Dynamical Analysis of a Variable Stiffness Active Magnetic Bearing System with 1:2 Internal Resonance under PD Control

Extensive research has solved the dynamic problems of electromagnetic bearings under 1:1 internal resonance, but the complex nonlinear behavior of a 12-pole variable stiffness system under 1:2 internal resonance has not been explored to a large extent. This paper aims to bridge this gap by providing comprehensive dynamic analysis and PD control strategies for such systems. First, the expression for the electromagnetic force in the system is derived based on electromagnetic theory, applying Newton’s second law and considering the effect of rotor gravity. The resulting differential equations governing the dynamics and control of the magnetic bearing, which include quadratic and cubic terms, are then derived. Next, the relationships between the first-order and second-order natural frequencies are analyzed, taking into account 1:2 internal resonance, primary parameter resonance, and 1/2 subharmonic resonance. A perturbation analysis of the system is performed using the method of multiple time scales, yielding the four-dimensional averaged equations in both polar and Cartesian coordinates, as well as the amplitude–frequency response equations. Finally, numerical simulations reveal distinct vibration modulation patterns: parametric excitation dominates the first-order modal response, whereas external forcing preferentially excites the second-order mode, resulting in anisotropic oscillations along orthogonal axes. Moreover, the differential gain is demonstrated to be a critical factor in the suppression of chaos and bifurcations. These results significantly advance the understanding of the nonlinear dynamic behaviors inherent in active electromagnetic bearing systems.