Method for Calculating the Trajectory of Projectiles and Rockets Shooting from Moving Carrier

Abstract

¤ Ʉɨɪɨɥɟɜ ɋ. Ⱥ., Ɋɭɫɹɤ ɂ. Ƚ., ɋɭɮɢɹɧɨɜ ȼ. Ƚ., 2016 1 ɇɢɠɟ ɜ ɤɚɱɟɫɬɜɟ ɨɛɴɟɤɬɚ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɜɵɫɬɪɟɥ ɫɧɚɪɹɞɚ ɢɡ ɨɪɭɞɢɹ. Ɉɫɨɛɟɧɧɨɫɬɢ ɦɨɞɟɥɢɪɨɜɚɧɢɹ, ɫɜɹɡɚɧɧɵɟ ɫ ɩɭɫ-ɤɨɦ ɪɚɤɟɬ, ɜ ɬɟɤɫɬɟ ɨɬɦɟɱɟɧɵ ɩɪɢ ɧɟɨɛɯɨɞɢɦɨɫɬɢ. Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɜɧɟɲɧɹɹ ɛɚɥɥɢɫɬɢɤɚ, ɬɪɚɟɤɬɨɪɢɹ, ɚɷɪɨɞɢɧɚɦɢɱɟɫɤɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɩɨɞɜɢɠɧɵɣ ɧɨɫɢɬɟɥɶ, ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɫɬɪɟɥɶɛɵ, ɧɟɫɭɳɢɣ ɜɢɧɬ ɜɟɪɬɨɥɟɬɚ, ɢɧɞɭɤɬɢɜɧɚɹ ɫɤɨɪɨɫɬɶ. ȼɜɟɞɟɧɢɟ ɉɪɢ ɫɬɪɟɥɶɛɟ ɫ ɩɨɞɜɢɠɧɨɝɨ ɧɨɫɢɬɟɥɹ (ɉɇ) ɧɟɨɛ-ɯɨɞɢɦɨ ɭɱɢɬɵɜɚɬɶ ɦɧɨɠɟɫɬɜɨ ɮɚɤɬɨɪɨɜ, ɜɥɢɹɸɳɢɯ ɧɚ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɫɬɪɟɥɶɛɵ ɢ ɧɚɱɚɥɶɧɵɣ ɭɱɚɫɬɨɤ ɬɪɚɟɤɬɨɪɢɢ ɫɧɚɪɹɞɨɜ ɢ ɪɚɤɟɬ. Ɉɫɧɨɜɧɵɦɢ ɜɨɡɦɭ-ɳɚɸɳɢɦɢ ɮɚɤɬɨɪɚɦɢ ɩɪɢ ɫɬɪɟɥɶɛɟ ɫ ɜɟɪɬɨɥɟɬɚ ɹɜ-ɥɹɸɬɫɹ ɩɚɪɚɦɟɬɪɵ ɞɜɢɠɟɧɢɹ ɜɟɪɬɨɥɟɬɚ ɢ ɩɨɬɨɤ ɜɨɡ-ɞɭɯɚ, ɫɨɡɞɚɜɚɟɦɵɣ ɧɟɫɭɳɢɦ ɜɢɧɬɨɦ. Ⱦɥɹ ɢɫɫɥɟɞɨɜɚ-ɧɢɹ ɩɪɢɜɟɞɟɧɧɵɯ ɮɚɤɬɨɪɨɜ ɪɚɡɪɚɛɨɬɚɧɚ ɦɟɬɨɞɢɤɚ ɪɚɫɱɟɬɚ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɫɧɚɪɹɞɨɜ ɢ ɪɚɤɟɬ ɩɪɢ ɫɬɪɟɥɶɛɟ ɫ ɩɨɞɜɢɠɧɨɝɨ ɧɨɫɢɬɟɥɹ.Ɋɚɫɱɟɬ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɫɬɪɟɥɶɛɵ ɫ ɩɨɞɜɢɠɧɨɝɨ ɧɨɫɢɬɟɥɹ Ox y z ɫ ɧɚɱɚɥɨɦ ɜ ɰɟɧɬɪɟ ɦɚɫɫ ɉɇ ɢ ɨɫɹɦɢ ɜ ɜ ɜ , , x y z , ɫɨɪɢɟɧɬɢɪɨɜɚɧ-ɧɵɦɢ ɜɞɨɥɶ ɩɪɨɞɨɥɶɧɨɣ, ɜɟɪɬɢɤɚɥɶɧɨɣ ɢ ɩɨɩɟɪɟɱɧɨɣ ɨɫɟɣ ɉɇ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ɉɪɢɟɧɬɚɰɢɹ ɫɜɹɡɚɧɧɨɣ ɫɢɫ-ɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɨɪɦɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɝɥɚɦɢ ɬɚɧɝɚɠɚɊɢɫ. 1. Ɉɪɢɟɧɬɚɰɢɹ ɩɨɞɜɢɠɧɨɝɨ ɧɨɫɢɬɟɥɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ Ʉɨɦɩɨɧɟɧɬɵ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɜ ɧɨɪɦɚɥɶɧɨɣ ɫɢɫ-ɬɟɦɟ ɤɨɨɪɞɢɧɚɬɄɨɦɩɨɧɟɧɬɵ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɜ ɫɜɹɡɚɧɧɨɣ ɫɢɫ-ɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɭɬɟɦ ɩɨɜɨɪɨɬɚ ɧɚ ɭɝɥɵ, , 0, , 0 , 0, .