МАТЕМАТИЧНА МОДЕЛЬ ПОЛЬОТУ СНАРЯДА ЯК МОДИФІКОВАНА МОДЕЛЬ МАТЕРІАЛЬНОЇ ТОЧКИ У ЯВНІЙ ФОРМІ

Abstract

A promising direction for improving the accuracy and operational efficiency of firing artillery systems is the development of ballistic computers that implement procedures for determining settings using the solution of a system of differential equations (mathematical model) that describe the movement of a projectile in the air. The main problem in the development of mathematical models is the possibility of determining with a given accuracy the coefficients of aerodynamic forces (moments) that are in the right-hand sides of differential equations. An urgent issue in this direction is the study of the possibility of restoring the coefficients of aerodynamic forces (moments) by solving the inverse problem of external ballistics using mathematical models of projectile flight. The movement of projectiles is usually described using one of three mathematical models, which differ from each other in the main level of complexity and, accordingly, the level of adequacy to the real process of projectile movement in the air. Therefore, the complexity of recovery procedures is determined by the number of coefficients of aerodynamic forces (moments) and differential equations included in mathematical models. A modified model of a material point is presented as a mathematical model of projectile flight in which all aerodynamic forces are taken into account, the orientation of the projectile is characterized by taking into account the nutation angle, and the kinetic energy of the rotational movement is taken into account through the angular velocity of the projectile around its axis of symmetry. It is shown that the modified material point model is determined by adding an implicit ordinary differential equation - an equation that describes the nutational oscillations of the projectile, which depend on the acceleration of its flight. Procedures for converting the system of differential equations of the modified material point model to an explicit form are disclosed, which allows them to be solved on the basis of standard numerical methods.