## 2 Theoretical Discussion
### Key Dynamic Stability Evaluation Parameters Review  
#### 2.1.1 Derailment Coefficient and Wheel Load Fluctuation  
The derailment coefficient is defined as the ratio of lateral pressure (the force exerted by the wheels on the rail at a horizontal direction) and the wheel load (force exerted on the rail at vertical direction) of a railroad vehicle. The lateral force is commonly expressed as Q, while the vertical rail load expressed as P, and the derailment coefficient is expressed Q/P. When the lateral force increases or the vertical wheel load decreases, the deviation coefficient (Q/P) increases, resulting in the increase in the probability of derailment.  
\[\frac{\text{Q}}{\text{P}}=\text{Derailment Coefficient(Q/P)}\]where:  
Q: Lateral wheel load (kN);  
P: Vertical wheel load (kN).  
During the train operation, the hunting oscillation movement of the wheels, in addition to the specification load of the vehicle, exerts a vertical force and lateral force on the rail simultaneously. The analysis of the derailment coefficient is divided into a static analysis and a dynamic analysis. The static analysis in the state shown in Figure 1 holds the following equation considering the force balance at the contact point.  
The evaluation principle is the same in multiple countries, but the requirement criteria (safety limit) is different with some nations. Refer to Table 1 for details.  
With the calculated derailment coefficient from the trackside measurement, the rate of wheel load reduction is generally used to evaluate ride safety with regards to the dynamics of railway vehicles [13]. The static wheel load is determined in accordance to the vehicle's specification weight (static load) where the effect from the vibration or hunting oscillation of the vehicle is not applied. In most cases, this is obtained by the averaged wheel load of the vehicle operating at a speed of about 5 km/h in a flat straight section (the method for deriving static wheel load or quasi static wheel load is different based on national specifications or codes).  
The reduction of the wheel weight occurs due to vibration of the vehicle, knitting of the center, incorrect planarity of the track and vehicle, cant and centrifugal force in the curve, and wind pressure. The ratio of the static wheel weight (V) to the wheel weight reduction value (\(\Delta\)V) is called the wheel  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Nations (Standards or Codes)** & **Evaluation Criteria of Q/P** \\ \hline Union of Railways (UIC) & \(\leq\)1.2 \\ German ICE High Speed Train regulation & \(\leq\)0.8 \\ North American regulation & \(\leq\)1.0 \\ Korean Railway Code & \(\leq\)1.2 \\ Chinese Standard (GB5599-85) & \(\leq\)1.2 (Limit 1, acceptable), \(\leq\)1.0 (increased safety margin) \\ Japanese Railway Construction Standard & \(\leq\)0.8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Derailment coefficient standard criteria for different nations [11; 12].  
Figure 1: Interaction forces between the wheel and rail.  
load fluctuation ratio; if this value exceeds the allowable limit, there is a risk of derailment. Wheel load fluctuation is calculated with the following equation, and is calculated as an absolute value:  
\[\left|\text{{{{Wheel load fluctuation coefficient}}}}\right|=\ \frac{\text{Dynamic Load}\left({{V_{ Dynamic}}}\right)-\text{Static Load}\ ({{V_{static}}})}{{{V_{Static}}}} \tag{1}\]