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.@en

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.
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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.@en

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.@en

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.@en

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.@en

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.@en

ransition zones in railway tracks experience strong amplification of stress and strain fields due to the passage of train over inhomogeneity. The inhomogeneity in these zones can be attributed to changes in mechanical properties of material along the longitudinal direction of the track, and to displacement/traction discontinuities at interfaces leading to an amplified response in railway transition zones (TZ) with respect to the open tracks. In this paper, different kinds of inhomogeneities are considered in isolation and in combination to study the effects on railway track components in transition zone. The first type of inhomogeneity considered is non-uniformity of materials at various levels of the track along the longitudinal direction. The second type of inhomogeneity that will be considered arises from displacement and traction discontinuities at the interface of soil and structure and at the interface of sleepers and ballast (hanging sleepers). The results provide necessary insight for the design of effective mitigation measures to prevent the amplified response in railway TZ.@en

A novel nonlinear gradient elasticity model for predicting seismic wave behavior was proposed by the authors in previous studies, introducing higher-order gradient terms to account for small-scale soil heterogeneity and micro-structure. This work delves into various characteristics of the proposed model, focusing on the system’s response under initial conditions that induce different levels of nonlinearity. The results reveal that high nonlinearity in the initial conditions can lead to distinctive wave shapes and a non-zero plateau trailing behind the propagating wave. The greater the initial nonlinearity, the more pronounced this plateau becomes. These findings provide valuable insights into the behavior of the proposed nonlinear gradient elasticity model.@en

This paper studies the effectiveness of adding auxiliary rails as a mitigation measure for degradation in transition zones of railway tracks. More specifically, it investigates the settlement mechanisms counteracted by the additional rails. Results show that when the system’s response is in the quasi-static regime, adding auxiliary rails over the soft part of the transition zone is beneficial while adding them over the whole transition zone is not. Furthermore, the auxiliary rails have a beneficial impact also when the system’s response is in the dynamic regime; the beneficial effect is caused by the improved load distribution to the supporting structure and not from counteracting the dynamic response amplification that occurs at transition zones. While this mitigation measure has been previously investigated, the contribution of this study lies in a more in-depth analysis of the mechanism through which auxiliary rails can mitigate the degradation at transition zones.@en

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.@en

Hyperloop is an emerging transportation system that minimises the air resistance by having the vehicle travel inside a de-pressurised tube and eliminates the wheel-rail contact friction by using an electro-magnetic suspension system. Due to the very large target velocities, one of its challenges will be ensuring the system stability. This study aims to determine the velocity regimes in which the system is unstable and, more specifically, is focusing on the interplay between two fundamentally different instability sources, namely (i) the electro-magnetic suspension and (ii) wave-induced instability. To this end, unlike previous studies, the frequency and velocity dependent reaction force provided by the infinite guideway is properly accounted for. Results show that the interplay between the two instability mechanisms does not lead to significant qualitative changes compared to considering each one separately. This study can help Hyperloop engineers avoid excessive vibrations that can cause fatigue problems and, in extreme cases, derailment.@en

The study of periodic systems under the action of moving loads is of high practical importance in railway, road, and bridge engineering, among others. Even though plenty of studies focus on periodic systems, few of them are dedicated to the influence of a local inhomogeneous region, a so-called transition zone, on the dynamic response. In railway engineering, these transition zones are prone to significant degradation, leading to more maintenance requirements than the rest of the structure. This study aims to identify and investigate phenomena that arise due to the combination of periodicity and local inhomogeneity in a system acted upon by a moving load. To study such phenomena in their purest form, a one-dimensional model is formulated consisting of a constant moving load acting on an infinite string periodically supported by discrete springs and dashpots, with a finite domain in which the stiffness and damping of the supports is larger than for the rest of the infinite domain; this model is representative of a catenary system (overhead wires in railway tracks). The identified phenomena can be considered as additional constraints for the design parameters at transition zones such that dynamic amplifications are avoided.

