Dynamics of the railway track and the underlying soil: the boundary-element solution,theoretical results and their experimental verification

A combined finite-element boundary-element method is presented in detail to calculate the dynamic interaction of the railway track and the underlying soil. A number of results are shown for ballasted and slab track, demonstrating the influence of the stiffness of the soil and the rail pads on the vertical compliance of the track. The compliance of the track is combined with a simple model of the vehicle giving the transfer function of vehicle–track interaction. An experimental verification of the theoretical results is achieved by harmonic and impulse excitation with and without static (train-) load and by combined measurements of train–track–soil interaction. A clear vehicle–track resonance is found for the slab track with elastic rail pads and for higher frequencies at highspeed traffic, the dynamic axle loads due to sleeper passage are reduced.     

Effects of rail thermal stress on the dynamic response of vehicle and track

A new method is proposed to obtain the dynamic responses of the vehicle–track coupling system under the conditions of rail thermal stress changes in high-speed railways. Exact models are established with different rail longitudinal forces, in which multibody dynamic models are used for vehicles and the direct stiffness method for structures. In order to provide a general, simple and flexible formulation to express longitudinal stress distribution, the accurate model of long slab track consists of many small units with parameters which can be initialised separately. The exact analytical equation of track frequency and modal function was obtained by the transition matrix method, which can be used in calculating the dynamic response of wheel–rail coupling model. The proposed model is verified through comparisons with other classical solutions. Under the influence of train velocities and track irregularities, the specific vibration performances that frequency shifted and amplitude peak enhanced with thermal force are demonstrated through examples. The results show that the response analyses of vehicle and track have great application potentiality for fast estimation of the rail longitudinal stress.     

Low-frequency vibration control of floating slab tracks using dynamic vibration absorbers

This study aims to effectively and robustly suppress the low-frequency vibrations of floating slab tracks (FSTs) using dynamic vibration absorbers (DVAs). First, the optimal locations where the DVAs are attached are determined by modal analysis with a finite element model of the FST. Further, by identifying the equivalent mass of the concerned modes, the optimal stiffness and damping coefficient of each DVA are obtained to minimise the resonant vibration amplitudes based on fixed-point theory. Finally, a three-dimensional coupled dynamic model of a metro vehicle and the FST with the DVAs is developed based on the nonlinear Hertzian contact theory and the modified Kalker linear creep theory. The track irregularities are included and generated by means of a time–frequency transformation technique. The effect of the DVAs on the vibration absorption of the FST subjected to the vehicle dynamic loads is evaluated with the help of the insertion loss in one-third octave frequency bands. The sensitivities of the mass ratio of DVAs and the damping ratio of steel-springs under the floating slab are discussed as well, which provided engineers with the DVA's adjustable room for vibration mitigation. The numerical results show that the proposed DVAs could effectively suppress low-frequency vibrations of the FST when tuned correctly and attached properly. The insertion loss due to the attachment of DVAs increases as the mass ratio increases, whereas it decreases with the increase in the damping ratio of steel-springs.     

Dynamics of the railway track and the underlying soil: the boundary-element solution, theoretical results and their experimental verification

A combined finite-element boundary-element method is presented in detail to calculate the dynamic interaction of the railway track and the underlying soil. A number of results are shown for ballasted and slab track, demonstrating the influence of the stiffness of the soil and the rail pads on the vertical compliance of the track. The compliance of the track is combined with a simple model of the vehicle giving the transfer function of vehicle-track interaction. An experimental verification of the theoretical results is achieved by harmonic and impulse excitation with and without static (train-) load and by combined measurements of train-track-soil interaction. A clear vehicle-track resonance is found for the slab track with elastic rail pads and for higher frequencies at highspeed traffic, the dynamic axle loads due to sleeper passage are reduced.     

