On the Stability of Pendulum-type Motions in the Approximate Problem of Dynamics of a Lagrange Top with a Vibrating Suspension Point

Abstract

This paper addresses the motion of a Lagrange top in a homogeneous gravitational field under the assumption that the suspension point of the top undergoes high-frequency vibrations with small amplitude in three-dimensional space. The laws of motion of the suspension point are supposed to allow vertical relative equilibria of the top's symmetry axis. Within the framework of an approximate autonomous system of differential equations of motion written in canonical Hamiltonian form, pendulum-type motions of the top are considered. For these motions, its symmetry axis performs oscillations of pendulum type near the lower, upper or inclined relative equilibrium positions, rotations or asymptotic motions. Integration of the equation of pendulum motion of the top is carried out in the whole range of the problem parameters. The question of their orbital linear stability with respect to spatial perturbations is considered on the isoenergetic level corresponding to the unperturbed motions. The stability evolution of oscillations and rotations of the Lagrange top depending on the ratios between the intensities of the vertical, horizontal longitudinal and horizontal transverse components of vibration is described.

On the Dynamics of a Rigid Body in the Hess Case at High-Frequency Vibrations of a Suspension Point

“…When θ is replaced by π − θ, this Hamiltonian coincides with the Hamiltonian function describing the pendulum-type motions of the Lagrange top with a vibrating suspension point, considered in [27]. We briefly describe the motion of this model system.…”

“…This analogy is known for bodies in the Lagrange and Hess cases with fixed suspension points. The problem of the motion of the Lagrange top with a vibrating suspension was investigated in various formulations in [22][23][24][25][26][27]. Taking into account the analogy of two problems, some results obtained earlier in the dynamics of the top are used in the Hess case study.…”

The Dynamics of a Spherical Robot of Combined Type by Periodic Control Actions

“…A large number of works are devoted to pendulum systems with a vibrating point of suspension [4][5][6][7][8] in which the dynamics of systems and the conditions for their stability are investigated. The results obtained in these studies can be used to create more complex systems, including mobile robots.…”