Dynamics of the Tippe Top on a Vibrating Base

This paper studies the conditions under which the tippe top inverts in the presence of vibrations of the base along the vertical. A vibrational potential is constructed by averaging and it is shown that, when this potential is added to the system, the Jellett integral is preserved. This makes it possible to apply the modified Routh method and to find the effective potential to whose critical points permanent rotations or regular precessions of the tippe top correspond. Tippe top inversion is possible for a sufficiently large initial angular velocity under the condition that spinning with the lowest position of the center of gravity is unstable, spinning with the highest position of the center of gravity is stable, and that there are no precessions. Cases are found in which there is no inversion in the absence of vibrations, but it can be brought about by a suitable choice of the mean value of the squared velocity of the base. In particular, this type includes a ball with a spherical cavity filled with a denser substance.