An alternative interpretation of the Grioli gyroscope suspension points

TL;DR

This paper analyzes the possible regular precessions of a heavy asymmetric body with a fixed point, revealing conditions for their existence and showing that precession frequencies are quantized, similar to quantum spin phenomena.

Contribution

It provides a detailed, matrix-based analysis of regular precessions in asymmetric bodies, highlighting new geometric conditions for their occurrence and demonstrating frequency quantization.

Findings

Regular precessions occur along specific lines in the principal plane.

Precession frequencies are quantized, fixed by two specific values.

Conditions depend on the interchange of moments of inertia.

Abstract

We present a detailed analysis of all possible regular precessions of a heavy asymmetric body with a fixed point not coinciding with the center of mass. The calculations are done in terms of the rotation matrix, by writing the Euler-Poisson equations with all involved vectors parameterized in the Laboratory frame. It is shown that a regular precession is possible if the suspension point is chosen on the straight lines (lying in the principal plane) which are frontiers of the regions where, as the distance from the center of mass increases, the interchange of the intermediate and largest moments of inertia occurs. Like the spin of an electron in quantum mechanics, the frequency of regular precession in classical mechanics turns out to be rigidly fixed by two values, i.e., quantized.