# Is it possible to reproduce tennis racket theorem instability with a gymbal under earth gravitational field?

The tennis racket theorem (whose effect is also known as Dzhanibekov effect), states that the rotational motion around the intermediate principal axis is not stable (that is, any small disturbance will cause the object to cyclically flip). Such effect is usually demonstrated in absence of gravity or with objects in free fall filmed in slow motion.

I wonder if such effect can also be reproduced using a gymbal (aka cardanic suspension) hosting a body in a way such that it can be kickstarted easily to rotate not on its principal axis but on the secondary one. For instance an ellipsoid mounted on the inner ring so that the axle traverses its secondary axis.

I.e. does the gymbal work as the body was hold fixed in space by its center of gravity? (assuming the gyroscope is symmetric enough so that its center of mass coincides with the center of the ring) Or do other reaction forces come into play that prevent the tennis racket effect from being observable? I assume to perform the experiment under a gravitational field.

• The principle of a gimbal mounting is that rather than supporting a particular axis the gimbal mounting is in effect supporting a single point. The idea is to support the center of mass of the gyro wheel, so that the gyro wheel can change orientation as freely as if the mechanical suspension isn't there. But you are referring to a scenario such that the orientation is constrained. As the question stands it is unclear what you have in mind when you describe the motion as 'constrained to rotate not on its principle axis'. Commented Jul 7 at 4:50
• Also, the dynamic property described by the intermediate axis theorem is a property of motion of objects with three different moments of inertia along each of the three principle axes. You are referring to an 'ellipsoid of revolution'. That would be an object with an axial symmetry, which is an object with two different moments of inertia, instead of three different moments of inertia. Commented Jul 7 at 4:57
• Incidentally, the motion of a spinning object with three different moments of inertia is cyclic, and the cycle repeats with a fixed period. That is, the motion of a spinning object with three different moments of inertia is deterministic. The expression 'not stable' is usually meant for a context where motion is to a high degree unpredictable, such as in the case of turbulence. Commented Jul 7 at 5:04
• Recommended resource: a set of two youtube videos by David Brown (channel: Physics unsimplified) Rigid body motion and the Dzhanibekov effect David Brown has proceeded as follows. He used Lagrange multipliers to implement a simulation. He ran the simulation with various different parameters, such as ratio of moment of inertia. The resulting animations show the motion pattern in such a detailed way that many subtleties are visually clear. That resource takes exposition of the Dzhanibekov effect to a whole new level. Commented Jul 7 at 11:19
• Among the most interesting instances of Dhzanibekov effect: polhode motion of the gyro spheres of the gravity probe B experiment. Those gyros were the roundest objects ever manufactured. Their polhode motion, which was anticipated, has a period of months. (As it turned out there was a mode of kinetic energy dissipation that was not anticipated. So they had to model that dissipation to arcseconds per year precision.) Commented Jul 7 at 11:29

A gimbal suspension for the purpose of demonstrating the Dzhanibekov effect (when weightlessness is not available) presents special challenges.

In the simulation that he created David shows both the angular momentum vector, and the angular velocity vector. The angular momentum vector remains pointing in the same direction, as there is no external torque. But due to the presence of three different moments of inertia we have that continuously momentum is internally relocated. The angular velocity vector is defined relative to the outside shape; so the angular velocity vector is moving around.

The inner ring of a gimbal mounting connects to the outside shape of the object. That means that in order for the object to be free to display the Dzhanibekov effect the gimbal rings must move in accordance with the angular velocity vector. That's a tall order.

Historically the most demanding case of gimbal suspension was for the gyroscopes for the Apollo space program:
The Apollo Command Module and the Lunar Module both had a Primary Guidance, Navigation, and Control System

As we know: the whole design was under severe weight constraint; every gram had to count.

As you can see on the picture in the wikipedia article: the mechanical gyroscope system was massive.

Here is the engineering problem they had to overcome:
Under any change of orientation of the spacecraft the gyro wheel inside must remain free of external torque.

The gimbal rings have their own inertia. If the spaceship changes orientation then in order to allow the gyro wheel to stay in the same orientation the gimbal rings must move accordingly. There are ranges of angle where two gimbal rings must move in coordination. There are ranges of angle where one gimbal ring must actually move faster than the change of orientation of the spacecraft.

Low friction bearings go a long way, but because of the inertia of the gimbal rings there will still be transient torque on the gyro wheel: not acceptable.

To keep change of orientation of the gyro wheel to within specification the gimbal rings were equiped with sensors and electromotors. When the sensors sense the that the gimbal ring needs to move the electromotors move the gimbal ring at the rate that it needs to move so as to allow the gyro wheel to keep the same orientation.

The astronauts still had to perform scheduled re-calibration of the gyro wheel orientation (using a distant star as reference). The objective was to bring down the number of times that re-calibration had to be performed.

Returning to the problem of creating a gimbal suspension for a Dzhanibekov demonstration:

The gimbal rings need to be very nimble, otherwise they will affect the orientation of the angular momentum vector, thus compromising the clarity of the demonstration

So:
I expect that building a gimbal suspension for a Dzhanibekov effect demonstration will be very challenging.

• ok thanks so the short answer is "yes, in theory", but probably a real gymbal would not be able to let the object flip while rotating? Also, I suspect that, in spite of the stable case, the gymbal would suffer for more internal stresses, right? Commented Jul 8 at 12:07

I'm posting a new answer for the following reason: a totally different way of implementing a Dzhanibekov effect demonstration occurred to me.

Manufacture an object that is spherical on the outside, but internally consisting of different materials, in such a way that the object has three different moments of inertia.

For instance, an internal object can be made out of a high density metal, padded out with a transparent plastic to a spherical shape.

Create a bowl, spherical on the inside, matching the spherical object, so that the sphere fits in with just a tiny gap. With the bowl: use a distribution of small holes to flow air into the gap so that the sphere is suspended on an air cushion.

The heavier the object, the closer to the idealized case the demonstration will be. The friction will correlate with area, but the inertial mass increases with the volume of the object.

So that is my suggestion for achieving a form of suspension that - without weightlessness - will allow demonstration of the Dzanibekov effect

• Me likes. I hope somebody with the right tooling will build that. Commented Jul 11 at 0:44