The use of control to eliminate subharmonic and chaotic impacting motions of a driven beam

Abstract

Results of a numerical investigation are presented of the behaviour of a mathematical model which approximates the response of a thin metal strip, or beam, held vertically and clamped at its base, which is driven to impact against a motion limiting constraint. Previous theoretical analyses, in conjunction with numerical studies, have highlighted the complicated dynamics that the system exhibits following a single or series of grazing bifurcations. Under certain conditions subharmonic solutions and chaotic impacting motions are unavoidable. These solutions, which involve several impacts per orbital period introducing relatively large impact velocities, have been viewed experimentally. Meanwhile research into the control of chaos, in which unstable orbits embedded within chaotic motions are stabilized, has advanced over recent years. New control methods have been developed and experimentally implemented. The present work brings these two fields together for the first time. Numerical evidence is given for the use of control to eliminate complicated motions by tracking lower periodic solutions. The stabilized motions near the grazing incidence have lower impact velocities than the naturally existing stable solutions leading to a significant reduction of the impact force which may have significant engineering relevance. © 1997 Academic Press Limited.