Prediction of vibrations induced by underground railway traffic in Beijing

This paper examines the problem of subway induced vibrations on line 4 of Beijing metro, which is currently under construction and is planned to pass in close proximity of the Physics Laboratory of Beijing University. The laboratory has a lot of equipment that is very sensitive to traffic induced vibrations and future operation of metro line 4 is a matter of concern. Hence, it is important to study the influence of subway induced vibrations inside the laboratory and to propose a viable solution to mitigate the vibrations. In this paper, the tunnel north of Chengfulu station is modelled using a coupled periodic FE-BE model and the free-field response due to moving trains is predicted. In addition, vibration measurements have been performed on the site of the Physics Laboratory to estimate the existing vibration levels due to road traffic. The predicted and measured vibrations are superimposed to assess the vibrations due to the combined effect of road and railway traffic in the vicinity of the Physics Laboratory. Apart from the numerical investigations, vibration measurements have also been performed on a similar site at line 1 of Beijing metro to substantiate the estimated results on metro line 4. Finally, it is studied how the vibrations can be controlled using a floating slab track, which is widely used as an effective measure of vibration isolation in tunnels. The efficiency of a 7.9 Hz floating slab track as a vibration countermeasure is assessed in this paper. This study demonstrates the applicability of the numerical model for the relevant assessment of subway induced vibrations and its use to study the performance of different track structures in the tunnel.     

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 precise integration method for solving coupled vehicle–track dynamics with nonlinear wheel–rail contact

A new method is proposed as a solution for the large-scale coupled vehicle–track dynamic model with nonlinear wheel–rail contact. The vehicle is simplified as a multi-rigid-body model, and the track is treated as a three-layer beam model. In the track model, the rail is assumed to be an Euler-Bernoulli beam supported by discrete sleepers. The vehicle model and the track model are coupled using Hertzian nonlinear contact theory, and the contact forces of the vehicle subsystem and the track subsystem are approximated by the Lagrange interpolation polynomial. The response of the large-scale coupled vehicle–track model is calculated using the precise integration method. A more efficient algorithm based on the periodic property of the track is applied to calculate the exponential matrix and certain matrices related to the solution of the track subsystem. Numerical examples demonstrate the computational accuracy and efficiency of the proposed method.     

Symplectic analysis of vertical random vibration for coupled vehicle–track systems

A computational model for random vibration analysis of vehicle–track systems is proposed and solutions use the pseudo excitation method (PEM) and the symplectic method. The vehicle is modelled as a mass, spring and damping system with 10 degrees of freedom (dofs) which consist of vertical and pitching motion for the vehicle body and its two bogies and vertical motion for the four wheelsets. The track is treated as an infinite Bernoulli–Euler beam connected to sleepers and hence to ballast and is regarded as a periodic structure. Linear springs couple the vehicle and the track. Hence, the coupled vehicle–track system has only 26 dofs. A fixed excitation model is used, i.e. the vehicle does not move along the track but instead the track irregularity profile moves backwards at the vehicle velocity. This irregularity is assumed to be a stationary random process. Random vibration theory is used to obtain the response power spectral densities (PSDs), by using PEM to transform this random multiexcitation problem into a deterministic harmonic excitation one and then applying symplectic solution methodology. Numerical results for an example include verification of the proposed method by comparing with finite element method (FEM) results; comparison between the present model and the traditional rigid track model and; discussion of the influences of track damping and vehicle velocity.     

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.     

The influence of track modelling options on the simulation of rail vehicle dynamics

This paper investigates the effect of different models for track flexibility on the simulation of railway vehicle running dynamics on tangent and curved track. To this end, a multi-body model of the rail vehicle is defined including track flexibility effects on three levels of detail: a perfectly rigid pair of rails, a sectional track model and a three-dimensional finite element track model. The influence of the track model on the calculation of the nonlinear critical speed is pointed out and it is shown that neglecting the effect of track flexibility results in an overestimation of the critical speed by more than 10%. Vehicle response to stochastic excitation from track irregularity is also investigated, analysing the effect of track flexibility models on the vertical and lateral wheel–rail contact forces. Finally, the effect of the track model on the calculation of dynamic forces produced by wheel out-of-roundness is analysed, showing that peak dynamic loads are very sensitive to the track model used in the simulation.     

Fluctuation mechanism of formation of discontinuous tracks by fast ions in solids

The effect of one-electron and many-electron charge-exchange processes involving fast heavy ions on the formation of tracks in crystals is considered. The proposed model of fluctuations of charge distributions gives reasonable estimates for the length of defects regions and the distance between defects in discontinuous tracks. The modified model of a thermal peak predicts the track radius. The validity of the model for explaining all the properties of formation of discontinuous and continuous tracks is demonstrated by the example of InP crystals irradiated by 250-MeV xenon ions.     