Effect of curved track support failure on vehicle derailment

In order to investigate the effect of curved track support failure on railway vehicle derailment, a coupled vehicle–track dynamic model is put forward. In the model, the vehicle and the structure under rails are, respectively, modelled as a multi-body system, and the rail is modelled with a Timoshenko beam rested on the discrete sleepers. The lateral, vertical, and torsional deformations of the beam are taken into account. The model also considers the effect of the discrete support by sleepers on the coupling dynamics of the vehicle and track. The sleepers are assumed to move backward at a constant speed to simulate the vehicle running along the track at the same speed. In the calculation of the coupled vehicle and track dynamics, the normal forces of the wheels/rails are calculated using the Hertzian contact theory and their creep forces are determined with the nonlinear creep theory by Shen et al [Z.Y. Shen, J.K. Hedrick, and J.A. Elkins, A comparison of alternative creep-force models for rail vehicle dynamic analysis, Proceedings of the 8th IAVSD Symposium, Cambridge, MA, 1984, pp. 591–605]. The motion equations of the vehicle/track are solved by means of an explicit integration method. The failure of the components of the curved track is simulated by changing the track stiffness and damping along the track. The cases where zero to six supports of the curved rails fail are considered. The transient derailment coefficients are calculated. They are, respectively, the ratio of the wheel/rail lateral force to the vertical force and the wheel load reduction. The contact points of the wheels/rails are in detail analysed and used to evaluate the risk of the vehicle derailment. Also, the present work investigates the effect of friction coefficient, axle load and vehicle speed on the derailments under the condition of track failure. The numerical results obtained indicate that the failure of track supports has a great influence on the whole vehicle running safety.     

Investigation of the effects of sleeper-passing impacts on the high-speed train

The sleeper-passing impact has always been considered negligible in normal conditions, while the experimental data obtained from a High-speed train in a cold weather expressed significant sleeper-passing impacts on the axle box, bogie frame and car body. Therefore, in this study, a vertical coupled vehicle/track dynamic model was developed to investigate the sleeper-passing impacts and its effects on the dynamic performance of the high-speed train. In the model, the dynamic model of vehicle is established with 10 degrees of freedom. The track model is formulated with two rails supported on the discrete supports through the finite element method. The contact forces between the wheel and rail are estimated using the non-linear Hertz contact theory. The parametric studies are conducted to analyse effects of both the vehicle speeds and the discrete support stiffness on the sleeper-passing impacts. The results show that the sleeper-passing impacts become extremely significant with the increased support stiffness of track, especially when the frequencies of sleeper-passing impacts approach to the resonance frequencies of wheel/track system. The damping of primary suspension can effectively lower the magnitude of impacts in the resonance speed ranges, but has little effect on other speed ranges. Finally, a more comprehensively coupled vehicle/track dynamic model integrating with a flexible wheel set is developed to discuss the sleeper-passing-induced flexible vibration of wheel set.     

Simulation of vertical dynamic vehicle–track interaction using a two-dimensional slab track model

The vertical dynamic interaction between a railway vehicle and a slab track is simulated in the time domain using an extended state-space vector approach in combination with a complex-valued modal superposition technique for the linear, time-invariant and two-dimensional track model. Wheel–rail contact forces, bending moments in the concrete panel and load distributions on the supporting foundation are evaluated. Two generic slab track models including one or two layers of concrete slabs are presented. The upper layer containing the discrete slab panels is described by decoupled beams of finite length, while the lower layer is a continuous beam. Both the rail and concrete layers are modelled using Rayleigh–Timoshenko beam theory. Rail receptances for the two slab track models are compared with the receptance of a traditional ballasted track. The described procedure is demonstrated by two application examples involving: (i) the periodic response due to the rail seat passing frequency as influenced by the vehicle speed and a foundation stiffness gradient and (ii) the transient response due to a local rail irregularity (dipped welded joint).     

Influence of some vehicle and track parameters on the environmental vibrations induced by railway traffic

A study is performed on the influence of some typical railway vehicle and track parameters on the level of ground vibrations induced in the neighbourhood. The results are obtained from a previously validated simulation framework considering in a first step the vehicle/track subsystem and, in a second step, the response of the soil to the forces resulting from the first analysis. The vehicle is reduced to a simple vertical 3-dof model, corresponding to the superposition of the wheelset, the bogie and the car body. The rail is modelled as a succession of beam elements elastically supported by the sleepers, lying themselves on a flexible foundation representing the ballast and the subgrade. The connection between the wheels and the rails is realised through a non-linear Hertzian contact. The soil motion is obtained from a finite/infinite element model. The investigated vehicle parameters are its type (urban, high speed, freight, etc.) and its speed. For the track, the rail flexural stiffness, the railpad stiffness, the spacing between sleepers and the rail and sleeper masses are considered. In all cases, the parameter value range is defined from a bibliographic browsing. At the end, the paper proposes a table summarising the influence of each studied parameter on three indicators: the vehicle acceleration, the rail velocity and the soil velocity. It namely turns out that the vehicle has a serious influence on the vibration level and should be considered in prediction models.     

