Nutation 

Nutation is a steady state motion of the bodies k axis about the K axis with constant angle between them, it is observed as a fast wobbling motion.

By substituting Euler's angles into the second of the gyroscope equations the following expression is obtained.

The solution to this quadratic is given by 


For fast spin  and so 
One solution to the above equation is



This motion is known as precession. Precession occurs in the presence of an applied couple and occurs at a slow rate in comparison to the rate of spin.

Another root of the equation for fast spin is given by

 
This motion occurs at a constant rate even when the applied couple is zero. This motion is called nutation.

Video of nutation

When a gyroscope is released from rest as in the above video cupsidal nutation is observed. The motion is depicted in the figure below

 

During this motion moment of momentum is conserved about the vertical axis (assuming frictionless bearings) .The component of moment of momentum about the vertical axis is zero to start with and so remains zero during the motion. During the motion the loss in potential energy is balanced by a gain in potential energy.

Nutation can be seen to be a damped motion and after a while the observed oscillations die away. The angle of steady precession is equal to  half