interaction
### Investigation of Tread Deformation
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Tire Type** & **Band Number** & **Length \(a\), mm** & **Width \(b\), mm** & \(\beta\) **Band** \\ \hline A & 1 & 39 & 29 & 3.83 \\ \cline{2-5}  & 2 & 32 & 29.6 & 3.67 \\ \hline B & 1 & 37 & 29 & 3.75 \\ \cline{2-5}  & 2 & 31 & 29 & 3.67 \\ \cline{2-5}  & 3 & 239 & 24 & 4.58 \\ \hline C & 1 & 33 & 20 & 2.62 \\ \cline{2-5}  & 2 & 36 & 18 & 2.08 \\ \hline D & 1 & 116 & 42 & 3.33 \\ \cline{2-5}  & 2 & 90 & 41 & 3.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The values of the stiffening coefficient of the tread elements for individual tread bands. Both models offer different characteristics: the modulus of elasticity \(E_{1}\) or deformation \(\Delta\delta\) corresponding to the radial load \(R_{z}\) value. For comparison of the results, the generalized characteristic describing the relationship between load and deformation--stiffness \(C_{1}\) was used:  
\[\Delta\delta=C_{1}\cdot R_{z} \tag{16}\]  
where \(C_{1}\)--tread stiffness and \(R_{z}\)--radial load of the tire. When determining the measure \(C_{1}\) according to the first or second models, it should be taken into the account that the road contact area and the volume of the deformable tread part depend on the load. The following assumptions were used when determining the area of the contact with the road:  
* the contact with the road has an elliptical shape and is limited by the tire width and length of the contact with the road;
* the layers under the tread are sufficiently stiff, and their length does not change during deformation of the tire;
* there is the linear dependency between the radial deformation and radial load;
* the pressure is distributed evenly across the contact surface. The provided assumptions enable determination of the pressure in the contact area. Nonlinear \(q(R_{z})\) dependence was obtained for all the tested tires (Figure 8). The tire load is calculated using the nominal load of the tire \(R_{znom}\), which is specified in the catalogs, by setting the limits \(R_{zmin}=0.2R_{znom}\) and \(R_{zmax}=1.4R_{znom}\). The both models were compared to the first model by calculating the following:  
\[\Delta\delta=q\cdot\delta/E_{1} \tag{17}\]  
Nonlinear dependence was observed \(C_{1a}(R_{z})\) and was related to nonlinear dependence between \(R_{z}\) and the length of the contact with the road. This generated the nonlinear \(q(R_{z})\) dependence that reduced tread deformation in the area of higher loads. The specifics of the tire-road contact are explained by the fact that the type of tire \(C_{1a}(R_{z})\) has no significant effect on the nature of the dependence (Figure 9). The influence of tread wear on its stiffness was determined using the second model that provides more accurate reflection of the deformation conditions of short rubber prisms [36]. The effect of prism end constraint was not assessed when calculating tread stiffness. For the calculation of the tread stiffness of type A tire, the tread height was 8.1 mm for a new tire and 1.6 mm for a worn tire, and an intermediate wear level (4.9 mm) was provided. The tread wear changed the \(C\)-\(R_{z}\) dependence slightly (Figure 10) but had a significant effect on the stiffness value. The effect was not monotonic as the stiffness would be subject to particular change at the beginning of the wear. For the tested tires, tread wear could increase the stiffness of the tread in relation to the nominal load by up to 1.3 times. Figure 8: Radial loads of the tires (marking according to Table 2). Depending on the constraint conditions, the second model provided very different results. For more accurate assessment, the effect of the latter constraint, given that the properties of the ring consisting of the cord and breaker are anisotropic, and the influence of deformation characteristics of the tire ring consisting of the tread, breaker, cord and protective layer, on the tread properties were determined, and the road-to-tread contact model was adjusted. The tape for Nokian passenger car tires 175/70R13 82T was modeled.