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

Abstract

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.

Figures (21)

• Figure 1: Pendulum-magnet embedded into a variable magnetic field invoked by two current-carrying solenoids.
• Figure 2: Location of the equilibrium points obtained for $\alpha=\frac{\pi}{6}$ rad, and a) for $I_1=-1.0$ A, while b) for $I_1=1.0$ A.
• Figure 3: Real parts of the eigenvalues of the corresponding Jacobian matrix obtained for $\alpha=\frac{\pi}{6}$ rad and a) for $I_1=-1.5$ A, b) for $I_1=1.5$ A.
• Figure 4: Shape of the potential $U$ for different values of the coil currents $I_1$, $I_2$ and for $\alpha=\frac{\pi}{6}$ rad. a) $I_1=I_2=-1.0$ A. b) $I_1=-I_2=-1.0$ A. c) $I_1=-I_2=1.0$ A. d) $I_1=I_2=1.0$ A.
• Figure 5: Basins of attraction of the stationary solutions for different values of the currents $I_1$ and $I_2$ and for $\alpha=\frac{\pi}{6}$ rad: a) $I_1=I_2=-1.0$ A; b) $I_1=-I_2=-1.0$ A; c) $I_1=-I_2=1.0$ A; d) $I_1=I_2=1.0$ A.