(4) ȼ ɤɚɱɟɫɬɜɟ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɜɧɟɲɧɟɣ ɛɚɥɥɢɫɬɢɤɢ ɡɚɞɚɸɬɫɹ ɦɨɞɭɥɶ ɜɟɤɬɨɪɚ ɧɚ-ɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɫɧɚɪɹɞɚ ɫ ɭɱɟɬɨɦ ɞɜɢɠɟɧɢɹ ɉɇ:ɧɨɣ ɫɤɨɪɨɫɬɢ ɜ ɫɬɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɨɩɪɟ-ɞɟɥɹɟɦɵɟ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ (4). ɉɨɫɤɨɥɶɤɭ ɩɪɢ ɞɜɢɠɟɧɢɢ ɉɇ ɨɫɶ ɨɪɭɞɢɹ ɢ, ɫɨɨɬ-ɜɟɬɫɬɜɟɧɧɨ, ɨɫɶ ɫɢɦɦɟɬɪɢɢ ɫɧɚɪɹɞɚ ɧɟ ɫɨɜɩɚɞɚɸɬ ɫ ɫɭɦɦɚɪɧɵɦ ɜɟɤɬɨɪɨɦ ɫɤɨɪɨɫɬɢ ɫɧɚɪɹɞɚ, ɬɨ ɜɨɡɧɢɤɚ-ɸɬ ɭɝɥɵ ɧɭɬɚɰɢɢ, ɤɨɬɨɪɵɟ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟ-ɦɟɧɢ ɪɚɜɧɵ:ɇɚɱɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɢ ɜɟɪɬɢ-ɤɚɥɶɧɨɣ ɫɨɫɬɚɜɥɹɸɳɢɯ ɷɤɜɚɬɨɪɢɚɥɶɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨ-ɪɨɫɬɢ ɫɧɚɪɹɞɚ ɨɩɪɟɞɟɥɹɸɬɫɹ ɱɟɪɟɡ ɤɨɦɩɨɧɟɧɬɵ ɭɝɥɨ-ɜɨɣ ɫɤɨɪɨɫɬɢ ɉɇ: Ɋɚɫɱɟɬ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɫɧɚɪɹɞɚ (ɪɚɤɟɬɵ)Ʉɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɦɚɫɫ ɫɧɚɪɹɞɚ ɜ ɫɬɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɨɩɪɟɞɟɥɹɸɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ [2]:ɝɞɟ ɫ x -ɞɚɥɶɧɨɫɬɶ; ɫ y -ɜɵɫɨɬɚ ɩɨɥɟɬɚ; ɫ z -ɛɨɤɨ-ɜɨɟ ɨɬɤɥɨɧɟɧɢɟ ɜ ɫɬɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ; ɤ V -ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɦɚɫɫ ɫɧɚɪɹɞɚ (ɪɢɫ. 2). ɇɚɱɚɥɶɧɵɟ Ox y z , ɫɜɹɡɚɧɧɨɣ ɫ ɰɟɧɬɪɨɦ ɦɚɫɫ ɫɧɚɪɹɞɚ ɢ ɨɪɢɟɧɬɢɪɨɜɚɧɧɨɣ ɩɨ ɜɟɤ-ɬɨɪɭ ɫɤɨɪɨɫɬɢ (ɫɦ. ɪɢɫ. 2):Ɂɞɟɫɶ g -ɭɫɤɨɪɟɧɢɟ ɫɢɥɵ ɬɹɠɟɫɬɢ; (10)- (12) ɡɚɞɚ-ɸɬɫɹ ɫɨɨɬɧɨɲɟɧɢɹɦɢ (5)-(6).