# Selected Problems in a Two-Vehicle Impact Collision Modeling
## 2 Simulation of a Collision between Two Passenger Vehicles  
The simulation of a collision of two passenger vehicles (no. 1 and 2 in Figure 1) has been prepared for the locked front wheels during the simulation. While preparing the simulations, some general assumptions were made in accordance with, e.g., [21]:  
* Both vehicle models were considered linear and their mass-inertia parameters were adopted according to the V-SIM's database. * The vehicles performed a planar motion during the impact, which took place on a dry road surface with a coefficient of adhesion \(\mu\) = 0.8. * Before the impact, the vehicles were moving at 70 km/h (19.44 m/s, vehicle no. 1) and 90 km/h (25 m/s, vehicle no. 2). The total mass of vehicle no. 1 was set at 1410 kg, while the total mass of vehicle no. 2 was at 1125 kg. The coefficient of restitution was assumed by the program at R = 0.04. The simulated collisions of the vehicles were frontal and oblique, so the occurrence of the impulses of the colliding forces in a tangential direction could not be omitted. The impact plane is understood to be perpendicular to the road surface and, at the same time, tangential to both the surfaces of the colliding vehicles [21]. The simulation of a collision between two passenger vehicles was prepared as a case of a collision with the front wheels locked during the crash. In Table 1, the selected results of the locked simulation are presented. This set contains both the translational and the angular velocities before and after the collision. Additionally, the masses and the moments of inertia of both vehicles are presented as the parameters necessary in the analysis in a mathematical sense. On this basis, the authors attempted to show that parameters from a mathematical point of view could be useful for a forensic expert to validate accident reconstructions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Vehicle** & **2** & **2** \\ \hline mass & 1410 kg & 1125 kg \\ moment of inertia relative to the longitudinal axis \(O_{x}\) passing & 529 kg\(\cdot\)m\({}^{2}\) & 410 kg\(\cdot\)m\({}^{2}\) \\ through the center of mass of a vehicle & 2343 kg\(\cdot\)m\({}^{2}\) & 1654 kg\(\cdot\)m\({}^{2}\) \\ moment of inertia relative to the vertical axis \(O_{z}\) passing & 2466 kg\(\cdot\)m\({}^{2}\) & 1576 kg\(\cdot\)m\({}^{2}\) \\ through the center of mass of a vehicle & 0.04 & 0.04 \\ coefficient of restitution & 0.04 & 0.04 \\ before the collision & & \\ forward velocity \(V_{x}\) & 19.43 m/s & 24.99 m/s \\ lateral velocity \(V_{y}\) & 0 m/s & 0 m/s \\ vertical velocity \(V_{z}\) & 0.02 m/s & 0.01 m/s \\ angular velocity around the longitudinal \(O_{x}\) axis & 0 & 0 \\ angular velocity around the lateral \(O_{y}\) axis & 0 & 0 \\ impulse of the impact force & 27,400 Ns & 27,400 Ns \\ after the collision & & \\ forward velocity \(V_{x}\) & 0.87 m/s & 0.68 m/s \\ lateral velocity \(V_{y}\) & \(-\)5.72 m/s & 0.45 m/s \\ vertical velocity \(V_{z}\) & \(-\)0.17 m/s & 0.19 m/s \\ angular velocity around the longitudinal \(O_{x}\) axis & \(-\)1.44 rad/s & 0.22 rad/s \\ angular velocity around the lateral \(O_{y}\) axis & 1.04 rad/s & 1.39 rad/s \\ angular velocity around the vertical \(O_{z}\) axis & 0.63 rad/s & 4.03 rad/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of a frontal oblique collision simulations of a collision between two passenger vehicles with locked front wheels. Figure 1: The mutual location of both vehicles at the beginning of the collision. Based on the obtained results (Table 1), it is necessary to mention that the post-collision velocities can be decomposed into two components in the case of a planar motion performed by the vehicles (along the normal and the tangential axes of the \(O_{nt}\) local coordinate frame), where \(O\) is the geometric center of the collision. Initially, the velocities obtained as a result of the simulation were presented along the axes of the global and inertial \(O^{\prime}{}_{xy}\) frame (Figure 2).