# Flow-Induced Vibration Analysis by Simulating a High-Speed Train Pantograph
## 3 Pantograph Structural Modal Simulation Analysis
### Pantograph Modal Simulation Analysis Results  
The modal characteristics of the pantograph were analyzed under two different conditions: without prestress and with prestress. Since the pantograph can be considered as fixed to the top of the train, its boundary conditions were set as follows: the three mounting seats of the base frame were fixed constraints, eliminating all six degrees of freedom, as shown in Figure 13 (the planes where the red dots are located represent fixed constraints). For critical hinge locations, revolute constraints contact settings were used (such as the connection between the upper arm and the bow head system, the connection between the ends of the pull rod, the connection between the upper arm and the balancing rod, the connection between the balancing rod and the base frame system, and the connection between the lower arm and the base frame system), as illustrated in Figure 14 (the locations marked with red circles indicate hinge constraints). To replicate the modal test constraint conditions, a vertical spring constraint was added at the midpoint of the carbon strip on the bow head to simulate the catenary contact. Research [38] has shown that with increasing train speeds, for dynamic simulations of the catenary-pentograph interaction at speeds exceeding 120 km/h, using a fixed stiffness spring to model the catenary contact is more reasonable. To simulate the effect of the pantograph's airbag [39], a spring constraint of 3300 N/m was added between the lower arm and the base frame, as shown in Figure 15. For the condition without prestress, the initial catenary contact force was not considered; for the condition with prestress, a lifting torque was applied so that the vertical spring reaction force on the carbon strip of the bow head was 70 N (to describe the static contact force between the catenary and the pantograph), as illustrated in Figure 16.  
Figure 12: The simplified modal dynamics simulation model of the pantograph. Figure 13: Simulation and analysis diagram of fixed constraint base. Figure 14: Simulation and analysis diagram of hinge constraints on key connecting parts. (**a**) the connection between the upper arm and the bow head system. (**b**) the connection between the ends of the pull rod. (**c**) the connection between the ends the stabilizer rod. (**d**) the connection between the lower arm and the base frame system. Figure 15: Simulation and analysis diagram of spring constraint. The material parameters for the main components of the pantograph are set as shown in Table 6.  
Through finite element software simulation, the first 10 natural frequencies were obtained, as shown in Table 7.  
The results from the modal analysis, which reveal the natural frequencies, show that for both the pantographs without prestress and with prestress, the first 10 modal frequencies are within 40 Hz, and the first five frequencies are all below 12 Hz. Therefore, it can be considered that the pantograph is a typical system with a dense modal distribution and that its frequencies are relatively low. The influence of prestress on the natural frequencies of the pantograph is minor and does not alter the trend of the modal shapes for each order. The first to fifth modal shape diagrams of the entire pantograph structure, obtained through simulation software, as shown in Figure 17. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Component** & **Material Type** & \(\rho\) **[kg/m\({}^{3}\)]** & **E [GPa]** & **N [-]** \\ \hline Pantograph head & Carbon alloy & 1000 & 180 & 0.3 \\ Pantograph head bracket & Stainless steel & 7930 & 193 & 0.247 \\ Upper arm & Stainless steel & 7930 & 193 & 0.247 \\ Link arm & Aluminum alloy & 2700 & 71 & 0.33 \\ Lower arm & Stainless steel & 7930 & 193 & 0.247 \\ Stabilizer & Aluminum alloy & 2700 & 71 & 0.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Material parameters of the main components of the pantograph. Figure 16: Simulation and analysis diagram of preload.