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Poinsot’s Paper Spaceship #2 – Zeppelin: Unstable Rotation About the Intermediate Axis.

Poinsot’s Paper Spaceship #2 – Zeppelin: Unstable Rotation About the Intermediate Axis.

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Subtitle: Why UFOs are more stable than zeppelins.

This UFO…

Geometric drawing of an ellipsoid intersecting a aphere, inspired by Louis Poinsot's geometric theory of the rotation of rigid bodies. Orthographic projection showing all three straight-on views. Lines used to create a tesselation of the drawing surface, colored in the style of stained glass. Watercolor pencils on 21cmx21cm watercolor paper. By elkement 2025.
Poinsot’s Paper Spaceship by elkement 2025

… tells a story of rotation.

The yellow ellipsoid encapsulates the main features of a rotating rigid body: Its moments of inertia, that is: how masses are distributed in relation to three principal axes.

The moment of inertia is larger, if more mass is positioned farther from the axis of rotation. The rotational energy of a spinning top – E – is connected to its inertia tensor – Θ

2E = \vec{\omega} \Theta \vec{\omega}

with ω being the angular velocity.

Θ is a matrix that encodes the distribution of masses. Angular momentum L is connected to angular velocity via this matrix:

\vec{L} = \Theta \vec{\omega}

… so the vector L does not need to be parallel to vector ω.

If there is no external force, L is fixed in space and ω is moving. But from the perspective of the spinning top L is moving. This motion is fully determined by constant energy, the moments of inertia, and the magnitude of angular momentum:

L cannot not change its length: When seen from the gyroscope’s system it cannot be distorted. So, its size (or its square) are fixed:

L_1^2 + L_2^2 + L_3^2 = \vec{L}^2

Angular velocity ω can be substituted in the expression for energy:

\frac{L_1^2}{\theta_1} + \frac{L_2^2}{\theta_2} + \frac{L_3^2}{\theta_3} = 2E

The parameters in the denominator – \theta_1, \theta_2, \theta_3 – are moments of inertia for rotations about axes 1,2,3 (I am not calling them x,y,z as these axes 1,2,3 spin with the rotating body). In the 1,2,3 system this matrix is diagonal, and the indexed thetas are its eigenvalues.

These are two equations for three variables – the components of angular momentum. There is one parameter left – so, the solution is a 1-dimensional curve: This curve is the intersection between the “energy ellipsoid” and the “angular momentum sphere” … the shapes which make up my UFO!

My UFO uses an elliptical saucer, so the intersection curve is not exactly a circle. If the saucer was a circular disc, also the path of L would be circular. Then the rotating body would also be symmetrical, and we could use these cones to envisage the motion also in the usual frame of reference, not attached to he gyroscope.

When the radius of the angular momentum sphere is equal to the largest or the smallest axis of the ellipsoid, the ellipsoid either fully embraces the sphere – or the the sphere embraces the ellipsoid. Reduce the radius of the UFO’s sphere and it will be completely inside the saucer part. In both of these cases, the intersection curves shrinks down to two points: There are only two possible solutions for the angular momentum vector – “up” and “down”.

In the system attached to the spinning top’s principal axes, the other components of L are then zero. For example, if the solutions are on the “1” axis, vector L has only a “1” component. If you multiply this vector with the inertia tensor in its diagonal form, also angular momentum has only a “1” component. For rotations about the principal axes, angular momentum and angular velocity are parallel.

This should also be the case for the intermediate axis: The argument about L having only a single component is the same.

But the relationship of the sphere and the ellipsoid is different.

This is celebrated in Poinsot’s Paper Spaceship – Zeppelin – a drawing showing the ellipsoid with its “intermediate sphere” from four different viewpoints:

Geometric drawing of an ellipsoid intersecting a aphere, inspired by Louis Poinsot's geometric theory of the rotation of rigid bodies. The angular momentum sphere's radius is equal to the intermediate axis of the energy ellipsoid, indicating unstable rotation about the intermediate axis. Orthographic projection showing all three straight-on views. Lines used to create a tesselation of the drawing surface, colored in the style of stained glass. Watercolor pencils on 21cmx21cm watercolor paper. By elkement 2025.
Poinsot’s Paper Spaceship – Zeppelin by elkement 2025

The yellow sphere’s radius is exactly equal to the intermediate axis of the zeppelin. Would the sphere be a bit smaller or larger, the X-shaped intersection curve would fragment into two parts. The closer the radius gets to either the largest or the smallest axis, the more the intersection curves resemble circles.

When the intersection curve is a small circle, the rotation is stable in case L would deviate just a little bit from the axis. The figure axis would move around L, similar to the perfect rotation of the symmetrical top, encapsulated in the picture of the dancing cones.

The “more intermediate” the axis becomes, the more pronounced the difference. A small deviation of L would make it wander away quickly and turn “upside down”, adding an “unexpected” rotation. It can be demonstrated using for example a tennis racket: Trying to rotate racket about the intermediate axis will nearly always also add a rotation by 180° about the axis parallel to the handle. The angular moment vector follows one branch of the X-shaped intersection curve all the way to the other side of the ellipsoid.

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elkemental Force
elkemental Force
@elkement.art@elkement.art

Art inspired by physics.

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Comments

2 responses to “Poinsot’s Paper Spaceship #2 – Zeppelin: Unstable Rotation About the Intermediate Axis.”

  1. Andrea Avatar
    Andrea

    This is a great post!

    As a science fiction author and aviation aficionado, I’ve often contemplated the physics of UFOs and their stability. The problems arise when they need to change direction or simply change orientation. They’d have to have a sophisticated thrust control system for applying vectors, as the thrusters would be spinning too. It would be like in a helicopter, where the thrust needs to be applied at 90° to the direction you want the spinning system to change. Also, in aviation, especially in fighter aircraft, stability isn’t always the goal.

    Reply
    1. elkement Avatar
      elkement

      You are probably the most quallified person I know to comment on UFOs and aviation – so thanks a lot for the comment, Andrea! Admittedly, I did not really think about the real-life challenges of actually moving an UFO through space … it was more of a silly delight to see things in this rather “Platonic” shapes :)

      Reply

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