Gyroscopic effect of a sphere

Question

I am looking at: https://en.wikipedia.org/wiki/Euler%27s_equations_(rigid_body_dynamics)

\begin{align} I_1\dot{\omega}_{1}+(I_3-I_2)\omega_2\omega_3 &= M_{1}\\ I_2\dot{\omega}_{2}+(I_1-I_3)\omega_3\omega_1 &= M_{2}\\ I_3\dot{\omega}_{3}+(I_2-I_1)\omega_1\omega_2 &= M_{3} \end{align}

and think of a sphere, where \begin{align} I_1 = I_2 = I_3 = I \end{align}

thus \begin{align} I_1\dot{\omega}_{1} &= M_{1}\\ I_2\dot{\omega}_{2} &= M_{2}\\ I_3\dot{\omega}_{3} &= M_{3} \end{align}

Assuming my model has inputs as: \begin{align} {\omega}_1 = 1 && \dot{\omega}_1 = 0\\{\omega}_2 = 0 && \dot{\omega}_2 = 0\\ {\omega}_3 = 0 && \dot{\omega}_3 = 1 \end{align}

We see: \begin{align} {M}_1 &= 0 \\ {M}_1 &= 0 \\ {M}_3 &= I \dot{\omega}_3 \end{align}

which is only the effect of inertia as if no rotation about 1 had been applied.

This indicates, that the gyroscopic effect vanishes in case of a sphere, and inverts when a disk turns into a rod (I_1 < I_3), where 1 is the axis of constant rotational speed.

I wasn't able to find any evidence of my conclusion on the internet. Am I missing something?

Answer 1

Without looking at the mathematical expressions, consider the following:

A sphere can be thought of as a stack of disks. A disk has (compared to other shapes) a forceful gyroscopic response because most of the mass is closer to the perimeter than to the axis of rotation, the mass distribution is close to planar.

Given that a sphere can be thought of as a stack of disks we can already infer that a spinning sphere will have a gyroscopic response. For a disk and a sphere with the same total mass the response of the sphere will be in comparison less forceful, but it will be there.

Apart from the above there is another consideration.
In the case of a disk there is an obvious optimal spin axis; the axis that perpendicular to the disk, throught the center. It's optimal in the sense: with that axis as spin axis you get the strongest possible gyroscopic response (relative to the total mass of the disk).

In the case of a sphere there is no optimal axis. With a sphere you can spin up around any axis, and the forcefulness of the gyroscopic response will be the same in all cases.