Multiple resonance of the ferromagnetic thin plate with axial velocity subjected to a harmonic excitation in an air-gap magnetic field

Studying the magnetoelastic coupling vibration of moving plates under the influence of complex air-gap magnetic fields is of theoretical significance for modeling multi-field dynamics and solving nonlinear dynamic issues. This paper considers the scenario where an armature wall acts on one side of an axially moving ferromagnetic thin plate and investigates the multiple resonance of the plate subjected to a harmonic excitation in the air-gap magnetic field. First, based on electromagnetic theory combined with boundary conditions of the magnetic field, the distribution of the air-gap magnetic field is determined by solving the Laplace equation of the magnetic scalar potential function. Next, expressions for the magnetic and Lorentz forces acting on the moving ferromagnetic plate are derived. Then, the thin plate energy expression considering geometric nonlinearity and the virtual work expressions of the forces are obtained by applying the Kirchhoff plate theory and the principle of virtual work. Subsequently, the nonlinear vibration equation of the system with magnetoelastic coupling is established using the Hamiltonian variational principle. Utilizing the established mechanical model, the principal-internal resonance is analytically solved employing the Galerkin and multiple scales methods, with stability criteria for steady-state motion solutions determined using Lyapunov stability theory. Finally, the nonlinear dynamics of the system are analyzed through analytical and numerical examples, and the influences of different physical parameter variations on the dynamic behavior are discussed. Results indicate that variations in physical parameters affect the amplitude, resonance region, and regions of multiple solutions of the system. Moreover, the system may transition to chaos through quasiperiodic torus breakdown or period-doubling bifurcation due to changes in physical parameters. The results of this study provide theoretical references for research in magnetoelastic coupling nonlinear dynamics in fields of magnetic drive and electromagnetic control, as well as for product design and optimization.