Simulation of track-ground vibrations due to a high-speed train: the case of X-2000 at Ledsgard

This paper attempts to demonstrate an application of the author's simulation model for predicting the train-track and nearby ground-borne vibrations by the Swedish high-speed train X-2000 at Ledsgard. The validation of the computation results are tested against the available field measurement data at the site. In this way, the theoretical prediction of the model can be verified whilst also providing a clear-cut explanation of the observed data. The findings are stated as follows:The train-induced vibrations at the track differ significantly depending on train geometry and speed. At low speeds the response is quasi-static so that the track response due to train axle loads appears mostly downward at the point of their action. On the other hand, at high speeds the train-induced response becomes dynamic due to the inertia generated in the track-ground system, so that the track vibrations appear evenly in both upward and downward directions. As the results of the soft soil deposits at Ledsgard, the high train speed is almost in the trans-Rayleigh wave state so that a large amplified track response appeared due to the resonance between the track behavior and the Rayleigh wave propagation in the ground. This is explained by the frequency-wavenumber spectrum.To provide useful engineering information relating to vibration mitigation at the train track and nearby ground, a preliminary investigation was carried out by simulating the effect of constructing of a wave impeding barrier (WIB) at the site. The aforementioned frequency-wavenumber spectrum showed that the stiffening effects by the WIB installation into soft layers led a shift of response from a large dynamic one at high train speeds to a small, quasi-static one.     

Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load

The present paper investigates the convergence of the Galerkin method for the dynamic response of an elastic beam resting on a nonlinear foundation with viscous damping subjected to a moving concentrated load. It also studies the effect of different boundary conditions and span length on the convergence and dynamic response. A train–track or vehicle–pavement system is modeled as a force moving along a finite length Euler–Bernoulli beam on a nonlinear foundation. Nonlinear foundation is assumed to be cubic. The Galerkin method is utilized in order to discretize the nonlinear partial differential governing equation of the forced vibration. The dynamic response of the beam is obtained via the fourth-order Runge–Kutta method. Three types of the conventional boundary conditions are investigated. The railway tracks on stiff soil foundation running the train and the asphalt pavement on soft soil foundation moving the vehicle are treated as examples. The dependence of the convergence of the Galerkin method on boundary conditions, span length and other system parameters are studied.     

Vertical dynamic response of non-uniform motion of high-speed rails

In this paper, a computational study using the moving element method (MEM) is carried out to investigate the dynamic response of a high-speed rail (HSR) traveling at non-uniform speeds. A new and exact formulation for calculating the generalized mass, damping and stiffness matrices of the moving element is proposed. Two wheel–rail contact models are examined. One is linear and the other nonlinear. A parametric study is carried out to understand the effects of various factors on the dynamic amplification factor (DAF) in contact force between the wheel and rail such as the amplitude of acceleration/deceleration of the train, the severity of railhead roughness and the wheel load. Resonance in the vibration response can possibly occur at various stages of the journey of the HSR when the speed of the train matches the resonance speed. As to be expected, the DAF in contact force peaks when resonance occurs. The effects of the severity of railhead roughness and the wheel load on the occurrence of the jumping wheel phenomenon, which occurs when there is a momentary loss of contact between the wheel and track, are investigated.     

Theoretical modeling and characteristic analysis of moving-train induced ground vibrations

An integrated train-track-subsoil dynamic interaction model of moving-train induced ground vibration is developed on the basis of vehicle dynamics, track dynamics and the Green's functions of subsoil. The model takes account of the vibrations of vehicle components, the quasi-static axle loads and the dynamic excitations between the wheels and track. The analyzed results from an example show that the ground vibration characteristics have a close relationship with train speed and soil properties; the dynamic responses excited by wheel-track irregularity have big influence on the high frequency components of ground vibration; with the increase of distance to the track, the ground acceleration has the tendency of decrease, and the relevance of acceleration curves and train excitation becomes less obvious; the intersections of moving load speed-lines and subsoil dispersion curves are some resonance frequencies that cause the amplification of ground vibrations; there exists a critical speed for moving train that is close to the minimum velocity of the Rayleigh's wave in the subsoil.     

A multi-stage high-speed railroad vibration isolation system with “negative” stiffness

The intensity of mechanical vibrations generated by a moving train increases at high speeds. Conventional railroad vibration isolation systems are either insensitive or inefficient in the infra-low frequency range, which is the range most hazardous to humans. In this work, an approach for upgrading a multi-stage system for engineers, crew, passengers, and people living near railroads is presented. The approach involves the design of geometrically similar redundant mechanisms with “negative” stiffness that could be compactly inserted into all system stages without degrading structure and safety. The approach is implemented in the design of seat suspensions. Finally, a concept relevant to the design of mechanisms for vehicle spring hangers and track bed elements is discussed in terms of the dimensional analysis.     

