MATHEMATICAL MODEL OF ANALYSIS OF TRAJECTORIES OF SMALL-CALIBER ANTI-AIRCRAFT ARTILLERY SHELLS

The expediency of performance improvement of anti-aircraft gun systems provided in terms of military-technical assistance to Ukraine is considered by all available ammunition, including the newest type with programmable airburst. A simplified mathematical model for determining the trajectory of small-calibre anti-aircraft shells is presented to evaluate anti-aircraft ammunition efficiency by computer simulation. The model is based on the equation of projectile centre of gravity motion under the influence of the force of gravity and air drag with a set of assumptions that are essential combat conditions for short-range anti-aircraft gun systems. The correspondence algorithm of calculations is based on solving linear scalar differential equations set by numerical integration with a fixed integration step. Calculations have been simplified by the representation form, which allows simultaneous solving of all equations of set and can be provided by embedded numerical integration solvers of standard computer simulation packages. The reliability of the mathematical model was confirmed by a comparative analysis of obtained calculation results and reference trajectories derived from statistical processing of test firings of Gepard 1 A2 self-propelled anti-aircraft gun. At the same time, the values of the relative error positions of the projectile in the vertical plane did not exceed 2 %. Such values allow the utilisation of the model in different shell performance evaluation tasks. For example, the calculated trajectories of PMD 044 armour-piercing-incendiary tracer and PMD 060 programmable airburst shells of 35 mm × 228 mm ammunitions have been obtained and presented for the reference firing conditions. Further improvement of the mathematical model should include clarifying the projectile drag coefficient and considering the impact of deviation of firing conditions from the reference ones.