Railway ground vibrations induced by wheel and rail singular defects

Railway local irregularities are a growing source of ground-borne vibration and can cause negative environmental impacts, particularly in urban areas. Therefore, this paper analyses the effect of railway track singular defects (discontinuities) on ground vibration generation and propagation. A vehicle/track/soil numerical railway model is presented, capable of accurately predicting vibration levels. The prediction model is composed of a multibody vehicle model, a flexible track model and a finite/infinite element soil model. Firstly, analysis is undertaken to assess the ability of wheel/rail contact models to accurately simulate the force generation at the wheel/rail contact, in the presence of a singular defect. It is found that, although linear contact models are sufficient for modelling ground vibration on smooth tracks, when singular defects are present higher accuracy wheel/rail models are required. Furthermore, it is found that the variation in wheel/rail force during the singular defect contact depends on the track flexibility, and thus requires a fully coupled vehicle/track/foundation model. Next, a parametric study of ground vibrations generated by singular rail and wheel defects is undertaken. Six shapes of discontinuity are modelled, representing various defect types such as transition zones, switches, crossings, rail joints and wheel flats. The vehicle is modelled as an AM96 train set and it is found that ground vibration levels are highly sensitive to defect height, length and shape.     

Dynamic responses of a high-speed railway car due to wheel polygonalisation

The polygonal wear around the wheel circumference could pose highly adverse influences on the wheel/rail interactions and thereby the performance of the vehicle system. In this study, the effects of wheel polygonalisation on the dynamic responses of a high-speed rail vehicle are investigated through development and simulations of a comprehensive coupled vehicle/track dynamic model. The model integrates flexible slab track, wheelsets and axle boxes subsystem models so as to account for elastic deformations caused by impact loads induced by the wheel polygonalisation. A field-test programme was undertaken to acquire the polygonal wear profile and axle box acceleration response of a high-speed train, and the data are used to demonstrate the validity of the coupled vehicle/track system model. Subsequently, the effects of wheel polygonalisation are evaluated in terms of wheel/rail impact forces, axle box vertical acceleration and dynamic stress developed in the axle considering different amplitudes and harmonic orders of the polygonal wear. The results suggest that the high-order wheel polygonalisation can give rise to high-frequency impact loads at the wheel/rail interface, and excite some of the vibration modes of the wheelset and the axle box leading to high-magnitude axle box acceleration and dynamic stress in the wheelset axle.     

Probabilistic assessment of railway vehicle-curved track systems considering track random irregularities

The randomness of track irregularities directly leads to the random vibration of the vehicle–track systems. To assess the dynamic performance of a railway system in more comprehensive and practical ways, a framework for probabilistic assessment of vehicle-curved track systems is developed by effectively integrating a vehicle–track coupled model (VTCM), a track irregularity probabilistic model (TIPM) with a probability density evolution method (PDEM). In VTCM, the railway vehicle and the curved track are coupled by the nonlinear wheel–rail interaction forces, and through TIPM, the ergodic properties of random track irregularities on amplitudes, wavelengths and probabilities can be properly considered in the dynamic calculations. Lastly, PDEM, a newly developed method for solving probabilistic transmissions between stochastic excitations and deterministic dynamic responses, is introduced to this probabilistic assessment model. Numerical examples validate the correctness and practicability of the proposed models. In this paper, the results of probabilistic assessment are presented to illustrate the dynamic behaviours of a high-speed railway vehicle subject to curved tracks with various radii, and to demonstrate the importance of considering the actual status of wheel–rail contacts and curve negotiation effects in vehicle-curved track interactions.     

On the interest of integrating vehicle dynamics for the ground propagation of vibrations: the case of urban railway traffic

This paper presents a complete numerical model for studying the vertical dynamics of the vehicle/track interaction and its impact on the surrounding soil, with the emphasis on vehicle modelling. A decoupling between the track and the soil is proposed, due to the difficulty of considering all the subsystem components. The train/track model is based on a multibody model (for the vehicle) and a finite element model (for the track). The soil is modelled using an infinite/finite element approach. Simulations of both models are carried out in the time domain, which is better able to simulate the propagation of the vibration waves and to take into account the possible nonlinearity of a component. The methodology is applied in the case of an urban tram track and validated with the available experimental data. Models for the tram, the track and the soil are described. Results from the complete model of the vehicle and a simple model, based on an axle load, are compared with experimental results and the benefits of a complete model in the simulation of the ground vibration propagation induced by railway vehicles are demonstrated. Moreover, a parametric study of the vehicle wheel type is conducted, which shows the advantage of a resilient wheel, for various rail defects.     

