Control of Suppression of Radial Vibrations of a Two-mass System with its Simultaneous Spinning-up

Abstract

The object of research in this work is a two-mass controlled mechanical system consisting of a carrier disk rotating about its axis, fixed in space, and a carried ring connected to the disk by means of weightless elastic elements. There are no dampers in the system. The process of suppression of radial oscillations is considered from the perspective of the theory of optimal control. On sufficiently large time intervals, Newton’s numerical method is used to solve the boundary value problem of the Pontryagin’s maximum principle. The properties of phase trajectories of the system are studied depending on the initial states of the disk and ring and the number of springs in a complex model of elastic interaction. It is shown how, under certain initial conditions and parameters of the system, due to the radiality of the elastic force and the law of conservation of angular momentum, the trajectory of the center of mass of the ring tends to a circle. The specified tendency to enter the circular motion mode is not uniform and depends on the number of springs. It is shown that with a small number of elastic elements, the trajectory of the ring does not take the form of a circle, but almost complete damping of radial vibrations occurs. It has been established that with the parameters of the system considered during the numerical experiment, the control is relay with a fairly large number of switchings. In this case, the entire system is simultaneously spinning-up.