Suppressing parametric resonance of a Hyperloop vehicle using a parametric force

TL;DR

This paper analyzes the stability of a Hyperloop vehicle model under electromagnetic and aeroelastic forces, demonstrating how parametric forces can be modulated to suppress resonance and improve system stability.

Contribution

It introduces a method to suppress parametric resonance in Hyperloop systems by modulating aeroelastic forces, supported by analytical stability boundary expressions.

Findings

Parametric resonance can be suppressed by modulating aeroelastic forces.

Stability boundaries are analytically derived and depend on phase shift.

Parametric electromagnetic forces influence the system's oscillations.

Abstract

In this paper, we study the stability of a simple model of a Hyperloop vehicle resulting from the interaction between electromagnetic and aeroelastic forces for both constant and periodically varying coefficients (i.e., parametric excitation). For the constant coefficients, through linear stability analysis, we analytically identify three distinct regions for the physically significant equilibrium point. Further inspection reveals that the system exhibits limit-cycle vibrations in one of these regions. Using the harmonic balance method, we determine the properties of the limit cycle, thereby unravelling the frequency and amplitude that characterize the periodic oscillations of the system's variables. For the varying coefficients case, the stability is studied using Floquet analysis and Hills determinant method. The part of the stability boundary related to parametric resonance has an elliptical shape, while the remaining part remains unchanged. One of the major findings is that a linear parametric force, can suppress or amplify the parametric resonance induced by another parametric force depending on the amplitude of the former. In the context of the Hyperloop system, this means that parametric resonance caused by base excitation-in other words by the linearized parametric electromagnetic force can be suppressed by modulating the coefficient of the aeroelastic force in the same frequency. The effectiveness is highly dependent on the phase difference between the modulation and the base excitation. The origin of the suppression is attributed to the stabilizing character of the parametric aeroelastic force as revealed through energy analysis. We provide analytical expressions for the stability boundaries and for the stability's dependence on the phase shift of the modulation.