Impact Loads due to Wheel Flats and Shells

A Finite Element (FE) model of vehicle-track system is employed to duplicate the experiments carried out by British Rail and CP Rail System. The theoretical results of the wheel/rail contact forces, rail-pad forces and strains in the rail showed very good correlation to the experimental data. Extensive results are compared with experimental data in the time domain for through validation of the developed model. The characteristics of the impact loads due to wheel flats and shells are investigated based on the validated FE model. The study shows that the shape and size of flat or shell, axle load, vehicle speed and rail-pad stiffness mainly affect the impact loads. Adding elastomeric shear pads on the wheelset bearing does not reduce the wheel/rail dynamic contact force but it may reduce the dynamic force on the bearing. Reducing rail-pad stiffness to a certain level on a concrete-tie track may significantly reduce the dynamic load and the force transmitted to the concrete tie.     

On the effect of unsupported sleepers on the dynamic behaviour of a railway track

The effect of unsupported sleepers on the dynamic behaviour of a railway track is studied based on vehicle–track dynamic interaction theory, using a model of the track as a Timoshenko beam supported on a periodic elastic foundation. Considering the vehicle's running speed and the number of unsupported sleepers, the track dynamic characteristics are investigated and verified in the time and frequency domains by experiments on a 1:5 scale model wheel–rail test rig. The results show that when hanging sleepers are present, leading to a discontinuous and irregular track support, additional wheel–rail interaction forces are generated. These forces increase as further sleepers become unsupported and as the vehicle's running speed increases. The adjacent supports experience increased dynamic forces which will lead to further deterioration of track quality and the formation of long wavelength track irregularities, which worsen the vehicles’ running stability and riding comfort. Stationary transfer functions measurements of the dynamic behaviour of the track are also presented to support the findings.     

Influence of polygonal wear of railway wheels on the wheel set axle stress

The coupled vehicle/track dynamic model with the flexible wheel set was developed to investigate the effects of polygonal wear on the dynamic stresses of the wheel set axle. In the model, the railway vehicle was modelled by the rigid multibody dynamics. The wheel set was established by the finite element method to analyse the high-frequency oscillation and dynamic stress of wheel set axle induced by the polygonal wear based on the modal stress recovery method. The slab track model was taken into account in which the rail was described by the Timoshenko beam and the three-dimensional solid finite element was employed to establish the concrete slab. Furthermore, the modal superposition method was adopted to calculate the dynamic response of the track. The wheel/rail normal forces and the tangent forces were, respectively, determined by the Hertz nonlinear contact theory and the Shen–Hedrick–Elkins model. Using the coupled vehicle/track dynamic model, the dynamic stresses of wheel set axle with consideration of the ideal polygonal wear and measured polygonal wear were investigated. The results show that the amplitude of wheel/rail normal forces and the dynamic stress of wheel set axle increase as the vehicle speeds rise. Moreover, the impact loads induced by the polygonal wear could excite the resonance of wheel set axle. In the resonance region, the amplitude of the dynamic stress for the wheel set axle would increase considerably comparing with the normal conditions.     

Influence of uneven rail irregularities on the dynamic response of the railway track using a three-dimensional model of the vehicle–track system

A mathematical model of the vehicle–track interaction is developed to investigate the coupled behaviour of vehicle–track system, in the presence of uneven irregularities at left/right rails. The railway vehicle is simplified as a 3D multi-rigid-body model, and the track is treated as the two parallel beams on a layered discrete support system. Besides the car-body, the bogies and the wheel sets, the sleepers are assumed to have roll degree of freedom, in order to simulate the in-plane rotation of the components. The wheel–rail interface is treated using a nonlinear Hertzian contact model, coupling the mathematical equations of the vehicle–track systems. The dynamic interaction of the entire system is numerically studied in time domain, employing Newmark's integration method. The track irregularity spectra of both the left/right rails are taken into account, as the inputs of dynamic excitations. The dynamic responses of the track system induced by such irregularities are obtained, particularly in terms of the vertical (bounce) and roll displacements. The numerical model of the present research is validated using several benchmark models reported in the literature, for both the smooth and unsmooth track conditions. Four sample profiles of the measured rail irregularities are considered as the case studies of excitation sources, examining their influences on the dynamic behaviour of the coupled system. The results of numerical simulations demonstrate that the motion of track system is significantly influenced by the presence of uneven irregularities in left/right rails. Dynamic response of the sleepers in the roll direction becomes more sensitive to the rail irregularities, as the unevenness severity of the parallel profiles (quantitative difference between left and right rail spectra) is increased. The severe geometric deformation of the track in the bounce–pitch–roll directions is mainly related to such profile unevenness (cross-level) in left/right rails.     

