# Study on the Frontal Collision Safety of Trains Based on Collision Dynamics
## 2 Three-Dimensional Train Collision Dynamic Model  
Based on the traditional three-dimensional collision dynamic model, a three-dimensional train collision dynamic model applicable to higher collision velocities was established, as shown in Figure 1, by considering the vehicle's actual structure and the functioning mechanisms of its components during high-speed collision. To accurately describe the vehicle's motion during a high-speed collision, a three-dimensional vehicle model with 38 DOF was developed, including one carbody, two bogie frames, and four wheelsets. The carbody and bogie frame possess six degrees of freedom (longitudinal, lateral, vertical, roll, pitch, and yaw), and the wheelsets have five degrees of freedom (longitudinal, lateral, vertical, roll, and yaw). The carbody is connected to the frames via secondary suspension, and the frames are connected to the wheelsets via primary suspension. The dynamic equation of the model is as follows:  
\[M\ddot{u}+C\dot{u}+K\mathbf{u}=\mathbf{F} \tag{1}\]where \(M\), \(C\), and \(K\) represent the mass matrix, damping matrix, and stiffness matrix, respectively. \(u\) and \(F\) denote the displacement vector and force vector, respectively. By organizing and analyzing the dynamic equations of the vehicle model, the mass matrix of the vehicle model and the displacement vector can be obtained by Equations (2) and (3), respectively. The stiffness matrix is a sparse matrix of size 38 \(\times\) 38, as shown in Table 1. The stiffness terms in the stiffness matrix are replaced with damping terms to obtain the damping matrix. \[ \begin{split}\mathbf{M}_{\rm v}=\mathrm{diag}(m_{\rm cb},m_{\rm bg},m_{\rm bg},m_{\rm w},m_{\rm w},m_{\rm w},m_{\rm w},I_{\rm cby},I_{\rm bgy},I_{ \rm bgy},m_{\rm cb},m_{\rm bg},m_{\rm bg},\\ m_{\rm w},m_{\rm w},m_{\rm w},m_{\rm w},m_{\rm cb},m_{\rm bg},m_{ \rm w},m_{\rm w},m_{\rm w},m_{\rm w},m_{\rm w},I_{\rm bgx},\\ l_{\rm bgy},I_{\rm wx},I_{\rm wx},I_{\rm wx},I_{\rm wx},I_{\rm bz },I_{\rm bgz},I_{\rm bgz},I_{\rm wz},I_{\rm wz},I_{\rm wz},I_{\rm wz})\end{split}  \tag{2}\]  
where \(m\) and \(I\) are mass and moments of inertia, respectively. The subscripts cb, bg, and w denote the carbody, bogie frames, and wheels, respectively. \[ \begin{split}\mathbf{u}_{\rm v}=&[z_{\rm cb},z_{\rm bg1},z_{\rm bg2},z_{\rm w1},z_{\rm w2}.z_{\rm w3},z_{\rm w4},\beta_{\rm cb},\beta_{ \rm bg1},\beta_{\rm bg2},x_{\rm cb},x_{\rm bg1},x_{\rm bg2},x_{\rm w1},\\ & x_{\rm w2},x_{\rm w3},x_{\rm w4},x_{\rm cb},y_{\rm bg1},y_{\rm bg 2},y_{\rm w1},y_{\rm w2},y_{\rm w3},y_{\rm w4},a_{\rm cb},a_{\rm bg1},a_{\rm bg 2},\\ &\alpha_{\rm w1},\alpha_{\rm w2},\alpha_{\rm w3},\alpha_{\rm w4},\gamma_{\rm cb},\gamma_{\rm bg1},\gamma_{\rm bg2},\gamma_{\rm w1},\gamma_{ \rm w2},\gamma_{\rm w3},\gamma_{\rm w4}]^{\rm T}\end{split}  \tag{3}\]  
where \(z\), \(\beta\), \(x\), \(y\), \(\alpha\), and \(\gamma\) represent the longitudinal, lateral, vertical, roll, pitch, and yaw motion, respectively. The track model utilizes a fixed-track model with a three-layer discrete elastic support. The rail is simplified as a finite-length Euler beam supported by continuous elastic point supports. Below, are the sleeper and ground, which form a three-layer vibration model using distributed spring-damper units. Compared to the moving track model, this model can more accurately simulate the wheel-rail interaction force after high-speed collisions. The rail and sleeper have translational degrees of freedom in the vertical and lateral directions. If \(E\), \(I\), \(l\), \(m_{r}\), \(N_{k}\), \(m_{s}\), and \(n_{s}\) denote the elastic modulus of the rail, moments of inertia of the rail section, length of each section of the rail, rail mass, maximum mode order of vibration for the rail, sleeper mass, and sleeper number, respectively, according to the  
Figure 1: Three-dimensional train collision dynamic model. differential equation governing the Euler beam vibration, the mass matrix and stiffness matrix of the track model can be obtained as follows:  
\[M_{\rm R}=