Theoretical and numerical analysis of regular one-side oscillations in a single pendulum system driven by a magnetic field

Abstract

This paper presents a theoretical and numerical analysis of one-side oscillation in a single magnetic pendulum. The system is composed of a physical pendulum with a neodymium magnet fixed to the end of the rod. The pendulum is driven by a pulsating, repulsive magnetic field generated by an electric coil placed underneath. The pendulum pivot is damped by an elastic element. The excitation current signal has a pulsating rectangular waveform with a controlled frequency and duty cycle. In general, magnetic interaction weakens with increasing distance between the magnet and the coil, making excitation in this system not only time-dependent but also position-dependent. The specific type of solution, referred to as “one-side oscillation” is analysed in terms of a frequency and a duty cycle of the current signal. By one-side oscillations, we mean oscillations of the pendulum characterized by the same sign of angular displacement, without passing through the lowest and the highest vertical equilibrium positions. The analysis is based mainly on the assumption that outside some “active zone” the influence of magnetic interaction on the pendulum dynamics is negligible, and the system can be discretized into two states: with and without magnetic force. The results were confirmed by experimental data showing the different periodicity of one-side oscillations. Limitations of the proposed analysis were found in the case of some variants of the analysed solution type. An overview of the system dynamics is presented in the form of bifurcation diagrams obtained numerically and verified by experimental estimates, which show the existence of chaotic behaviour and multiperiodicity for various values of the frequency of the current signal.