Study of transient analysis of a conductive beam carrying an electrical current subjected to magnetic field with elastically restrained ends

The present study considers the nonlinear vibration behavior of a beam with general boundary conditions that carry an electrical current in the magnetic field. This paper discusses the magnetic couple, the transverse magnetic force, the electrical current, and the damper. By contrast, the magnetic field is selected as an arbitrary function of time. Under certain hypotheses, Hamilton's principle is used along with Maxwell's equations to derive the governing equation. An elastically restrained beam carrying an electrical current is also solved using Galerkin's method under a magnetic field. Thus, the effect of the rotational and the translational support flexibilities, the magnetic field, and other parameters are evaluated. For a more detailed investigation, some numerical examples are investigated to present the simplicity and efficiency of this formulation. Based on the numerical results, it is clear that the natural frequency of the ferromagnetic beam is sensitive to the angle and magnetic field. By increasing magnetic field intensity, the magnitude of the natural frequency of the beam increases. But with the increase of the angle, the frequency value decreases. Therefore, at larger angles, the impact of the intensity of the magnetic field will be less. Also, it is determined from the results that the beam deflection in various magnetic fields indicates a significant effect of the boundary conditions, not only on the dynamic response of a damped beam but also on the rate of damping of the response. The dynamic response under the magnetic field is decreased when the beam experiences a stiffer constant in its support. The results are shown that the effect of stiffening for the transitional support is more significant than that of the rotational support. Also, the influence of the boundary constraints becomes smaller when the magnetic field becomes smaller.