Stability of Vertical Rotations of an Axisymmetric Ellipsoid on a Vibrating Plane

Abstract

In this paper, we address the problem of an ellipsoid with axisymmetric mass distribution rolling on a horizontal absolutely rough plane under the assumption that the supporting plane performs periodic vertical oscillations. In the general case, the problem reduces to a system with one and a half degrees of freedom. In this paper, instead of considering exact equations, we use a vibrational potential that describes approximately the dynamics of a rigid body on a vibrating plane. Since the vibrational potential is invariant under rotation about the vertical, the resulting problem with the additional potential is integrable. For this problem, we analyze the influence of vibrations on the linear stability of vertical rotations of the ellipsoid.

Numerical Analysis of a Drop-Shaped Aquatic Robot

“…The study of finite-dimensional models is also of interest for the theory of dynamical systems. Despite their simplicity, such models can exhibite various dynamical effects: asymptotic stability with respect to part of variables [24,25], Fermi acceleration under the periodic controls [26][27][28][29], stabilization or loss of stability due to periodic disturbances [30], chaotic scattering of trajectories [23,31], the transition to chaos according to Feigenbaum's scenario [32,33] and through a series of doublings of the invariant tori [34][35][36], etc.…”