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Magnetoelastic nonlinear free vibration of an annular plate under a nonuniform magnetic field of the long straight current‐carrying wire

In this article, the magnetoelastic nonlinear free vibration of a thin conductive annular plate under the nonuniform toroidal magnetic field is investigated, where the magnetic field is generated by a long straight current‐carrying wire. Based on electromagnetic theory, expressions of the nonuniform magnetic field, electromagnetic forces are deduced. The nonlinear free vibration equation of plate is derived by using the Hamilton principle. Considering different boundary conditions, the axisymmetric vibration differential equation is achieved by the Galerkin method. The method of multiple scales is employed to obtain the second‐order approximate analytical solutions and natural frequency. In numerical calculation, the characteristics of nonlinear natural vibration and singular points varying with different parameters, for example, current intensity and plate size, are presented, respectively. The results indicate that, with the increase of current, natural frequency increases and then stabilizes in the inner‐clamped and out‐free (C‐F) boundary, but decreases and stabilizes in the cases of the inner‐clamped and out‐clamped (C‐C) boundary or inner‐simply and out‐simply (S‐S) boundary. Moreover, when wire current is off, natural frequency maintains a constant, where the equilibrium solution of the system is a center. In addition, for all boundary conditions, natural frequency increases significantly with increase of internal radius and thickness, but decreases with the increase of outer radius.