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To analyze the motion laws of a magnetic and elastic coupling system under the influence of various factors, this paper proposes a magnetic coupling pendulum based on spring pieces and magnets—a magnetic–mechanical oscillator. By fixing spring pieces onto two non-magnetic bases and attaching magnets to their upper ends, which repel each other, the potential energy during oscillation is expanded using Fourier series. Subsequently, Lagrange equations are solved to study the effects of the first two terms of potential energy. spring piece Using algebraic simplification and transformation, as well as model experiments, its dynamic behaviors, such as its motion frequency and phase characteristics, were explored according to different influencing factors, including the magnetic potential energy, the elastic modulus of the spring, and the external environment. We analyzed the impact of the simplified system's main parameters on its vibration modes, deriving the model formulas and the experimental results. The conclusions from the formula derivation indicate that the motion frequency of the simplified system is a superposition of two frequency. Under certain conditions, these frequencies can be separated, simplifying the system's dynamics. Equally, the experimental results corroborated the findings from the formula derivation, providing a reference for the design and study of dynamic behaviors in magnetic coupling systems.
Zhao et al. (Mon,) studied this question.