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

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 Kirchhoff’s 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 Hill’s 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. Moreover, we found that in all cases, the relative sizes of the ellipses are independent of the excitation frequency, when normalized by the local width of the stable domain. Additionally, the impact of using hybrid magnets in the supports—combining electromagnets with permanent magnets—on the parametric resonance is analysed, showing that they preserve the same stability boundaries while changing the steady-state response. Results reveal critical conditions under which each type of resonance dominates, offering key insights into the safe design and operation of magnetically suspended vehicles.