Chapter 8: 2015 Problem 13: Magnetic Pendulum

In this solution, we study the stable oscillations of a magnet pendulum driven by electromagnet connected to an AC power source. The driving force is calculated when considering the electromagnet as a magnetic dipole. The dynamic equation of the system is derived. Based on numerical analysis and experimental observations, the motion of the pendulum has been investigated. Under appropriate driving voltage U, the pendulum can do stable large oscillations when the driving frequencies are odd multiples of the pendulum natural frequency, i.e. f ≈ (2k + 1) f₀. Phase trajectories and phase diagrams are presented. The phase diagram has up to three discrete limit cycles: one for small oscillation, others for big oscillations. The theoretical conditions we found are consistent with our experiment results.