Motion analysis of magnetic spring pendulum

In order to analyze the motion characteristics of the spring pendulum under the action of magnetic field force, the motion of the spring pendulum will be studied by applying a uniform magnetic field in the vertical direction. Firstly, a first-order approximate solution is given by studying the micro-vibration around its equilibrium point. And an approximate solution similar to the Foucault pendulum is also presented in the case of a soft spring with strong ductility. Then, according to the resonance conditions of mechanical vibration, the internal resonance phenomenon of magnetic spring pendulum is discovered, and then the conclusion that the energy of the system is cyclically transmitted between the three modes of breathing, oscillating and deflection is presented subsequently. Finally, the influence of magnetic field strength on the motion stability of the spring pendulum is explored, and not only the bifurcation phenomenon at its equilibrium point is found, but also the complex dynamic behavior including chaotic motion occurs.