Bifurcation and chaos in axially moving ferromagnetic thin plates with imperfections under magnetic field and line load

This paper explores the complex dynamic behavior of axially moving ferromagnetic thin plates with initial geometric imperfection, subject to a magnetic field and a line load. The equation of motion for the imperfect plate is derived using Hamilton's variational principle, based on Kirchhoff's thin plate theory and the magnetoelastic theory of ferromagnetic materials, and discretized via the Galerkin method. When system stiffness becomes negative under specific parameters, leading to structural instability, the Melnikov method is used to predict chaos analytically and to establish the necessary conditions for its onset. Based on these predictions, a chaotic distribution map is generated in the parameter space to visualize chaotic regions. Numerical simulations are performed to produce bifurcation diagrams, maximum Lyapunov exponent plots, and system response curves, providing a dynamic analysis of the system's behavior under varying bifurcation parameters. The results indicate a complex nonlinear coupling between the plate's physical and geometric properties and the external magnetic field conditions. Variations in magnetic field intensity, axial velocity, and frequency ratio result in complex nonlinear dynamics, including bifurcations and transitions between periodic and chaotic states. This study offers critical insights into the dynamic characteristics and evolution of nonlinear systems, with significant implications for controlling and predicting chaotic vibrations in engineering applications.