Theory of electrodynamic levitation with a continuous sheet track — Part I

Exact analytical expressions for forces on moving rectangular current-carrying coils above and below an infinite conducting sheet track of arbitrary thickness are developed. These general expressions for the lift and drag forces acting on the excitation coils as functions of speed are investigated for normal flux, null flux and brake flux systems and discussed with the aid of numerical calculations. A concise parameter study is also made. The system velocity characteristics with constant load (technologically more relevant) are given for the first time for normal and null flux systems.     

Fully three-dimensional analysis of high-speed train-track-soil-structure dynamic interaction

In this paper, a general and fully three dimensional multi-body-finite element-boundary element model, formulated in the time domain to predict vibrations due to train passage at the vehicle, the track and the free field, is presented. The vehicle is modelled as a multi-body system and, therefore, the quasi-static and the dynamic excitation mechanisms due to train passage can be considered. The track is modelled using finite elements. The soil is considered as a homogeneous half-space by the boundary element method. This methodology could be used to take into account local soil discontinuities, underground constructions such as underpasses, and coupling with nearby structures that break the uniformity of the geometry along the track line. The nonlinear behaviour of the structures could be also considered. In the present paper, in order to test the model, vibrations induced by high-speed train passage are evaluated for a ballasted track. The quasi-static and dynamic load components are studied and the influence of the suspended mass on the vertical loads is analyzed. The numerical model is validated by comparison with experimental records from two HST lines. Finally, the dynamic behaviour of a transition zone between a ballast track and a slab track is analyzed and the obtained results from the proposed model are compared with those obtained from a model with invariant geometry with respect to the track direction.     

Reducing slab track vibration into bridge using elastic materials in high speed railway

In this paper, the problem of vibration transmission from slab track structures into bridge is studied by theoretical analysis. A vehicle-track-bridge coupling system dynamics model is established based on a multibody dynamics theory and a finite element method. The system model consists of vehicle model, track-bridge model and wheel/rail interaction model. The vehicle model is established based on the multibody dynamics theory, and the tack-bridge model is established by the finite element method. The vehicle model and track-bridge model are coupled through wheel/rail interaction model, and the track irregularities are included. The system dynamic responses are calculated, and the effectiveness of elastic materials in vibration reducing is discussed. The results demonstrate that elastic materials like slab mat layer inserted between slab track and bridge can reduce vibration transmitted from track into the bridge. Some suggestions for the design and application of slab mat are provided in the end of the paper.     

Synchronization in arrays of coupled self-induced friction oscillators

We investigate synchronization phenomena in systems of self-induced dry friction oscillators with kinematic excitation coupled by linear springs. Friction force is modelled according to exponential model. Initially, a single degree of freedom mass-spring system on a moving belt is considered to check the type of motion of the system (periodic, non-periodic). Then the system is coupled in chain of identical oscillators starting from two, up to four oscillators. A reference probe of two coupled oscillators is applied in order to detect synchronization thresholds for both periodic and non-periodic motion of the system. The master stability function is applied to predict the synchronization thresholds for longer chains of oscillators basing on two oscillator probe. It is shown that synchronization is possible both for three and four coupled oscillators under certain circumstances. Our results confirmed that this technique can be also applied for the systems with discontinuities.     

Electromechanical tailoring of structure with periodic piezoelectric circuitry

In this paper, we study the integration of piezoelectric transducers and electrical circuitry onto a mechanical host structure to form an electromechanical periodic system. This scheme enables us to take advantage of the unique wave propagation characteristics of spatially periodic systems and thus manipulate the energy flow/distribution in the host mechanical structure. Different from most previous investigations where spatial periodicity is typically due to the periodic discontinuities in the mechanical structure, in the proposed scheme the mechanical structure can remain to be continuous without being altered. Instead, the electrical circuitry consists of periodic branches which, when coupled to the mechanical structure through the piezoelectric transducers, yields repetitive subsystems in the electromechanical integrated system. The circuitry elements can be online tunable, effectively making the energy manipulation adaptive to excitation variations. A transfer matrix-based approach is adopted in the dynamic analysis of the integrated system, where each subsystem is represented by two state vectors with a transfer matrix relating them. Given the repetitive nature of the subsystems due to periodicity, analysis on the entire system is conducted by focusing on one single subsystem. Systematic analysis is carried out to demonstrate the characteristics of wave propagation and attenuation in terms of propagation constants as well as the mechanical and electrical responses. Effects of circuitry tuning are also discussed through detailed parametric studies. Implications and potential applications of the proposed scheme are illustrated.     

