Quantifying periodic, multi-periodic, hidden and unstable regimes of a magnetic pendulum via semi-analytical, numerical and experimental methods

The dynamics of a magnetic pendulum mounted on a shaft with an elastic element is considered. The impulsive excitation of the pendulum is performed through the interaction of a strong neodymium magnet and an electric coil. The mathematical model of the pendulum is represented by the non-autonomous ordinary nonlinear differential equation. The model is verified in a wide range of motion regimes by the laboratory experiment. Periodic motions are considered in more detail by the semi-analytic approximate method utilizing the averaging ideas. The amplitude and phase shift of system solutions possess highly nonlinear features which we propose to approximate by the saw-tooth functions. This new approach allows one to improve the results of the application of the conventional averaging method and is promising for the application and analysis of nonlinear systems with a strong nonlinearity.