Application of computational methods in problems of aeroballistics. determination of conjugate throwing angles and construction of ballistic trajectories

Abstract

The problem of aeroballistics of non-reactive artillery projectiles is studied by means of a simplified mathematical model. The following problems are considered as separate subproblems: initial value problem, determination of the horizontal range of a projectile (4th order Runge-Kutta method with modification of polynomial interpolation); optimization problem (Powell's method), determination of the angles of maximum range — the throwing angles at which the maximum horizontal range is achieved; the boundary value problem (shooting method with Newton-Raphson method/ secant method/ polynomial interpolation), determination of throwing angles at a given distance and the problem of finding conjugate trajectories — the low and high angled trajectories, which achieve the same horizontal range of the projectile at different throwing angles; the inverse geodesy problem (Vincenty's formulae), determination of the geodesic distance between two geographical points on the WGS-84 non-spherical Earth model. The following characteristics are graphically illustrated as a function of throwing angles: horizontal and vertical ranges, maximum vertical and horizontal velocity components, magnitudes of terminal velocities, impact angle, and flight time of projectiles. The existence of conjugate trajectories is established and the strategy of sequential firing of projectiles with the aim of simultaneously hitting the target along different trajectories is determined. The programming of the numerical methods, the algorithm for solving the problem, and the visualization elements were implemented using the MATLAB application package, and the developed methodology and software have demonstrated their effectiveness and the possibility of their practical application.