Efficient calculation of earth penetrating projectile trajectories

Abstract

Currently, two methods exist to determine trajectory of a ballistic penetrator: Poncelet Analysis and Differential Area Force Law (DAFL) methods. An exact solution for the Poncelet Equation exists; making for easy computation. However, the one dimensional nature of the equation fails to capture the intricate three-dimensional nature of real world ballistic penetrator trajectories. The DAFL methods employ empirically derived stress algorithms to calculate to forces acting on a differential area of a projectile. These stresses are then used to determine the forces and moments acting on the differential areas. These forces and moments are then used to solve the equations of motion to determine the trajectory of the ballistic penetrator. The DAFL methods accurately capture the three dimensional nature of the penetrator's trajectory, but are computationally intensive which make them slow. The Integrated Force Law (IFL) method combines the computational ease of the Poncelet Analysis with the accuracy of the DAFL methods. In IFL, the projectile shape is modeled as a polynomial. The stress algorithms used in the DAFL methods are them numerically integrated over the top and bottom surfaces of the projectile to determine the force and moment acting on the top and bottom half of the weapon. These two forces and moments are then used to solve the equations of motion. J-hook trajectories are solved in less than 40 seconds and stable trajectories are solved in less than three seconds.