# A Method of Predicting Critical Derailment Speed and Analysis of Derailment Process
## 3 Calculation Methods of Critical Speed
### Progressive Trial Method  
The progressive trial method is to calculate the derail speed of the vehicle when it overspeeds through the curve. During the process of the speed increases regularly and steadily, the speed at which the train derailment occurs is defined as the critical derailment speed.  
Figure 2: Relationship between traverse and speed.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Limit Cycle Method** & **Root Locus Method** \\ \hline \hline Consider non-linear factors & Consider linear factors \\ \hline \hline Perform integral operation in time domain & Perform integral operation in frequency domain \\ \hline The vehicle passes through the exciting line at a & The instability speed of the vehicle is \\ certain speed, and then runs on the smooth line & determined based on the inherent vibration \\ to observe whether the lateral displacement of & frequency of each rigid body at different \\ the wheelset converges. When the lateral & vehicle speeds. The minimum damping value \\ that is, the lateral displacement diverges, the & less than 0 represents the \\ speed at the moment is defined as the & system destabilization. \\ instability critical speed. & \\ \hline \hline \end{tabular}
\end{table}
Table 3: The limit cycle method and root locus method.  
The running speed of the vehicle on the curve is shown in Figure 3. The initial speed when the train enters the curve safely is low. After traveling a certain distance, the speed accelerates to a certain value and the vehicle runs steadily. The process of acceleration and maintenance of speed is repeated until the vehicle derails.  
In this method, the definition of the critical derailment state comes from the geometric state of wheel-rail contact, and the train derailment is determined according to the wheel lift, which is also the most direct way to judge derailment. As shown in Figure 4, \(\mathrm{I_{w}}\) is the lateral displacement of the wheelset; \(\mathrm{h_{p}}\) is the wheel lifting height, which is the vertical distance between the nominal contact point of the wheel tread and the highest point of the rail; \(\mathrm{h_{i}}\) is the height of the wheel flange; and \(\mathrm{h_{j}}\) is the height of the wheelset jumps.  
When the bottom of the flange is completely climbed onto the rail, \(\mathrm{h_{p}}=\mathrm{h_{i}}\). When the wheelset jumps from the rail, \(\mathrm{h_{p}}=\mathrm{h_{j}}>\mathrm{h_{i}}\). It shows that when the wheel lifting height \(\mathrm{h_{p}}\) exceeds the flange height \(\mathrm{h_{j}}\), and the lateral displacement of the wheelset \(\mathrm{I_{w}}\) is greater than the width of the rail head, it can be determined that the wheel is separated from the rail; the wheel speed at this time is defined as the critical derailment speed.  
The critical derailment speeds under different radius curves are shown in Figure 5. The critical derailment speed is positively correlated with the radius. When the radius of the curve increases, the critical speed increases and the growth rate becomes slower. For example, when the curve radius increases from 800 m to 1000 m, the critical speed increases 30 km/h. When the curve radius increases from 1600 m to 1800 m, the critical speed only increases 20 km/h. It can be seen that the larger the radius of the curve is, the smaller the influence of this factor on the increase in critical derailment speed.  
Figure 4: Geometric state of wheel–rail during derailment.  
Figure 3: The running speed of the vehicle on the curve.