Mathematical modeling of speed change of vehicles at emergency braking

Abstract

The object of research is the reconstruction of the development of the mechanism and circumstances of the road accident. The key point of research is establishment of the car speed at different moments in the development of a traffic accident. Precisely the accuracy in determining the car speed affects the nature of the main findings of the examination. When calculating the car speed, the expert solves the inverse problem, that is, determines the speed according to the braking performance of the car and the length of the trail of braking. For the last two decades, the design of the car's brake system has changed. The brakes are equipped with antilock braking systems. At the same time, the efficiency of vehicle braking has improved significantly. But it turned out that experts can’t objectively determine the car speed, which is equipped with an anti-lock braking system, because such brakes do not leave traces of braking on the road surface. When developing and solving this problem by determining the vehicle speed, the vehicle is equipped with an anti-lock braking system, methods of differentiating and integrating a complex function are used. With a comparative analysis of the existing and proposed methods for calculating the speed of the car in the process of emergency application braking, the graphical method. According to the research results, a mathematical model for determining the speed of the car is developed, which is equipped with an anti-lock braking system. This model allows to take into account the effect of the forces of air resistance, resistance to movement and resistance to recovery not only in the steady phase of braking, but also during the reaction of the driver and the timing of the brakes. An analysis of the mathematical model shows that during these time intervals a certain deceleration will act on the car, which will depend on the speed of movement and the state of the car. Moreover, the action of the drag resistance force can significantly increase this deceleration and influence the change in the car speed. The proposed mathematical model more accurately displays the actual process of emergency braking of the car and provides a reduction in the error in calculating the speed of the car by 4–8 % compared with existing calculation methods.