Suitable Mass Density Function for an Artificial Satellite to Prevent Chaotic Motion after Collision with Space Debris

Abstract

Artificial satellites are widely used in different areas such as communication, position systems, and agriculture. The number of satellites orbiting Earth is becoming huge, and many are set to be launched soon. This huge number of satellites in addition to space debris are sources of concern. Indeed, some incidents have occurred either between satellites or because of space debris. These incidents are a threat for the hit satellite and can be a source of irreversible damages. A hit satellite may diverge to a chaotic motion with all the entailed consequences. The inertia moment of a satellite is a main factor to determine if the hit satellite is heading toward a chaotic motion or not. The inertia moment is determined over the mass density function. In this paper, a circularly orbiting artificial satellite was modeled as a thin rotating rod. The objective was to determine a suitable mass density function for this satellite allowing the prevention as much as possible of the chaotic motion after being hit. This unknown density mass function satisfies a system of equations reflecting some physical constraints. Conventional procedures are not applicable to solve this system of equations. The presented resolution method is based on several mathematical transformations, allowing converting this system into a highly nonlinear one with several unknowns. Several mathematical techniques were applied, and an analytical solution was obtained. Finally, from the mechanical engineering point of view, the obtained mass density function corresponds to a Functionally Graded Material (FGM).