The attraction basin in the generalized Kapitsa’s problem

Abstract

The attraction basin of a stable vertical position of rod under vertical vibration of the support in the Kapitsa’s problem and its generalizations is studied. A long enough flexible rod with a free upper end and a clumped lower end under its weight is shown to be unstable. The support is assumed to be subjected to harmonic vibrations. In the resent works it is established that under some level of vibrations the vertical position becomes stable. Here the attraction basin of this position is discussed. As a first step the attraction basin in the classic Kapitsa’s problem is found. Then a rigid rod with the elastic support of a lower end is studied. The last problem is a model of a flexible rod with the clumped end. The asymptotic method of two-scaled expansions is used. It is established that a transition in the vertical position essentially depends on the initial phase of vibrations. As a result it occurs that the attraction basin consists of two parts. In one of them this transition does not depend on the initial phase, and in the other part a dependence on the initial phase has place.