Keywords: Intermediate axis theorem, Tennis racket theorem, Dzhanibekov effect, Coriolis force, Euler equations
In 1988 I witnesses the instability of the rotation about the intermediate axis of a foam brick.
Since then I have been fascinated by this effect. It was one of the many experiments which enriched a lecture series on kinetics and on that day Euler equations were on the agenda. Colored surfaces of the brick made it possible to observe the effect without micro gravity and slow-motion equipment.
This post is about reproducing an “intuitive” visualization of an explanation of the effect by Terry Tao from 2011 using 4 rigidly connected point masses. 8 years later the explanation was animated in a YouTube video (The Bizarre Behavior of Rotating Bodies) and considered to be the “best intuitive” explanation.
Motivated by the video, I wondered whether a similar animation with acting forces is possible with MapleSoft products and whether there might be a better intuitive explanation without the use of centrifugal forces. Initially I saw this more as a good test of MapleSim’s visualization capabilities. Finally, it took over 3 years and numerous attempts (mostly during vacation, kind of a substitute for drawing circles in the sand...) to come to a conclusion on the effect.
Intermediate_axis_theoreme_with_3_point_masses.msim
About the model:
Unlike the YouTube video, I decided to simulate 3 identical point masses because a 3-mass model fits better to a T-handle (overlayed in the animation above), video footage from space experiments and discussions in this forum (221298, 225760, 228066).
The movement of the model generates acceleration forces on each mass. The clip displays the corresponding opposing forces that act in the model (i.e. act on the massless T-structure). The blue mass, which is not perfectly centered on the axis of rotation at the start of the simulation perturbs the orbits of the red and the green masses. That was my initial intuitive attempt to explain the effect.
The 3 masses form an isosceles triangle. Here it is helpful to think of a rotating arrowhead where the shape determines stability of the rotation. The aspect ratio (the ratio of the height to the base length) of the triangle determines the stability of rotation about the mirror symmetry axis of the triangle (i.e. the symmetry axis of the T-structure). An obtuse triangle (“blunt”, aspect ratio < sqrt(3)/2) is unstable when rotating about an axis that is slightly inclined with respect to this axis of symmetry. The inclination can be in the plane of the triangle or out of plane. An acute (“pointy”) triangle only wobbles.
About the MapleSim model:
A supplementary rigid body component without mass and rotational inertia is used at the center of mass of the three masses to impose initial conditions. Rotating the triangle at the start of the simulation about the center of mass of the 3 masses prevents the triangle from drifting laterally away from its initial position. This effect of lateral drift is visible in video footage from space with the T-handle.
The rotational inertia of the other rigid body components is set to zero. Without rotational inertia it could be assumed that only Newtonian mechanics are used in the simulation (i.e. no Euler equations are integrated). This is however wrong. MapleSim generates automatically from a system with 3x6=18 coordinates a system with 3 Newtonian equations for translation and 3 Euler equations for rotation.
Forces and moments are measured with sensor components. Visualization is done with force and moment visualization components. These components are “abused” to display the following other physical quantities:
The angular momentum of the masses
The vectors of the angular velocity and the angular acceleration
Moments of the forces with respect to the center of mass
Moments of the forces with respect to the center of the base of the triangle
For a clean model, sensor components and mathematical components to calculate physical quantities are grouped in three subsystems (one per mass, indicated with a colored dot in the image below).
The model contains parameter sets for in plane and out of plane inclination of the axis of the T with respect to the initial axis of rotation (the x-axis).
