Stability investigation of the steady motions of an isolated system, carrying out plane motion

Abstract

The paper investigates the conditional stability of steady motions of a flat model of an isolated system consisting of a rotating LB, material point, which creates its static imbalance, and two identical mathematical pendulums, mounted on the longitudinal axis of the LB and moving in the plane of the static imbalance, the relative motion of which is prevented by the viscous resistance. It was found that in the case where there is imbalance and pendulums can eliminate it with a certain reserve, there is one basic motion; in the absence of imbalance, there is a one-parameter family of basic motions; in the case of maximum imbalance, which can be eliminated by pendulums, there is one basic motion, but it generates pseudo-family of basic motions. Also, it was found that some basic motions are conditionally asymptotically stable, if they, or family, or pseudo-family of basic motions are isolated. In the absence of imbalance, the presence of a single zero root of the characteristic equation does not affect the stability of the one-parameter family of basic motions, and is responsible for the transition from one to another steady motion of the family. In the case of maximum imbalance, the presence of a single zero root of the characteristic equation does not affect the stability of the basic motion, and is responsible for the transition from one to another steady motion of pseudo-family. Transients, depending on the system parameters can be aperiodic or oscillatory-damped. It was found that the side motions are unstable.