Abstract:
A spherical rigid body rolling without sliding on a horizontal support is considered.
The body is axially symmetric but unbalanced (tippe top). The support performs highfrequency
oscillations with small amplitude. To implement the standard averaging procedure,
we present equations of motion in quasi-coordinates in Hamiltonian form with additional terms
of nonholonomicity [16] and introduce a new fast time variable. The averaged system is similar
to the initial one with an additional term, known as vibrational potential [8, 9, 18]. This
term depends on the single variable — the nutation angle $\theta$, and according to the work
of Chaplygin [5], the averaged system is integrable. Some examples exhibit the influence of
vibrations on the dynamics.
References