## 4 Conclusions  
A model of a car wheel was proposed which incorporates the detailed geometry of a rubber tyre with metallic reinforcement, a metallic rim and the road surface. The nonlinear dynamics of this model were analysed using the finite element method. The numerical  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{5}{c}{**Internal Pressure, MPa**} \\  & **1.8** & \multicolumn{5}{c}{**2.0**} & **2.2** & **Load, kN** \\ \hline Direction & x & z & x & z & x & z \\ \hline \(\mu_{z}\) skewness & 0.0132 & 0.0179 & 0.0211 & 0.0127 & 0.0219 & 0.0098 & \\ \hline \(F_{\text{n}}\) asymmetry & 1.101 & - & 1.092 & - & 1.089 & - & 4 \\ \hline \(F_{\text{tx}}\) asymmetry & 0.979 & - & 0.985 & - & 1.015 & - & \\ \hline \(\mu_{z}\) skewness & 0.0004 & 0.0193 & 0.0144 & 0.0246 & 0.0148 & 0.0255 & \\ \hline \(F_{\text{n}}\) asymmetry & 1.070 & - & 1.068 & - & 1.064 & - & 6 \\ \hline \(F_{\text{tx}}\) asymmetry & 1.105 & - & 1.084 & - & 1.071 & - & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Skewness of friction coefficients and normal force asymmetry.  
Figure 9: Friction \(\mu_{z}\), normal \(F_{\text{n}}\) and tangential \(F_{\text{tx}}\) force distribution on the tyre-asphal interface for pressures 0.2 (**a**), 0.18 (**b**) and 0.22 (**c**) MPa under a \(\sim\)6 kN load.  
investigation of tyre deflection for normal and overloaded initial conditions led to the following conclusions.  
1. In the case of a slight asymmetry in the tread pattern of the tyre, in the contact area between the tyre and the road surface, the slip velocities of the treads at opposite points in the contact area (11 and 15) in the radial direction of the tyre were different in size and opposite signs. This suggests that the treads on different sides of the contact zone would generate different frictional forces and have different amounts of tread wear. Tread slip rates in the longitudinal contact zone of the tyre in the transverse section of the contact zone were almost the same. This trend of slip speeds was observed for different tyre pressures.
2. In the case of a slight asymmetry in the tread pattern of the tyre, it was found that, in the area of contact between the tyre and the road surface, the vertical velocity distributions of the treads in the transverse direction of the tyre (11 to 15 treads) were almost the same.
3. Due to the slight asymmetry of the tyre tread pattern, the contact normal and tangential tread forces at the edges of the tyre were higher than at the centre of contact of the tyre, and the right and left tread forces were different, a trend that persisted at different wheel vertical loads of 4 kN and 6 kN.
4. Finally, when talking about the safety of a moving car, we should keep in mind the asymmetry of tyre tread patterns, which can influence instability due to asymmetric tangential forces in the tyre contact zone during braking when the tyre is in an overdeflected state.  
Conceptualisation, A.M., M.B. and R.K.; methodology, R.K.; software, O.A.; validation, O.A., A.M. and M.B.; formal analysis, A.M.; investigation, O.A.; resources, A.M.; data curation, M.B.; writing--original draft preparation, A.M.; writing--review and editing, M.B.; visualisation, A.M.; supervision, M.B.; project administration, R.K.; funding acquisition, R.K. All authors have read and agreed to the published version of the manuscript.  
This research was funded by a Lithuanian Research Council grant (project registration Nr. MIP-17-233).  
Not applicable.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.