Use of control to maintain period-1 motions during wind-up or wind-down operations of an impacting driven beam

We consider the dynamical response of a thin beam held fixed at one end while excited by an external driving force. A motion limiting constraint, or stop, causes the beam to impact. During wind-up or wind-down operations, in which the driving frequency is continuously altered, the system can undergo complicated motions close to the value of frequency at which impacts may first occur, the grazing bifurcation. In this region, the beam may experience several impacts within a long period-repeating solution or even chaotic behavior which, in practical terms, may be undesirable to the long-term integrity of the system. The first task is to identify the zones in the space of parameters (forcing amplitude or, alternatively, the gap between the beam and the stop) in which period-1 motions can be guaranteed. In this paper, in the areas in which complicated or chaotic motion occurs, a control strategy is proposed which stabilises unstable period-1 motions. As a consequence, numerical simulations indicate that, for any choice of parameter in the range, simple period-1 motions can be maintained, limiting the number of impacts (together with their velocity).