Nonlinear Magnetoelastic Vibration Equations and Resonance Analysis of A Current-conducting Thin Plate

Based on the Maxwell equations, the electromagnetic constitutive relations and boundary condition, the electrodynamic equation and the electromagnetic force expressions in an electromagnetic field are derived. Using the principle of virtual work, the basic set of equations for nonlinear electromagnetic elasticity vibration expressed by the displacement of a thin plate in a longitudinal and a transverse magnetic field is obtained, respectively. In addition, we study the nonlinear principal resonance and the solution stability of a thin plate with two opposite sides simply supported and subjected to a mechanical live load and in a constant transverse magnetic field. By the method of multiple scales, the amplitude frequency response equation and the approximate analytic solution in steady motion are also derived. According to the characteristic of singularity and the Lyapunov stability theory, the stability of the solution is analyzed and the critical condition of stability is determined. Finally, by means of numerical calculations, the amplitude frequency response curves, time history response plots and phase charts of the magnetoelasticity vibration are obtained.