This animation was done in Blender.
The equations of motion given in [1] were modified and numerically integrated to generate the top’s movement. A ‘wet’ viscous friction model was used. The modifications to the equations of motion were:
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A spin decay term ( -kS ) was added to the spin time derivative equation. This term accounts for spinning friction at the contact point and aerodynamic drag. These two effects were not modeled in [1].
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As the top starts to lean over the angle the top’s symmetry axis makes with vertical (theta) begins to increase. At a critical value of theta the rim of the top’s disk makes contact with the horizontal surface. Assuming some compliance between the top and the horizontal surface there is a small region of theta were the top has two contact points. To simplify the dynamics a single virtual contact point interpolated between the two actual contact points was used. When theta is outside the compliance region the top has a contact point either at the spherical tip or the rim based on if theta is less than or greater than the compliance region respectively.
[1] Moffatt, H. K., Shimomura, Y. & Branicki, M. 2004, “Dynamics of an axismmetric
body spinning on a horizontal surface. I. Stability and the gyroscopic
approximation”, Proc. R. Soc. A. 460, 3643-3672.