Mathematical model for estimation the stability of the vehicle's motion on overturning

Abstract

Problem. During the reconstruction of the circumstances of road traffic accidents with vehicles overturning, difficulties arise with determining the parameters of vehicles in the process of their overturning. This is due to the fact that the recommended calculation methods are often simplified. The main focus of such techniques is to determine the minimum speed of vehicles, which leads to their overturning. In fact, the speed of vehicles before overturning can be significantly higher. Goal. This paper is dedicated to developing mathematical model of overturning vehicles that makes possible to determine not only the conditions for overturning vehicles, but also other parameters of the vehicle movement in the process of overturning. Methodology. The overturning of the vehicle occurs as a result of the action of inertial forces after collision with an immovable side obstacle. In this case, the moment from the force of gravity of the vehicle keeps it from overturning. In the process of overturning the vehicle, the moment from the force of gravity decreases due to the decrease in the arm of the force of gravity. To compile a mathematical model, the basic equation of dynamics during rotational motion was used. The mathematical model of a vehicle overturning is written in the form of a nonlinear homogeneous second order differential equation. An analytical solution of this equation is obtained. Results. Developed mathematical model makes possible to determine not only the conditions for overturning vehicles, but also other parameters of the vehicle movement from the moment the center of mass begins to rise to the moment of its maximum rise in the process of overturning. For a particular case, when the critical speed of a vehicle during its overturning is determined, the developed mathematical model fully corresponds to the mathematical model based on the law of conservation of energy. For a specific vehicle, numerical results were obtained that fully correspond to the physics of the overturning process.