Dynamics of the Final Deflections of a Flexible Kapitsa Pendulum on a Hinged Support

Abstract The problem of a pendulum oscillating in a gravity field about its upper equilibrium position under the influence of vertical vibrations of the foundation with a finite angle of deviation from the vertical—the classic Kapitsa problem—is solved in a generalized formulation for a flexible compressible pendulum as a homogeneous deformable rod using the Bernoulli-Euler beam model. Geometrically nonlinear motions of a hinged pendulum are considered, taking into account small transverse and longitudinal oscillations of the pendulum axis. Transverse and longitudinal deformations are approximated by functions obtained by solving the corresponding linear problems of transverse and longitudinal deformations. The problem is reduced to a three-degree-of-freedom system with five dimensionless problem parameters, including the initial phase of foundation vibrations. Taking into account the smallness of the dimensionless amplitude of foundation vibrations allows for the application of the asymptotic method of two-scale expansions. An explicit expression for the boundaries of the basin of attraction of a stable solution to the problem is obtained. For both inextensible and extensible rods, the effect of transverse vibrations on stability and the boundaries of the attraction domain of a stable solution to the problem was determined in the region before the first longitudinal resonance. It was shown that accounting for bending deformations of the pendulum affects the stability boundaries: increasing the flexibility of the pendulum from an absolutely rigid state to the first resonance of bending vibrations reduces stability, and for an even more flexible pendulum, the stability boundaries in the region beyond the resonance are wider than for an inflexible one. A numerical solution of the unaveraged system for certain parameters yielded three more resonances.