# Research on Response Postures of Subway Train in Straight Line Collision
## 2 The Subway Train Numerical Model
### Model Validation  
#### 2.4.1 One-Dimensional Collision Dynamics Model for Trains  
To verify the accuracy of numerical simulation results, this section constructs a one-dimensional mathematical model to simulate the longitudinal collision process of trains. To achieve a balance between accuracy and precision, the following basic assumptions are made when constructing and simplifying the model:  
1. The car bodies are treated as rigid bodies, and their masses are assumed to be concentrated at their respective geometric centers. 2. Each carriage is divided into three rigid bodies, which are connected by springs. The deformation of the carriages is equivalent to the deformation of these springs. 3. For the couplers and springs inside the carriages, only their equivalent stiffness characteristics are considered. Based on the above assumptions, the longitudinal collision model of the train can be simplified as a multi-rigid-body mass-spring system, where each carriage is simplified into three rigid body segments and the energy absorption devices are simplified into a nonlinear spring system. The longitudinal collision condition is set as head-on collision. The carriage numbering and condition diagram are shown in Figure 6.  
The elastic loading phase of the coupler buffer device is simplified to a line segment with a constant slope, and the plastic loading phase is simplified to a horizontal line segment, representing its steady-state force, as shown in Figure 7.  
The axial force of the coupler can be expressed as:  
\[F=p(k_{1},f_{\mathrm{s}},k_{\mathrm{u}},x_{\mathrm{u}})= \begin{cases}k_{1} \Delta x&\Delta\dot{x}>0,\Delta x\leq x_{1}\\ f_{\mathrm{s}}&\Delta\dot{x}>0,x_{1}<\Delta x\leq x_{\mathrm{u}}\\ k_{\mathrm{u}}\Delta x&\Delta\dot{x}<0,x_{\mathrm{u}}-\frac{f_{\mathrm{u}}}{ k_{\mathrm{u}}}<\Delta x<x_{\mathrm{u}}\\ 0&\Delta\dot{x}<0,0<\Delta x\leq x_{\mathrm{u}}-\frac{f_{\mathrm{u}}}{k_{ \mathrm{u}}}\end{cases}  \tag{7}\]  
Figure 5: Total internal energy time history curve. Figure 6: Simplified diagram of train collision. where \(p\) represents the loading and unloading function of the coupler; \(k_{\rm{l}}\) represents the compression stiffness; \(f_{\rm{s}}\) steady-state force; \(k_{\rm{u}}\) represents the unloading stiffness; \(x_{\rm{u}}\) represents the compression stroke. Take a certain carriage as a separate body and analyze its stress situation, as shown in Figure 8.  
Based on the force analysis of a single carriage, the longitudinal collision motion of the train can be described using a system of nonlinear differential equations:  
\[\left\{ \begin{array}{l}m_{11}\ddot{x}_{11}=-F_{11}\\ m_{11}\ddot{x}_{12}=F_{11}-F_{12}\\ \vdots\\ m_{22}\ddot{x}_{i2}=F_{i1}-F_{i2}\\ \vdots\\ m_{n2}\ddot{x}_{n2}=F_{n1}-F_{n2}\\ m_{n3}\ddot{x}_{n3}=F_{n2}\end{array} \right. \tag{8}\]  
where \(m\) is the mass of the rigid bodies of the carriages. \(F_{i}\) are the collision force between the two carriages, which is function of the relative displacement between the two carriages. Carriage 1 is subjected to a longitudinal load on only one end, so \(F_{0}\) = 0. The collision force between carriages is expressed as:  
\[F_{i}=k_{i}(x_{i3}-x_{i+1,1})+b_{i} \tag{9}\]  
The internal collision force within the carriage is expressed as:  
\[F_{i1}=k_{i1}(x_{i1}-x_{i,2})+b_{i1},\ F_{i2}=k_{i2}(x_{i2}-x_{i,3})+b_{i2} \tag{10}\]  
Substitute the above equations into Equation (8):  
Figure 8: Force situation of a single carriage. Figure 7: Simplified force–displacement curve of coupler buffer devices. \[\left\{ \begin{array}{l}m_{11}\ddot{x}_{11}=-k_{11}(x_{11}-x_{12})-b_{11}\\ m_{12}\ddot{x}_{12}=k_{11}(x_{11}-x_{12})+b_{11}-k_{12}(x_{12}-x_{13})-b_{12}\\ \vdots\\ m_{22}\ddot{x}_{i2}=k_{i1}(x_{i1}-x_{i2})+b_{i1}-k_{i2}(x_{i2}-x_{i3})-b_{i2}\\ \vdots\\ m_{n2}\ddot{x}_{n2}=k_{n1}(x_{n1}-x_{n2})+b_{n1}-k_{n2}(x_{n2}-x_{n3})-b_{n2}\\ m_{n3}\ddot{x}_{n3}=k_{n2}(x_{n2}-x_{n3})+b_{n2}\end{array} \right.