A numerical model for calculating vibration from a railway tunnel embedded in a full-space

Vibration generated by underground railways transmits to nearby buildings causing annoyance to inhabitants and malfunctioning to sensitive equipment. Vibration can be isolated through countermeasures by reducing the stiffness of railpads, using floating-slab tracks and/or supporting buildings on springs. Modelling of vibration from underground railways has recently gained more importance on account of the need to evaluate accurately the performance of vibration countermeasures before these are implemented.This paper develops an existing model, reported by Forrest and Hunt, for calculating vibration from underground railways. The model, known as the Pipe-in-Pipe model, has been developed in this paper to account for anti-symmetrical inputs and therefore to model tangential forces at the tunnel wall. Moreover, three different arrangements of supports are considered for floating-slab tracks, one which can be used to model directly-fixed slabs. The paper also investigates the wave-guided solution of the track, the tunnel, the surrounding soil and the coupled system. It is shown that the dynamics of the track have significant effect on the results calculated in the wavenumber-frequency domain and therefore an important role on controlling vibration from underground railways.     

A comparison of a theoretical model for quasi-statically and dynamically induced environmental vibration from trains with measurements

This paper presents comparisons between a theoretical ground vibration model and measured data at three sites. The model, which is briefly outlined here, encompasses both the quasi-static and dynamic mechanisms of excitation. The vertical dynamics of a number of vehicles travelling at a constant speed on an infinite track are coupled to a semi-analytical model for a three-dimensional layered ground. This model is also used to demonstrate the roles of the two components of vibration at different frequencies and for train speeds below and above the lowest ground wave speed. It is found that, in most practical cases, the dynamic component gives rise to the higher level of vibration.     

Prediction and measurements of vibrations from a railway track lying on a peaty ground

This paper introduces a two-dimensional model for the response of the ground surface due to vibrations generated by a railway traffic. A semi-analytical wave propagation model is introduced which is subjected to a set of harmonic moving loads and based on a calculation method of the dynamic stiffness matrix of the ground. In order to model a complete railway system, the effect of a simple track model is taken into account including rails, sleepers and ballast especially designed for the study of low vibration frequencies. The priority has been given to a simple formulation based on the principle of spatial Fourier transforms compatible with good numerical efficiency and yet providing quick solutions. In addition, in situ measurements for a soft soil near a railway track were carried out and will be used to validate the numerical implementation. The numerical and experimental results constitute a significant body of useful data to, on the one hand, characterize the response of the environment of tracks and, on the other hand, appreciate the importance of the speed and weight on the behaviour of the structure.     

Numerical analysis of isolation of the vibration due to moving loads using pile rows

The isolation of the vibration due to moving loads using pile rows embedded in a poroelastic half-space is investigated in this study. Based on Biot's theory and integral transform method, the free field solution for a moving load applied on the surface of a poroelastic half-space and the fundamental solution for a harmonic circular patch load applied in the poroelastic half-space are derived first. Using Muki and Sternberg's method and the fundamental solution for the circular patch load as well as the obtained free field solution for the moving load, the second kind of Fredholm integral equations in the frequency domain describing the dynamic interaction between pile rows and the poroelastic half-space is developed. Numerical solution of the frequency domain integral equations and numerical inversion of the Fourier transform yield the time domain response of the pile–soil system. Comparison of our results with some known results shows that our results are in a good agreement with existing ones. Numerical results of this study show that velocity of moving loads has an important impact on the vibration isolation effect of pile rows. The same pile row has a better vibration isolation effect for the lower speed moving loads than for the higher speed ones. Also, the optimal length of piles for higher speed moving loads is shorter than that for lower speed moving loads. Moreover, stiff pile rows tend to produce a better vibration isolation effect than flexible pile rows do.     

A new track inspection car based on a laser camera system

We develop and build a new type of inspection car.A beam that is not rigidly connected to the train axle boxes and can absorb the vibration and impact caused by the high speed train is used,and a laser-camera measurement system based on the machine vision method is adopted.This method projects structural light onto the track and measures gauge and longitudinal irregularity.The measurement principle and model are discussed.Through numerous practical experiments,the rebuilt car is found to considerably eliminate the measurement errors caused by vibration and impact,thereby increasing measurement stability under high speeds.This new kind of inspection cars have been used in several Chinese administration bureaus.     

