Hyperloop is a high-speed transportation mode which operates by sending magnetically levitated capsule-like vehicles through a near vacuum tunnel. Due to the reduced air resistance and friction, speeds exceeding those of modern aircraft should become possible. When the system is moved underground,
the stability of vehicle vibrations may become problematic as, especially in soft soils, the Hyperloop pod velocities could easily surpass the propagation speeds of waves in the soil. In that case, the radiation of anomalous Doppler waves into the system may lead to instability of vehicle vibrations, which means that the amplitude of the vibrations would grow exponentially. The aim of this study is to evaluate the system’s stability and its sensitivity to changes in the model by using analytical methods. To that end, the study is motivated by two central research questions: (1) How can the system be modelled and described, taking into account the interaction between soil, tunnel and vehicle and how does the model description change when the magnetic levitation suspension system is introduced? (2) What is the influence of the model parameters on the system’s stability and what impact do modifications in the vehicle suspension have?
The first research question is answered by modelling the underground Hyperloop system as a two-mass oscillator moving uniformly along an infinitely long Euler-Bernoulli beam embedded in a visco-elastic half-plane. Through definition of the governing equations of motion, boundary and interface conditions and subsequent application of Laplace and Fourier integral transforms, the model is reduced to a lumped model for which the characteristic equation has been derived. In the latter model, the reaction of the beam-half-plane system in the point of contact with the moving load is represented by an equivalent dynamic spring stiffness. The introduction of the magnetic levitation adds a magnetic spring to the lumped model. This spring is placed in series with the equivalent spring representing the beam-half-plane stiffness. The second research question is answered by first studying the velocity-dependent equivalent dynamic stiffness of the supporting structure in the point of contact with the moving oscillator as a function of the frequency of the oscillator vibrations. When the imaginary part of this stiffness is negative, instability of vertical oscillator vibrations may occur due to so-called negative radiation damping. Then, based on the D-decomposition method, the instability domain is found for the base parameters of the system, whereupon this instability zone is parametrically studied. The influence of the magnetic levitation suspension system is established by deriving the instability domain for two different models of a concrete tunnel and various magnitudes of the viscous damping in the vehicle. It is found that the model parameters significantly influence the system’s stability. The oscillator’s viscosity in particular has an important stabilizing effect. For both the non-magnetically and magnetically levitated vehicle, a limited magnitude of the oscillator’s viscous damping stabilizes the system for all velocities up to 300 m/s. Therefore, instability should not be considered as a real danger for the Hyperloop vehicle. Yet, when the electrodynamic levitation suspension system is accounted for, the total mass of the Hyperloop pod and the parameters of the non-contact suspension system should be determined with care as a change in their values may influence the instability domain considerably.