Mathematical model to simulate the trajectory elements of an artillery projectile proof shot

In external ballistics of a conventional spin-stabilised artillery projectile, there are a number of trajectory models developed for computing trajectory elements having varying degrees of complexity. The present study attempts to propose a single mathematical model, viz., simplified point-mass/simple particle trajectory model to simulate the trajectory elements of a typical spinstabilised flat-head artillery projectile proof shot. Due to difficulties in the projectile shape and size, and the complicated nature of air resistance, an accurate mathematical prediction of the trajectory is difficult. To simplify the computations, the governing equations of motion of the projectile have been simplified and assumed that the projectile is a particle and the only forces acting on the projectile are drag and gravity. With this model, trajectory elements have been generated and compared with experimental results obtained in the field test. The measuring instrument used in this case is a Doppler radar.

ρ

Average density of the air

Reference area/maximum cross-sectional area of the projectile = πr² = πd²/4

Mass of the projectile

Length of the projectile

Diameter of the projectile

Drag coefficient (dimensionless number)

Acceleration due to gravity at sea level

Step size time

Horizontal range of the projectile

Velocity of the projectile wrt the ground coordinate system at any time t

Initial/muzzle velocity of the projectile

Horizontal component of the velocity

Vertical component of the velocity

Angle of inclination of trajectory to horizontal at time t

Initial angle of inclination of trajectory

Velocity of sound at sea level

Angle of sight (= tan-1(y/x))

Angle of fall

Angle of yaw

Coordinates of the CG of the projectile at time t

Coordinates of the origin of the trajectory, i.e., at t = 0

Range

Maximum vertex altitude/height

Point of fall, i.e., the end-point of trajectory

Position of the projectile at any time t

Flight time to any point along the trajectory from (x0,y0)

Total time of flight

Drag force on the projectile = ½ ρSV²CD

Drift of the projectile = K,t².cos ε, where K is a constant