On stability of relative equilibria of a double pendulum with vibrating suspension point

We consider the motions of a double pendulum consisting of two hinged identical rods. The pendulum suspension point is assumed to perform harmonic vibrations of arbitrary frequency and arbitrary amplitude in the vertical direction. We carry out a complete nonlinear analysis of the stability of the four pendulum relative equilibria on the vertical. The problem on the stability of the relative equilibria of the mathematical pendulum in the case where the suspension point performs vertical harmonic vibrations of arbitrary frequency and arbitrary amplitude was considered in a linear setting 1–3] and a nonlinear setting 4, 5]. In the case of small-amplitude rapid vertical vibrations of the suspension point, linear and (mathematically not fully rigorous) nonlinear stability analysis of the relative equilibria was carried out for an ordinary pendulum 6–9] and a double pendulum 10, 11]. In 12], for the same case of rapid vibrations, stability conditions in the linear approximation were obtained for the four relative equilibria of a system consisting of two physical pendulums. In the special case of a system consisting of two identical rods, the problem was solved in the nonlinear setting.