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.

The increasing size of offshore wind turbine foundations necessitates innovative approaches for monopile installation. Traditionally performed through impact driving, the challenges of large stresses induced on the monopile and high levels of underwater noise emissions have driven a shift toward vibratory installation methods. This study investigates the vibro-installation process of steel tubular piles in dense saturated sand through controlled lab-scale experiments. The experiments systematically varied penetration rates and driving frequencies to analyze the interaction between the piles and the surrounding soil. The results reveal critical insights into the influence of vibratory parameters on soil resistance and pile drivability, with a specific focus on the response of the pile tip and shaft under different conditions. These findings contribute to improved predictive models for monopile installation, addressing data gaps in offshore conditions and supporting the optimization of vibratory techniques for sustainable and cost-effective wind energy development.

Maglev and the newer Hyperloop technologies are advanced transportation systems that eliminate wheel–rail friction using electromagnetic suspension/levitation. The electromagnetic suspension is inherently unstable and requires a control strategy for safe operation, which has been previously studied in the context of Maglev. However, the interaction between electromagnetic instability and another instability mechanism, known as wave-induced instability, occurring at high vehicle velocities, has not been explored. This interaction between two distinct instability mechanisms is the focus of this study. From a practical perspective, this study examines the stability of magnetically suspended vehicles (e.g., Maglev or Hyperloop) in relation to vehicle velocity and control gains. To account for this, this study properly includes the infinite guideway, thus allowing vehicle velocity to influence system stability. The results show that at sub-critical velocities, the guideway's reaction force helps suppress perturbations and stabilize the system, with instability driven solely by improper electromagnetic control. However, at super-critical velocities, wave-induced instability drastically reduces the stable parameter space. This study further proposes a methodology to distinguish the contribution of each instability mechanism to the overall system stability, which is important for efficient mitigation measures. The findings reveal that beyond a certain super-critical velocity, wave-induced instability dominates much of the control-gain plane, with the control strategy effective in only limited regions. In conclusion, the study recommends revising control design strategies, as solely focusing on maximizing energy dissipation through control can trigger wave-induced instability. A more effective approach balances energy dissipation with avoiding the activation of wave-induced instability by steering clear of problematic vibration frequencies. These insights provide guidance for improving control strategies.

Transition zones, characterized by significant variation in track properties (e.g., foundation stiffness) near rigid structures like bridges and tunnels, necessitate more frequent maintenance compared to standard track sections due to higher levels of differential settlements observed at transition zones. Field measurements on one-way tracks reveal asymmetric settlement patterns (i.e., different settlement in the soft-to-stiff vs. stiff-to-soft transitions), yet existing literature often investigate either one or the other transition type without investigating the potential limited validity of results. This study investigates the similar aspects as well as the dissimilar ones regarding the behaviour of soft-to-stiff and stiff-to-soft transitions. Modelling results show that the behaviour of the two transitions can be considerably different. These results strongly suggest that for a mitigation measure to be efficient, it may be necessary to have different designs for the two types of transition wherever possible (i.e., in one-way tracks). This study can help researchers and engineers understand the different degradation patterns obtained using more complex models or from field measurements.

The Hyperloop, a developing transportation system, reduces air resistance by housing the vehicle within a depressurized tube and eliminates contact friction by using an electro-magnetic suspension/levitation system. Maintaining system stability poses a challenge due to the exceptionally high target velocities. Consequently, it is important to know apriori the velocity regimes in which the system can be unstable. The authors have recently investigated this aspect by, unlike previous studies, properly accounting for the frequency and velocity dependent reaction force provided by the infinite guideway. Furthermore, that study focused on the interplay between two fundamentally different instability sources, namely (i) the electro-magnetic suspension, and (ii) wave-induced instability, showing that stability domains drastically change above a certain vehicle velocity. The current study presents a methodology to distinguish the contribution of each instability mechanism to the overall system stability, and demonstrates that the wave-induced instability mechanism is causing the drastic stability change at large vehicle velocities. This investigation offers physical insight into the mechanisms that can cause instability in the Maglev/Hyperloop systems, and can help engineers that develop this novel transportation system to avoid excessive vibrations and, in extreme cases, derailment.

