Experiments and Simulation of a Gyroscope with Friction
The equations of motion for the heavy top (a gyroscope anchored at a point along its central axis acting under the influence of gravity) are well known, but how to model the effects of friction is not simple.
We attached a cardboard disk around the equator of the Super Precision Gyroscope purchased from www.gyroscope.com. We filmed the gyrosope and used image-proessing software to ttrack the edge of the disk and used this to determine the orientation of the gyroscope as a function of time.
We then simulated the equations of motion of the gyroscope with ad-hoc damping terms and got the following picture that qualitatively matches our experiment:
Both
graphs show the angle theta between the rotation axis and the vertical.
Theta increases as the gyroscope falls. (The y-axis in the bottom
figure has accidentally been shifted by pi/2.)