Dynamics and stability of Kapitsa's two-link pendulum

The object of study. The upper upward position of the pendulum subjected to vibration of the pendulum base is known to be stable for some parameters of the base vibration. The research is devoted to dynamics of the model of a two-link inverted pendulum in a general nonlinear formulation. The goal. The boundaries of the parameters of a given base vibration, under which the inverted mode is stable, are assumed to be known. The goal is to find the regions of the initial conditions of the problem, namely, the initial non-small angles of deviation of the pendulum links from the vertical that result in stable oscillation in the inverted position. We intend to reveal the impact of the rod compressibility on the oscillation mode, as well as the influence of the resonance on stability in the framework of more complex formulation of the problem which involves account for small elastic axial deformation in the rods. Methods. By applying the laws of dynamics to the moving elements of the structure, we derive the complete nonlinear system of equations of the pendulum motion in two formulations: (i) for a system with two and (ii) four degrees of freedom, respectively. The equations include the parameter of small base vibration amplitude, which makes it possible to apply the two-scale asymptotic expansion method. The method leads to a system of averaged equations of motion which is convenient for the benchmark study of parameters. Results. The modes and eigenfrequencies of small oscillations of the pendulum are found depending on the dimensionless parameter of the problem. In the nonlinear formulation, the maximum deviations of the pendulum links are calculated which ensure a stable solution to the problem for zero initial angular velocities. Depending on the initial phase of vibration of the base, the boundaries of absolute and partial zones of stability of vibrations are obtained. In the absolute zone, stable oscillations are realized for any value of the initial phase of the base vibration. In the partial region, stable oscillation occurs at least for one set of initial condition. The dynamics of the pendulum is compared with and without account for rod the compressibility. The results are presented in the graphs.