# Assessment of Active Wheelset Steering System Using Computer Simulations and Roller Rig Tests
## 3 Computer Simulations
### Track Radius Estimation
Consequently, the angles of rotation of the first and second bogie differ, and vary in dependence of many parameters such as wheel and rail profiles, creep coefficients, value of unbalanced lateral acceleration, or torque caused by lateral deflection of flexi-coil springs. Typically, the leading bogie exhibits smaller  
Figure 8: Track curvature estimated by bogie to car body rotation and its derivative. Figure 9: Comparison of track curvature estimation error. angle and trailing bogie larger angle than is expected by ideal alignment of wheelsets and the track centre. The maximum deviation of the angles of rotation can be expressed by:  
\[\Delta\alpha=\frac{2\sigma}{a} \tag{3}\]  
where \(2\sigma\) is gauge clearance and \(a\) is the wheelbase of the bogie. For the rail profile UIC 60, rail inclination 1:40, wheel profile S1002, and nominal values of wheelset and rail gauge is the gauge clearance \(2\sigma=13\) mm. Wheelbase of the locomotive under consideration \(a=2500\) mm. According to Equation (3), it is then \(\Delta\alpha=0.0052\) rad \(=0.30\) deg. Thus, due to uncertain lateral position of wheelsets the error of estimated track curve curvature can reach tens of percent (see Table 3). In order to determine the probable position of the wheelsets within the gauge clearance the set of simulations was performed. The simulation parameters are summarized in Table 4, where \(p\) is superelevation of rails and \(a_{n}\) is the value of uncompensated lateral acceleration. The simulations were performed for the ideal track without irregularities and friction coefficient 0.4. The quasistatic value of bogies rotation towards car body were observed. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Manoeuvre \(\mathbf{\mathrm{N}}\)** & \(\mathbf{R}\) & \(\mathbf{p}\) & \(\mathbf{\
u}\) & \(\mathbf{a}_{n}\) \\ \hline
[-] & **[m]** & **[m]** & **[km/h]** & **[m/s\({}^{-2}\)]** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 4: Simulations parameters. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & **Bogie 1** & & & **Bogie 2** \\ \hline \(R\) & \(\rho\) & \(\mathbf{\alpha}\) & \(\mathbf{\alpha}_{max}\) & \(\rho_{max}\) & \(\Delta\rho\) & \(\mathbf{\alpha}_{min}\) & \(\rho_{min}\) & \(\Delta\rho\) \\ \hline
**[m]** & **[1/m]** & **[rad]** & **[rad]** & **[1/m]** & **[\%]** & **[rad]** & **[1/m]** & **[\%]** \\ \hline
250 & 0.0040 & 0.0174 & 0.0226 & 0.0052 & 29.9 & 0.0122 & 0.0028 & \(-\)29.9 \\
300 & 0.0033 & 0.0145 & 0.0197 & 0.0045 & 35.9 & 0.0093 & 0.0021 & \(-\)35.9 \\
400 & 0.0025 & 0.0109 & 0.0161 & 0.0037 & 47.8 & 0.0057 & 0.0013 & \(-\)47.8 \\
500 & 0.0020 & 0.0087 & 0.0139 & 0.0032 & 59.8 & 0.0035 & 0.0008 & \(-\)59.8 \\
600 & 0.0017 & 0.0073 & 0.0125 & 0.0029 & 71.7 & 0.0021 & 0.0005 & \(-\)71.7 \\
800 & 0.0013 & 0.0054 & 0.0106 & 0.0024 & 95.6 & 0.0002 & 0.0001 & \(-\)95.6 \\
1000 & 0.0010 & 0.0044 & 0.0096 & 0.0022 & 119.5 & \(-\)0.0009 & \(-\)0.0002 & \(-\)119.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Track curvature estimation error for extreme positions of wheelsets. The results confirm underestimation of track curvature on leading bogie and overestimation of track curvature on trailing bogie. This phenomenon is more significant for negative values of uncompensated lateral acceleration, i.e., in low speeds. In order to eliminate it, a linear approximation of relation between calculated and real track curvature was constructed. The approximation is shown on Figure 10 by yellow curve and mathematically can be expressed by:  
\[ \begin{array}{l}\rho_{real}=a_{1}+b_{1}\rho_{1}\\ \rho_{real}=a_{2}+b_{2}\rho_{2}\end{array}  \tag{4}\]  
where \(\rho_{real}\) is real track curvature, \(\rho_{1}\) and \(\rho_{2}\) are track curvatures calculated on leading and trailing bogie, respectively, and \(a_{1}\), \(a_{2}\), \(b_{1}\), and \(b_{2}\) are coefficients of the linear approximation. Combining the relations Equations (2) and (4) the final relation for calculation of the track curvature is obtained:  
\[