A survey of wheel–rail contact models for rail vehicles

Accurate and efficient contact models for wheel–rail interaction are essential for the study of the dynamic behaviour of a railway vehicle. Assessment of the contact forces and moments, as well as contact geometry provide a fundamental foundation for such tasks as design of braking and traction control systems, prediction of wheel and rail wear, and evaluation of ride safety and comfort. This paper discusses the evolution and the current state of the theories for solving the wheel–rail contact problem for rolling stock. The well-known theories for modelling both normal contact (Hertzian and non-Hertzian) and tangential contact (Kalker's linear theory, FASTSIM, CONTACT, Polach's theory, etc.) are reviewed. The paper discusses the simplifying assumptions for developing these models and compares their functionality. The experimental studies for evaluation of contact models are also reviewed. This paper concludes with discussing open areas in contact mechanics that require further research for developing better models to represent the wheel–rail interaction.     

Sensitivity of the wheel–rail contact interactions and Dang Van Fatigue Index in the rail with respect to irregularities of the track geometry

This paper investigates the effects of the track geometry irregularities on the wheel–rail dynamic interactions and the rail fatigue initiation through the application of the Dang Van criterion, that supposes an elastic shakedown of the structure. The irregularities are modelled, using experimental data, as a stochastic field which is representative of the considered railway network. The tracks thus generated are introduced as the input of a railway dynamics software to characterise the stochastic contact patch and the parameters on which it depends: contact forces and wheelset–rail relative position. A variance-based global sensitivity analysis is performed on quantities of interest representative of the dynamic behaviour of the system, with respect to the stochastic geometry irregularities and for different curve radius classes and operating conditions. The estimation of the internal stresses and the fatigue index being more time-consuming than the dynamical simulations, the sensitivity analysis is performed through a metamodel, whose input parameters are the wheel–rail relative position and velocity. The coefficient of variation of the number of fatigue cycles, when the simulations are performed with random geometry irregularities, varies between 0.13 and 0.28. In a large radius curve, the most influent irregularity is the horizontal curvature, while, in a tight curve, the gauge becomes more important.     

Design and optimisation of wheel–rail profiles for adhesion improvement

This paper describes a study for the optimisation of the wheel profile in the wheel–rail system to increase the overall level of adhesion available at the contact interface, in particular to investigate how the wheel and rail profile combination may be designed to ensure the improved delivery of tractive/braking forces even in poor contact conditions. The research focuses on the geometric combination of both wheel and rail profiles to establish how the contact interface may be optimised to increase the adhesion level, but also to investigate how the change in the property of the contact mechanics at the wheel–rail interface may also lead to changes in the vehicle dynamic behaviour.     

Systems-on-chip approach for real-time simulation of wheel–rail contact laws

This paper presents the development of a systems-on-chip approach to speed up the simulation of wheel–rail contact laws, which can be used to reduce the requirement for high-performance computers and enable simulation in real time for the use of hardware-in-loop for experimental studies of the latest vehicle dynamic and control technologies. The wheel–rail contact laws are implemented using a field programmable gate array (FPGA) device with a design that substantially outperforms modern general-purpose PC platforms or fixed architecture digital signal processor devices in terms of processing time, configuration flexibility and cost. In order to utilise the FPGA's parallel-processing capability, the operations in the contact laws algorithms are arranged in a parallel manner and multi-contact patches are tackled simultaneously in the design. The interface between the FPGA device and the host PC is achieved by using a high-throughput and low-latency Ethernet link. The development is based on FASTSIM algorithms, although the design can be adapted and expanded for even more computationally demanding tasks.     

A robust non-Hertzian contact method for wheel–rail normal contact analysis

A modified Kik–Piotrowski (MKP) model is proposed in this paper for an accurate and robust calculation of wheel–rail normal contact problem. The presented method is able to consider the relationship between the elastic deformation of a line and the normal pressure distribution within the contact patch. A novel shape correction method is put forward to correctly describe the elastic deformation of the contact patch. Taking the results estimated by Kalker’s variational method and Kik–Piotrowski method as references, the proposed method is validated by three contact cases, including the assumed standardised non-Hertzian contact and the two-point contact, as well as the contact behaviours based on three actual wheel–rail profiles. The simulation results indicate that, compared with Kik–Piotrowski method, the proposed MKP method achieves better agreement with Kalker’s variational method. Moreover, the MKP method can avoid the abrupt change of wheel–rail normal force due to the sudden transfer of the contact point, which contributes to a better computational stability.     

Strategies to simulate wheel–rail adhesion in degraded conditions using a roller-rig

Railway vehicles adopt mechatronic devices to maximize the traction/braking effort. These systems often use complex algorithms that require long experimental validation tests on track. The possibility to perform the same tests on a roller-rig gives the opportunity to simplify the validation activity, under safe conditions, and to reduce the costs. The main challenge is to be able to reproduce the same adhesion conditions on a roller-rig with respect to the track, especially in degraded conditions. First, the paper shows experiments performed to reproduce degraded adhesion on a conventional roller-rig for a single wheelset. Then, an innovative roller-rig is proposed in order to reproduce the effect of the passage of several wheelsets on the track.     

