# Braking Performance and Response Analysis of Trains on Bridges Under Seismic Excitation
## 2 Materials and Methods
### Subway-Bridge Coupled Systems  
The studied bridge is a bidirectional simply supported concrete girder bridge with a single span of 35 m. The overall bridge is simplified to a representative multi-span simply supported girder bridge structure in order to be closer to the actual operating conditions of the subway line. In order to accurately simulate the response characteristics of the bridge under train operation and seismic loading, a track-bridge subsystem model is constructed based on the finite element method, in which the track, bridge, and abutment parts are discretely modeled by beam units to fully consider the dynamic coupling effects of the structure in the longitudinal and vertical directions. The dynamic behavior of the bridge structure is described by the established dynamic equations, whose mathematical expression form is shown in Equation (3). In order to ensure that the high-speed running subway train can realize complete deceleration and stopping on the bridge under seismic action, after reasonable speed-distance calculation and analysis, the bridge is finally set as a 10-span simply supported girder bridge, and the total length can meet the space required for the vehicle to decelerate to a complete stop. At the same time, considering the spatial stiffness of the bridge structure and the demand for force transmission, the height of the bridge pier is taken as 20 m. The structural form of the bridge and the train running on the bridge are shown in Figure 1, which provides a structural basis for the subsequent coupled dynamic response analysis.  
\[M_{RB}\ddot{X}_{RB}+C_{RB}\dot{X}_{RB}+K_{RB}X_{RB}=F_{RB} \tag{3}\]  
where \(M_{RB}\), \(C_{RB}\), and \(K_{RB}\) denote the mass, damping, and stiffness matrices of the rail-bridge structure, respectively; \(\ddot{X}_{RB}\), \(\dot{X}_{RB}\), and \(X_{RB}\) denote the acceleration, velocity, and displacement vectors of the rail-bridge structure, respectively; and \(F_{RB}\) is the force vector on the rail-bridge structure.  
The dynamic coupling between subway trains and bridges is realized through the wheel-rail relationship, forming a unified vehicle-bridge coupling system. In this study, the wheel-rail contact relationship is modeled by [PERSON]'s linear creep theory, which is used to simulate the creep force caused by micro-slip between the wheelset and the rail, so as to more realistically reflect the wheel-rail interaction during train operation. Through this method, the subway train and the bridge system are coupled into a whole multi-degree-of-freedom dynamic system model, and its dynamic control equations can be expressed as Equation (3). During the interaction between the train and the bridge, the external forces on the system include various types, such as the inertia force of the subway train body, the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Project** & **Numerical Value** & **Unit** \\ \hline Body mass & 21,920 & kg \\ Frame quality & 2550 & kg \\ Wheelest mass & 1420 & kg \\ Nodal moment of inertia of wheelset & 97 & kg - m\({}^{2}\) \\ Frame nodding moment of inertia & 1750 & kg - m\({}^{2}\) \\ Body nodding moment of inertia & 617,310 & kg - m\({}^{2}\) \\ Axle weight & 14 & t \\ Crew & 240 & people \\ \hline \hline \end{tabular}
\end{table}
Table 1: Technical parameters of metro B vehicles.  
elastic force caused by the structural deformation, the longitudinal braking force generated during operation, and the seismic force input under seismic excitation. These forces act together on the axle coupling system, making its response have obvious nonlinear and time-varying characteristics, which have an important impact on the dynamic performance and safety of the system.