钢弹簧浮置板轨道位移实测与分析

为评价钢弹簧浮置板轨道钢轨与浮置板位移的合理性，通过现场实测与动力学仿真计算，对比分析钢轨与浮置板在列车以不同速度通过时的位移变化，并且模拟了地铁正式运营后的最不利情况。研究结果表明:车速的改变对钢弹簧浮置板轨道钢轨与浮置板的垂向位移没有大的影响。列车荷载的增加及不平顺的恶化会导致轮轨之间的作用力加强，进而导致钢轨与浮置板的垂向位移增大。     

Switch Panel wear loading – a parametric study regarding governing train operational factors

The acting forces and resulting material degradation at the running surfaces of wheels and rail are determined by vehicle, track, interface and operational characteristics. To effectively manage the experienced wear, plastic deformation and crack development at wheels and rail, the interaction between vehicle and track demands a system approach both in maintenance and in design. This requires insight into the impact of train operational parameters on rail- and wheel degradation, in particular at switches and crossings due to the complex dynamic behaviour of a railway vehicle at a turnout. A parametric study was carried out by means of vehicle-track simulations within the VAMPIRE® multibody simulation software, performing a sensitivity analysis regarding operational factors and their impact on expected switch panel wear loading. Additionally, theoretical concepts were cross-checked with operational practices by means of a case study in response to a dramatic change in lateral rail wear development at specific switches in Dutch track. Data from train operation, track maintenance and track inspection were analysed, providing further insight into the operational dependencies. From the simulations performed in this study, it was found that switch rail lateral wear loading at the diverging route of a 1:9 type turnout is significantly influenced by the level of wheel–rail friction and to a lesser extent by the direction of travel (facing or trailing). The influence of other investigated parameters, being vehicle speed, traction, gauge widening and track layout is found to be small. Findings from the case study further confirm the simulation outcome. This research clearly demonstrates the contribution flange lubrication can have in preventing abnormal lateral wear at locations where the wheel–rail interface is heavily loaded.     

A linear complementarity method for the solution of vertical vehicle–track interaction

A new method is proposed for the solution of the vertical vehicle–track interaction including a separation between wheel and rail. The vehicle is modelled as a multi-body system using rigid bodies, and the track is treated as a three-layer beam model in which the rail is considered as an Euler-Bernoulli beam and both the sleepers and the ballast are represented by lumped masses. A linear complementarity formulation is directly established using a combination of the wheel–rail normal contact condition and the generalised-α method. This linear complementarity problem is solved using the Lemke algorithm, and the wheel–rail contact force can be obtained. Then the dynamic responses of the vehicle and the track are solved without iteration based on the generalised-α method. The same equations of motion for the vehicle and track are adopted at the different wheel–rail contact situations. This method can remove some restrictions, that is, time-dependent mass, damping and stiffness matrices of the coupled system, multiple equations of motion for the different contact situations and the effect of the contact stiffness. Numerical results demonstrate that the proposed method is effective for simulating the vehicle–track interaction including a separation between wheel and rail.     

Prediction of the interaction between a simple moving vehicle and an infinite periodically supported rail – Green's functions approach

This paper herein describes the interaction between a simple moving vehicle and an infinite periodically supported rail, in order to signalise the basic features of the vehicle/track vibration behaviour in general, and wheel/rail vibration, in particular. The rail is modelled as an infinite Timoshenko beam resting on semi-sleepers via three-directional rail pads and ballast. The time-domain analysis was performed applying Green's matrix of the track method. This method allows taking into account the nonlinearities of the wheel/rail contact and the Doppler effect. The numerical analysis is dedicated to the wheel/rail response due to two types of excitation: the steady-state interaction and rail irregularities. The study points out to certain aspects regarding the parametric resonance, the amplitude-modulated vibration due to corrugation and the Doppler effect.