Assessment of a semi-Hertzian method for determination of wheel–rail contact patch

Wheel–rail contact calculations are essential for simulating railway vehicle dynamic behavior. Currently, these simulations usually use the Hertz contact theory to calculate normal forces and Kalker's ‘FASTSIM’ program to evaluate tangential stresses. Since 1996, new methods called semi-Hertzian have appeared: 5 Kik, W. and Piotrowski, J. A fast approximate method to calculate normal load at contact between wheel and rail and creep forces during rolling. Paper presented at the 2nd Mini-conference on Contact Mechanics and Wear of Rail/Wheel Systems. July29–31, Budapest.  [Google Scholar] 7 Ayasse, J. B., Chollet, H. and Maupu, J. L. 2000. Paramètres caractéristiques du contact roue-rail. Rapport de Recherche INRETS n225, ISSN 0768–9756 (in French) [Google Scholar] (STRIPES). These methods attempt to estimate the non-elliptical contact patches with a discrete extension of the Hertz theory. As a continuation of 2 Ayasse, J. B and Chollet, H. 2005. Determination of the wheel–rail contact patch in semi-Hertzian conditions. Vehicle System Dynamics, 43(3) [Google Scholar], a validation of the STRIPES method for normal problem computing on three test cases is proposed in this article. The test cases do not fulfill the hypothesis required for the Hertz theory. Then, the Kalker's FASTSIM algorithm is adapted to STRIPES patch calculus to perform tangential forces computation. This adaptation is assessed using Kalker's CONTACT algorithm.     

An alternative to FASTSIM for tangential solution of the wheel–rail contact

In most rail vehicle dynamics simulation packages, tangential solution of the wheel–rail contact is gained by means of Kalker's FASTSIM algorithm. While 5–25% error is expected for creep force estimation, the errors of shear stress distribution, needed for wheel–rail damage analysis, may rise above 30% due to the parabolic traction bound. Therefore, a novel algorithm named FaStrip is proposed as an alternative to FASTSIM. It is based on the strip theory which extends the two-dimensional rolling contact solution to three-dimensional contacts. To form FaStrip, the original strip theory is amended to obtain accurate estimations for any contact ellipse size and it is combined by a numerical algorithm to handle spin. The comparison between the two algorithms shows that using FaStrip improves the accuracy of the estimated shear stress distribution and the creep force estimation in all studied cases. In combined lateral creepage and spin cases, for instance, the error in force estimation reduces from 18% to less than 2%. The estimation of the slip velocities in the slip zone, needed for wear analysis, is also studied. Since FaStrip is as fast as FASTSIM, it can be an alternative for tangential solution of the wheel–rail contact in simulation packages.     

A new method for wheel–rail contact force continuous measurement using instrumented wheelset

With the aim of improving the continuous measurement of wheel–rail contact force by instrumented wheelset, instead of solving the non-linear equations, we proposed a new method based on state space theory. With this new method, the wheel–rail contact force can be calculated by the recurrence relation and the signals from strain gauge bridges on wheel web. The implementation of continuous instrumented wheelset is quite general and simplified, due to the specific bridging scheme is not necessary. It means that continuous measurement of the contact force could be realised with a simple bridging scheme, even as simple as discrete instrumented scheme. In this work, we first demonstrated and discussed the effectiveness and accuracy of this new method by estimation results with the numerical simulations, and we also applied this new method to two field tests, including one was conducted in a loop test line using a high-speed train and the other one was conducted in an urban line with a light rail vehicle. In a word, this new method is proved to be an effective way to monitor the wheel–rail contact force of rail vehicle track system.     

ADAMS/rail虚拟样机技术在车辆系统建模及仿真分析中的应用

利用美国MDI公司的ADAMS／rail软件，建立车辆系统动力学的三维模型，利用标准界面并以走行部、车体及辅助设备的模板为基础建立组装和系统，并可在标准界面下对其进行动力学分析。     

Physical processes in wheel–rail contact and its implications on vehicle–track interaction

Friction within the wheel–rail contact highly influences all aspects of vehicle–track interaction. Models describing this frictional behaviour are of high relevance, for example, for reliable predictions on drive train dynamics. It has been shown by experiments, that the friction at a certain position on rail is not describable by only one number for the coefficient of friction. Beside the contact conditions (existence of liquids, solid third bodies, etc.) the vehicle speed, normal loading and contact geometry are further influencing factors. State-of-the-art models are not able to account for this sufficiently. Thus, an Extended-Creep-Force-Model was developed taking into account effects from third body layers. This model is able to describe all considered effects. In this way, a significant improvement of the prediction quality with respect to all aspects of vehicle–track interaction is expected.     

