The inverse problem of stabilization of a spherical pendulum in a given position under oblique vibration

Abstract

The inverse problem is posed of stabilizing a spherical pendulum (a mass point at the end of a weightless solid rod of length l ) in a given position using high-frequency vibration of the suspension point. The position of the pendulum is determined by the angle between the pendulum rod and the gravity acceleration vector. For any given position of the pendulum, a series of oblique vibration parameters (amplitude of the vibration velocity and the angle between the vibration velocity vector and the vertical) were found that stabilize the pendulum in this position. From the obtained series of solutions, the parameters of optimal vibration (vibration with a minimum amplitude of velocity) are selected depending on the position of the pendulum. The region of initial conditions is studied, of which the optimal vibration leads the pendulum to a predetermined stable position after a sufficiently long time. This area, following N. F.Morozov et al., called the area of attraction.