Nonlinear free vibration analysis of an elastically supported annular plate in nonuniform induced magnetic field

The aim of this article is to investigate nonlinear free vibration and electromagnetic mechanical behavior of a conductive annular plate with elastic boundary in induced non-uniform magnetic field. On basis of magnetoelastic vibration control equation of conductive plates, displacement solution for plates under complex elastic constraints is first solved, and the axisymmetric nonlinear differential equation for plate is derived by combining Galerkin method. Introducing the multi-scale method to solve the differential equation of elastic boundary plate in induced magnetic field to achieve the analytical expression for natural frequency including current and elastic stiffness terms, and the variation of system singularity is analyzed through stability discriminant. Through numerical method, the nonlinear frequency variation curves, electromagnetic characteristic curves, stability area and phase trajectory graphics of singularities with various control parameters are presented. The result accuracy verification was also conducted in comparison to relevant literature and numerical solutions. The interesting and key findings revealed that, under internal diameter clamped and outer diameter elastically supported boundary, the variation of natural frequency with current, time, the nonlinear soft and hard characteristics are completely opposite to rest two boundaries. The elastic stiffness may significantly increase natural frequency, but does not change the soft or hard characteristics of system. With influence of elastic boundaries, the distribution of electromagnetic force and torque in radial direction of plate becomes quite complex under non-uniform magnetic field. In addition, wire current may change the type of system singularity as current increases.