Modified Linear Theory for Spinning or Non-Spinning Projectiles

Static and dynamic stability are the most important phenomena for stable flight atmospheric motion of spin and fin stabilized projectiles. If the aerodynamic forces and moments and the initial conditions are accurately known, an essentially exact simulation of the projectile’s synthesized pitching and yawing motion can be readily obtained by numerical methods. A modified trajectory linear theory of the same problem implies an approximate solution. INTRODUCTION More than 80 years ago, English ballisticians [1] constructed the first rigid six-degree-of-freedom projectile exterior ballistic model. Their model contained a reasonably complete aerodynamic force and moment expansion for a spinning shell and included aerodynamic damping along with Magnus force and moment. Guided by an extensive set of yaw card firings, these researchers also created the first approximate analytic solution of the six-degree-of-freedom projectile equations of motion by introducing a set of simplifications based on clever linearization by artificially separating the dynamic equations into uncoupled groups. The resulting theory is commonly called projectile linear theory. Kent [2], Neilson and Synge [3], Kelley and McShane [4], and Kelley et al. [5] made refinements and improvements to projectile linear theory. Projectile linear theory has proved an invaluable tool in understanding basic dynamic characteristics of projectiles in atmospheric flight, for establishing stability criteria for finand spin-stabilized projectiles, and for extracting projectile aerodynamic loads from spark range data. In the present work, the full six degrees of freedom (6DOF) projectile flight dynamics atmospheric model is considered for the accurate prediction of short and long range trajectories of high spin and fin-stabilized projectiles. It takes into consideration the influence of the most significant forces and moments, in addition to gravity. Projectiles, which are inherently aerodynamically unstable, can be stabilized with spin. For this condition, the spin rate must be high enough to develop a gyroscopic moment, which overcomes the aerodynamic instability, and the projectile is said to be gyroscopically stable. This is the case for the most gun launched projectiles (handguns, rifles, cannons, *Address correspondence to this author at the Laboratory of Firearms and Tool Marks Section, Criminal Investigation Division, Hellenic Police, and Post graduate Student, Mechanical Engineering and Aeronautics Department, University of Patras, Greece; E-mail: [email protected] etc.) where the rifling of the barrel provides the required axial spin to projectile. In describing this condition, a gyroscopic stability factor can be applied, which is obtained from the roots of the modified linear theory in the equations of projectile motion. Also, dynamic stability is defined as the condition where a system is perturbed and the ensuing oscillatory has a tendency to either decrease or increase. Note that this definition assumes that the static stability is present, otherwise the oscillatory motion would not occur. PROJECTILE MODEL The present analysis considers two different types of representative projectiles: a spin-stabilized of 105mm and a mortar fin-stabilized of 120 mm. Basic physical and geometrical characteristics data of the above-mentioned 105 mm HE M1 and the non-rolling, finned 120 mm HE mortar projectiles are illustrated briefly in Table 1. Table 1. Physical and Geometrical Data of 105 mm and 120mm Projectiles Types Characteristics 105 mm HE M1 projectile 120 mm HE mortar projectile Reference diameter, mm 114.1 119.56 Total Length, mm 494.7 704.98 Total mass, kg 15.00 13.585 Axial moment of inertia, kg-m 2326x1