Attraction Basin in the Generalized Kapitsa Problem with a Finite or Countable Number of Internal Degrees of Freedom

The problem of pendulum oscillation in a gravitational field around the upper equilibrium position under the influence of vertical vibrations of the base with a finite angle of deviation from the vertical is the classical Kapitsa problem, which is solved for a generalized model of a compressible pendulum. The effect of taking into account the compressibility of the generalized Kapitsa pendulum on the stability and attraction basins of the stable equilibrium position is considered. The problem is solved in various formulations. In the discrete model, the pendulum is taken as an incompressible rod with weights on springs that can move along the pendulum axis. A rod deformed in the longitudinal direction, the transverse deformations of which can be neglected, is the continuous model. An equation is obtained that describes the dynamics of the system in the general case of these two models. To determine the basin of attraction of the stable solution, the asymptotic method of two-scale expansions is used. The asymptotically averaged equation includes the parameters of the pendulum base oscillations, as well as oscillations in the internal degrees of freedom. Taking into account the oscillations in the internal degrees of freedom gives a correction to the dynamic picture of slow pendulum swings and affects both the stability and the basin of attraction of the upper equilibrium position. It turned out that the desired correction is affected by one integral parameter of the system associated with dynamic displacement of the center of mass of the pendulum. The results are shown in graphs.