Dynamical behavior of a vertical magnetic pendulum driven by one coil magnet excited by a voltage source

A vertical magnetic pendulum driven by one coil magnet excited by a voltage source is considered in this work. The mathematical model of the system is derived taking into consideration the geometrical properties of the magnets and the induced electromotive force in the coil magnet. The equilibrium points are obtained and their stability is investigated. The shape of the mechanical potential is that of a symmetric tristable potential or that of a symmetric tetrastable potential when the voltage across the coil magnet is negative or positive respectively. Analytical calculations using the harmonic balance method are carried out to predict the amplitude of the angular displacement and the amplitude of the current through the coil magnet when an alternating voltage is applied. Considering the magnitude and frequency of the external voltage as the control parameters, the two-parameter bifurcation diagrams are plotted showing the different dynamical behaviors of the system. Hysteresis and multistability are also observed in this magnetic pendulum. Excellent agreements are found between the analytical and simulation results and also between the simulation results and the experimental ones.