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.     

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).     

Experimental and numerical investigation of the effect of rail corrugation on the behaviour of rail fastenings

This paper presents the results of a detailed investigation of the effects of rail corrugation on the dynamic behaviour of metro rail fastenings, obtained from extensive experiments conducted on site and from simulations of train–track dynamics. The results of tests conducted with a metro train operating on corrugated tracks are presented and discussed first. A three-dimensional (3D) model of the metro train and a slab track was developed using multi-body dynamics modelling and the finite element method to simulate the effect of rail corrugation on the dynamic behaviour of rail fastenings. In the model, the metro train is modelled as a multi-rigid body system, and the slab track is modelled as a discrete elastic support system consisting of two Timoshenko beams for the rails, a 3D solid finite element (FE) model for the slabs, periodic discrete viscoelastic elements for the rail fastenings that connect the rails to the slabs, and uniformly viscoelastic elements for the subgrade beneath the slabs. The proposed train–track model was used to investigate the effects of rail corrugation on the dynamic behaviour of the metro track system and fastenings. An FE model for the rail fastenings was also developed and was used to calculate the stresses in the clips, some of which rupture under the excitation of rail corrugation. The results of the field experiments and dynamics simulations provide an insight into the root causes of the fracture of the clips, and several remedies are suggested for mitigating strong vibrations and failure of metro rail fastening systems.     

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.     

Effect of temperature- and frequency-dependent dynamic properties of rail pads on high-speed vehicle–track coupled vibrations

The soft under baseplate pad of WJ-8 rail fastener frequently used in China’s high-speed railways was taken as the study subject, and a laboratory test was performed to measure its temperature and frequency-dependent dynamic performance at 0.3?Hz and at ?60°C to 20°C with intervals of 2.5°C. Its higher frequency-dependent results at different temperatures were then further predicted based on the time–temperature superposition (TTS) and Williams–Landel–Ferry (WLF) formula. The fractional derivative Kelvin–Voigt (FDKV) model was used to represent the temperature- and frequency-dependent dynamic properties of the tested rail pad. By means of the FDKV model for rail pads and vehicle–track coupled dynamic theory, high-speed vehicle–track coupled vibrations due to temperature- and frequency-dependent dynamic properties of rail pads was investigated. Finally, further combining with the measured frequency-dependent dynamic performance of vehicle’s rubber primary suspension, the high-speed vehicle–track coupled vibration responses were discussed. It is found that the storage stiffness and loss factor of the tested rail pad are sensitive to low temperatures or high frequencies. The proposed FDKV model for the frequency-dependent storage stiffness and loss factors of the tested rail pad can basically meet the fitting precision, especially at ordinary temperatures. The numerical simulation results indicate that the vertical vibration levels of high-speed vehicle–track coupled systems calculated with the FDKV model for rail pads in time domain are higher than those calculated with the ordinary Kelvin–Voigt (KV) model for rail pads. Additionally, the temperature- and frequency-dependent dynamic properties of the tested rail pads would alter the vertical vibration acceleration levels (VALs) of the car body and bogie in 1/3 octave frequencies above 31.5?Hz, especially enlarge the vertical VALs of the wheel set and rail in 1/3 octave frequencies of 31.5–100?Hz and above 315?Hz, which are the dominant frequencies of ground vibration acceleration and rolling noise (or bridge noise) caused by high-speed railways respectively. Since the fractional derivative value of the adopted rubber primary suspension, unlike the tested rail pad, is very close to 1, its frequency-dependent dynamic performance has little effect on high-speed vehicle–track coupled vibration responses.     

On vehicle/track impact at connection between a floating slab and ballasted track and floating slab track's effectiveness of force isolation

When a vehicle runs over the connection between a floating slab track (FST) and ballasted track, wheel/rail impact may occur because of the stiffness difference in the two kinds of track, and thus a transition sector is usually included at the connection to smoothen the stiffness change. This phenomenon is studied by numerical simulation using a time-domain model for an idealised case without such a transition to determine whether it is actually necessary. Calculation results show that the wheel/rail impact load is moderate for a light FST and increases with the vehicle speed or decreasing the natural frequency of the FST. From simulation the wheel/rail parametric excitation is observed, as a result of variation in the stiffness of the FST with the period of the single slab length. The wheel/rail load due to the parametric excitation also increases with the vehicle speed. In addition, good performance of vibration isolation can be seen for the FST in terms of the force transmitted to the infrastructure.     

