Simple 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 purely sinusoidal base excitation caused by surface irregularities or vibration of the support induced by external noise. The narrow airgap between the vehicle and the support makes the system sensitive to the motion of the support, as small amplitudes of the latter create significant excitation. The vehicle is modelled as a three-degree-offreedom model where the vehicle is suspended via two identical electromagnetic actuators from rigid supports that oscillate. 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 simple 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 the PD control parameter space, and the influence of system parameters on these boundaries is characterized. For a mean phase shift between the base excitations, the ratio of the sizes of the two simple-parametric resonance ellipses is three to one, whereas for the combination parametric resonance ellipses, the ratio is fourteen to one. One of the ellipses associated with the combination parametric resonance is the largest in that situation. 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 resonances is analysed, showing that they are equivalent to those of the electromagnet-only case; the ellipses only shift in accordance with an overall widening of the stable domain. Results reveal critical conditions under which each type of resonance dominates, offering key insights for safe design and operation of magnetically suspended vehicles.