# Prediction of Theoretical Derailments Caused by Cross-Winds with Frequency
## 4 Evaluation of Derailment Safety under Cross-Wind
### Derailment Safety According to the Cross-Wind Regulations  
The ratios of wheel unloading between simulations and the theoretical formula were compared under the driving conditions (Table 7), in the case that wind forces are applied in a horizontal direction (\(\varnothing=0^{\circ}\) in Figures 4 and 6) on the ground. Table 10 shows the ratio of wheel unloading when constant cross-winds act on the KTX power car on the straight track and curved tracks (lateral acc. 0.5 [m/s\({}^{2}\)]; this is the case of 1.0 [m/s\({}^{2}\)] referred to in the EN regulation [EN 14067]). Here it is assumed that there are no lateral forces transmitted to the wheelset due to rail irregularities. \begin{table}
\begin{tabular}{c c} \hline \hline
**Wind Speed [m/s]** & **Driving Conditions** \\ \hline
45– & Operation stop \\ \hline
40–45 & Under 90 [km/h] \\ \hline
30–40 & Under 170 [km/h] \\ \hline
30 & Decelerated operation \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The regulations on driving conditions according to wind speeds. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Country/Regulation** & **Method/Criteria** & **Note** \\ \hline EU & \((\text{P}/\text{Q})_{smu}<0.8\) & 2 m moving average of derailment coefficient \\ \hline UIC & P/Q \(<0.8\) & Derailment coefficient \\ \hline TSI & \(\Delta\)Q/Q\({}_{0}<0.9\) & Ratio of wheel unloading \\ \hline \multirow{4}{*}{North America} & P/Q \(<1.0\) & Derailment coefficient \\ \cline{2-3}  & \(\Sigma\) [P/Q] \(<1.0\)\(\sim 1.5\) & Weinstock coefficient \\ \cline{1-1} \cline{2-3}  & \(\Delta\)Q/Q\({}_{0}<0.9\) & Ratio of wheel unloading \\ \hline RTRI (Japan) & \(\Delta\)Q/Q\({}_{0}<1.0\) & Ratio of wheel unloading \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The various evaluation criteria of derailment. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** \\ \hline Aerodynamic side force coefficient & \((C_{y})\) & 0.6 \\ \hline Side area & (A) & 70.5 & [m\({}^{2}\)] \\ \hline Air density & \((\rho)\) & 1.275 & [kg/m\({}^{3}\)] \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The specifications of the high-speed train. Figure 10 shows the contact forces of wheel/rail and the ratio of wheel unloading. When the wind speed of about 65 [m/s] acts on the vehicle, the wheel unloading ratio exceeds the TSI criteria for the rollover derailment (\(\Delta\)Q/Q \(<\) 0.9). On the other hand, basic lateral forces acting on running wheelsets take place due to wheel conicity and rail irregularities, even though there is no cross-wind. In the previous study, the KTX has a measured coefficient of derailment (P/Q = 0.315) during normal running on a high-speed rail [36,37]. Using the measured coefficient of derailment and Equation (31), the P/Q can be converted to an additional lateral force acting on the wheelset, as given in Equation (32). The ratios of wheel unloading according to cross-wind speeds are compared in Table 11. These results were obtained considering both the wheel-rail interactions and the lateral forces due to cross-winds on the running vehicle. \[\frac{P}{Q}=\frac{(F_{LH1}+F_{RH1})(1+\mu\beta)-\mu(1+\gamma)F_{LV1}+\mu\gamma F _{RV1}}{(1+\gamma)F_{RV1}-\gamma F_{LV1}+\beta(F_{LH1}+F_{RH1})} \tag{31}\]