High-frequency random vibration analysis of a high-speed vehicle–track system with the frequency-dependent dynamic properties of rail pads using a hybrid SEM–SM method

A hybrid Spectral Element Method (SEM)–Symplectic Method(SM) method for high-efficiency computation of the high-frequency random vibrations of a high-speed vehicle–track system with the frequency-dependent dynamic properties of rail pads is presented. First, the Williams-Landel-Ferry (WLF) formula and Fractional Derivative Zener (FDZ) model were, respectively, applied for prediction and representation of the frequency-dependent dynamic properties of Vossloh 300 rail pads frequently used in China's high-speed railway. Then, the proposed hybrid SEM–SM method was used to investigate the influence of the frequency-dependent dynamic performance of Vossloh 300 rail pads on the high-frequency random vibrations of high-speed vehicle–track systems at various train speeds or different levels of rail surface roughness. The experimental results indicate that the storage stiffness and loss factors of Vossloh 300 rail pad increase with the decrease in dynamic loads or the increase in preloads within 0.1–10,000?Hz at 20°C, and basically linearly increase with frequency in a logarithmic coordinate system. The results computed by the hybrid SEM–SM method demonstrate that the frequency-dependent viscous damping of Vossloh 300 rail pads, compared with its constant viscous damping and frequency-dependent stiffness, has a much more conspicuous influence on the medium-frequency (i.e. 20–63?Hz) random vibrations of car bodies and rail fasteners, and on the mid- (i.e. 20–63?Hz) and high-frequency (i.e. 630–1250?Hz) random vibrations of bogies, wheels and rails, especially with the increase in train speeds or the deterioration of rail surface roughness. The two sensitive frequency bands can also be validated by frequency response function (FRF) analysis of the proposed infinite rail–fastener model. The mid and high frequencies influenced by the frequency-dependent viscous damping of rail pads are exactly the dominant frequencies of ground vibration acceleration and wheel rolling noise caused by high-speed railways, respectively. Even though the existing time-domain (or frequency-domain) finite track models associated with the time-domain (or frequency-domain) fractional derivative viscoelastic (FDV) models of rail pads can also be used to reach the same conclusions, the hybrid SEM–SM method in which only one element is required to compute the high-order vibration modes of infinite rail is more appropriate for high-efficiency analysis of the high-frequency random vibrations of high-speed vehicle–track systems.     

Influence of long-wavelength track irregularities on the motion of a high-speed train

Vertical track irregularities over viaducts in high-speed rail systems could be possibly caused by concrete creep if pre-stressed concrete bridges are used. For bridge spans that are almost uniformly distributed, track irregularity exhibits a near-regular wave profile that excites car bodies as a high-speed train moves over the bridge system. A long-wavelength irregularity induces low-frequency excitation that may be close to the natural frequencies of the train suspension system, thereby causing significant vibration of the car body. This paper investigates the relationship between the levels of car vibration, bridge vibration, track irregularity, and the train speed. First, this study investigates the vibration levels of a high-speed train and bridge system using 3D finite-element (FE) transient dynamic analysis, before and after adjustment of vertical track irregularities by means of installing shimming plates under rail pads. The analysis models are validated by in situ measurements and on-board measurement. Parametric studies of car body vibration and bridge vibration under three different levels of track irregularity at five train speeds and over two bridge span lengths are conducted using the FE model. Finally, a discontinuous shimming pattern is proposed to avoid vehicle suspension resonance.     

Dynamic Interaction between Wheel and Track - A Parametric Search towards an Optimal Design of Rail Structures

The dynamic vertical interaction between a moving rigid wheel and a flexible railway track is investigated. A round and smooth wheel tread and an initially straight and noncorrugated rail surface are assumed in the present optimization study. A symmetric linear three-dimensional beam structure model of a finite portion of the track is suggested including rail, pads, sleepers and ballast with spatially nonproportional damping. The full interaction problem is numerically solved by use of an extended state-space vector approach in conjunction with a complex modal superposition for the track. Transient bending stresses in sleepers and rail are calculated. The influence of seven selected track parameters on the dynamic behaviour of the track is investigated. A two-level fractional factorial design method is used in the search for a combination of numerical levels of these parameters making the maximum bending stresses a minimum.     

A New Wheel/Rail Spatially Dynamic Coupling Model and its Verification

Based on the theory of vehicle-track coupling dynamics, a new wheel/rail spatially dynamic coupling model is established in this paper. In consideration of rail lateral, vertical and torsion vibrations and track irregularities, the wheel/rail contact geometry, the wheel/rail normal contact force and the wheel/rail tangential creep force are solved in detail. In the new wheel/rail model, the assumption that wheel contacts rail rigidly and wheel always contacts rail is eliminated. Finally, by numeric simulation comparison with international well-known software NUCARS, comparison with vehicle-track vertical coupling model, and comparison with running test results by China Academy of Railway Sciences, the new wheel/rail spatially dynamic coupling model is shown to be correct and effective.     

