Double Resonance of Magnetism-Solid Coupling of in-Plane Moving Thin Plates With Linear Loads and Elastic Supports

Abstract

For the in-plane moving thin plates with linear loads and elastic supports in magnetic field, the potential energy, the kinetic energy and the electromagnetic force expressions of the system were given. Based on the Hamiltonian variational principle, the magnetism-solid coupling nonlinear vibration equation for the in-plane moving strip plate was deduced. For the clamped-hinged boundary condition, the variable separation method and the Galerkin method were employed to obtain the 2DOF nonlinear vibration differential equations containing the simple harmonic linear load and the electromagnetic damping force terms. The multiscale method was used to analytically solve the principal-internal resonance problem, and the 1st-order state equation and the resonance response characteristic equation for the system under the double joint resonance were obtained. Through numerical examples, the 1st- and 2nd-order resonance amplitude curves of the in-plane moving thin plate were obtained. The effects of different parameters and load positions on vibration characteristics of the system were analyzed. The results show that, for the principal-internal resonance occurring in the system, the multivaluedness and jumping phenomenon of the solution are obvious, and the effects of the elastic support and the linear load position on the resonance are significant. Additionally, the 1st- and 2nd-order resonance multivalued solution areas appear and disappear simultaneously, which reflects obvious internal resonance characteristics.