Ⱦɥɹ ɜɪɚɳɚɸɳɟɝɨɫɹ ɫɧɚɪɹɞɚ ɚɤɫɢɚɥɶɧɚɹ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɭɪɚɜɧɟɧɢɹɝɞɟ x m -ɤɨɷɮɮɢɰɢɟɧɬ ɚɷɪɨɞɢɧɚɦɢɱɟɫɤɨɝɨ ɚɤɫɢɚɥɶ-ɧɨɝɨ ɞɟɦɩɮɢɪɭɸɳɟɝɨ ɦɨɦɟɧɬɚ (ɦɨɦɟɧɬɚ ɬɪɟɧɢɹ) ɜ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ Oxyz , ɫɜɹɡɚɧɧɨɣ ɫ ɰɟɧɬɪɨɦ ɦɚɫɫ ɫɧɚɪɹɞɚ, ɝɞɟ ɨɫɢ , ,x y z ɧɚɩɪɚɜɥɟɧɵ ɜɞɨɥɶ ɩɪɨɞɨɥɶ-ɧɨɣ, ɜɟɪɬɢɤɚɥɶɧɨɣ ɢ ɩɨɩɟɪɟɱɧɨɣ ɨɫɟɣ ɫɧɚɪɹɞɚ, ɫɨɨɬ-ɜɟɬɫɬɜɟɧɧɨ; l -ɞɥɢɧɚ ɫɧɚɪɹɞɚ; x I -ɚɤɫɢɚɥɶɧɵɣ ɦɨ-ɦɟɧɬ ɢɧɟɪɰɢɢ; 0 x Z -ɚɤɫɢɚɥɶɧɚɹ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜ ɦɨɦɟɧɬ ɜɵɫɬɪɟɥɚ.Ƚɨɪɢɡɨɧɬɚɥɶɧɚɹ ɢ ɜɟɪɬɢɤɚɥɶɧɚɹ ɫɨɫɬɚɜɥɹɸɳɢɟ ɭɝɥɚ ɧɭɬɚɰɢɢ ɫɧɚɪɹɞɚ 1 2 , G G ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ ɫɢɫɬɟ-ɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ:ɝɞɟ 1 2 , Z Z -ɫɨɫɬɚɜɥɹɸɳɢɟ ɷɤɜɚɬɨɪɢɚɥɶɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɫɧɚɪɹɞɚ.Ⱦɥɹ ɨɩɪɟ...

Mathematical Modeling of Shooting from Moving Carrier

“…The motion trajectory of the projectiles and rockets is calculated based on the method for solving the problem of external ballistics when shooting from a moving carrier, presented in works [8,9], under initial conditions defined by Eqs (10)- (13).…”

“…To take into consideration the effect of the helicopter rotor on the movement of projectiles and rockets in the initial part of the trajectory, a method has been developed for calculating the parameters of the air flow generated by the screw rotation [12]. For a given flight mode of a helicopter, the average value of the inductive velocity ( )…”

“…The initial conditions for the system of differential Eqs (7) and (8) are determined from Eqs (3) and (4) for the initial coordinates and velocities of the points of the mechanical system at t = 0.…”

Simulation Method of Projectiles Dispersion

“…− расчет траектории движения снаряда [1,2]; − решение задачи внешнего обтекания и расчет аэродинамических коэффициентов снаряда заданной формы [3]; − моделирование факторов, влияющих на рассеивание снарядов; − методику проведения статистических испытаний и определения характеристик рассеивания.…”