# Prediction of Theoretical Derailments Caused by Cross-Winds with Frequency
## 4 Evaluation of Derailment Safety under Cross-Wind
### Derailment Safety Due to the Cross-Wind Frequencies
#### 4.4.1 The 2D Dynamic Vehicle Model without Dampers  
In the 2D dynamic vehicle model of Section 2, the dampers in suspension are not considered to formulate theoretical equations. Therefore, if the lateral natural frequency of the vehicle and the frequency of the cross-wind are matched, resonance phenomena will occur and the amplitude will be continuously increased as time goes by.  
The resonance phenomena can easily be described using the dynamic equation of a simple spring-mass system. When a periodic external force is applied to the system without dampers, the dynamic equation is shown in Equation (33), and its general solution is shown in Equation (34) [39].  
\[\mathrm{m}\ddot{x}+kx=F_{0}\cos\omega t \tag{33}\]  
\[\mathrm{x(t)}=\mathrm{A}\cos\omega_{0}t+\mathrm{B}\sin\omega_{0}t+\frac{F_{0} }{m\big{(}\omega_{0}^{2}-\omega^{2}\big{)}}\cos\omega t \tag{34}\]  
A and B are constants determined by the initial conditions and \(\omega_{0}\) is a natural frequency.  
\[\omega_{0}=\sqrt{k/m} \tag{35}\]  
If the frequency of external forces (\(\omega\)) is equal to or very near to that of the vehicle (\(\omega_{0}\)), the solution of the dynamic equation can be converted as shown in Equation (36).  
\[\mathrm{x(t)}=\mathrm{A}\cos\omega_{0}t+\mathrm{B}\sin\omega_{0}t+\frac{F_{0} }{2m\omega_{0}^{2}}t\sin\omega_{0}t \tag{36}\]  
Because of the term \(t\sin\omega_{0}t\), the solution continuously increases with the passage of time. This phenomenon is called a resonance as shown in Figure 11.  
However, the actual springs in a vehicle suspension must not be extended longer than the predesigned distance not to be permanently deformed or damaged over the design limit. So, the spring constant (k) must be assumed as a nonlinear over the linear elastic range of a normal spring, which means Equation (33) is invalid in an excessive amplitude condition.  
Figure 12 and Table 13 show the results that the ratios of the wheel unloading of the simulation and the current formula without dampers are rapidly increased at a resonance condition.  
Figure 11: The amplitude under a resonance condition.  
However, the results using [PERSON]'s formula are constant regardless of the cross-wind frequency because the formula cannot consider it. Also, the formula cannot take into account the change of ratio of wheel unloading by rail irregularities. Thus, when rail irregularity (P/Q = 0.315) is applied, as shown in Figure 12, the value of the ratio of wheel unloading is low compared to other results, even in low frequency.  
Even though the ratio of the wheel unloading can have 1 as the maximum value, the TSI criteria regards over 0.9 as a derailment. Here, the simulation contact forces were filtered using 1.5 [Hz] low-pass filter to remove high-frequency components caused by numerical factors.  
Although oscillatory lateral forces between wheel-tails are generated by the wheel conicity in the simulation, our formulations could not consider it. These oscillation effects gave higher values in simulation compared to the results of our formula in Table 13.  
\begin{table}
\begin{tabular}{c} **[PERSON]’s Formula** \\ **(P/Q = 0.315)** \\ \end{tabular} \\ \hline
0.1 & 0.488 & 0.355 & 0.528 \\ \hline
0.3 & 0.496 & 0.355 & 0.530 \\ \hline
0.5 & 0.510 & 0.355 & 0.532 \\ \hline
0.7 & 0.548 & 0.355 & 0.579 \\ \hline
0.9 & 0.908 & 0.355 & 1.000 \\ \hline
1.1 & 0.585 & 0.355 & 0.636 \\ \hline
1.3 & 0.525 & 0.355 & 0.580 \\ \hline
1.5 & 0.507 & 0.355 & 0.569 \\ \hline \hline \end{tabular}
\end{table}
Table 13: The comparison between the theoretical formulas (current study and [PERSON]’s formula) and simulations.  
Figure 12: The comparison of the ratios of wheel unloading according to the frequencies of a cross-wind.