A New Wheel/Rail Spatially Dynamic Coupling Model and its Verification

Based on the theory of vehicle-track coupling dynamics, a new wheel/rail spatially dynamic coupling model is established in this paper. In consideration of rail lateral, vertical and torsion vibrations and track irregularities, the wheel/rail contact geometry, the wheel/rail normal contact force and the wheel/rail tangential creep force are solved in detail. In the new wheel/rail model, the assumption that wheel contacts rail rigidly and wheel always contacts rail is eliminated. Finally, by numeric simulation comparison with international well-known software NUCARS, comparison with vehicle-track vertical coupling model, and comparison with running test results by China Academy of Railway Sciences, the new wheel/rail spatially dynamic coupling model is shown to be correct and effective.     

Dynamic Interaction between Wheel and Track - A Parametric Search towards an Optimal Design of Rail Structures

SUMMARY

The dynamic vertical interaction between a moving rigid wheel and a flexible railway track is investigated. A round and smooth wheel tread and an initially straight and noncorrugated rail surface are assumed in the present optimization study. A symmetric linear three-dimensional beam structure model of a finite portion of the track is suggested including rail, pads, sleepers and ballast with spatially nonproportional damping. The full interaction problem is numerically solved by use of an extended state-space vector approach in conjunction with a complex modal superposition for the track. Transient bending stresses in sleepers and rail are calculated. The influence of seven selected track parameters on the dynamic behaviour of the track is investigated. A two-level fractional factorial design method is used in the search for a combination of numerical levels of these parameters making the maximum bending stresses a minimum.     

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.     

Study on rail corrugation of a metro tangential track with Cologne-egg type fasteners

In Chinese metro lines, rail corrugation on both tangential and tight curved tracks with Cologne-egg type fasteners is very severe. Based on the viewpoint of friction-induced vibration causing rail corrugation, the rail corrugation on a tangential track with Cologne-egg type fasteners is studied in this paper. A vibration model of an elastic multiple-wheelset-track system with Cologne-egg type fasteners is established. Both the complex eigenvalue analysis and the transient dynamic analysis are performed to study the stability and the dynamic performance of the wheelset-track system. The simulation results show that a low rail support stiffness value is responsible for rail corrugation on the tangential track. When the Cologne-egg fasteners characterised by a lower stiffness value are replaced with the DTVI2 fasteners characterised by a higher stiffness value, rail corrugation disappears. However, rail corrugation on tight curved tracks cannot be suppressed using the same replacement. The above conclusions are consistent with the corrugation occurrences in actual metro tracks.     

Dynamic Train-Track Interaction: Variability Attributable to Scatter in the Track Properties

A numerical method to simulate vertical dynamic interaction between a rolling train and a railway track has been used to investigate the influence of stochastic properties of the track structure. A perturbation technique has been used to investigate the influence of the scatter in selected track properties. The train-track interaction problem has been numerically solved by use of an extended state-space vector approach in conjunction with a complex modal superposition for the whole track structure. All numerical simulations have been carried out in the time-domain with a moving mass model. Properties such as rail pad stiffness, ballast stiffness, dynamic ballast-subgrade mass and sleeper spacing have been studied. To obtain sufficient statistical information from track structures, full-scale measurements in the field and laboratory measurements have been carried out. The influence of scatter in the track properties on the maximum contact force between the rail and the wheel, the maximum magnitude of the vertical wheelset acceleration, and the maximum sleeper displacement have been studied. Mean values and standard deviations of these quantities have been calculated. The effects of the variation of the investigated track properties are discussed.     

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.     

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.     

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.     

Fundamentals of vehicle–track coupled dynamics

This paper presents a framework to investigate the dynamics of overall vehicle–track systems with emphasis on theoretical modelling, numerical simulation and experimental validation. A three-dimensional vehicle–track coupled dynamics model is developed in which a typical railway passenger vehicle is modelled as a 35-degree-of-freedom multi-body system. A traditional ballasted track is modelled as two parallel continuous beams supported by a discrete-elastic foundation of three layers with sleepers and ballasts included. The non-ballasted slab track is modelled as two parallel continuous beams supported by a series of elastic rectangle plates on a viscoelastic foundation. The vehicle subsystem and the track subsystem are coupled through a wheel–rail spatial coupling model that considers rail vibrations in vertical, lateral and torsional directions. Random track irregularities expressed by track spectra are considered as system excitations by means of a time–frequency transformation technique. A fast explicit integration method is applied to solve the large nonlinear equations of motion of the system in the time domain. A computer program named TTISIM is developed to predict the vertical and lateral dynamic responses of the vehicle–track coupled system. The theoretical model is validated by full-scale field experiments, including the speed-up test on the Beijing–Qinhuangdao line and the high-speed running test on the Qinhuangdao–Shenyang line. Differences in the dynamic responses analysed by the vehicle–track coupled dynamics and by the classical vehicle dynamics are ascertained in the case of vehicles passing through curved tracks.