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 unraveling 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 Hill's 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 also 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. Finally, we emphasize that suppressing parametric resonance through an added, linear state-dependent force with the coefficient having the same period as the original force can be achieved in other physical systems too. ... Read more

Hyperloop is a new high-speed transportation system currently in development. Utilising an electromagnetic suspension within a low-pressure tube, this approach eliminates the traditional wheel-track friction and drastically reduces the air resistance, targeting speeds up to 1000 km/h. Compared to traditional trains and aeroplanes, the Hyperloop is a more sustainable alternative as it produces no greenhouse gas emissions and uses ten times less energy than road and aviation transportation. The Hyperloop, therefore, has great potential to contribute to the goal of achieving climate neutrality by 2050.

Ensuring the stability of the Hyperloop is a critical challenge in its realisation. Multiple instability mechanisms are present in the system: (1) electromagnetic instability, (2) wave-induced instability, and (3) aeroelastic effects, such as galloping. Additionally, the imperfections in the guideway, make the Hyperloop system prone to parametric instability. This thesis primarily investigates how aeroelastic forces and parametric resonance interact with the electromagnetic and wave-induced instability mechanisms, impacting the overall stability of the Hyperloop system.

Based on the results, it can be concluded that the aeroelastic force destabilises the system, where its impact increases as the velocity rises. As the aeroelastic force continuously injects energy into the system, the unstable domain expands. Furthermore, parametric resonance, represented by ellipse-shaped indentations in the stability planes, is observed when the excitation frequency is twice the natural frequency of the system. Based on the control parameter range used for maglev trains, it is found that the stable domain is narrow, with only a small region prone to parametric resonance. Therefore, choosing the control parameters precisely is essential to prevent instability and parametric resonance.

When examining the combined effect of aeroelastic force and the irregular guideway profile, the results show that the aeroelastic force shifts the overall position of the stability boundaries but it does not affect the parametric resonance regions. Instead, the amplitude of the guideway’s irregular profile influences the size of the ellipses and the stable domain. A larger amplitude leads to an expansion of the unstable domain and an increase in the size of the ellipse; therefore, it is advised to maintain a smooth guideway.

This thesis further evaluates the capabilities of analytical expressions for the simpler 1.5 degree-of-freedom (DOF) electromagnetically suspended mass system by comparing it to systems that incorporate the beam dynamics. The results show that the analytical expressions from the 1.5 DOF system cannot approximate the position of the ellipses for the more complicated systems across all velocities. However, beyond the velocities of $1.3v_{cr}$, the analytical expressions effectively approximate the ellipse size. Notably, while the position and size of the ellipse are influenced by the guideway’s surface roughness amplitude, only the size of the ellipse is affected by the amplitude of the oscillations in the 1.5 DOF system. For the more complicated systems, the analytical expressions of the ellipse size relative to the amplitude provide accurate results, making the simplified system a valuable approach for estimating the ellipse size of parametric resonance in systems with beam dynamics. ... Read more