# Prediction of Theoretical Derailments Caused by Cross-Winds with Frequency
## Vehicle Behaviors under a Constant Cross-Wind
### Dynamic Equations for a Cyclical Cross-Wind Acting on the Body  
Winds in nature can generally blow at a constant speed but sometimes in the form of momentary gusts in the EN regulation. However, a cross-wind with various frequency components acts on railway vehicles by means of wayside structures repeatedly positioned or a cross-running train from the opposite direction. In this study, we assumed that the cyclical cross-wind consists of one sine component. We evaluated how it affects derailment of a railway vehicle. We can consider a wind force with frequency as the summation of a constant wind force of \(P_{t}\) and an alternating wind force of \(F_{w}\sin\omega t\) as shown in Figure 5.  
In order to derive the dynamic equations for the cross-wind with frequency, the free body diagram in term of a deformed shape is shown in Figure 6. In this step, the centrifugal force and gravity were not considered because their effects were considered in the previous step in Figure 4.  
And then, the final forces of suspensions and the behavior of a train can be calculated by superimposing the two results obtained in the previous step (Figure 4) and this step (Figure 6). Here we assumed the cyclic effects are small compared to those of the constant force. The equation of motion for the car body:  
\[\sum F_{x2}=F_{t}(\sin\omega t)\cos\varnothing-F_{RH2}-F_{LH2}=m_{2}\bar{x}_{2} \tag{19}\]  
\[\sum F_{y2}=-F_{t}(\sin\omega t)\sin\varnothing-F_{RV2}-F_{LV2}=m_{2}\bar{y}_{2} \tag{20}\]  
\[\sum M_{G2}=F_{t}(d\cos\varnothing+l\sin\omega t)+h_{2}(F_{RH2}+F_{LH2})+e_{2} (F_{LV2}-F_{RV2})=l_{2}\bar{\theta}_{2} \tag{21}\]  
Figure 5: The wind force with a frequency component. Figure 6: The free body diagram under the cyclic wind force with a specific frequency. The forces of the secondary suspension springs generated by relative displacements between the body and the bogie can be obtained by the Equations (10)-(12), but the following assumptions are made to simplify these equations. \[k_{1l}\gg k_{2l},\;k_{1v}\gg k_{2v}\;\rightarrow\;x_{2}\gg x_{1},\;y_{2}\gg y _{1},\;\theta_{2}\gg\theta_{1}\]  
From the Equations (19) and (21), the natural frequencies (\(w_{2a}\), \(w_{2b}\)) for the lateral and rolling directions of the body are derived as the following Equations (22) and (23). In general, since most of the cross-wind acting on a vehicle has low frequencies, the first natural frequency of the car body on suspensions becomes an important factor to assess derailment safety. \[w_{2a}^{2}=\frac{1}{2}\left\{ \begin{array}{c}\left(\frac{2k_{2y}}{m_{2}}+ \frac{2k_{2y}l_{2}^{2}}{l_{2}}+\frac{2k_{2y}c_{2}^{2}}{l_{2}}\right)\\ -\sqrt{\left(\frac{2k_{2y}}{m_{2}}+\frac{2k_{2y}h_{2}^{2}}{l_{2}}+\frac{2k_{2y }c_{2}^{2}}{l_{2}}\right)-4\frac{2k_{2y}}{m_{2}}\frac{2k_{2y}c_{2}^{2}}{l_{2} }}\end{array} \right\} \tag{22}\]  
\[w_{2b}^{2}=\frac{1}{2}\left\{ \begin{array}{c}\left(\frac{2k_{2y}}{m_{2}}+ \frac{2k_{2y}h_{2}^{2}}{l_{2}}+\frac{2k_{2y}c_{2}^{2}}{l_{2}}\right)\\ +\sqrt{\left(\frac{2k_{2y}}{m_{2}}+\frac{2k_{2y}h_{2}^{2}}{l_{2}}+\frac{2k_{2y }c_{2}^{2}}{l_{2}}\right)-4\frac{2k_{2y}}{m_{2}}\frac{2k_{2y}c_{2}^{2}}{l_{2} }}\end{array} \right\} \tag{23}\]  
From the free body diagram of Figure 6, the equations of motion of the bogie and the wheelset can be derived as follows. * The equation of motion for the bogie frame: \[\sum F_{x1}=F_{RH2}+F_{LH2}-F_{RH1}-F_{LH1}=m_{1}\ddot{x}_{1}\] (24) \[\sum F_{y1}=F_{RV2}+F_{LV2}-F_{RV1}-F_{LV1}=m_{1}\ddot{y}_{1}\] (25) \[\sum M_{G1}=h_{1u}(F_{RH2}+F_{LH2})+h_{1l}(F_{RH1}+F_{LH1})+e_{2}(F_{RV2}-F_{ LV2})+e_{1}(F_{LV1}-F_{RV1})=l_{1}\ddot{\theta}_{1}\] (26)
* The equation of motion for the wheelset: \[\sum F_{x0}=F_{RH1}+F_{LH1}-R_{RH}=m_{0}\ddot{x}_{0}\] (27) \[\sum F_{y0}=F_{RV1}+F_{LV1}-R_{RV}-R_{LV}=m_{0}\ddot{y}_{0}\] (28) \[\sum M_{G1}=R_{RH}h_{0}+e_{1}(F_{RV1}-F_{LV1})+e_{0}(F_{LV}-F_{RV})=l_{0}\ddot{ \theta}_{0}\] (29)  
It is possible to predict the forces of suspensions and wheel/rails, and the behavior of a vehicle by solving the simultaneous differential Equations (19)-(21), and (24)-(29).