The effect of critically moving loads on the vibrations of soft soils and isolated railway tracks

The dynamic response of the railway track is strongly influenced by the underlying soil. For a soft soil and very high train speeds or for a very soft soil and regular train speeds, the train speed can be close to the speed of elastic waves in the soil. This paper presents a detailed study of the so-called “moving-load effect”, i.e. an amplification of the dynamic response due to the load movement, for the tracks on soft soil. The analysis is carried out by evaluating the related integrals in the wavenumber domain. The influence of the load speed is quantified for a large set of parameters, showing that the effect on the soil vibration is reduced with increase of the frequency, track width and inverse wave velocity. Therefore, the moving-load effect associated with vibratory train loads is negligible whereas the amplification associated with the moving dead weight of the train can be significant. The strong moving-load effect on a perfectly homogeneous soil, however, can be strongly diminished by a layered or randomly varying soil situation. This theoretical result is affirmed by measurements at a test site in Germany where the trains run on a very soft soil at a near-critical speed. The results for soft soils are compared with experimental and theoretical results for a stiff soil. It is found that the influence of the stiffness of the soil is much stronger than the moving-load effect. This holds for the soil vibration as well as for the track vibration which both show a minor dependence on the load speed but a considerable dependence on the soil stiffness in theory and experiment.Railway tracks can include soft isolation elements such as rail pads, sleeper shoes and ballast mats. For these types of isolation elements and normal soil conditions, the influence of the load speed is usually negligible. There is only one isolation measure for which the moving load may be effective: a track which is constructed as a heavy mass-spring system. The resonance of this track system is shifted to lower frequencies and amplitudes for increasing train speed. A critical train speed can be reached if the mass-spring system has a marginal bending stiffness along the track.     

A rational fraction polynomials model to study vertical dynamic wheel–rail interaction

This paper presents a model designed to study vertical interactions between wheel and rail when the wheel moves over a rail welding. The model focuses on the spatial domain, and is drawn up in a simple fashion from track receptances. The paper obtains the receptances from a full track model in the frequency domain already developed by the authors, which includes deformation of the rail section and propagation of bending, elongation and torsional waves along an infinite track. Transformation between domains was secured by applying a modified rational fraction polynomials method. This obtains a track model with very few degrees of freedom, and thus with minimum time consumption for integration, with a good match to the original model over a sufficiently broad range of frequencies. Wheel–rail interaction is modelled on a non-linear Hertzian spring, and consideration is given to parametric excitation caused by the wheel moving over a sleeper, since this is a moving wheel model and not a moving irregularity model. The model is used to study the dynamic loads and displacements emerging at the wheel–rail contact passing over a welding defect at different speeds.     

A railway track dynamics model based on modal substructuring and a cyclic boundary condition

This article presents a technique for modelling the coupled dynamics of a railway vehicle and the track. The method is especially useful for simulating the dynamics of high speed trains running on nonlinear tracks. The main hypothesis is a cyclic system: an infinite track on which there is an infinite set of identical vehicles spaced at a regular interval of distance. Thus the main problems of the finite-length track models (e.g. the waves that reflect at the end of the track and interact with the vehicle; and the time interval of integration must be shorter than the track length divided by the velocity) are avoided. The flexibility of the method can be observed from the case studies presented in the present work: a vehicle passing over a hanging sleeper, and the vehicle-track dynamics for different ballast compaction cases. The results show the influence of the hanging sleeper gap on the wheel-rail contact forces, and the bending moment at the sleeper for different ballast compaction cases.     

Prediction of rail corrugation using a rotating flexible wheelset coupled with a flexible track model and a non-Hertzian/non-steady contact model

This paper presents a model for simulating vehicle–track interaction at high frequencies for investigations of rail roughness growth. The dynamic interaction model developed employs a substructuring technique and the whole system consists of a number of substructures that can be modelled independently. The systems are coupled through the forces at the wheel–rail contact and the railpad. A coupled, rotating flexible wheelset, a flexible track model and a non-Hertzian/non-steady contact model have been implemented and results are presented here for a free wheelset on a symmetrical track system with initial random and sinusoidal roughness. Both rigid and flexible wheelsets are considered.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

Noise-induced asymptotic periodicity in a piecewise linear map

We examine asymptotically periodic density evolution in one-dimensional maps perturbed by noise, associating the macroscopic state of these dynamical systems with a phase space density. For asymptotically periodic systems density evolution becomes periodic in time, as do some macroscopic properties calculated from them. The general formalism of asymptotic periodicity is examined and used to calculate time correlations along trajectories of these maps as well as their limiting conditional entropy. The time correlation is shown to naturally decouple into periodic and stochastic components. Finally, asymptotic periodicity is studied in a noise-perturbed piecewise linear map, focusing on how the variation of noise amplitude can cause a transition from asymptotic periodicity to asymptotic stability in the density evolution of this system.     

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.