Nonlinear principal resonance of a soft ferromagnetic rectangular plate under a transverse magnetic field

Abstract

In this paper, a model is presented for a soft ferromagnetic r ectangular thin plate which is coupled with a transverse magnetic field and is further under a transverse uniformly di stributed load. The four edges of the plate are simply supported. Using the multiple scale method, the amplitude-frequency response equation in steady state is obtained. The amplitude- frequency response curves and the relevant time history curves are given by Matlab programs. Furthermore, the influence of the plate thickness, magnetic field and magnetize coefficien t on the vibration behaviour of the system is analyzed. For the soft ferromagnetic material, since there is a magnetization in the magnetic field, an eddy current is produced when there is a relative motion between the structure and the magnetic field. Based on a quantitative analysis of the vibration of a beam-plate with two hinged ends, Liu (1) discussed the factors that affect the vibration frequency and amplitude of the ferromagnetic material plate. The numerical simulation results showed that the introducing geometrical nonlinearity increases the vibration frequency whilst the electric conductivity and the magnetic permeability decrease the frequency. The nonlinear magnetoelastic vibration equations and the electro-magnetic field equations of a current-conducting thin plate in magnetic fields were considered in (2,3) where the ma gnetoelastic combination resonances and stabilities of the thin plate in a transverse magnetic field w ere derived. Furthermore, the amplitude frequency response equation was obtained so that the stabilities of the solutions were also analyzed using the multiple scales method. Gao (4) studied the magneto-elastic-plastic dynamic responses and the dynamic instability of a ferromagnetic beam-plate under some magnetic pulses. Based on the sandwich beam model in elastic-plastic analysis and the Gerlerkin's method, Gao derived the magneto-elastic- plastic dynamic governing equations of the ferromagnetic beam-plate with unmovable simple supports. However, little attention has been paid to the dynamic resonance of the soft ferromagnetic structure. For the soft ferromagnetic material, the magnetization effect can not be neglected. Therefore, in this paper, we study the nonlinear principal resonance of a ferromagnetic rectangular plate, considering the magnetic force caused by the magnetization and eddy current but neglecting the effect of the electric polarization, magnetostriction and magnetic hysteresis during the magnetization process.