# Simulation Study on Ground Vibration Reduction Measures of the Elevated Subway Line
## 2 Numerical Model
### Vibration Source Model  
In this paper, the SIMPACK (v.9.10) and ABAQUES (v.18.0) software programs are used to establish the train-track-viaduct rigid-flexible coupling dynamics model, as shown in Figure 1.  
A train is established by the SIMPACK multi-body dynamics software and the vehicle visualization model is shown in Figure 2 below. The whole vehicle system consists of seven components, which include a body, two bogies, and four wheelsets. The body is connected to the bogies through a secondary suspension and the bogies are connected to the wheelsets through a primary suspension. The parameters of each part of the system are shown in Table 1.  
Figure 1: Train–track–viaduct rigid–flexible coupling dynamics model. Figure 2: Vehicle system dynamics visualization model. Among them, wheelsets, rails, U-beams, piers, and other components are modeled as flexible bodies, and the property parameters of each component are shown in Table 2. The finite element model of each component is shown in Figure 3. According to the mass matrix M, stiffness matrix K, and the corresponding shape function U of the finite element model of the flexible structure, the modal mass matrix M and modal stiffness matrix K of the structure and the generalized vector p of the interaction between the wheel and the structure can be obtained, and the formulas are expressed as follows:  
\[\left\{ \begin{array}{l}\overline{M}=U^{T}MU\\ \overline{K}=U^{T}KU\\ p=U^{T}p\\ U=(u_{1}\;u_{2}\cdots u_{3}\cdots u_{n})\end{array} \right. \tag{1}\]  
where \(u_{l}=(l=1,2,\ldots,n)\) is the lth order modal shape of the structure, where \(n\) is the modal degree of freedom; \(p\) is the force between the wheel and the structure. For orthogonal shape functions, the intrinsic vibration patterns are orthogonalized static vibration patterns, and the equations of motion of the elastic structure can be simplified to a system of n second-order differential equations:  
\[m_{l}\ddot{z}_{l}+d_{l}\dot{z}_{l}+k_{l}z_{l}=\overline{p}_{l} \tag{2}\]  
where \(m_{l}\), \(d_{l}\), \(k_{l}\), and \(p_{l}\) are the modal mass, modal damping, modal stiffness, and generalized force corresponding to each vibration \(u_{l}\), respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Name** & **Numerical Value** & **Unit** \\ \hline Mass of car body & 49.52 & t \\ Mass moment of inertia of car body roll & 80 & t-m\({}^{2}\) \\ Mass moment of inertia of car body pitch & 1836 & t-m\({}^{2}\) \\ Mass moment of inertia of car body yaw & 1835 & t-m\({}^{2}\) \\ Car body center of gravity coordinates & 2.056 & m \\ Mass of bogie & 4387 & kg \\ Mass moment of inertia of car body yaw & 1847.4 & kg\(\cdot\)m\({}^{2}\) \\ Mass moment of inertia of car body yaw & 3903.8 & kg\(\cdot\)m\({}^{2}\) \\ Mass moment of inertia of car body yaw & 5583.4 & kg\(\cdot\)m\({}^{2}\) \\ Mass of wheelset & 1443 & kg \\ Longitudinal stiffness of primary suspension & 5.2 & MN/m \\ Lateral stiffness of primary suspension & 5.2 & MN/m \\ Vertical stiffness of primary suspension & 0.95 & MN/m \\ Longitudinal damping of primary suspension & 15 & kN-s/m \\ Lateral damping of primary suspension & 2 & kN-s/m \\ Vertical damping of primary suspension & 20 & kN-s/m \\ Longitudinal stiffness of secondary suspension & 0.148 & MN/m \\ Lateral stiffness of secondary suspension & 0.148 & MN/m \\ Vertical stiffness of secondary suspension & 0.452 & MN/m \\ Longitudinal damping of secondary suspension & 50 & kN-s/m \\ Lateral damping of secondary suspension & 32 & kN-s/m \\ Vertical damping of secondary suspension & 32 & kN-s/m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vehicle structural parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & **Density (kgm\({}^{-3}\))** & **Elastic Modulus (GPa)** & **Poisson’s Ratio** \\ \hline Wheelset & 7800 & 206 & 0.3 \\ Steel rail & 7800 & 206 & 0.3 \\ U-shaped bridge & 2450 & 37 & 0.2 \\ Pier & 2450 & 37 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Component parameters.