## 7 Conclusions  
(1) The aerodynamic uplift force of the pantograph is primarily distributed on the pantograph head structure, where the aerodynamic uplift forces on the upper framework are all negative, while the absolute values of the aerodynamic uplift forces on the lower arm rods are the smallest. The aerodynamic uplift force on the pantograph head gradually decreases with increasing wind direction angle. Within the range of wind directions from \(10^{\circ}\) to \(50^{\circ}\), under the same crosswind angle, the aerodynamic uplift force on the pantograph head increases with increasing crosswind speed. However, when the wind direction angle reaches \(60^{\circ}\) and the crosswind speed is \(30\) m/s, the aerodynamic uplift force on the pantograph head sharply decreases. Additionally, the aerodynamic uplift force of the pantograph gradually increases with the increasing pantograph operating speed or crosswind speed. This result indicates that the bow collector should be reinforced in pantograph designs to withstand high uplift forces, especially during high-speed operations in windy conditions. On the other hand, the negative lift on the upper frame could be exploited to stabilize the pantograph and reduce the overall aerodynamic load. Design optimization should focus on improving the structural strength of the bow collector while possibly streamlining the upper frame to minimize negative lift-induced instability. (2) The operating speed of the pantograph and the wind direction angle have a significant impact on the aerodynamic uplift force of the main components, while the crosswind speed has a relatively minor effect on both the pantograph and main components' aerodynamic uplift forces. When the wind direction angle is small or the pantograph operating speed is high, the aerodynamic uplift force reaches higher values, increasing the likelihood of pantograph catenary contact and even bow scraping accidents. Conversely, in conditions with high crosswind speeds and low pantograph operating speeds, the pantograph is prone to experiencing insufficient aerodynamic uplift force, leading to arcing and disconnection phenomena. This finding challenges the conventional design assumption that uplift forces always increase with wind speed. In practical terms, this suggests that pantograph designs need to account for specific wind angles and speeds rather than relying on linear aerodynamic models. For instance, safety protocols can be developed to monitor critical wind angles and speeds, allowing for dynamic adjustment or emergency braking in high-wind scenarios to prevent damage or detachment from the catenary. (3) Through a comprehensive analysis of the factors influencing the aerodynamic uplift force of the pantograph, a relationship formula is derived that considers the aerodynamic uplift force of the pantograph in relation to the pantograph operating speed \(V_{t}\), crosswind speed \(V_{w}\), and wind direction angle \(\theta\). Furthermore, adjustments are made to empirical formulas relating to pantograph contact force and train operating speed. This formula provides a practical tool for railway engineers to forecast aerodynamic uplift forces under a wide range of operational conditions. It can be integrated into simulation software or used for real-time monitoring of pantograph performance during high-speed train operations, enabling engineers to make informed decisions about speed adjustments, pantograph positioning, and even maintenance schedules based on predicted aerodynamic loads. Conceptualization, M.Z. ; formal analysis, Y.D. and Z.J. ; data curation, Z.J. ; investigation, X.L. ; project administration, M.Z. ; resources, Z.J. ; software, Y.D. ; supervision, X.L. ; validation, M.Z. ; writing--original draft, M.Z. ; writing--review and editing, Y.D. All authors have read and agreed to the published version of the manuscript.