The noise behavior of the wheel/rail-system— some supplementary results

Acoustical measurements were carried out on railroad coaches, on standard tracks and in the free field during test runs. In particular the influences of noise parameters like train speed, track condition, wheel type or locomotive propulsion were examined. Among other things, it appeared that the track conditions can vary considerably, a fact that has a great influence on all measurement values. One obtains a kind of “track profile” relatively independent of the train speed. Measurements both on the parts of the rail and in the free field during the pass-by of a train wheel, just as do the measurements of the wheel levels at the same time, indicate that the rail in the frequency range between 500 and 1200 Hz is the most important factor with regard to sound radiation. Only above this range is the wheel the essential radiator, mainly in the range around 2000 Hz. Further it could be ascertained that the total acceleration levels of the wheel rim have a greater speed exponent than the total acceleration levels of the rail. This can be important if one makes an extrapolation for high train speeds. Additional damping of coach wheels results in a greater noise reduction not only for the radiated sound but also for the structure-borne sound at the rails. This fact indicates the relatively strong coupling between rail and wheel. Furthermore it was ascertained that the levels generated by a locomotive in the upper frequency range are similar to those produced by damped coach wheels. A propulsion influence of an electrical locomotive on the radiated total sound level could not be ascertained. In the last section possible noise generating mechanisms are pointed out with regard to their importance as indicated by our present state of knowledge.     

Effect on vehicle and track interaction of installation faults in the concrete bearing surface of a direct-fixation track

Various installation faults can occur in fasteners in the construction of a direct-fixation track using the top-down method. In extreme cases, these faults may cause excessive interaction between the train and track, compromise the running safety of the train, and cause damage to the track components. Therefore, these faults need to be kept within the allowable level through an investigation of their effects on the interactions between the train and track. In this study, the vertical dynamic stiffness of fasteners in installation faults was measured based on a dynamic stiffness test by means of an experimental apparatus that was devised to feasibly reproduce installation faults with an arbitrary shape. This study proposes an effective analytical model for a train–track interaction system in which most elements, except the nonlinear wheel–rail contact and some components that behave bilinearly, exhibit linear behavior. To investigate the effect of the behavior of fasteners in installation faults in a direct-fixation track on the vehicle and track, vehicle–track interaction analyses were carried out, targeting key review parameters such as the wheel load reduction factor, vertical rail displacement, wheel load, and mean stress of the elastomer. From the results, it was noted that it is more important for the installation faults in the concrete bearing surface of a direct-fixation track to be limited for the sake of the long-term durability of the elastomer rather than for the running safety of the train or the structural safety of the track.     

菱形HSLDS隔振器负刚度机构质量及摩擦力影响分析

以菱形负刚度机构HSLDS(high static low dynamic stiffness)隔振器(简称菱形HSLDS隔振器)为研究目标,采用虚功法建立负刚度机构等效摩擦力模型,并以拉格朗日方法建立包含负刚度机构质量及摩擦力因素的动力学方程;利用谐波平衡法(HBM)求解动力学方程,分析了负刚度机构质量及摩擦力对隔振...     

Ground-borne vibration due to static and dynamic axle loads of InterCity and high-speed trains

In predictions of railway-induced vibrations, a distinction is generally made between the quasi-static and dynamic excitation. The quasi-static excitation is related to the static component of the axle loads. The dynamic excitation is due to dynamic train–track interaction, which is generated by a large number of excitation mechanisms, such as the spatial variation of the support stiffness and the wheel and track unevenness. In the present paper, the quasi-static excitation and the dynamic excitation due to random track unevenness are evaluated by means of numerical predictions. A solution strategy is presented that allows for the evaluation of the second-order statistics of the response due to dynamic excitation based on the power spectral density function of the track unevenness. Due to the motion of the train, the second-order statistics of the response at a fixed point in the free field are non-stationary and an appropriate solution procedure is required. The quasi-static and dynamic contribution to the track and free-field response are analysed for the case of InterCity and high-speed trains running at a subcritical train speed. It is shown how the train speed affects the quasi-static and dynamic contribution. Finally, results of numerical predictions for different train speeds are compared with field measurements that have been performed at a site along the high-speed line L2 Brussels–Köln within the frame of homologation tests.     

Modelling of continuous and discontinuous floating slab tracks in a tunnel using a periodic approach

This paper presents a periodic approach to couple a track and a tunnel-soil system of different periodicity. The periodicity of the track and the tunnel-soil system is exploited using the Floquet transform to efficiently formulate the problem in the frequency-wavenumber domain as well as to limit the discretization effort to a reference cell. The track and the tunnel-soil system are modelled as two separate systems of different periodicity and are coupled in the frequency-wavenumber domain. A coupled periodic finite element-boundary element method is used to model the tunnel-soil system, while a periodic finite element model or an analytical approach is used to model the track.A general analytical formulation to compute the response of three-dimensional periodic media that are excited by moving loads is discussed. It is shown that the response due to moving loads on the track can be calculated from the transfer function of the track-tunnel-soil system and the axle loads.A methodology for computing the transfer functions of the coupled track-tunnel-soil system as well as the computation of dynamic forces accounting for the interaction between the moving vehicle and the periodic track are described. The model accounts for quasi-static forces as well as dynamic forces due to parametric excitation and unevenness excitation.The methodology has been used to assess the vibration isolation efficiency of continuous and discontinuous floating slab tracks. It is concluded that both continuous and discontinuous floating slab tracks have a similar efficiency in the frequency range well above the isolation frequency of the slabs, which is usually higher than the slab passage frequency. In case of discontinuous slab tracks, the parametric excitation is found to be important, which results in a poorer performance of the track at low frequencies.     

