Nonlinear vibrations of conductive beams exposed to magnetic traction forces and eddy currents

Abstract

This study explores the nonlinear vibration behavior of conductive beams influenced by magnetic traction forces and eddy currents. Novel relationships for electromagnetic interaction forces with ferromagnetic materials are derived from Maxwell’s equations and Lorentz forces. By employing the Euler–Bernoulli beam theory and the von Karman theory of large deformations, nonlinear differential governing equations are formulated. Utilizing the Galerkin method, these equations are discretized and analytically solved via the variational iteration method. Traditional perturbation methods fail due to the large coefficients and heterogeneity of the nonlinear equations, necessitating the introduction of a method of successive modifications for analytical solutions. The investigation examines the impact of various parameters on beam vibration characteristics, including time response, frequency response, and instantaneous frequency. Numerical results demonstrate that increasing magnetic field intensity leads to a reduction in oscillation amplitude, with stabilization times of 50.6 s for 2 T and 23.6 s for 4 T. Furthermore, the oscillation frequency of the system reaches saturation at 9.87 kHz under the two values of field intensity after 50 s and 20 s, respectively. Chaotic behavior, period-2, and period-3 motions can be realized under various magnetic field intensities, further confirmed by the self-similarity of Poincaré characteristic of chaotic systems. That would provide certain insights into the dynamic response of conductive beams under electromagnetic force and nonlinear damping caused by Lorentz forces.