## 4 Results and Discussion
### Frequency-Domain Characteristics  
Power spectral density curves of the contact force between the pantograph and catenary, the vertical displacement of the contact strip, and the vertical acceleration of the contact strip as the train passes through the tunnel at different speeds are shown in Figures 15-17, respectively. Running speed does not appear to change the laws of the vibration characteristics. The basic frequency of the contact force between the pantograph and catenary is 2.11 Hz and the amplitudes are larger at the basic frequency versus multiple frequencies of 3 times, 4 times, 6 times, 10 times, 11 times, 15 times, 20 times, 25 times, 30 times,  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Maximum** & **Minimum** & **Mean** & **Standard** & **Standard** & **Mean** \\  & **(m)** & **(m)** & **(m)** & **Deviation** & **Deviation** & **Difference** \\ \hline
280 km/h & 0.01982 & 0.00383 & 0.01092 & 0.00246 & - & - \\
320 km/h & 0.01947 & 0.00282 & 0.01252 & 0.00276 & 12.20\% & 14.65\% \\
360 km/h & 0.0265 & 0.00469 & 0.01468 & 0.00331 & 34.55\% & 34.43\% \\
400 km/h & 0.02477 & 0.00208 & 0.01071 & 0.00384 & 56.10\% & \(-1.92\%\) \\
420 km/h & 0.02498 & 0.00398 & 0.01293 & 0.00387 & 57.32\% & 18.41\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Statistical results of vertical displacement of the contact strip passing through tunnel at different speeds.  
Figure 14: Comparison curves of vertical acceleration of the contact strip passing through tunnel at different speeds.  
or 35 times, where higher frequencies appear. The line test data after 20 Hz filtering is insufficient for the high-speed pantograph. Valleys appear immediately after the peaks at many frequencies (e.g., 5 times, 10 times, 15 times). At running speeds of 360 km/h, 400 km/h and 420 km/h, larger amplitudes appear at low frequencies (e.g., around 1.75 Hz, 8.8 Hz, 12.26 Hz) and are not multiple frequencies of the basic frequency.  
The basic frequency of contact strip vertical displacement is 2.11 Hz and the amplitudes are larger at the basic frequency and multiple frequencies of two times, three times, and five times. The main frequencies are mainly concentrated at frequencies below 25 Hz. At running speeds of 360 km/h, 400 km/h, and 420 km/h, larger amplitudes appear at lower frequencies (e.g., around 1.75 Hz) and are not multiple frequencies of the basic frequency. Above 250 Hz, the vertical displacement amplitude of the contact strip at 420 km/h drops rapidly and there is a large oscillation approaching 500 Hz.  
The basic frequency of contact strip vertical acceleration is 2.11 hz, and the amplitudes are larger at the basic frequency and multiple frequencies of two times, three times, five times, six times, 10 times, and 15 times. At running speeds of 360 km/h, 400 km/h, and 420 km/h, larger amplitudes appear at low frequencies (e.g., around 1.75 Hz, 4.98 Hz, 5.6 Hz, 7.1 Hz, and 8.8 Hz) and are not multiple frequencies of the basic frequency. The main frequencies are not obvious over 50 Hz. Until reaching 500 Hz, the amplitudes do not approach very small values.  
Figure 15: Power spectral density curves of contact force between pantograph and catenary passing through tunnel at different speeds.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Maximum** & **Minimum** & **Mean** & **Standard** & **Standard** & **Minimum** \\  & **(m/s2)** & **(m/s2)** & **(m/s2)** & **Deviation** & **Deviation** & **Difference** \\ \hline
280 km/h & 10.6145 & \(-\)7.42214 & \(-\)0.2054 & 2.51441 & - & - \\
320 km/h & 10.243 & \(-\)11.7679 & 0.01795 & 2.6966 & 7.25\% & 58.55\% \\
360 km/h & 12.1125 & \(-\)15.0945 & 0.0103 & 3.10342 & 23.43\% & 103.37\% \\
400 km/h & 10.3612 & \(-\)8.02659 & 0.04289 & 2.75807 & 9.69\% & 8.14\% \\
420 km/h & 10.0105 & \(-\)8.3469 & 0.02085 & 2.9433 & 17.06\% & 12.46\% \\ \hline \hline \end{tabular}
\end{table}
Table 5: Statistical results of vertical acceleration of the contact strip passing through tunnel at different speeds.