Attraction Basins and Longitudinal Resonances in the Generalized Kapitsa Problem for the Inflexible Longitudinally Deformable Rod

The effect of stability of a pendulum in the upper equilibrium position under the influence of vertical vibration of its base was first described by A. Stephenson and studied in more detail by P.L. Kapitsa. The pendulum rod in these experiments is under the influence of a high-intensity longitudinal load, and for a sufficiently soft material of the pendulum, resonant oscillation modes are possible. For a model of a pendulum with one internal degree of freedom in the form of a point mass on a spring, the influence of resonance was previously studied. In this paper, longitudinal vibrations of a pendulum are considered without taking into account bending deformations and a continuum model of a compressible pendulum rod is taken, which has a countable number of longitudinal resonances. The influence of tension-compression waves of a deformable pendulum rod on its stability was found. Using the asymptotic method of two-scale expansions, taking into account the small amplitude of oscillations of the base of the pendulum, an averaged equation was obtained that describes the movement of the pendulum. In this case, longitudinal deformations are considered linear and small, and the angle of deviation of the pendulum axis from the vertical is considered finite. An explicit formula is obtained for the critical value of the pendulum’s deflection angle, which gives stable oscillations in its upper position. Stable and unstable regions are observed near each resonance. As the resonance number increases, the size of the corresponding stable region decreases. By adding a non-deformable rod component with a mass of at least one third of the mass of the deformable component to the pendulum model, it is possible to increase the size of stable regions near higher resonances.