# Selected Problems in a Two-Vehicle Impact Collision Modeling
## 3 Identification of the Parameters Crucial for the Analytical Calculations  
In order to properly proceed with the analytical calculations of the discussed collision, it seems appropriate to transform the global \(O^{\prime}xy\) frame in which the velocities were determined by V-SIM to a local \(O_{nt}\) frame, which seems more useful in mathematical modeling due to the fact that its origin \(O\) is at the point of an initial contact between the colliding vehicles. However, this may not be so obvious because of a variety of possibilities in which the axes of the \(O_{nt}\) frame may be located, as the contact area between the vehicles is usually not a single point. Therefore, in the presented example, point \(O\) was adopted in the middle of the section to determine the initial contact between the vehicles (Figure 4). The axes \(O_{t}\) and \(O_{n}\) could also be located differently versus both of the colliding vehicles. Usually, the software used in simulations adopts the point of initial contact between the vehicles, and so it was in this case. Then, it was adopted such that the \(O_{t}\) axis is located as easily as possible, i.e., along the plane of one of the vehicles. Hence, the axis \(O_{n}\) must be perpendicular to \(O_{t}\) and the proposed configuration of the adopted \(O_{nt}\) frame is shown in Figure 4. Although the adoption of both axes relates only to the considered example, the method for evaluating the results obtained in the simulation can be valid for other cases. To properly describe the collision in the aspect of mathematical modeling is also necessary for making certain assumptions regarding the simplifications of the collision modeling which will not affect the considered phenomena. The quasi-stiff models of the vehicles have been assumed with the specified mass and stiffness, which did not change during the collision. The motion of the vehicles was considered on a flat road surface. However, in further research, the possibility of both uneven and sloping roads can be taken into account, which could lead to a resultant motion. In more complicated cases, a non-linear suspension may be included, but it seems that during the collision it does not have a significant influence on the analyzed phenomena. Additionally, the additional impulses due to the locked wheel have been omitted and will be included in further studies. It seems convenient to present the adopted method graphically so that all the necessary trigonometric transformations can be easily applied. Therefore, in Figure 4, the mutual location of both vehicles is presented at the moment of initial contact, along with the  
Figure 4: The necessary angles in transforming the \(O^{\prime}xy\) into the \(O_{nt}\) frame along with the numbers of the vehicles involved. necessary dimensions and the adopted \(O_{nt}\) coordinate frame. At first, only the planar view is necessary, as the planar motion will be considered. To determine the velocity components along the normal and the tangential directions versus the plane of collision, an angle between the speed vector and the longitudinal axis of each vehicle was selected from the collision protocol provided by V-SIM. This angle was then used to distribute the speed into a normal and a tangential velocity. Therefore, it seems necessary to relate the applied local \(O_{nt}\) frame to the global \(O^{\prime}{}_{xy}\) with the specific angles which are marked in Figure 4 and are as follows:  
* \(\alpha_{1}\)--the angle between the speed of vehicle no. 1 and the normal axis (\(n\)) of the adopted \(O_{nt}\) frame, which determines a division of the speed into two components (normal and tangential). * \(\beta\)--the angle between the speed of vehicle no. 1 and the x axis of the global \(O^{\prime}{}_{xy}\) frame, which determines its initial position versus the \(O^{\prime}{}_{xy}\) frame. * \(\gamma\)--the angle between the speed of vehicle no. 2 and the x axis of the global \(O^{\prime}{}_{xy}\) frame, which determines its initial position versus the \(O^{\prime}{}_{xy}\) frame.