Parametric and primary resonances of a rotating shaft under the harmonic electromagnetic loads

This study presents a comprehensive investigation of the nonlinear dynamic responses of an eccentric rotating shaft characterized by geometric and inertial nonlinearities subjected to harmonic electromagnetic excitation. The governing nonlinear partial differential equations are reduced to ordinary differential equations through the Galerkin method, allowing for a streamlined analysis of the system's behavior. To enhance the understanding of the system's dynamics, the method of multiple scales is utilized to derive solvability conditions, leading to the formulation of frequency-amplitude curves across primary and parametric resonance scenarios. Our detailed analysis reveals the intricate interplay of various parameters, including the magnetic load parameter and the amplitude and frequency of the electromagnetic excitation. Notably, numerical simulations demonstrate the emergence and disappearance of the double jump phenomenon in both forward and backward modes under resonance conditions. Additionally, Hopf bifurcation points are identified, indicating transitions in the system's stability and the emergence of periodic solutions.