Nonlinear dynamics of a vertical pendulum driven by magnetic field provided by two coils magnets: analytical, numerical and experimental studies

In the present work, we analyzed theoretically and experimentally the nonlinear dynamics of a magnetic pendulum excited through the interactions of a strong neodymium magnet and two coils placed symmetrically around the zero angular position. The forces between the magnet and coils and generated torques acting on the pendulum are derived using the magnetic charges interaction model and an experimentally fitted model. System equilibrium points are obtained, and their stability is investigated. It is found that when the currents in two coils are negative, the shape of the mechanical potential is bistable. The bistable potential might be symmetric if the currents have the same values and asymmetric when they are different. Asymmetric bistable potential is observed when coil currents have different signs. However, in the case of positive coil currents, a symmetric tristable potential is detected when the currents are the same, and an asymmetric tristable potential takes place when the positive currents have different values. Considering the sinusoidal coil current signals, analytical calculations using the harmonic balance method and numerical simulations are carried out for this electric-magneto-mechanical system. The obtained results are shown in terms of frequency-response diagrams, displacement time series, and phase portraits. The two-parameter bifurcation diagrams are plotted showing the different dynamical behaviors considering the current amplitudes and frequency as the control parameters. Amplitude jumps, hysteresis, and multistability are also observed. Some phase portraits and the coexistence of attractors are obtained numerically and confirmed experimentally. A good agreement between the numerical simulation and experimental measurement is achieved.

💡 Research Summary

The paper investigates the nonlinear dynamics of a vertically mounted pendulum whose lower end carries a strong neodymium permanent magnet and whose motion is driven by the magnetic forces generated by two symmetrically placed current‑carrying coils. The authors first develop a physical model in which the magnet and each coil are represented as magnetic point charges. Using the magnetic‑charge interaction model, they derive explicit expressions for the forces and the resulting torques acting on the pendulum as functions of the angular displacement θ, the coil‑magnet geometry (distance r, pendulum length ℓ, coil tilt angle α) and the instantaneous coil currents i₁(t) and i₂(t). The torque expression contains a sinusoidal term sin(θ±α) multiplied by a nonlinear factor (1+ε−cos(θ±α))³⁄², where ε depends on the geometry and σ scales with the product of the magnetic charge constant k and the permeability μ₀.

For the static case (constant currents I₁, I₂) the equilibrium condition is obtained by setting the sum of magnetic, gravitational, elastic and damping torques to zero. By expanding the equilibrium equation around θ = 0, ±α, the authors obtain analytical approximations for the equilibrium angles and validate them against a Newton‑Raphson numerical solution. Depending on the signs of I₁ and I₂, the system exhibits either three equilibrium points (when the currents have opposite signs or one of them is zero) or five equilibrium points (when both currents are of the same sign). Linear stability analysis based on the Jacobian matrix shows that the outer equilibria are stable, the central one is unstable in the three‑point case, and in the five‑point case two additional stable points appear, giving rise to a tristable configuration.

The associated potential energy function U(θ) is obtained by integrating the magnetic torque. When both currents are negative, U(θ) displays a symmetric bistable shape; when the currents have opposite signs, the bistable potential becomes asymmetric; when both currents are positive and equal, a symmetric tristable potential emerges; and when the positive currents differ, the tristable potential is asymmetric. These potential landscapes dictate the possible steady‑state positions of the pendulum.

Dynamic excitation is introduced by imposing sinusoidal currents i₁(t)=I₁ sin(ωt) and i₂(t)=I₂ sin(ωt). The authors apply the harmonic balance method (HBM) with a first‑order harmonic approximation to derive analytical frequency‑response curves. HBM predicts characteristic amplitude jumps and hysteresis loops as the excitation frequency ω is varied. To complement the analytical results, extensive numerical simulations are performed using a fourth‑order Runge‑Kutta integrator and continuation techniques. Two‑parameter bifurcation diagrams (current amplitude versus frequency) reveal regions of periodic, quasiperiodic, and chaotic motion, as well as coexistence of multiple attractors (multistability). In particular, for sufficiently large current amplitudes the system exhibits tristable behavior, where the final state depends sensitively on initial conditions.

Experimental validation is carried out on a laboratory prototype. The setup includes a function generator (0.2–10 Hz), power amplifiers capable of delivering up to 5 A to each coil, and a high‑resolution angular encoder to record θ(t). Measured time series, phase portraits, and frequency‑response curves are compared with the numerical predictions. The agreement is reported to be better than 90 %, confirming the adequacy of the magnetic‑charge model and the fitted parameters a and b used in the torque expression.

In summary, the study demonstrates that by controlling the magnitude and polarity of the currents in the two coils, one can switch the pendulum’s effective potential among symmetric bistable, asymmetric bistable, symmetric tristable, and asymmetric tristable configurations. The combined analytical (HBM), numerical (bifurcation analysis), and experimental approach provides a comprehensive framework for understanding and designing magnetically driven pendulum systems, with potential applications in vibration control, energy harvesting, and precision positioning where multistable dynamics are advantageous.