Existence and Stability Analysis on Circular Motion of Pendulums With Uniformly Rotating Pivots

Authough with rich dynamic meanings, the particular circular motion and the stability of pendulums with horizontally uniformly rotating pivots have been seldom studied. Firstly, for the pendulum moving in vacuum and within a medium respectively, the general motion equations under gravity and disturbing force were established. Newton’s second law in a noninertial reference frame was used through introduction of a fictitious inertia force. Secondly, the existence of the particular motion was converted into the root finding of a quartic equation. According to Descartes’ rule of signs and analysis of the monotonicity of quartic polynomials, the relationships between the number of solutions and the physical parameters of the pendulum were given. In vacuum, the number of particular motion solution is 0, 1 or 2, and within a medium, the number is either 1 or 3. Their judging criteria were also given. Thirdly, Lyapunov’s first approximation theory was used to investigate the nonlinear stability. The motion equation was linearized around the particular solution, the stability of the particular motion was judged by the signs of the real parts of the eigenvalues related to the linear differential equation. The subsequent quartic characteristic equations were skillfully converted into quadratic equations. Thus, the linearly stability conditions in vacuum and the asymptotic stability conditions within a medium were deduced. Finally, numerical simulations were given to verify and confirm the theoretical conclusions.