## 2 Vehicle System Modeling
### Longitudinal Dynamics of Railway Vehicles  
In this study, an eight-car urban electric train, configured as TC-M1-M2-T1-T2-M1-M2-TC, was selected as the research subject vehicle system. Here, TC represents the driver's cab car, M1 and M2 are locomotives, and T1 and T2 are trailer cars. During the modeling process of the electric train system, for the sake of convenience and practicality in this study, the elasticity effects within the vehicle were excluded, and the vehicle system was assumed to be a rigid body system. Consequently, the longitudinal dynamic equations of motion for the electric train, as shown in Figure 1, can be expressed as follows:\[\left\{ \begin{array}{c}m_{1}\dot{v}_{1}=f_{1}-f_{01}-F_{1}^{R}\\ m_{2}\dot{v}_{2}=f_{d1}+f_{2}-f_{1}-f_{12}-F_{2}^{R}\\ m_{3}\dot{v}_{3}=f_{d2}+f_{3}-f_{2}-f_{3}-F_{3}^{R}\\ m_{4}\dot{v}_{4}=f_{4}-f_{3}-f_{44}-F_{3}^{R}\\ m_{5}\dot{v}_{5}=f_{5}-f_{4}-f_{5}-F_{5}^{R}\\ m_{6}\dot{v}_{6}=f_{A3}+f_{6}-f_{5}-f_{6}-F_{6}^{R}\\ m_{7}\dot{v}_{7}=f_{A4}+f_{7}-f_{6}-f_{67}-F_{7}^{R}\\ m_{8}\dot{v}_{8}=f_{8}-f_{7}-f_{8}-F_{8}^{R}\end{array} \right. \tag{1}\]  
Here, \(m_{i}\) and \(v_{i}\) represent the mass and speed of the \(i\)-th car, respectively; \(F_{bi}\) and \(F_{di}\) denote the mechanical braking force and driving force applied to the \(i\)-th car; \(f_{i}\) is the internal force acting between the \(i\)-th car and the \((i+1)\)-th car; and \(F_{i}^{R}\) represents the running resistance of the \(i\)-th car (including rolling resistance, air resistance, and gradient resistance) [3; 4; 5; 6]. From a macroscopic perspective, it is assumed that the relative motion characteristics of the cars, as expressed in Figure 1 and Equation (1), are negligible, allowing the speed of all cars to be represented by a single, representative speed, \(v\). Under this assumption, the eight degrees of freedom expressed in the eight equations of motion in Equation (1) can be reduced to a single degree of freedom by eliminating the eight internal forces \(f_{i}\). During this transformation, it is assumed that traction forces act on the four locomotive cars, while braking and running resistance forces act on all cars. \[\sum_{i=1}^{8}m_{i}\cdot\dot{v}=\sum_{i=2,3,6,7}(F_{di}-F_{bi})-\sum_{i=1,4,5,8 }F_{bi}-\sum_{i=1}^{8}F_{i}^{R} \tag{2}\]  
Here, it is assumed that the combined driving and braking forces generated by the four locomotive cars are uniform, represented by the value \(F_{t}\), and that the mechanical braking forces for the trailing electric cars are also uniform, denoted as \(F_{b}\). Additionally, the driving load forces \(F_{i}^{R}\) acting on each car influence the vehicle's behavior in the form of a total resultant force; thus, we define and represent the total driving load as \(F^{R}\). Under these assumptions, the equations of motion in Equation (2) can be approximated as follows:  
\[M\dot{v}=4F_{t}-4F_{b}-F^{R} \tag{3}\]  
Here, \(M\) represents the total mass of the vehicle, \(F_{t}\) denotes the average effective driving force of a single locomotive car, \(F_{b}\) represents the average mechanical braking force of a single trailing car, and \(F^{R}\) denotes the total driving load of the eight-car train. Figure 1: Longitudinal dynamic modeling of train vehicles. \[\left\{ \begin{array}{c}M=\sum\limits_{i=1}^{8}m_{i}\\ 4F_{t}=\sum\limits_{i=2,3,6,7}(F_{di}-F_{bi})\\ 4F_{b}=\sum\limits_{i=1,4,5,8}F_{bi}\\ F^{R}=\sum\limits_{i=1}^{8}F_{i}^{R}\end{array} \right. \tag{4}\]  
The total resistance force \(F^{R}\) can be divided into the running resistance \(F_{r}(v)\), gradient resistance \(F_{g}(\theta)\), and curve resistance \(F_{c}\). However, this study considers only straight-line driving and disregards curve resistance. The running resistance \(F_{r}(v)\) is expressed by the running resistance equation, known as the Davis equation, shown in Equation (6) [7].