Geometry of motion and nutation stability of free axisymmetric variable mass systems

TL;DR

This paper develops a geometric approach to analyze the rotational motion and nutation stability of free axisymmetric variable mass systems, providing analytical solutions and insights into their stability under mass loss.

Contribution

It introduces a geometric method to study the motion of variable mass systems, deriving an analytical solution for nutation without solving complex differential equations.

Findings

Analytical solution for the second Euler angle of nutation.

Certain configurations exhibit unbounded angular speed growth.

All configurations studied show nutational stability despite instability in angular speeds.

Abstract

In classical mechanics, the 'geometry of motion' refers to a development to visualize the motion of freely spinning bodies. In this paper, such an approach of studying the rotational motion of axisymmetric variable mass systems is developed. An analytic solution to the second Euler angle characterising nutation naturally falls out of this method, without explicitly solving the nonlinear differential equations of motion. This is used to examine the coning motion of a free axisymmetric cylinder subject to three idealized models of mass loss and new insight into their rotational stability is presented. It is seen that the angular speeds for some configurations of these cylinders grow without bounds. In spite of this phenomenon, all configurations explored here are seen to exhibit nutational stability, a desirable property in solid rocket motors.