# Has anyone tested the Dzhanibekov Effect to see if the frequency of the 180 degree flip changes based on the objects speed or mass?

Rotation around axis 2, the intermediate axis of an object that is not perfectly symmetrical will flip 180 degrees while continuing to spin in the same direction. I'm curious if there is a way to calculate the frequency of the flip based on the speed of the spin and/or the mass of what is spinning. The article below is why I am asking. I wonder if this has been studied and if so, if there may be a way to calculate the time between core reversals based on the speed of earths rotation and the mass of the core.

Stated in the article: "With a radius of almost 2,200 miles, Earth’s core is about the size of Mars. It consists mostly of iron and nickel, and contains about about one-third of Earth’s mass." What the article doesn't clarify is that they are discussing the inner core.

If the inner core has stopped spinning or just slowed to match the speed of earth, as suggested in the article that would be about 1670 Kilometers at the equator.

If there is a way to calculate this, it could help to confirm either a 70-80 year inner core reversal or a 20-30 year inner core reversal.

The magnetic reversal of Earth will happen when the molecules of liquid iron in the outer core of the Earth start moving in the opposite direction to the solid iron molecules of the inner core. Studies show the polar shift occurring once every 450,000 years or so, with the last one being about 780,000 ago. If we can calculate and confirm via multiple disciplines the movement of the inner core, perhaps we could then apply those calculations to the outer core, giving us another indicator of when a polar shift may occur.

https://www.cnn.com/2023/01/25/world/earth-core-turning-scli-scn-intl/index.html

I'm very sorry if this is a stupid question, but I could not find answers through search engines.

• Recommended resources for the Dzhanibekov effect: - 2020 Youtube video by David Brown The Dzhanibekov effect, equations and simulations - 2018 Article by Nicholas Mecholsky Analytic formula for the geometric phase of an Asymmetric top Commented Aug 27, 2023 at 4:38
• seems to me the answer is yes, here is a demonstration for the classroom researchgate.net/publication/… Commented Aug 27, 2023 at 4:48
• Related: physics.stackexchange.com/q/17504/2451 and links therein. Commented Aug 27, 2023 at 6:50
• Is this related to the Tippe Top ? Commented Aug 27, 2023 at 8:09
• @PM2Ring The tippe top is not an instance of the intermediate axis theorem. In the case of the tippe top: out of the three moments of inertia two are very close to each otther. In order to obtain a vivid Dzhabenikov effect the three moments of inetia must be al lthree signiticantly different. With two moments of inertia very close: any Dzhabenikov effect will be very slow. Commented Aug 27, 2023 at 8:33

I need to address some misunderstandings here.

Demonstrations of the Dzhanibekov effect are designed to be vivid, so the spinning object is chosen/designed for maximal effect.

If the moments of inertia are much closer together then the effect is still there, but much slower.

The strongest example of that is the motion of the gyroscopes of the Gravity Probe B experiment.

Those spheres were exceedingly close to perfect spheres. The designers of the experimental setup knew in advance that the axis of rotation of the spheres would move around. This shifting of the rotation axis is referred to as polhode motion, it is perfectly predictable.

The spheres of the Gravity Probe B experiment were so close to perfectly spherical that the period of the polhode motion was in the order of months. In the idealized case of perfectly frictionless motion the period of the polhode motion is perfectly periodic. That is to say: the polhode motion is not a sign of some form of instabililty. Think of the polhode motion as a form of oscillation; an oscillation with a predictable period.

The amplitude of the polhode motion depends on the random initial alignment at the time of spinning up the gyroscope.

The most fortunate case is that the initial spin axis is aligned with the axis of largest moment of inertia. Then there will be no polhode motion.

The most unlucky initial condition is that at initial spin up the spin axis happens to be close to the axis of intermediate moment of inertia. That results in the largest possible amplitude of the polhode motion.

In the case of the Gravity Probe B experiment: it turned out there was an energy dissipation mode that had not been anticipated. That energy dissipation mode had a number of effects: it caused the gyroscopes to slow down faster than anticipated, and it drained the kinetic energy correlated with the polhode motion, so over time the amplitude of the polhode motion was shrinking.

If the experiment could have run longer then eventually all four gyros would have settled on rotation around the axis of largest moment of inertia.

Summerizing:
The Dzhanibekov effect only has a fast frequency when the object used for demonstrating it is chosen/designed to display a dramatic effect.
In physically relevant cases the three moments of inertia are closly matching (but not perfectly matching) and then the frequency of the polhode motion is very slow.

I recommend particularly: study the video by David Brown that I had already linked to in a comment: Dzhanibekov effect, with equations and video footage of the simulation runs he created.

• I'm just a curious layman, so most of this goes into things way over my head. However, if I'm understanding, it seems the Berry Phase or Euler equations are able to predict the flip? If that is the case, where can I read about those predictions? Thank you, again. Commented Aug 28, 2023 at 0:11