Optimal Rocking and Damping of a Swing

Abstract

The problem of maximising or minimising the inclination of a swing to the vertical at its highest point is considered. The swing control problem is of interest from the viewpoint of theoretical mechanics, control theory, oscillations theory. It is closely related to the problem of using extensible rods to damp the oscillations of a satellite around its centre of mass in gravitational field, and with control problems for some sporting motions. The system of a swing and a person on it is modelled by a pendulum and a material point. This point can be shifted within some limits along the straight line, passing through the pivot and the mass centre of the pendulum. The viscous damping because of air resistance opposing the motion of the swing, and also in the pivot are taken into account together with dry friction in the pivot.