Necessary conditions for tumbling in the rotational motion

Abstract

The goal of this work is the investigation of the necessary conditions for the possible existence of tumbling in rotational motion of rigid bodies. In a stable spinning satellite, tumbling may occur by sufficient strong action of external impulses, when the conical movement characteristic of the stable attitude is de-characterized. For this purpose a methodology is chosen to simplify the study of rotational motions with great amplitude, for example free bodies in space, allowing an extension of the analysis to non-conservative systems. In the case of a satellite in space, the projection of the angular velocity along the principal axes of inertia must be known, defining completely the initial conditions of motion for stability investigations. In this paper, the coordinate systems are established according to the initial condition in order to allow a simple analytical work on the equations of motion. Also it will be proposed the definition of a parameter, calling it tumbling coefficient, to measure the intensity of the tumbling and the amplitude of the motion when crossing limits of stability in the concept of Lyapunov. Tumbling in the motion of bodies in space is not possible when this coefficient is positive. Magnus Triangle representation will be used to represent the geometry of the body, establishing regions of stability/instability for possible initial conditions of motion. In the study of nonconservative systems for an oblate body, one sufficient condition will be enough to assure damped motion, and this condition is checked for a motion damped by viscous torques. This paper seeks to highlight the physical understanding of the phenomena and the influence of various parameters that are important in the process.