Comments on ‘the Kalker book of tables for non-Hertzian contact of wheel and rail’

The ‘simple double-elliptical contact’ (SDEC) approach by Piotrowski et al. [The Kalker book of tables for non-Hertzian contact of wheel and rail. Vehicle Syst Dyn. 2017;55:875–901] generates a-symmetrical contact patches in an elegant way. This allows to extend the table-based approach for the wheel–rail creep force calculation towards non-Hertzian contact geometry. This is an important line of research, because FASTSIM is intricate for non-Hertzian contacts, whereas CONTACT requires long calculation times.

Here, we comment on the further motivation that's provided for the approach. According to the authors, ‘the spin creepage generates longitudinal creep force in non-symmetric, non-elliptical contacts’, which is ‘completely lost’ when using elliptical regularisation. We demonstrate that this mainly depends on the choice of contact origin, and that the interaction is much reduced if different choices are made. This suggests that elliptical regularisation may be viable still, if the details are properly worked out. Furthermore, we introduce the spin center and the free-rolling position as means to extend the table-based approach towards more general non-Hertzian circumstances.     

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.     

A non-Hertzian method for solving wheel–rail normal contact problem taking into account the effect of yaw

A novel approach is proposed in this paper to deal with non-Hertzian normal contact in wheel–rail interface, extending the widely used Kik–Piotrowski method. The new approach is able to consider the effect of the yaw angle of the wheelset against the rail on the shape of the contact patch and on pressure distribution. Furthermore, the method considers the variation of profile curvature across the contact patch, enhancing the correspondence to CONTACT for highly non-Hertzian contact conditions. The simulation results show that the proposed method can provide more accurate estimation than the original algorithm compared to Kalker’s CONTACT, and that the influence of yaw on the contact results is significant under certain circumstances.     

A method to determine the two-point contact zone and transfer of wheel–rail forces in a turnout

A practical method to determine the zone of two contact points and the transfer of wheel–rail forces between two rails in a turnout is presented in this paper. The method is based on a wheel–rail elastic penetration assumption and used to study a turnout system for a 200 km/h high-speed railway in China. Rail profiles in a number of key sections in the turnout are identified first, and profiles in other sections are then obtained by interpolation between key sections. The track is modelled as flexible with rails and sleepers represented by beams and the interaction between the vehicle and turnout is simulated for cases of the vehicle passing the turnout. Results are mainly presented for two-point contact positions and the characteristics of the wheel–rail forces transference. It is found that the heights of the switch and crossing rail top have significant effects on the wheel–rail contact forces. Finally, the optimised top height for the crossing rails is proposed to reduce the system dynamic force in the turnout system.     

An efficient approach to the analysis of rail surface irregularities accounting for dynamic train–track interaction and inelastic deformations

A two-dimensional computational model for assessment of rolling contact fatigue induced by discrete rail surface irregularities, especially in the context of so-called squats, is presented. Dynamic excitation in a wide frequency range is considered in computationally efficient time-domain simulations of high-frequency dynamic vehicle–track interaction accounting for transient non-Hertzian wheel–rail contact. Results from dynamic simulations are mapped onto a finite element model to resolve the cyclic, elastoplastic stress response in the rail. Ratcheting under multiple wheel passages is quantified. In addition, low cycle fatigue impact is quantified using the Jiang–Sehitoglu fatigue parameter. The functionality of the model is demonstrated by numerical examples.     

Dynamic vehicle–track interaction in switches and crossings and the influence of rail pad stiffness – field measurements and validation of a simulation model

A model for simulation of dynamic interaction between a railway vehicle and a turnout (switch and crossing, S&C) is validated versus field measurements. In particular, the implementation and accuracy of viscously damped track models with different complexities are assessed. The validation data come from full-scale field measurements of dynamic track stiffness and wheel–rail contact forces in a demonstrator turnout that was installed as part of the INNOTRACK project with funding from the European Union Sixth Framework Programme. Vertical track stiffness at nominal wheel loads, in the frequency range up to 20?Hz, was measured using a rolling stiffness measurement vehicle (RSMV). Vertical and lateral wheel–rail contact forces were measured by an instrumented wheel set mounted in a freight car featuring Y25 bogies. The measurements were performed for traffic in both the through and diverging routes, and in the facing and trailing moves. The full set of test runs was repeated with different types of rail pad to investigate the influence of rail pad stiffness on track stiffness and contact forces. It is concluded that impact loads on the crossing can be reduced by using more resilient rail pads. To allow for vehicle dynamics simulations at low computational cost, the track models are discretised space-variant mass–spring–damper models that are moving with each wheel set of the vehicle model. Acceptable agreement between simulated and measured vertical contact forces at the crossing can be obtained when the standard GENSYS track model is extended with one ballast/subgrade mass under each rail. This model can be tuned to capture the large phase delay in dynamic track stiffness at low frequencies, as measured by the RSMV, while remaining sufficiently resilient at higher frequencies.