Analysis of motion of the body of a motor car hit on its side by another passenger car

Based on an analysis of the course of a few experimental crash tests, a physical model and afterwards a mathematical model were prepared to describe the motion of bodies of the vehicles involved during the phase of impact. The motion was analysed in a global coordinate system attached to the road surface. Local coordinate systems were also adopted with their origins being placed at the centres of mass of the vehicles. Equations of motion of the model were derived. The calculation results enabled defining the influence of the location of the point of impact against the vehicle side on e.g. the following: - time history of the impact force exerted by the impacting car (A) on the impacted car (B) as well as characteristic values of this force and of the impulse of the impact force; - time histories showing changes in the velocity of the centre of vehicle mass and in the angle of deviation of the velocity vector from the direction of motion of the impacted vehicle before the collision; - trajectory of the centre of mass and angle of rotation of the body of the impacted vehicle. The calculations were focused on the initial period of motion of the body of the impacted vehicle, up to the instant of 200 ms from the start of the collision process. After this time, the vehicles separate from each other and move independently. The results obtained from the calculations covering this initial period make it possible to determine the starting-point values of the parameters to be taken for further calculations of the free post-impact motion of the cars.


Introduction
Right-angle collisions of motor vehicles amount to more than 25 % of road accidents in Poland. Knowledge of the course of accidents of this kind and, first of all, exploration of the course of the dynamic interactions that take place during the right-angle collision of two motor vehicles will enable further improvement of traffic incident modelling and reconstruction methods. The identification of the dynamic interactions between the vehicles in the culminating phase of the collision process will also make it possible to analyse the processes of injury formation in vehicle occupants and will provide a basis for the creation of new personal protection and safety means.
Many articles dealing with side collisions of motor vehicles have been published. Predominantly, they present analyses of the deformation of the side of a motor vehicle body and of the distribution of forces on the barrier face [2,3] during a right-angle and oblique impact [7]. The results of such research works are oriented at improving the vehicle occupant protection system and designing a Mobile Deformable Barrier (MDB) [5,6,9]. In publication [1], the displacement of a motor car hit on its side has been presented, with highlighting the influence of the stiffness of the load-bearing vehicle structure on this process. Interesting are the results of an analysis of road accidents in Japan, where the occurrence of injuries was linked with the speed and type of the vehicle hit on its side [8]. There is a definite lack of works where the issue of impact-caused interactions between vehicles involved in a right-angle collision would be addressed. An attempt to analyse this problem has been presented in [4]. The objective of this work was to determine the influence of the place of a frontal impact of a motor car against the side of another car on the course of the impact-caused interaction between the two vehicles and on the change in the velocity vector and trajectory of the body of the impacted vehicle. In this work, experimental testing and mathematical modelling were combined together. The results presented have been obtained for the time interval 0-0.2 s. This is the period after which the vehicles separate from each other and move independently in accordance with their individual kinetic energy. Based on the experimental test results having been collected, a physical model and afterwards a mathematical model were prepared to describe the motion of bodies of the vehicles involved during the phase of impact. The motion was analysed in a global coordinate system attached to the road surface. Local coordinate systems were also adopted, with their origins being located at the centres of mass of the vehicles. The local systems were used for presenting the measurement results and for deriving the equations of motion of the model.

Preparation and carrying out of experimental tests
Six crash tests were performed at the Automotive Industry Institute (PIMOT) in Warsaw with the use of 12 passenger cars of the same make and model. In each test, the front of car A crashed into the left side of car B (Fig. 1). The pre-impact speed of car A was about 50 km/h and it was twice as high as that of car B. The crash tests were carried out on a test yard with dry concrete surface, in good weather conditions. During the tests, the steering wheel of each car was left free and car wheels were not braked. The relative positions of cars A and B were changed in successive crash tests. The location of the point of impact on the side of vehicle B was defined by the distance LAB between the longitudinal plane of symmetry of car A and the front wheel axis of car B (see Fig. 1). The values of the characteristic parameters of successive crash tests have been specified in Table 1.  The tests were carried out with the use of Honda Accord cars manufactured in 2000-2002. The cars were in good technical condition and had undamaged and non-corroded bodies, which had not been previously repaired. At the centre of mass of each car, there was a three-axial acceleration sensor, installed together with sensors measuring the angular velocity of the vehicle body in relation to the three coordinate axes, whose directions and senses were consistent with those of the local coordinate systems attached to the cars.

