Understanding the Dzhanibekov Effect

  • The Dzhanibekov Effect is also known as the Tennis Racquet Theorem because it is perfectly illustrated using a tennis racquet. Try holding the racquet face up and spin it - you will see the Dzhanibekov Effect before your eyes.
  • The axes with greatest and smallest moments of inertia will spin perfectly, the intermediate axis will spin wildly.
  • Like so many elements of mechanical engineering numerous examples of the Dzhanibekov Effect/Tennis Racquet Theorem are happening before our eyes every single day of our lives - but we take them for granted.
Dzhanibekov Effect

The Dzhanibekov Effect is also commonly known as the “Tennis Racquet Theorem”

The Dzhanibekov Effect phenomenon, also known as the Tennis Racquet Theorem, is a result in standard mechanics which describes the movement of a rigid object with the help of three distinctive key moments of inertia. It is correspondingly dubbed as the Dzhanibekov Effect, after the Russian astronaut named Vladimir Dzhanibekov who discovered the consequences of the theorem when he was in space in 1985. It is also know as the Tennis Racquet Theorem because it is perfectly demonstrated when we toss a tennis racket in the air.

Three main axes of rotation

A rigid object has three main axes of rotation – in effect 3 different ways in which you spin/rotate the object.  The object will be stable only near axes one and axes three but will be unstable about the axis number two. Axis two is also known as the intermediate axis.  The odd-looking effect takes place when the body is rotated about an axis which is near the intermediate axis. For a rigid body which is spinning, moments of inertia around the space-frame axes are continually fluctuating as time changes and the body rotates. To simplify the description of this motion, you can also choose a co-ordinate system instead which rotates alongside the body. The Euler’s Equations of Motion which results in moments of inertia which are time-independent.

A highly interesting use of Euler’s Equations can be seen in the derivation of the astonishing Dzhanibekov Effect. This effect or theorem applies to all cases where moments of inertia regarding principal axes are spaced: I1 << I2 << I3. The theorem states clearly that rotations around axes l and 3 are more stable than they are around axis 2. Despite the fact that I2 might be very close to I3 in value. The theorem or effect can easily be demonstrated by using a tennis racket, hence the name.

See the Dzhanibekov Effect for Yourself

To better understand the phenomenon it is better that people see it for themselves. For this, you will need a body which has a constant density and is in the shape of a rectangular prism, or a box. Almost everybody has an item which is similar to this type of a body, for instance, a book!

It is also characteristically dimensioned so that all the three pairs of the body sides are rectangles. This makes the values of the moments of inertia all distinctive and easy to determine.  You need to make the body rigid, for which you can make use of rubber bands. This will help the body to spin around without the pages getting turned. The order of the discovered values of the moment of inertia for any axis will correspond to the length of the crosswise for the face of the book in which the principal axis is perpendicular.  Consequently, for the book with the biggest dimension of height, middle width and smallest thickness, the axis which has the largest value of the moment of inertia is the axis, when if you look at the cover and spin it the cover always faces you.

It can definitely be seen that through spinning the book along with all three of these axes, the spins for the largest and the smallest are steady, whilst the ones for the middle is unstable, with the spinning quickly turning into a mess.

YouTube Video Link

To have a better understanding regarding what the Dzhanibekov Effect is all about, you can also have a look at the following YouTube video. The video explains with the help of graphics and images in great detail, elaborating how the principle works. For all those who have trouble understanding the formula mathematically this link could be of great help:

Link: https://en.wikipedia.org/wiki/Tennis_racket_theorem
Link: http://www.tart-aria.info/sdvig-polyusov-chast-1-fizika-protsessa/

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