In recent times, railway transportation has received increasing attention, particularly for its ability to operate entirely on electricity sourced from renewable sources. However, the growing demand for railway services has transformed previously acceptable issues into significant challenges, disrupting normal traffic operations. One such issue is ground-borne vibration especially in urban and inter-urban locations. This study explores the efficacy of a novel mitigation technique, termed a "metawedge," in reducing ground-borne vibration at the receiving end. The metawedge consists of a series of periodically arranged barriers that act as resonators. Unlike traditional metamaterials, each resonator within the metawedge possesses slightly different natural frequencies compared to its neighbours. With an appropriate choice of this variation, incoming Rayleigh (surface) waves are converted into body waves, redirecting energy deeper into the ground. Simulation results demonstrate that the metawedge can significantly diminish vibration levels with just a few resonators. Additionally, unlike conventional single trenches, which effectively mitigate vibrations only at specific angles of incoming waves (outside the critical cone), the metawedge remains efficient within this cone. While a theoretical proof-of-concept has been previously presented by the authors, this study makes a step forward by proposing a realizable design. Consequently, this work showcases the potential and feasibility of metamaterials to address present and future challenges in railway transportation.

A novel nonlinear 1-D gradient model has been previously proposed by the authors, combining (i) the higher-order gradient terms that capture the influence of material micro-structure and (ii) a nonlinear softening material behavior through the use of a hyperbolic constitutive model. While the previous study focused on the existence and properties of solitary-type waves, the current study focuses on the characteristics of the transient wave propagation in the proposed model. Findings show that as nonlinearity increases, the bulk of the wave slows down, and its shape becomes more distorted in comparison to the response of the linear system. The energy analysis reveals that, unlike the linear system, the nonlinear one continuously exchanges energy, in which the kinetic energy decreases over time while the potential one increases. Furthermore, the spectral (wavenumber) energy density of the nonlinear-elastic system presents peaks at large wavenumbers. However, these are eliminated when a small amount of linear viscous damping is added indicating that they are not physically relevant. A notable feature that persists despite the presence of damping is the formation of small-amplitude waves traveling in the opposite direction to the main wave. Generalized continua, like gradient elasticity models, miss the small energy scatter by the micro-structure. This study shows that adding material nonlinearity to a homogeneous generalized continuum can capture reverse energy propagation, though at much smaller magnitudes than the main wave. These findings shed light on the characteristics of the transient wave propagation predicted by the proposed nonlinear 1-D gradient model and its applicability in, for example, predicting the seismic site response.

The current study is concerned with ground-borne vibrations induced by railways and their impact on nearby structures and inhabitants. More specifically, it explores the efficacy of the so-called metawedge, a novel mitigation measure, in reducing ground-borne vibrations along the propagation path. A metawedge comprises a series of periodically arranged resonators along the propagation direction positioned either on the ground surface or embedded into the soil at varying depths. The difference between the metawedge and a classical locally-resonant metamaterial is that the metawedge resonators have a smooth variation of the resonance frequency with longitudinal direction. This arrangement enables the conversion of incoming Rayleigh waves into body waves, effectively channeling the energy deeper into the ground. While a theoretical proof-of-concept has been previously presented by the authors, this study makes a step forward by proposing a realizable design. Simulations indicate that a metawedge with realistic properties can significantly diminish vibration levels. Unlike conventional single trenches, which are effective only against incoming waves beyond a specific angle (outside a critical cone), the metawedge proves efficient also within this cone. This work aims to showcase the potential and feasibility of metamaterials to address present and future challenges in railway transportation.

