## 4 Aerodynamic Lift of the Pantograph
### Aerodynamic Lods on the Pantograph
There is a clear separation between the different crosswind speeds, with higher crosswind speeds (e.g., 30 m/s) showing larger lateral force coefficients. The confidence intervals (shaded regions) show relatively narrow bands for lower wind speeds (10 m/s, 15 m/s, 20 m/s), suggesting a reliable model fit. The prediction bands show that variability is relatively low, even for higher wind speeds. Notably, the confidence bands for 30 m/s are wide, reflecting uncertainty in the lateral force coefficient at high crosswind speeds. Based on the numerical simulation, a comprehensive relationship equation for the aerodynamic forces and moment coefficients of the pantograph with respect to the train speed, crosswind speed, and wind direction angle is proposed. Taking the aerodynamic drag of the pantograph as an example, the aerodynamic drag \(F_{x}\) is related to the pantograph's operating speed (i.e., train speed) \(V_{t}\), crosswind speed \(V_{w}\), air density \(\rho\), dynamic  
Figure 16: Lateral force coefficient change curve linear fitting curve. Figure 14: Resistance coefficient change curve linear fitting curve. Figure 15: Left coefficient change curve linear fitting curve. viscosity coefficient \(\mu\), characteristic size of the pantograph \(h\), and surface roughness of the pantograph \(k_{s}\), and can be expressed by the following equation:  
\[f(F_{x},V_{t},V_{w},\theta,\rho,\mu,h,k_{s})=0 \tag{4}\]  
Based on \(Re=V_{t}\rho h/\mu\), therefore, as follows:  
\[\frac{F_{x}}{V_{t}^{2}\rho h^{2}}=f(\frac{V_{w}}{V_{t}},\frac{1}{Re},\frac{k_{ s}}{h},\theta) \tag{5}\]  
The drag coefficient of the pantograph is given by \(C_{x}=F_{x}/V^{2}\rho h^{2}S_{x}\) where \(V^{2}~{}=~{}V_{t}^{2}~{}+~{}V_{w}^{2}~{}+~{}2V_{t}V_{w}\cos\theta\), thus can derive, as follows:  
\[C_{x}=F_{x}/V^{2}\rho h^{2}S_{x},V^{2}=V_{t}^{2}~{}+~{}V_{w}^{2}~{}+~{}2V_{t}V _{w}\cos\theta \tag{6}\]  
Therefore,  
Figure 10:0  
Since the geometric dimensions of the pantograph remain constant, and \(k_{s}/h\) is a constant, the drag coefficient of the pantograph is only related to \(V_{w}/V_{t}\) and \(\theta\). Expressing \(\theta\) in the form of \(cos\theta\), where \(V_{t}\) and \(V_{w}\) represent the train speed and crosswind speed, respectively, and their ratio is dimensionless, defined as the wind speed to train speed ratio \(\lambda_{wt}\), \(\lambda_{wt}=V_{w}/V_{t}\). Thus, we have the following:  
Figure 10:1  
From the above equation, it is evident that the drag coefficient of the pantograph is only related to the wind speed to train speed ratio \(\lambda_{wt}\) and the cosine of the crosswind angle \(\cos\theta\). Based on the variation patterns of the aerodynamic force and moment coefficients of the pantograph, a functional form \(f(\lambda_{wt},\cos\theta)\) can be obtained, thus providing the specific expression of the drag coefficient. By fitting the data, the values of the coefficients in the equation can be determined, leading to the final comprehensive relationship between the drag coefficient and the train speed, crosswind speed, and wind direction angle. Let \(f(\lambda_{wt},\cos\theta)=a\lambda_{wt}^{b}(1~{}+~{}m\lambda_{wt}^{c}|\cos \theta|^{n})\), substituting it into Equation (7), we can obtain the specific expression of the drag coefficient, as follows:  
Figure 10:2  
In the above equation, a, b, c, m, and n are undetermined coefficients, which are solved through data fitting to obtain \(a=0.779\), \(b=0.128\), \(c=1.267\), \(m=1.951\), and \(n=0.683\), thereby obtaining the drag coefficient, as follows:  
Figure 10:3  
When the wind direction angle \(\theta=90^{\circ}\), the drag coefficient is \(C_{x}=0.779\lambda_{wt}^{0.128}/(1+\lambda_{wt}^{2})\); when the wind direction angle \(\theta=0^{\circ}\), the drag coefficient is \(C_{x}=(0.779\lambda_{wt}^{0.128}~{}+~{}1.519\lambda_{wt}^{1.395})/(1~{}+~{}2 \lambda_{wt}~{}+~{}\lambda_{wt}^{2})\). The above equation considers the special conditions when the wind direction is \(0^{\circ}\) and \(90^{\circ}\).