Dynamics of Pendulum Forced by a Magnetic Excitation with Position-Dependent Phase

TL;DR

The paper addresses the nonlinear dynamics of a magnetic pendulum driven by time-varying magnetic excitation whose phase depends on the state. It develops a Newton–Euler based model with a magnetic torque $M_{mag}$ and a friction term $M_F$, then nondimensionalizes to a dimensionless ODE $y''+\beta y'+\alpha y+\gamma\sin{(\frac{1}{\gamma}y)}+[\delta+\zeta e^{\nu {y'}^2}]\tanh(\sigma y')=A_0 e^{-y^2} y \sin{(\Omega x+\phi_0)}$, and analyzes its dynamics via bifurcation and Lyapunov exponent methods. Validation against experiments confirms substantial agreement in phase diagrams, Lyapunov spectra, and trajectory shapes, demonstrating both chaotic and periodic regimes and coexisting attractors. The results highlight strong multistability and period-doubling routes to chaos, and show the system’s sensitivity to magnetic interaction, with potential applications in controlling resonance energy exchange in coupled magnetic pendulums. These findings advance understanding of nonlinear magneto-mechanical systems and inform design strategies for mechatronic devices leveraging phase-controlled forcing.

Abstract

This study investigates the dynamics of a magnetic pendulum under time-varying magnetic excitation with a position-dependent phase. The system exhibits complex chaotic and regular dynamics, validated through simulations and experiments. The mathematical model, based on a physical setup, includes a magnetic excitation torque with phase dependence on the dynamic variable. Bifurcation analyses confirm the rich multistability of the system, showcasing periodic attractors, period-doubling bifurcations, and chaotic behavior. Experimental validation demonstrates a high agreement between numerical and experimental results, supporting the efficacy of the proposed model. The study sheds light on the system's sensitivity to changes in magnetic interaction, providing insights into controlling resonance energy exchange in coupled magnetic pendulum systems.

Figures (5)

• Figure 1: (a) Experimental setup and (b) physical model. Description of the numerical labels can be found in the main text.
• Figure 2: The numerical (a) bifurcation diagram and (b) LLE diagram for the system described by Eq. (\ref{['eq:motion_dimensionless_8']}) are presented, with a fixed $A_0=81.82$, showcasing variations in $\Omega$ for both increasing and decreasing values. Simulation (red color) and experimental (gray color) phase planes were obtained for certain values (c,d).
• Figure 3: Charts illustrating the dynamical regimes of the system (\ref{['eq:motion_control_phase_10']}) under excitation with position-dependent and two different frequencies: (a) $\Omega=0.664$, (b) $\Omega=0.700$.
• Figure 4: Bifurcation diagram illustrating the route to chaos through a period-doubling sequence.
• Figure 5: Phase plots and Poincaré section (a, b) obtained for periodic and chaotic oscillations, and their frequency spectrums (c, d).