The increasing demand for renewable energy sources is driving a significant rise in the adoption of offshore wind energy. Offshore wind turbines are typically supported by large foundation piles driven into the seabed. The primary method of installation is by hydraulic impact hammers. However, this method generates excessive underwater noise among other drawbacks. Consequently, alternative techniques are being explored by both researchers and industry. One promising alternative is vibratory driving, which theoretically produces less underwater noise compared to impact driving. Nevertheless, there remains a substantial amount of energy transmitted into the water column, particularly from the higher harmonics of the driving frequency. While energy at the fundamental frequency is essential for efficient driving, the energy associated with higher harmonics does not contribute to this efficiency and can significantly increase radiated underwater noise. To address this issue, this study proposes a mitigation strategy to block energy transfer at super-harmonic frequencies in the pile-water-soil system. To selectively target these frequencies without affecting the fundamental one, a mitigation approach utilizing locally-resonant metamaterials is proposed. This involves integrating a transition piece between the vibratory hammer and the pile, that incorporates periodically inserted multiple-degrees-of-freedom systems. By manipulating the natural frequencies of these periodic inclusions, the transition piece forms band-gaps at the relevant super-harmonics. Initial findings indicate that this design has the potential to effectively mitigate noise and vibration at targeted frequencies. Nonetheless, further investigations employing more sophisticated models are necessary to confirm these outcomes.

In this paper, we delve into the dynamics of an electromagnetically suspended mass from a rigid support. The study employs a 1.5-degrees-of-freedom system which serves as a simplified model for a Hyperloop vehicle traveling in a tube. Through linear stability analysis, we analytically uncover three distinct regions for the physically significant equilibrium point. Further inspection reveals that the system exhibits limit-cycle vibrations in one of these regions. Employing the harmonic balance method, we determine the properties of the limit cycle, thus unravelling the frequency and amplitude characterizing the periodic oscillations of system’s variables. We also present preliminary findings regarding the influence of the steady aeroelastic force on the stability of the system.

The authors have previously introduced a novel gradient elasticity model for seismic wave predictions. The said model combines (i) the higher-order gradient terms that capture the influence of small-scale soil heterogeneity and/or microstructure and (ii) the nonlinear softening soil behaviour through the use of the hyperbolic soil model. The current study presents an in-depth analysis of the proposed model. The findings indicate that as nonlinearity increases, the bulk of the wave slows down, and its shape becomes more distorted in comparison to the response of the linear system. Furthermore, the wavenumber spectrum of the nonlinear-elastic response presents peaks at large wavenumbers. However, these are eliminated when a small amount of linear viscous damping is added indicating that they are not physically relevant. One model feature that does not disappear with the presence of damping is the formation of small-amplitude waves travelling in the opposite direction to the main wave. These findings shed light on the characteristics of the proposed nonlinear gradient elasticity model and its applicability for predicting the seismic site response.

Railway induced ground vibrations are of increasing importance for structures and inhabitants in the vicinity of railway tracks. This study investigates the capabilities of a novel mitigation measure, a so-called metawedge, in reducing the ground-borne vibration at the receiver end. A metawedge is series of barriers (i.e., resonators) arranged periodically in the longitudinal direction and each one is offset with respect to the others in depth direction (i.e., while the first barrier is completely on the surface, the last barrier can be completely embedded). The advantage of this countermeasure is that it can convert the incoming Rayleigh (surface) waves into body ones, redirecting the energy content deep into the ground. Simulation results show that the metawedge is capable of significantly reducing the vibration levels with as few as five resonators. Furthermore, while conventional single trenches are efficient as mitigation measures only at a certain angle of the incoming waves (outside the critical cone), the metawedge is efficient inside this cone. Although the metawedge solution is promising, this paper serves solely as a proof of concept, and additional studies are necessary to design realistic resonators that can comply with the low frequencies of the railway induced ground vibrations. Nonetheless, this study shows that metamaterials-inspired solutions can play an important role in addressing present and future challenges of the railway transportation.

