# Study on Gage Widening Methods for Small-Radius Curves
## 4 Adaptability Analysis of Gage Widening Standards
### Influence of Gage Widening on Curve Passing Performance
As shown in Figure 9, when the rail tilt angle was \(\theta\), the anti-tipping torque provided by the backing plate was obtained by  
\[M=2\int\limits_{0}^{L/2}\frac{c}{L}\cdot\theta\cdot x\cdot xdx=\frac{c\theta L ^{2}}{12}. \tag{13}\]  
The load formed by the compression of the backing plate during tipping was  
\[F=\int\limits_{0}^{L/2}\frac{c}{L}\cdot\theta\cdot xdx=\frac{c\theta L}{8}. \tag{14}\]  
The left and right sides of the centerline of the backing plate provided the same tipping torque, and the distance between the spring and the center of the backing plate was  
\[x_{c}=\frac{M}{2F.}=\frac{L}{3}. \tag{15}\]  
Figure 8: Fastener system model. Figure 7: Track dynamics model. As can be observed in the above equation, the larger the width of the backing plate, the higher the anti-tipping performance of the backing plate was. It can be seen that a complex multifocal model was established in this paper, which was closer to the actual operational situation of vehicles in curved sections and the actual force transmission mechanism inside the vehicle-track system. The conclusions obtained have more reference values. Moreover, based on existing computer capabilities, such multi-degree-of-freedom models can be calculated fairly quickly. Based on the established vehicle-track coupling dynamics model, the wheel or rail dynamic responses of the vehicle passing through curves with radii of 150, 200, 250, 300, 350, and 400 m curves were calculated, and the wheel or rail lateral force was used as the evaluation index to analyze the regularity of the influence of gage widening on curve passing. Each curve was divided into three working conditions: under-superelevation, balanced superelevation, and over-superelevation. The speed of the train passing through each curve is listed in Table 16.  
Figure 10 shows the regularity of the wheel or rail lateral force change corresponding to various radius curves with the widening of the gage. With the increase in the gage widening value, the wheel and rail lateral forces corresponding to various radius curves show a decreasing trend, and the rate of this trend also decreases. The 10 mm gage widening value is the changing point of the wheel and rail lateral force reduction rate, and \begin{table}
\begin{tabular}{c c c} \hline \hline
**Radius (m)** & **Superelevation (mm)** & **Speed (km/h)** \\ \hline \multirow{3}{*}{150} & 36 & \multirow{3}{*}{35} \\  & 96 & \\  & 156 & \\ \hline \multirow{3}{*}{200} & 34 & \multirow{3}{*}{40} \\  & 94 & \\  & 154 & \\ \hline \multirow{3}{*}{250} & 35 & \multirow{3}{*}{45} \\  & 95 & \\  & 155 & \\ \hline \multirow{3}{*}{300} & 38 & \multirow{3}{*}{50} \\  & 98 & \\  & 158 & \\ \hline \multirow{3}{*}{350} & 42 & \multirow{3}{*}{55} \\  & 102 & \\ \cline{1-1}  & 162 & \\ \hline \multirow{3}{*}{400} & 46 & \multirow{3}{*}{60} \\  & 106 & \\ \cline{1-1}  & 166 & \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Calculated working conditions. Figure 9: Schematic of rail tipping. the changing trend of the wheel and rail lateral forces gradually stabilizes after 10 mm. For a curve with a radius of less than 300 m, when the gage widening value increases from 0 to 10 mm, the wheel and rail lateral forces rapidly decrease under a range of 16-20%. For a curve with a radius exceeding 300 m, the degree of influence of the widening gage on the wheel and rail lateral forces is weakened. When the gage widening value is increased from 0 to 10 mm, the wheel and rail lateral forces reduction range is 10-15%. Therefore, in combination with the maintenance and repair conditions, appropriately expanding the curve radius range of gage widening can reduce wheel and rail lateral forces, thereby decelerating rail wear and prolonging the life of the rail. Figure 10: The influence of the gage widening value on wheel or rail lateral force. (**a**) R150 m; (**b**) R200 m; (**c**) R250 m; (**d**) R300 m; (**e**) R350 m; (**f**) R400 m.