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.

The hyperloop system is a new transportation mode, which consists a magnetic levitating capsule-like hyperloop pod and a vacuum tube. Due to small air hindrance, the hyperloop pod is conceived to have a maximum speed of 333 m/s. If such a hyperloop system is to be built underground in soft soils, the hyperloop speed can easily reach the wave propagation speeds in the soil. Strong wave radiation is expected when the hyperloop is travelling at wave propagation speeds, which are called the critical speeds.

The first objective is to analyse the dynamic influence from the hyperloop. A linear elastic half-space with an infinitely long concrete tunnel buried at a certain depth has been modeled. The excitation of the system is a hyperloop modeled as a moving constant load acting at the tunnel invert. In this thesis, a so-called indirect boundary element method (BEM) is applied. Indirect boundary integrals are formed which rely on the fundamental solutions for the interior medium, the two-and-a-half dimensional Green's functions. These 2.5D Green's functions are essentially the steady state solutions of the half-space subjected to a spatially varying line load. The space is assumed to be infinitely long and invariant in the direction parallel to the axis of the tunnel.

Before implementing the BEM model, two improvements have been made to the 2.5D Green's functions: a better convergence of the Green's function surface-related terms and a better satisfaction of stress-free boundary conditions at the free surface. The accuracy and correctness of the boundary element model using the improved Green's functions have been verified by intensive case studies. Firstly, the scattering of 3D harmonic seismic P waves by a cavity and a tunnel in a linear elastic half-space is analysed. Results are validated by comparing to those from literature. Secondly, the BEM model is employed for the moving load problem. The embedded concrete tunnel is modeled using the Donnell's theory for thin shells. A coupled form of the indirect boundary integrals is formulated. Using the same model parameters, the results obtained by the BEM are in good agreements with those from literature. Moreover, a parametric study has been conducted to study the effect of moving load velocity, tunnel depth and thickness of concrete lining on the dynamic response.

As a second objective of the current thesis work, the BEM model is compared with a finite element method (FEM) based model, developed by Movares B.V. The models are compared in both accuracy and computational efficiency. In the FEM model, the moving load is considered as a series of consecutive short pulses. The contributions from all the pulses are synthesized using a convolution. Furthermore, since the space is invariant in the direction parallel to the tunnel axis, it is possible to apply just one stationary impulse load in the finite element model. Using this method, a constant moving load and a moving load with acceleration are modeled. The FEM results are found to have close agreements with those by the BEM. Besides the Rayleigh wave speed in the soil, a second critical velocity which is related to the wave propagation in the tunnel is found. Furthermore, the case where a hyperloop runs constantly at the Rayleigh wave speed is more crucial than the case where the hyperloop accelerates and passes the Rayleigh wave speed.

A reciprocity-based direct boundary integral method and indirect boundary integral method have been introduced and used to obtain the two-dimensional response of a cylindrical cavity embedded in a uniform elastic half space subjected to SH wave. By introducing free wave field and the actual field with a cavity, the key point to solve for scattered wave field is cavity boundary integral. In order to do so, plane waves generated by free wave field and a 2D Green’s function for a half space are used. To overcome the limitations of direct boundary integral method, an indirect boundary integral method provided by a classical case study is used. In the end, results are calculated by the latter method, indirect boundary method, and are used to compare with the results from the classical case study.

When modelling a structure-soil system, interaction stresses at the interface of the structure and the surface of soil layer influence the dynamic behaviour of the system. These interaction stresses are not accounted for in many simplified models that evaluate the behaviour of the soil layer and the structure separately. Modelling a fully coupled system requires extensive computation that changes with every alteration to the model. Having a general framework of equations that can be easily adapted to each specific case can therefor be of great value.
In this thesis, we model the soil layer as a homogeneous, elastic continuum with two boundary conditions. One boundary condition is a kinematic excitation at the bottom of the soil layer, which is formulated in terms of a Fourier series with prescribed coefficients. The boundary condition at the top of the soil layer is a stress function, formulated as a Fourier series with unknown coefficients. The general solution of the equation of motion is then solved in terms of these known and unknown coefficients. The structure is modelled as an inextensible mass or as a mass-spring system, which is excited by an external force at the interface with the soil layer, formulated in the same Fourier terms and unknown coefficients as for the soil layer. The unknown coefficients are solved by means of an interface condition between the soil layer and the structure.
This computational method provides an equation of motion for the soil layer that depends on the applied structure model. When the structure model is altered, for example by a mass-spring system instead of a single mass, only the interface condition has to be reevaluated to find a solution for the equation of motion of the soil layer and the structure. This thesis shows that this computational model, where we write the specific solution to the equation of motion in terms of unknown Fourier series coefficients, does indeed work.
By analysing the stress distribution at the interface between the soil layer and the structure for different frequencies, results show that the stress is resonant at the natural frequencies of the system. The stress distribution is nearly uniform for most frequencies, but the stresses increase at the sides of the interface at the natural frequencies. It can also be shown that the interaction stresses increase with the frequency.
When analysing various transfer functions, the influence of the stress interaction between the soil layer and the structure is most visible. Computing a fully coupled system shows that the natural frequencies of the system are affected by the structure on the top of the soil layer. First, the natural frequencies are partially shifted to lower frequencies. Secondly, not all natural frequencies lead to infinite resonance: the transfer functions show an alternating pattern of finite and infinite responses to the excitation at the natural frequencies.
The transfer functions of added mass-spring systems, for example, used to model multi story buildings, also shows the influence of the interaction stress compared to the isolated model of the structure. The coupled system shows again that the natural frequencies are partially shifted and not all natural frequencies lead to infinite resonance.
It can be concluded that the interaction stress in a fully coupled system has a significant impact on the system and should be taken into account when modelling a structure-soil system. The tested computational method is good way to do so.