# A Method of Predicting Critical Derailment Speed and Analysis of Derailment Process
## 3 Calculation Methods of Critical Speed  
### The Limit Cycle Method and Root Locus Method  
The hunting instability critical speed is an important reference to determine the operating speed of high-speed trains. The hunting motion is a kind of lateral vibration when the train is running at high speed. Because of the tapered wheel tread, when two wheels are rolling on the rail with different rolling circles, the wheelset will produce a hunting motion that is serpentine-like. The excessive vibration amplitude will cause hunting instability when approaching the critical speed of hunting instability, and the safety of the vehicle operation will be greatly affected; in other words, there is a great risk of derailment. Two of the most common methods for calculating the hunting-instability critical speed in a numerical simulation are the limit cycle method and the root locus method. The premise of these two methods is that the vehicles travel along a straight track.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline Body mass/kg & 32,000 & Vehicle distance setting/m & 17.5 \\ Frame mass/kg & 3300 & Wheelbase/m & 2.5 \\ Wheel mass/kg & 1780 & tread rolling circle radius/m & 0.46 \\ Treat surface profile & LM & Rail profile & UIC60 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Vehicle–track model parameters.  
Figure 1: Vehicle–track system dynamics model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Alignment** & **Length (m)** & **Superelevation (mm)** & **Curve Radius (m)** \\ \hline Straight & 200 & 0 & \(\infty\) \\ Transition curve & 200 & 0–175 & \(\infty\)–R \\ Curve & 1000 & 175 & R \\ Transition curve & 200 & 175–0 & R–\(\infty\) \\ Straight & 200 & 0 & \(\infty\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Track alignment parameters.  
As shown in Table 3, the time-domain integral equation of the vehicle system of the limit cycle method is used, and the nonlinear instability critical speed of the vehicle is measured. The root locus method measures the linear instability critical speed of the vehicle through the integral operation in the frequency domain [17].  
Based on the limit cycle method, the variation curve of the lateral displacement of the wheelset with the speed is obtained, as shown in Figure 2, and it is determined that the nonlinear instability critical speed of the train is less than 320 km/h. According to the root locus method, it is determined that the linear instability critical speed of the train is about 380 km/h.  
The vehicle operating at the critical speed calculated by the above two methods is in a hunting instability state. When the speed is high enough to cause a hunting instability of the vehicle, derailment may not occur. Meanwhile, the speed which causes vehicle derailment may not induce hunting instability. Therefore, a new method is proposed to calculate the critical derailment speed when the vehicle is overspeeding along a curve, which is called the progressive trial method. The critical derailment velocity calculated by this method can ensure the inevitability of derailment.