## 3 Results
### Determination of the Tire Steering Resistance Torque of an Unlocked Steered Wheel in Static State
#### 3.2.3 Study of the Position of the Center of Relative Motion of the Contact Patch
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Tire Parameter**} & \multicolumn{6}{c}{**Parameter Values**} \\ \cline{2-7}  & \(l_{k}\) = 0 m & \(l_{k}\) = 0.225 m & \(l_{k}\) = 0.295 m & \(l_{k}\) = 0.365 m & \(l_{k}\) = 0.68 m & \(l_{k}\) = 0.82 m \\ \hline \(\Delta_{t}\), m (exper.) & 0 & 0.089 & 0.116 & 0.122 & 0.161 & 0.167 \\ \(R_{t}\), m (exper.) & 0 & 0.135 & 0.171 & 0.231 & 0.499 & 0.632 \\ \(\theta_{t}\), deg (42) & \(\infty\) & 24.41 & 18.85 & 13.95 & 6.46 & 5.1 \\ \(M_{\varphi}\), N\(\cdot\)m (exper.) & & 2434.5 & 2772.2 & 2825.7 & 2129.2 & 1395 \\ \(M_{\varphi}\), N\(\cdot\)m (45), (48) & 2062.2 & 2481 & 2731.9 & 2872.76 & 2045 & 1407 \\ \hline \end{tabular}
\end{table} 
Table 4: Results of calculations of the static tire steering resistance torque \(M_{\varphi}\) when changing the knuckle length \(l_{k}\) in the range from 0 to 0.82 m at \(p_{t}\) = 0.4 MPa, \(a\) = 0.225 m, \(b\) = 0.334 m, \(a_{k}\) = 9.5 deg, \(\beta_{k}\) = 0 deg, and \(\theta\) = 20 deg. Depending on the calculated values of the angle \(\theta_{t}\), the static tire steering resistance torque \(M_{\varphi}\) was calculated as follows: when the knuckle length is 0-0.365 m, according to dependence (45); when the knuckle length is 0.68 and 0.82 m, according to Expression (48). Figure 9 shows the dependence of the tire twisting angle \(\theta_{t}\) depending on the knuckle length when steering in static state at steer angle \(\theta=20\) deg. These data are approximated by means of MS Excel with a polynomial function of the third order:  
\[\theta_{t}=-108.49l_{k}^{3}+236.79l_{k}^{2}-180.76l_{k}+53.945,\,R^{2}=0.9989 \tag{60}\]  
Figure 10 shows the graph of dependence \(M_{\varphi}=f(l_{k})\) when steering in static state at a steer angle \(\theta=20\) deg, calculated according to Expressions (45) and (48) (solid line) and to experimentally obtained values of this torque (red dots). Figure 9: Dependencies of the angle of twisting of the tire \(\theta_{t}\) on the knuckle length on the support surface with the friction coefficient \(\varphi=0.7\). From an analysis of Figures 9 and 10, it can be seen that when the knuckle length changes from 0 to 0.362 m, the twisting angle decreases to 13.95 deg, and the static tire steering resistance torque is determined by Expression (45). If the knuckle has dimensions of 0.68 m and 0.82 m, then the static tire steering resistance torque is calculated according to Expression (48). An increase in the knuckle length from 0 to 0.362 m led simultaneously to a decrease in the twisting angle of the tire to 13.95 deg and an increase in the displacement \(\Delta_{t}\) from 0 to 0.122 m. The increase in the torque \(M_{\varphi\text{max}}\) is explained by an increase in the displacement of the tire contact patch relative motion center. In this range of changes in the knuckle length, there are sliding zones in the tire contact patch. As the knuckle length increases from 0.68 to 0.82 m, the twisting angle of the tire body decreases from 6.46 to 5.1 deg. In this case, there are adhesion and sliding zones in the tire contact patch. When the knuckle length increases, the area of the sliding zones decreases, the adhesion zones increase, which leads to a decrease in the torque \(M_{\varphi}\). From the analysis of the conducted studies, it follows that when steering in static state, the tire contact patch participates in the transport and relative motions. The transport motion center is the projection onto the support surface of the center of rotation of the wheel, and the dependencies for determining the position of this center are obtained. The center of relative motion is determined experimentally, and its position is within the tire contact patch. When the knuckle length increases, the center of relative motion shifts to the edge of the contact patch of the tire. At the same time, the static tire steering resistance torque depends on the value of the tire twisting angle and the position of the relative motion center.