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Dzhanibekov Effect or tennis racket theorem

Posted 4 years ago
20 Replies
18 Total Likes

The tennis racket theorem is a result in classical mechanics describing movement of a rigid body with three distinct principal moments of inertia. It is also dubbed the Dzhanibekov effect, after Russian astronaut Vladimir Dzhanibekov who discovered the theorem's consequences while in space in 1985.

The code with NDSolveValue is attached.

enter image description here

20 Replies

This is great, thanks for sharing! This can make a nice Demonstration. I am curious if this is also easy in SystemModeler.

I found on this site many dynamic simulation of physics.One of them is

  1. Rotation stability of 3D cube. Cube is stable when spinning around either the major or the minor principal axes. This demo illustrates this by solving Euler equations of motion in 3D for zero torque. Select the spin axes, then click on perturbe to see the effect.

enter image description here

Posted 4 years ago

Hello, Mariusz, I am working on a Wolfram Demonstrations based on your idea and the suggestion from Sam Carrettie. I am using a ping pong paddle to demonstrate the "Intermediate Axis Theorem" of which the Dzhanibekov Effect is another example. I am still working out the details but attached is a *.mov file to show the output. Thanks for you post.



Cool but,you did little mistake, you must rotate the Racket about 90 degrees with respect to the axis X.On this video see effect.

enter image description here

See atchment file they will help you.

Posted 4 years ago

I am aware of this video and indeed, the axis parallel to the paddle blade could also be an intermediate axis for some geometries or materials used in the paddle. In my example, I consider the paddle to be made of homogenous material of density 1 and with arbitrary dimensions that may not be realistic compared to real world ping pong equipment. In my example, the axes are crossing at the centroid: the intermediate axis is the phi axis perpendicular to the blade; in the video example it is more than likely the psi axis, parallel to the blade? Both are possible depending on the location of the centroid ( geometry, density distribution, etc...) You can see that the the two largest moments are very close compared to the minimum moment of inertia.enter image description here

I solved the problem. Animation is a solution of 6 equations.

enter image description here

Posted 4 years ago

Hello Mariusz I looked at your code and at the moments of inertia in your example... It is because in your specific case, the intermediate axis (due to the geometry and position of the centroid) the intermediate axis is the one parallel with the blade (the psi axis). And it does not give such a nice and smooth ride?? enter image description here

The angular speed plot on the left is from the paddle in my previous reply, the plot to the right is from your paddle You can see that in my case, the unstable rotation is around the theta axis (yellow curve), whereas in your case, the unstable rotation is around the psi axis (green curve) BTW, it is not recommended to use subscripted variables in Mathematica code. It sometimes gives problems. Thanks for yr interest in this case!

Hi Mariusz,

In your code you assume Ix=1, Iy=5, Iz=15

Assuming the shape as in your file, assuming density constant =1, we have

Ix = Int(y^2+z^2) = y2+z2

Iy = Int(x^2+z^2)=x2+z2

Iz = Int (x^2+y^2)=x2+y2

Where the integration is over the body, and I denoted x2,y2,z2 the corresponding integrals. From your conditions we should have



Eliminating x2 we get 9y2+11z2=0, which is clearly impossible since y2>0,z2>0. Can you make a realistic model?

Yes, a little messed up moments of inertia, but corrected the "code". I also did simulations in Blender 3D. The results are similar, so as correct. :)

Ps: Change the file extension "Tennis Racet.nb" to "Tennis Racet.blend" . To run the simulation in Blender 3D, press on the keyboard "Alt + A".

In the original model ihave changed the values Ix,Iy,Iz to realistic

  Ix = 5.666666666666667`;
   Iy = 33.666666666666664`;
   Iz = 38.666666666666664`;

Then there is a strange phenomenon - After restart colors Green and Blue change a while before the flip, like in precognition. I do not understand what is the reason? I am not sure if this effect is reproducible.

This strange phenomenon is associated only with animation,because is looped .

enter image description here


I have rewritten your program. Fixed the sign in Euler's equations, fixed the rendering of Euler angles, changed the geometry - we have three pairs of masses now, make it interactive, added drawing of the change in orientation. Probably it still have bugs.

Perhaps we can improve it, I think it can be a made into a nice Wolfram's demonstration.

enter image description here


This is really nice, @Arkadiusz Jadczyk! One demonstration already came out of this:

The Intermediate Axis Theorem Applied to a Ping-Pong Paddle Flip-Over

but I think your should try sending this to Demonstrations team to see if they consider it different enough to publish.

This demonstration now mentions this discussion:

Based on work by Mariusz Iwaniuk found at Dzhanibekov Effect or tennis racket theorem

Privet Vitaliy, I am going to change my demonstration using explicit solution formula with Jacobi elliptic functions nad quaternions. That may be new.

Very interesting. Did you find those with Mathematica or have some source like an article?

One good article is by Cushman, "No polar coordinates"[1]

Very cool. I was trying to recall the name of that effect for some time now. Here's another nice video of it in space:

enter image description here - another post of yours has been selected for the Staff Picks group, congratulations !

We are happy to see you at the top of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming!

Finally I have a tentative version of the demonstration based on explicit solution

Dzhanibekov Effect [1]

Excerpt on youtube:

Would appreciate comments and suggestions how to improve.

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