# Prediction of Theoretical Derailments Caused by Cross-Winds with Frequency
## 5 Conclusions  
Even though there was a lot of theoretical research to predict the risk of derailment due to cross-winds, most of them could not consider the variables like wheel-rail interactions and frequencies of cross-winds. So, there occurred some differences compared with actual results. Figure 16: The ratios of wheel unloading according to track gauges. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Parameters**} & \multicolumn{2}{c}{**Values**} \\ \hline \multirow{2}{*}{Wind} & Speed & 30 & [m/s] \\ \cline{2-4}  & Angle & 0 & [\({}^{\circ}\)] \\ \hline \multicolumn{2}{c}{Vehicle speed} & 300 & [km/h] \\ \hline \multicolumn{2}{c}{Rail irregularity (P/Q)} & 0.315 & \\ \hline \multicolumn{2}{c}{Track} & Straight & Curved & \\ \hline \multicolumn{2}{c}{Lateral acc.} & - & 1 & [m/s\({}^{2}\)] \\ \hline \hline \end{tabular}
\end{table} 
Table 15: The symbols used in the damped vibration equation. In this study, a new theoretical formula was proposed to predict the risk of derailment due to cross-winds with frequency considering wheel-rail interactions. The proposed formula was verified using MBD simulations. The following conclusions were obtained:  
1. The static equilibrium equations for the body, the bogie, and the wheelset were derived under a constant force of winds applied to the vehicle using the 2D free body diagram. The wheel-rail and suspension forces, and the displacements and rotation angle of the vehicle were obtained by solving the equations for each static variable. 2. The dynamic equations for the body, the bogie, and the wheelset were derived under a wind force with frequency components applied to the vehicle using the 2D free body diagram. The wheel-rail and suspension forces, and the displacements and rotation angle of the vehicle were obtained by solving the equations for each dynamic variable. In addition, the lateral and rolling natural frequencies of the vehicle were derived by solving the motion equations for the lateral and rolling directions. 3. The wheel-rail forces obtained from summing the dynamic and static ones were improved using a known derailment coefficient to consider wheel-rail interactions during a normal train running without winds. 4. The multibody dynamics software (Simpack) model was generated using the specifications of a Korean high-speed train (KTX) to evaluate our theoretical formula. For the operational regulations of the KTX under cross-wind conditions, the ratios of wheel unloading on the dynamic simulation and our theoretical formula were compared and well matched. 5. The ratio changes of wheel unloading were investigated under the presence or absence of suspension dampers considering the natural frequency of a vehicle and the cross-wind frequency. As a result, the ratios were found to exceed the TSI criteria regardless of the presence or absence of dampers if the natural frequency of the vehicle and the cross-wind frequency were matched. 6. The ratio changes of wheel unloading were investigated according to track gauges using the simulations, [PERSON]'s formula, and our theoretical one. Although all the results were similar on the straight and curved tracks, the wheel unloading ratios using [PERSON]'s formula were larger than the simulations and ours which are in a good agreement. So, our theoretical formula gives more accurate results than [PERSON]'s formula, because it considers various operational conditions like a cross-wind frequency and a normal running derailment coefficient due to wheel-rail interaction. Therefore, this formula has strong advantages over the previous studies or the commonly used [PERSON] or [PERSON]'s formula because it considers various variables, such as fluctuant cross-winds, rail irregularities, and dynamic derailment behaviors. It would be possible to predict well the derailment risk under actual operating conditions. It could be used for setting suspensions or railway vehicle specifications in the initial design stage.