On the motions of a double pendulum with vibrating suspension point

We consider the motions of a system consisting of two pivotally connected physical pendulums rotating about horizontal axes. We assume that the system suspension point, which coincides with the suspension point of one of the pendulums, performs harmonic vibrations of high frequency and small amplitude along the vertical. We also assume that the system has four relative equilibrium positions in which the suspension points and the pendulum centers of mass lie on one vertical line. We study the stability of these relative equilibria. For arbitrary physical pendulums, we obtain stability conditions in the linear approximation. For a system consisting of two identical rods, we solve the stability problem the in nonlinear setting. For the same system, we study the existence, bifurcations, and stability of high-frequency periodic motions of small amplitude other than the relative equilibria on the vertical line. The studies of dynamic stability augmentation in mechanical systems under the action of high-frequency perturbations was initiated in the paper 1, where it was shown that the unstable inverted equilibrium of a pendulum may become stable if the suspension point vibrates rapidly. This idea was developed in 2–10 and other papers, where several aspects of motion of a mathematical pendulum in the case of rapid small-amplitude vibrations of the suspension point were studied in the linear setting and also (without full mathematical rigor) in the nonlinear setting. The motions of the suspension point along an arbitrary oblique straight line 2, 4, 7, 8, along the vertical 3, 5, 6, along the horizontal 9, and in the case of damping 8 were considered. The monograph 10 deals with the stabilization of a pendulum or a system of pendulums under periodic and conditionally periodic vibrations of the suspension point along the vertical, along an oblique straight line, and along an ellipse. A rigorous nonlinear analysis of the existence and stability of periodic motions of the mathematical pendulum under horizontal and oblique vibrations of the suspension point at arbitrary frequencies and amplitudes can be found in 11, 12. For the case of vertical vibrations of the suspension point at an arbitrary frequency and amplitude, a rigorous stability analysis of the relative equilibria of the pendulum on the vertical was carried out in 13.