# Modeling Head-On Collisions: The Problem of Identifying Collision Parameters
## 2 Materials and Methods
The total force acting on the vehicle is the force of gravity, the forces of aerodynamics resistances, and the forces brought by the suspension of the vehicle wheels, which has been described with the following formula [11]:  
\[\overset{\rightarrow}{F}^{\prime}\ =\ \overset{\rightarrow}{F}^{\prime}_{g}+ \sum\overset{\rightarrow}{F}^{\prime}_{i}+\overset{\rightarrow}{F}^{ \prime}_{ax}+\overset{\rightarrow}{F}^{\prime}_{ay}. \tag{7}\]  
The gravitational force has been determined by transforming from the global system of reference according to the following formula [11]:  
\[\overset{\rightarrow}{F}_{g}\ =\ \left[ \begin{array}{c}0\\ 0\\ -m\cdot g\end{array} \right]\overset{\rightarrow}{x\to x^{\prime}}, \tag{8}\]  
where:  
m--mass of the vehicle, and  
g--standard acceleration due to gravity. The forces of aerodynamic resistance are determined irrespective of the frontal and side area of the vehicle from the speed of the vehicle, while considering wind speed \(\overset{\rightarrow}{w}\) according to the formulas provided below [11]:  
\[\overset{\rightarrow}{F}_{ax}^{\prime}\ =\ \left[ \begin{array}{c}\pm\rho/2 \cdot C_{x}\cdot A_{x}\cdot\left(\overset{\rightarrow}{F}_{x^{\prime}}-w_{x^ {\prime}}\right)^{2}\\ 0\\ 0\end{array} \right], \tag{9}\]  
\[\overset{\rightarrow}{F}_{ay}^{\prime}\ =\ \left[ \begin{array}{c}0\\ \pm\rho/2\cdot C_{y}\cdot A_{y}\cdot\left(\overset{\rightarrow}{F}_{y^{ \prime}}-w_{y^{\prime}}\right)^{2}\\ 0\end{array} \right], \tag{10}\]  
where:  
\(\rho\)--air density,  
\(C_{x},y\)--coefficients of the frontal and side air resistance,  
\(A_{x},y\)--frontal and side areas of the vehicle,  
\(\overset{\rightarrow}{r}_{x^{\prime}y^{\prime}}\)--linear velocities of the vehicle in the system connected with the vehicle, and  
\(w_{x},y\)--component wind speeds in the system connected with vehicle. The vehicle suspension model also considers the independent vertical movement for each wheel of the vehicle. From the positioning of the vehicle and the local configuration of the road surface,deflection of the suspension \(\mathrm{S_{i}}\) is determined, whereas the normal force of the response of the suspension considering the elasticity with progressive characteristics and the damping values is calculated from the momentary deflection and the velocity of its changes according to the formula [11]:  
\[F^{\prime}_{zi}\big{(}S_{i},\dot{S}_{i}\big{)}\ =\ \max\!\left(\max\!\left(C_{3i} \cdot S_{i}^{3}+C_{1i}\cdot S_{i}+F^{\prime}_{0i},0\right)+\left\{ \begin{array} []{l}D_{ci}\cdot\dot{S}_{i}^{\dot{S}_{i}\geq 0}\\ D_{ri}\cdot\dot{S}_{i}^{\dot{S}_{i}<0}\end{array} ,0\right), \tag{11}\]  
where:  
\(F^{\prime}_{zi}\)--vertical component of the response of the vehicle suspension,  
\(S_{i}\)--reduced value of deflection of the vehicle wheel suspension,  
\(C_{3i}\)--reduced coefficient of progression of stiffness of the vehicle wheel suspension,  
\(C_{1i}\)--reduced coefficient of stiffness of the vehicle wheel suspension,  
\(F^{\prime}_{0i}\)--normal force of the response of the vehicle wheel suspension,  
\(D_{ci}\)--reduced damping coefficient for the compression of the vehicle wheel suspension, and  
\(D_{ri}\)--reduced damping coefficient for the stretching of the vehicle wheel suspension. The program for determining the interaction of the tires with the road surface used a nonlinear model developed at the Highway Safety Research Institute (HSRI), University of Michigan, by the [PERSON] team and TM-Easy. The forces of the tire reactions are calculated in the system connected with the tangent point of the tire with the road surface, and the normal force of the tire is assumed as the normal force of the reaction of the suspension. The normal force of the tire, the rotational speed of the wheel, and the parameters of the road surface are used to determine the forces of the reaction of the tire in the other axes of the system. For each wheel of the vehicle, additional degree of freedom has also been considered, and rotational movement has been considered.