On a Theory of Swings

Abstract

The dynamics of a pendulum with controllable length, or swings, is considered. Possible variations of length are of bounded magnitude and arbitrary otherwise. A particular law of feedback-controlled stepwise variations, considered by Magnus to illustrate the basic swings effect, is shown to be the optimal one, as long as the goal is to maximize the growth rate of the energy of oscillations. Similarly, these temporal variations of length with the opposite sign, i.e. the “reversed swings” law, provide the maximal decay rate of the oscillation energy. The efficiency of the latter control law is predicted analytically for the case of a white-noise random external excitation of the pendulum, by calculating explicitly the expected steady-state response energy through direct application of Stochastic Differential Equations Calculus. For the case of a small bound R on relative variations of the length, the reversed swings effect is shown to be asymptotically equivalent to a linear viscous damping with damping ratio 3 R/π. Potential extensions of the considered control law are indicated, that is, of the “generalized reversed swings law”.