## 4 Study on the Length of EVA in the Pantograph-Catenary System  
### Influence of Different Vehicle Velocity and Contact Wire Tension on the Length of EVA  
In the catenary system, the wave velocity is determined by the vehicle velocity and contact wire tension [29], and the wave velocity also influences the propagation of vibration and energy distribution. Besides, the temperature change can also influence the contact wire tension. Therefore, the influence of vehicle velocity and contact wire tension on the energy distribution of the catenary is investigated. The Sweden stitched catenary system SYT7.0/9.8 interacting with a high-speed pantograph is formulated based on the present model, and their key parameters are obtained from [27]. The energy distribution of the catenary under the influence of different vehicle velocities \(V\) is firstly calculated based on Equation (20) and shown in Figure 4, where \(V\) is chosen from 150 km/h, 222 km/h, and 250 km/h. Figure 4 shows that \(V\) has an obvious influence on the energy distribution. As \(V\) increases from 150 km/h to 222 km/h, the maximum energy increases from 0.31 J to 0.78 J, and other peaks also increase. Most importantly, the energy of those places far from the pantograph also increases. For example, the energy of the catenary at \(x=18\) m increases from 0.001 J to 0.24 J, which means that the vibration propagated to this place at \(V=250\) km/h, and the EVA should also increase to include these vibrations. The energy distribution of the catenary under the influence of different contact wire tensions is then shown in Figure 5, where the contact wire tension changes from 9.8 kN to 15.8 kN. It can be seen from Figure 5 that the contact wire tension \(T_{S}\) also influences \begin{table}
\begin{tabular}{l l l l l} \hline Actual maximum of & & & & \\ contact force [N] & 197.6 (185–199) & 224.1 (203–252) & 227.6 (210–232) & 246.1 (239–255) \\ \hline Actual minimum of & & & & \\ contact force [N] & 97.1 (92–102) & 85.2 (56–88) & 122.3 (105–128) & 75.2 (43–78) \\ \hline Range of vertical & & & & \\ position of the point & 23.1 (18–25) & 33.5 (26–36) & 20.9 (13–23) & 59.1 (38–63) \\ \hline Maximum uplift at & & & & \\ support [mm] & 67.8 (55–79) & 73.1 (51–79) & 89.2 (74–95) & 90.2 (69–95) \\ \hline Percentage of loss of & & & & \\ contact [\%] & 0 (0) & 0 (0) & 0 (0) & 0 (0) \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 4: The energy distribution of the catenary at different vehicle velocities. the energy distribution, but its influence is smaller than that of the vehicle velocity. The maximum energy of the catenary decreases from 0.51 J to 0.23 J as the \(T_{S}\) increases from 9.8 kN to 15.8 kN, and the other peaks also decrease. However, the general shape of the energy distribution has no obvious changes, and the vibration propagation distance is reduced shortly. When \(T_{S}\) = 9.8 kN, the closest location with energy smaller than 0.01 J is \(x\) = 21.2 m, but it only changes to \(x\) = 25.7 m as \(T_{S}\) increases to 15.8 kN. The total energy of the EVA with respect to the length of the EVA under the influence of different \(V\) and \(T_{s}\) is then calculated based on Equation (21), as shown in Figure 6. It can be seen from Figure 6 that the total energy of the EVA converged to different values at different \(V\) and \(T_{s}\) values, and its corresponding length also changes. When \(V\) = 150 km/h, the total energy of the EVA converged to 46.2 J with \(d_{e}\) = 185.2 m, and it further increased to 118.4 J with \(d_{e}\) = 257.8 m and \(V\) = 250 km/h. When TS increases from 9.8 kN to 15.8 kN, the converged total energy of the EVA decreases from 105.1 J to 52.1 J as \(d_{e}\) changes from 215.3 m to 186.4 m. Based on this, the appropriate length of the EVA can be clearly determined by its energy distribution. The appropriate length of EVA with respect to different key parameters is finally analyzed.