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Hyperloop is an emerging high-speed transportation system in which air resistance is minimised by having the vehicle travel inside a de-pressurised tube supported by columns. This design leads to a strong periodic variation of the stiffness (among other parameters) experienced by the vehicle. Also, along its route, the Hyperloop will encounter so-called transition zones (e.g., junctions, bridges, etc.), where the properties (e.g., support stiffness) are different than for the rest of the structure. In railway engineering, increased degradation is seen in the vicinity of these transition zones, leading to increased frequency of maintenance. This work investigates response amplification mechanisms in a Hyperloop system that arise due to the combination of a transition zone and the structure having a periodic nature. The amplification mechanisms investigated here can help prevent degradation of the Hyperloop tube close to transition zones as well as fatigue and wear of the vehicle.@en

Under harmonic excitation, soil exhibits softening behaviour that can be captured through the so-called hyperbolic soil model. The response of systems with such a material model can elegantly be obtained using the classical Harmonic Balance Method (HBM). Soil also exhibits nonlinear hysteretic damping under harmonic excitation, feature which is not incorporated in the hyperbolic soil model. The response of a system that includes also the nonlinear hysteretic damping cannot be obtained using the classical HBM. This work demonstrates the application of an advanced HBM (more specifically, alternating frequency-time HBM) for finite and infinite systems that exhibit softening behaviour and nonlinear hysteretic damping. The purpose of this model is to, in the future, investigate the influence of the nonlinear hysteretic damping on the response of such systems, as opposed to linear viscous or hysteretic damping that is usually adopted. To conclude, we show that the advanced HBM is an effective tool for revealing fundamental characteristics of continuous systems with softening behaviour and nonlinear hysteretic damping whose stationary responses consist of either standing or propagating waves.@en

Transition zones in railway tracks are areas with considerable variation of track properties (i.e., foundation stiffness) encountered near structures such as bridges. Due to strong amplification of the railway track’s response, transition zones are prone to rapid degradation. To study this degradation, researchers and engineers have developed models ranging from simple 1-D models (e.g., beam on Winkler foundation) to complex 3-D models with accurate geometry and material behaviour. This study compares a 1-D model to a 2-D one with the aim of assessing if the degradation patterns predicted by the more simplistic model are accurate. We choose a very simple geometry for the 2-D model such that the comparison is restricted to (mainly) the influence of the soil layer (present in the 2-D model) on the predicted degradation at transition zones; incorporating the soil layer makes the response of the supporting structure frequency and wavenumber dependent as well as non-local, characteristics which are not usually incorporated in 1-D models. Preliminary results show that the degradation predicted by the 1-D model is significantly larger than the one in the 2-D model.@en

Locations in railway tracks where significant variations of the track properties occur are subject to increased track deterioration. To successfully mitigate this, the mechanisms leading to the increased deterioration need to be understood. To this end, this work presents a non-linear constitutive law for a lattice model able to describe the compaction behaviour of railway ballast. The parameters of the non-linear connections are tuned against lab experiments of cyclic loading tests and direct shear tests. The tuned lattice can be used with different foundation properties provided that the ballast in the track is equivalent to that of the tests. The non-linear lattice model is applied to the case of railway transitions, for which ballast compaction under train loading is studied as a cause of geometry degradation. It is observed that for the studied cases of a culvert crossing and of a ballast-slab transition, the operation-induced compaction converges monotonously to a stable situation, without leading to significant changes in the vehicle-track interaction. Ballast compaction is therefore insufficient as a stand-alone mechanism to explain a process of progressive degradation of the track geometry. Other mechanisms like autonomous differential settlement at the foundation level must be taken into account in such cases.@en

Transition zones in railway tracks are locations with a significant variation of track properties (i.e. foundation stiffness) encountered near structures such as bridges and tunnels. Due to strong amplification of the track’s response, transition zones are prone to rapid degradation. To investigate the degradation mechanisms in transition zones, researchers have developed a multitude of models, some of them being very complex. This study compares three solution methods, namely an integral-transform method, a time-domain method, and a hybrid method, with the goal of solving these systems efficiently. The methods are compared in terms of accuracy, computational efficiency, and feasibility of application to more complex systems. The model employed in this paper consists of an infinite, inhomogeneous, and piecewise-linear 1-D structure subjected to a moving constant load. Although the 1-D model is not particularly demanding computationally, it is used to make qualitative observations as to which method is most suitable for the 2-D and 3-D models, which could lead to significant gains. Results show that all three methods can reach similar accuracy levels, and in doing so, the time-domain method is most computationally efficient. The integral-transform method appears to be efficient in dealing with frequency-dependent parameters, while the time-domain and hybrid methods are efficient in dealing with a smooth nonlinearity. For multi-dimensional models, if nonlinearities and inhomogeneities are considered throughout the depth, the time-domain method is likely to be most efficient; however, if nonlinearities and inhomogeneities are limited to the surface layers, the integral-transform and hybrid methods have the potential to be more efficient than the time-domain one. Finally, although the 1-D model presented in this study is mainly used to assess the three methods, it can also be used for preliminary designs of transition zones in railway tracks.

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