Calculation of the impact force applied to the side of vehicle B
To analyse the processes that took place during the vehicle collision, a global coordinate system and local coordinate systems were adopted. A global coordinate system OGXGYGZG, attached to the road. The OGXGYG plane of this system is situated at the road surface level and the OGZG axis is pointing vertically upwards. The OGXG axis is parallel to the vector of pre-impact velocity of vehicle A and the OGYG axis is parallel to the vector of pre-impact velocity of vehicle B. Local levelled coordinate systems, attached to the centres of mass of vehicle A and B. The local coordinate system OAxPAyPAzPA has its origin OA situated at the centre of mass of vehicle A, the OAxPAyPA plane is parallel to the road surface plane, and the OAxPA axis is parallel to the longitudinal vehicle symmetry plane. The local coordinate system OBxPByPBzPB has its origin OB situated at the centre of mass of vehicle B, the OBxPByPB plane is parallel to the road surface plane, and the OBxPB axis is parallel to the longitudinal vehicle symmetry plane.      systems. The kinematic calculations, the results of which were used at the analysis of the post-impact motion of the vehicles, were carried out with taking into account a transformation of the measurement results from local coordinate systems of vehicles A and B to the global coordinate system. The transformation matrices (2, 3, and 4) have been presented below: (2) To determine the velocities and coordinates of the centre of mass of vehicle B in the global coordinate system, the following relation was used:

Results of the calculation of the impact force applied to vehicle B
Results of the calculation of the impact force FwB applied to car B and its normal component FnBXG in the global coordinate system have been shown in Fig. 4.  The time history of the impact force has the nature of a quick-changing process. Therefore, the extreme (instantaneous) values of this force do not have a decisive influence on the post-impact motion of vehicle B; they are strongly dependent upon the low-pass filter used. In this consideration, the maximum values obtained after averaging the function FnBXG (t) over time intervals of 3 ms, 5 ms, and 10 ms have been given in Table 2. Such a method of load presentation links the maximum values of the impact force with the time of duration of the resulting load. The load duration time is of decisive importance for the assessment of the effects of a side impact. Table 3 shows values of the impulse of the impact force and average values of the impact force, calculated for the time interval in which the highest values of the force FnBXG were recorded.

Vehicle motion caused by the impact force
Below are shown the primary effects caused by the impact force, i.e.: time histories of the velocity of the centre of mass of vehicle B and of the angle of deflection of the velocity vector from its pre-impact direction (Fig. 5); trajectories of the centre of mass of vehicle B and changes in the vehicle body yaw angle ΨB, i.e. the angle of rotation of the vehicle body around the vertical axis (Fig. 6).

Vehicle body tilt versus reactions on wheels
The side impact force is balanced by the force of inertia of the impacted vehicle and the tangential reactions developing at the contact between the vehicle wheels and the road surface. Of particular interest are the lateral tangential reactions. The course of changes in their values strongly depends on the roll angle ΦB resulting from the side impact. Fig. 7 shows the lateral tilt (roll) of the body of vehicle B at several instants in one of the crash tests (LAB = 1.34 m). A time history of the roll angle ΦB has been presented in Fig. 8, where the curves plotted have been based on measurement results obtained from six crash tests. Noteworthy are the very high values of the vehicle body roll angle, observed not only in the contact phase of the collision process but also after the vehicle separation. A vehicle body roll angle of about 7-8 deg causes the wheel suspension on one of the vehicle sides to be fully compressed. Since the roll angle values are observed to exceed this limit, they show that the vehicle body not only tilts by rotating around the longitudinal axis OBxB but also rotates around an instantaneous axis going through the centres of the areas of contact between the wheels and the road on one of the vehicle sides (left or right), as it can be seen in Fig. 7. In the case illustrated there, such a rotation caused the right wheels of vehicle B to be lifted off the road surface.

Recapitulation and conclusions
The test results obtained and the test result analysis carried out show, first of all, characteristics of the process of generating dynamic effects on the body of a motor car being struck on its side. The impactcaused interaction between vehicles (impact force) chiefly depends on the vehicle deformation process, development of the inertial resistance force exerted by the impacted vehicle, and development of the tangential reactions in the area of contact between the vehicle wheels and the road. Each of these factors has a complex influence on the time history of the impact force.
The calculations carried out were focused on the initial period of vehicle motion, up to the instant of 0.2 s from the start of the collision process. After this time, the vehicles separate from each other and move independently. The results obtained from the calculations covering this initial period make it possible to determine the starting-point values of the parameters to be taken for further calculations of the free post-impact motion of the cars.  (Tables 2 and 3) decrease with increasing distance LAB between the place of impact and the front axle of car B. 2. When car B is hit on a place close to its front axle, its pre-impact speed is raised by about 30 % ( The tests reported herein were carried out and their results were analysed within an authors' own research project No. N N509 559440.