# Influence of Metro Track Irregularities on Pantograph Vibration and Its Interaction with Catenary
## 3 Establishment and Validation of the Analysis Model  
### Numerical Model  
The vehicles operating on Z City Metro line 1 are metro type B vehicles with a maximum running speed of 80 km/h, designed by CRRC Zhuzhou Locomotive CO., Ltd. (Zhuzhou, China). A two-stage suspension system is used in the vehicle, in which the rotating-arm positioning measure is used in the wheelset, and the rotating arm and the double springs form the primary suspension system, which provides the vertical stiffness and the longitudinal and lateral stiffness required to ensure the running stability of the vehicle. In addition, the primary suspension also includes vertical hydraulic dampers to absorb the vertical vibration. The secondary suspension is mainly composed of air springs, lateral hydraulic dampers, vertical hydraulic dampers, etc. Moreover, in order to control the rolling motion of the car body, an anti-roll torsion bar is also installed between the car body and the bogie.  
Based on the structural parameters of the B-type metro vehicle and the bogie ZMC080, a vehicle dynamic model is established by using the multibody dynamics method (shown in Figure 4). The pantograph strip, upper frame, lower frame, car body, bogie, and wheelset are included in the model, connected by spring-damping systems. This manuscript focuses on the lateral and vertical vibrations of the pantograph, so the lateral and vertical degrees of freedom are considered. The car body, bogie, and wheelset are rigid bodies with 6 degrees of freedom [26]. The wheelset and the bogie are connected by the primary suspension, and  
Figure 3: Comparison of track irregularity spectra: (**a**) the comparison of the PSD of the level irregularity; (**b**) the comparison of the PSD of the alignment irregularity.  
the bogie and the vehicle body are connected by the secondary suspension. The vehicle dynamics model expression is as follows:  
\[\mathbf{M}_{V}\ddot{\mathbf{x}}+\mathbf{D}_{V}\dot{\mathbf{x}}+\mathbf{K}_{V}\mathbf{x}=0 \tag{1}\]  
The mass matrix can be expressed as:  
\[\mathbf{M}_{V}=\begin{pmatrix}\mathbf{M}_{P}&&\\ &\mathbf{M}_{C}&&\\ &&\mathbf{M}_{B}&\\ &&&\mathbf{M}_{W}\end{pmatrix} \tag{2}\]  
The components of the vehicle are connected by a two-stage suspension system containing springs and dampers. The nonlinear behavior of the lateral stops and lateral vibration absorber is considered. In the manuscript, the nonlinear characteristic curves of the primary vertical damping and secondary lateral damping of the vehicle are given in Figure 5. The stiffness and damping matrices can be expressed as:  
\[\mathbf{D}_{V}=\begin{pmatrix}\mathbf{D}_{PP}&\mathbf{D}_{PC}&&\\ \mathbf{D}_{CP}&\mathbf{D}_{CC}&\mathbf{D}_{CB}&\\ &\mathbf{D}_{BC}&\mathbf{D}_{BB}&\mathbf{D}_{BW}\\ &&&\mathbf{D}_{WB}&\mathbf{D}_{WW}\end{pmatrix},\;\mathbf{K}_{V}=\begin{pmatrix}\mathbf{K}_{PP} &\mathbf{K}_{PC}&&\\ \mathbf{K}_{CP}&\mathbf{K}_{CC}&\mathbf{K}_{CB}&\\ &\mathbf{K}_{BC}&\mathbf{K}_{BB}&\mathbf{K}_{BW}\\ &&&\mathbf{K}_{WB}&\mathbf{K}_{WW}\end{pmatrix} \tag{3}\]  
The subscripts \(P\), \(C\), \(B\), and \(W\) in Equation (2) represent the pantograph, the car body, the bogie frame, and the wheelset, respectively. Some parameters are shown in Table 1.  
In terms of the wheel-rail contact part, the Hertzian contact algorithm is used to calculate the wheel-rail interaction point and force [27]. The polygonal contact model (PCM) is used to calculate the PC interaction [14].  
Figure 4: Vehicle dynamic model.