# Study on the Frontal Collision Safety of Trains Based on Collision Dynamics
## 3 The Nonlinear Factors and Their Mathematical Models in a Multibody System
### Mathematical Model for Car-End Contact
#### 3.2.2 Stiffness Weakening Coefficient and Steady-State Coefficient  
If the car-end crush zone of a high-speed train is simplified as a thin-walled, hollow polyhedron, the weakening law of the stiffness of the thin-walled, hollow polyhedron compared to its corresponding solid polyhedron, as well as the relationship between the volume ratio and the steady-state coefficient, can be established from the perspective of volume ratio. A finite element model was established in LS-DYNA, as shown in Figure 3a, to simulate the mechanical characteristics of the thin-walled structure after impact at different volume ratios. The model comprises a rigid wall, impact bodies with different volume ratios, and a fixed base. The fixed base and rigid wall are modelled using rigid material, while the impact bodies are modelled using elastic-plastic material. All components are modelled using solid elements. The simulation's impact scenario involves a rigid wall with an initial velocity of 40 m/s impacting the aluminium alloy body with different volume ratios fixed on the base. The thin-walled body is a hexahedron with a square section. The side length of the front square section is 200 mm, the side length of the rear square section is 400 mm, and the overall length of the hexahedron is 800 mm. To investigate the relationship between different volume ratios and the stiffness weakening coefficient and steady-state coefficient, finite element numerical simulations were conducted using various thicknesses of 100, 80, 60, 40, 20, 10, 5, and 1 mm. The simulations aimed to obtain the load-deformation relationship under dynamic impact conditions. The load-deformation relationship for the impact bodies with thicknesses of 100 mm (solid) and 60 mm subjected to impact loading from a rigid wall is shown in Figure 4.  
Figure 3: Finite element model and dimensions. (**a**) Sketch of the impact model; (**b**) sketch of the validation model. After conducting multiple sets of numerical simulation experiments, the relationships between the stiffness weakening coefficient and steady-state coefficient with the volume ratio were obtained through polynomial fitting as follows:  
\[\left\{ \begin{array}{rl}\lambda=-4.668 Vol_{\rm r}^{6}+13.523 Vol_{\rm r}^{5}-16.5 Vol_{\rm r}^{4}+8.829 Vol_{\rm r}^{3}-\\ \\ 1.016 Vol_{\rm r}^{2}+0.828 Vol_{\rm r}+0.002\\ \\ \eta=0.439 Vol_{\rm r}^{4}+0.629 Vol_{\rm r}^{3}-1.948 Vol_{\rm r}^{2}+1.294 Vol_{ \rm r}+0.487\end{array} \right. \tag{16}\]  
where \(Vol_{\rm r}\) represents the volume ratio. The obtained relationships are based on a specific model. To enhance their applicability, velocity and section variation factors are introduced to modify the stiffness weakening and steady-state coefficients. Assuming that there is no coupling relationship between the modification factors and no coupling relationship between the modification factors and the volume ratio, the individual effects of each modification factor on the stiffness weakening coefficient and the steady-state coefficient can be analyzed separately. Therefore, a thin-walled body with a volume ratio 0.24 was selected to study the modification factors. The mechanical characteristics of the structure will exhibit significant differences under the impact of different velocities. Based on the original model, numerical experiments were conducted by varying the impact velocity with impact velocities set at 40, 35, 30, 25, 20, 15, and 10 m/s, respectively. It enabled the evaluation of the effects of different velocities on the two coefficients. After fitting the data, the relationship between the velocity modification factor and velocity can be obtained as follows:  
\[\xi_{\lambda}^{v}=0.896v_{\rm r}^{3}-2.721v_{\rm r}^{2}+2.841v_{\rm r}-0.015, \xi_{\eta}^{v}=\left\{ \begin{array}{rl}-3.36v_{\rm r}+2.49,v_{\rm r}<0.375\\ \\ -v_{\rm r}+1.605,0.375\leq v_{\rm r}<0.625\\ \\ 1,v_{\rm r}\geq 0.625\end{array} \right.