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.

中文翻译：

---

磁场驱动单摆系统规律单侧振荡的理论与数值分析

摘要 本文提出了单磁摆单侧振荡的理论和数值分析。该系统由一个物理钟摆和一个固定在杆端的钕磁铁组成。钟摆由放置在下方的电线圈产生的脉动排斥磁场驱动。摆式枢轴由弹性元件阻尼。励磁电流信号具有频率和占空比受控的脉动矩形波形。一般来说，磁相互作用随着磁铁和线圈之间距离的增加而减弱，使得该系统中的励磁不仅与时间有关，而且与位置有关。根据电流信号的频率和占空比来分析称为“单侧振荡”的特定类型的解决方案。通过一侧振荡，我们的意思是钟摆的振荡以相同的角位移符号为特征，而不通过最低和最高垂直平衡位置。分析主要基于这样的假设，即在某个“活动区”之外，磁相互作用对摆动力学的影响可以忽略不计，系统可以离散为两种状态：有磁力和无磁力。结果通过显示单侧振荡的不同周期性的实验数据得到证实。在所分析的解决方案类型的某些变体的情况下，发现了所提议的分析的局限性。系统动力学的概述以数值获得并通过实验估计验证的分岔图的形式呈现，