Ground vibrations from heavy freight trains

Ground vibration from heavy freight trains on good quality welded track are found to have only a weak dependence on train speed above 30 km/h. At the site on which these tests were carried out a critical speed was found at which the vibration reached a peak. The frequencies of vibration produced appear to be functions of track and vehicle dimensions and the critical speed occurs at the coincidence of sleeper passage frequency and the total vehicle on track resonance frequency.     

On the dynamics of interaction between a moving mass and an infinite one-dimensional elastic structure at the stability limit

The paper herein deals with the study of the dynamic behaviour generated by the instability of the vibration of a loaded mass, uniformly moving along an Euler-Bernoulli beam on a viscoelastic foundation, induced by the anomalous Doppler waves excited in the beam. This issue is relevant for the case of modern trains travelling along a track with soft soil when the trains speed exceeds the phase velocity of the waves induced in the track. The model corresponds to a railway vehicle reduced to a loaded wheel running along a (half) track. The beam takes account of the bending stiffness of the rail and the mass of the track, including the mass of the rail, semi-sleepers and half of the ballast layer, where the viscoelastic foundation represents the subgrade. The model includes the wheel/rail Hertzian contact and it allows the simulation of the possibility of contact loss. The nonlinear equations of motion are integrated using a numerical approach based on the Green’s function method. When the vibration becomes unstable, the system evolution is a limit cycle characterised by a succession of shocks, due to the action of two opposite factors: the anomalous Doppler waves that pump energy at the interface between the moving mass and the beam, thus forcing the mass to take off, and the static load that push the mass downwards. The frequency of the shocks increases at higher velocity and the magnitude of the impact force decreases; the most dangerous velocity is the critical one, which represents the stability limit of the linear approximation of the motion equations. The transient behaviour that precedes the limit cycle appearance is being analysed. The Hertzian contact influences the time history of the limit cycle and the magnitude of the impact force and, therefore, it is essential to be included in the model. To the authors’ knowledge, this problem has never been dealt with.     

一种光电载荷非线性隔振装置的研究

针对光电载荷对隔振性能的需求,提出一种采用菱形连杆机构作为负刚度组件,具有高静、低动刚度特点的非线性隔振器（简称菱形HSLDS隔振器）。采用静力学分析方法,建立了隔振器数学模型,研究了刚度参数设定以及非线性调节方法;利用谐波平衡法（HBM）求解动力学方程,分析了各参数对隔振性能的影响关系;采用动力学仿真软件ADAMS及实物样机对理论模型与结论进行了验证。测试结果表明:菱形HSLDS隔振器具有较方便的参数调整能力,零位刚度及刚度非线性可通过拉簧参数与连杆参数进行设定、优化,隔振的刚度非线性优化程度受主隔振器阻尼以及零位刚度参数影响。相比于传统线性隔振器,菱形HSLDS具有显著的非线性隔振优势,可较好地满足光电载荷隔振需求。     

Prediction of environmental noise from fast electric trains

A model is presented for the environmental noise of fast electric trains on continuous welded rails, based on a consideration of the possible modes of vibration of the wheels. Evidence is presented to support the idea that the wheels are the dominant sources of noise and radiate as resonant dipoles. The model is calibrated by the results of measurements and methods of predicting train noise level and noise energy (per unit area) are developed. The noise intensity is found to increase as the fourth power of the speed. Energy (per unit area) at an observer increases as the third power of the speed and is proportional to train length divided by the distance of the observer from the track. The relevance of the model as a basis for the calculation of train noise in terms of some noise indices is then discussed and it is shown that it can be used to predict the parameters required by some indices which may be relevant to the subjective effects of environmental train noise.     

弹性基础隔振系统的简化性能指标和有源控制力

系统研究了基础弹性对单层隔振系统、双层隔振系统及浮筏隔振系统隔振性能的影响。分析了不同隔振系统与不同弹性基础间的振动耦合特性,讨论了不同隔振系统的振级落差和力传递率特性,给出了振级落差和力传递率的简化计算方法。针对不同隔振系统的有源隔振问题,比较了不同作动器安装方式所需的控制力。研究表明,对于所有隔振系统,增加基础的刚度和阻尼有利于提高振级落差和力传递率;对于浮筏隔振系统,增加筏架的刚度和阻尼有利于提高隔振性能和减少有源隔振所需的控制力。