The increasing demand for renewable energy sources is driving a significant rise in the adoption of offshore wind energy. Offshore wind turbines are typically supported by large foundation piles driven into the seabed. The primary method of installation is by hydraulic impact hammers. However, this method generates excessive underwater noise among other drawbacks. Consequently, alternative techniques are being explored by both researchers and industry. One promising alternative is vibratory driving, which theoretically produces less underwater noise compared to impact driving. Nevertheless, there remains a substantial amount of energy transmitted into the water column, particularly from the higher harmonics of the driving frequency. While energy at the fundamental frequency is essential for efficient driving, energy associated with higher harmonics does not contribute to this efficiency and can significantly increase radiated underwater noise. To address this issue, this study proposes a mitigation strategy to block energy transfer at super-harmonic frequencies in the pile-water-soil system. To selectively target these frequencies without affecting the fundamental one, a mitigation approach utilizing locally-resonant metamaterials is proposed. This involves integrating a transition piece between the vibratory hammer and the pile, that incorporates periodically inserted multiple-degrees-of-freedom systems. By manipulating the natural frequencies of these periodic inclusions, the transition piece forms band-gaps at the relevant super-harmonics. Initial findings indicate that this design has the potential to effectively mitigate noise and vibration at targeted frequencies. Nonetheless, further investigations employing more sophisticated models are necessary to validate these outcomes.

While most recent models of railway tracks include the nonlocal nature of the foundation reaction force, few studies have investigated the influence of its nonlocal nature on the response. Accounting for the nonlocal nature of the foundation force is computationally expensive and increases the complexity of the model, thus, knowing when and when not to account for it is important. This paper aims to shed light on the influence of the nonlocal, both in time and space, reaction force provided by the foundation on the transient response at railway transition zones. To this end, a 2-D system in which the soil layer is modelled as a viscoelastic continuum is compared to an equivalent 1-D system with a local foundation reaction force (i.e., Winkler foundation). Results show that, in general, the response of the 2-D system with shallow and/or stiff soil layers can be well captured by the equivalent 1-D model. However, for medium-to-deep and/or soft soil layers, the nonlocality of the foundation reaction force is influential and the transient response at transition zones cannot be satisfactorily captured by 1-D models. Finally, the ballast settlement is also poorly captured for medium-to-deep and/or soft soil layers, with the main cause being the inability of the 1-D model to separate between ballast and soil stresses, and not the locality of the reaction force.

The Hyperloop, a developing transportation system, reduces air resistance by housing the vehicle within a depressurized tube and eliminates contact friction through an electro-magnetic suspension/levitation system. Maintaining system stability poses a challenge due to the exceptionally high target velocities. The interplay between electro-magnetic and wave-induced instability has been previously studied by the authors, showing that stability domains drastically change above a certain vehicle velocity. The current study demonstrates that the anomalous Doppler waves (i.e., wave-induced instability) are causing this drastic change. This investigation offers physical insight into the mechanisms that can cause instability in the Hyperloop system.

Transition zones, characterized by significant variation in track properties (e.g., foundation stiffness) near rigid structures like bridges and tunnels, necessitate more frequent maintenance compared to standard track sections due to higher levels of differential settlements observed at transition zones. Field measurements on one-way tracks reveal asymmetric settlement patterns (i.e., different settlement in the soft-to-stiff vs stiff-to-soft transitions), yet existing literature often investigate either one or the other transition type without investigating the potential limited validity of results. This study investigates the similar aspects as well as the dissimilar ones regarding the behaviour of soft-to-stiff and stiff-to-soft transitions. Modelling results show that the behaviour of the two transition can be considerably different. These results strongly suggest that for a mitigation measure to be efficient, it may be necessary to have different designs for the two types of transition wherever possible (i.e., in one-way tracks). This study can help researchers and engineers understand the different degradation patterns obtained using more complex models or from field measurements.

