Simple and Combination Parametric Resonances of an Electromagnetically Suspended Vehicle subject to Base Excitation

Abstract

This paper investigates the dynamic stability of an electromagnetically suspended vehicle, encountered in Hyperloop and Maglev systems, subject to periodic excitations caused by surface irregularities or vibration of the support induced by external noise. The narrow clearance between the vehicle and the support can make it highly sensitive to small oscillations, since the admissible amplitudes of the vehicle oscillations can be comparable to external excitation amplitude. The vehicle is modelled as a three-degree-of-freedom model where the vehicle is suspended via two identical electromagnetic actuators from a rigid support that oscillates. The governing equations are derived using force and torque balances, incorporating nonlinear electromagnetic forces, and Kirchhoffs law for the electromagnets with PD control strategy on the airgap. The equations of motion are linearized around the steady state induced by the surface oscillation, yielding a system with time-periodic coefficients. We analytically explore both principal and combination parametric resonances using an extended Hills method, and Floquet theory is used for numerical validation. The stability boundaries are obtained as ellipses in control gain parameter space, and the influence of system parameters on these boundaries is characterized. For the principal parametric resonance, the ratio of the sizes of the two obtained ellipses is three to one, whereas for the combination parametric resonance, the ratio is fourteen to one. When all ellipses are simultaneously present, one of the ellipses associated with the combination parametric resonance is the largest.