This study examines the impact of railway-induced ground-borne vibrations on nearby structures and residents, focusing on the effectiveness of the metawedge, a novel mitigation measure. The metawedge consists of a series of periodically arranged resonators along the propagation path, either placed on the ground surface or embedded at various depths. Unlike classical locally-resonant metamaterials, the metawedge features resonators with smoothly varying resonance frequencies in the longitudinal direction. Two metawedge designs, the forward and inverse metawedge, have been proposed in the literature. Despite their similarities, they operate on different principles: the forward metawedge decelerates incoming surface waves, localizing energy, while the inverse metawedge accelerates the waves, converting Rayleigh waves into body waves. This study compares the performance of both designs in mitigating train-induced ground-borne vibrations. Results indicate that both the forward and inverse metawedge exhibit remarkably similar performance for the specific design adopted. If this similarity holds across different designs, it offers engineers flexibility in choosing the appropriate measure based on practical needs. More generally, this work demonstrates the potential and feasibility of using metamaterials to address current and future challenges in railway transportation.

The Hyperloop is an innovative transportation system that is currently under development. It minimizes air resistance by enclosing the vehicle in a de-pressurized tube and eliminates wheel-rail contact friction through the use of an electromagnetic suspension/levitation, similar to Maglev trains. This design can potentially achieve much higher velocities compared to traditional railways, positioning the Hyperloop as an environmentally friendly alternative to air transportation.

A potential challenge for Hyperloop is ensuring the dynamic stability at large velocities, where multiple instability sources can be present. An apparent source is the electro-magnetic suspension (adopted by some designs) making a control strategy mandatory to ensure stability even at quasi-static velocities. A less obvious instability mechanism is that the vibration of a vehicle on an elastic guideway can become unstable when surpassing a critical velocity.

The authors have previously investigated the interplay between the electro-magnetic and wave-induced instability mechanisms and showed that the stability space changes significantly above a certain velocity. In other words, the control strategy can ensure the overall system stability only for a very limited range of its gains. The cause for this drastic change was attributed to the wave-induced instability mechanism. Metrikin demonstrated that this instability arises with the radiation of anomalous Doppler waves, which introduce more energy to the vehicle's vibration than normal Doppler waves radiate away from the vehicle. The current study demonstrates that the change of stability domain is indeed caused by the anomalous Doppler waves. While identifying unstable velocity regimes is practical for Hyperloop design, gaining insight into the contribution of individual instability mechanisms can be crucial for efficient mitigation.

The prediction of the so-called seismic site response (i.e., the response of the top soil layers induced by seismic waves) is important for designing structures in areas prone to earthquakes. For seismic loads that induce large soil strains, accounting for the nonlinear behaviour of the soil can be of importance for accurate predictions. In Ref. [1], the authors propose a nonlinear gradient elasticity model for predicting the seismic site response. In the said model, the nonlinear constitutive behaviour of the soil is governed by the hyperbolic soil model, in which the secant shear modulus is dependent on the shear strain through a non-polynomial (hyperbolic) relation. Moreover, the classical wave equation was extended to a nonlinear gradient elasticity model to capture the effects of small-scale heterogeneity/micro-structure. Compared to the classical continuum, higher-order gradient terms are introduced into the equation of motion, which lead to dispersive effects [2] prohibiting the formation of un-physical jumps in the response. The aforementioned model is used in this work too, in which a Gaussian pulse is imposed as an initial condition and the solution is determined using a novel finite difference scheme (see Ref. [1]). This work investigates the behaviour of the proposed model for different levels of initial nonlinearity (i.e., induced by the initial conditions). More specifically, we focus on explaining and studying the appearance of a non-zero plateau trailing behind as the initial shape propagates away. It is shown that the higher the initial nonlinearity, the more pronounced the plateau, indicating that the non-zero plateau is a characteristic of the system’s nonlinearity. The in-depth investigation of the proposed model's characteristics can be helpful when using it to accurately predict the seismic site response.

This short paper presents, in a summarized way, non-linear connections for a lattice model of a railway, with the intention of simulating ballast compaction at transition zones.