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.     

Influence of wheel–rail contact modelling on vehicle dynamic simulation

This paper presents a comparison of four models of rolling contact used for online contact force evaluation in rail vehicle dynamics. Until now only a few wheel–rail contact models have been used for online simulation in multibody software (MBS). Many more models exist and their behaviour has been studied offline, but a comparative study of the mutual influence between the calculation of the creep forces and the simulated vehicle dynamics seems to be missing. Such a comparison would help researchers with the assessment of accuracy and calculation time. The contact methods investigated in this paper are FASTSIM, Linder, Kik–Piotrowski and Stripes. They are compared through a coupling between an MBS for the vehicle simulation and Matlab for the contact models. This way the influence of the creep force calculation on the vehicle simulation is investigated. More specifically this study focuses on the influence of the contact model on the simulation of the hunting motion and on the curving behaviour.     

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.     

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 novel method to model wheel–rail normal contact in vehicle dynamics simulation

An approximate analytical method is proposed for calculating the contact patch and pressure distribution in the wheel–rail interface. The deformation of the surfaces in contact is approximated using the separation between them. This makes it possible to estimate the contact patch analytically. The contact pressure distribution in the rolling direction is assumed to be elliptic with its maximum calculated by applying Hertz' solution locally. The results are identical to Hertz's for elliptic cases. In non-elliptic cases good agreement is achieved in comparison to the more accurate but computationally expensive Kalker's variational method (CONTACT code). Compared to simplified non-elliptic contact methods based on virtual penetration, the calculated contact patch and pressure distribution are markedly improved. The computational cost of the proposed method is significantly lower than the more detailed methods, making it worthwhile to be applied to rolling contact in rail vehicle dynamics simulation. Such fast and accurate estimation of contact patch and pressure paves the way for on-line modelling of damage phenomena in dynamics simulation packages.     

The role of the contact geometry in wheel–rail impact due to wheel flats

A thorough investigation of wheel–rail impact due to wheel flats is presented, together with a quantitative characterization of the main mechanisms and parameters. A criterion for the speed with respect to contact loss between wheel and rail is derived. In the subcritical speed regime, the magnitude of the impact force is shown to be directly related to the geometry of the flat, whereas in the transcritical speed regime a fictitious or effective flat depth exists, which decreases with the second order of the speed. In this domain, the position of impact shifts towards the end of the flat with increasing speed. The impact force increases with the second order of the speed in the subcritical speed regime and approximately the first order of the speed in the transcritical speed regime. The magnitude of the impact force is inversely proportional to the minimum circumferential curvature of the wheel tread defect in the subcritical speed regime, and to the effective minimum curvature in the transcritical case. The variation with the flat depth is less; the impact varies with the square root of the flat depth. The presented theory is in accordance with measurements reported in the literature and explains characteristic features in them.     

Wheel-rail contact models for vehicle system dynamics including multi-point contact

Advanced modelling of rail vehicle dynamics requires realistic solutions of contact problems for wheels and rails that are able to describe contact singularities, encountered for wheels and rails. The basic singularities demonstrate themselves as double and multiple contact patches. The solutions of the contact problems have to be known practically in each step of the numerical integration of the differential equations of the model. The existing fast, approximate methods of solution to achieve this goal have been outlined. One way to do this is to replace a multi-point contact by a set of ellipses. The other methods are based on so-called virtual penetration. They allow calculating the non-elliptical, multiple contact patches and creep forces online, during integration of the model. This allows nearly real-time simulations. The methods are valid and applicable for so-called quasi-Hertzian cases, when the contact conditions do not deviate much from the assumptions of the Hertz theory. It is believed that it is worthwhile to use them in other cases too.     

Estimation of wheel–rail friction for vehicle certification

In certification of new rail vehicles with respect to running characteristics, a wide variety of operating conditions needs to be considered. However, in associated test runs the wheel–rail friction condition is difficult to handle because the friction coefficient needs to be fairly high and the friction is also generally hard to assess. This is an issue that has been studied in the European project DynoTRAIN and part of the results is presented in this paper. More specifically, an algorithm for estimating the wheel–rail friction coefficient at vehicle certification tests is proposed. Owing to lack of some measurement results, the algorithm here is evaluated in a simulation environment which is also an important step towards practical implementation. A quality measure of the friction estimate is suggested in terms of estimated wheel–rail spin and total creep. It is concluded that, tentatively, the total creep should exceed 0.006 and the spin should be less than 1.0 m^?1 for the algorithm to give a good friction estimate. Sensitivity analysis is carried out to imitate measurement errors, but should be expanded in further work.     

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.     

Short wavelength rail corrugation and non-steady-state contact mechanics

J. J. Kalker has been the first to consider non-steady-state or transient contact mechanics. Based on Kalker the second author developed a linear contact model for the non-steady-state rolling contact of a wheel running over slightly corrugated rails. The theoretical investigations are concentrated on linear, non-steady-state contact mechanics superimposed to a nonlinear reference state. The reference state is given by the running behaviour of a wheelset due to traction, curving or hunting. For the linear, non-steady-state analysis Kalker's theory has to be modified to predict wear rates in dependency of the corrugation wavelengths. As a result corrugations are only amplified in the range between 2 and 10 cm. Therefore, non-steady-state contact mechanics and wear are responsible for a wavelength fixing mechanism. Structural mechanics of the rail indicate that wavelength in this range is predominantly amplified.     

A rail roughness growth model for a wheelset with non-steady,non-Hertzian contact

This paper presents a model simulating rail roughness growth in which the interaction of a wheelset with the track is considered. The aim is to investigate any possible mechanism for roughness growth due to the coupling between the vertical dynamics, the torsional vibration across the axle of the wheelset and the non-steady contact mechanics. The time-domain simulations are carried out for a driven wheelset on tangent track. Both rigid and flexible are considered with parameter variations for moments of the wheelset, vehicle speeds and wavelengths of initial roughnesses. The 2D non-Hertzian and non-steady contact model used in simulations are based on influence coefficients obtained from a boundary element model. The nonlinear development of the rail roughnesses after millions of wheelset passages is also presented.     

Modelling of wheels and rail discontinuities in dynamic wheel–rail contact analysis

A classification of wheel–rail contact is given. Difference is made between modelling of a running wheel with continuous single-point-contact, as is common practice in wheel–rail contact analysis, and a wheel with transient double- or multi-point-contact, which may occur for rail irregularities with curvatures larger than that of the wheel circumference. It is shown that application of the first model for these irregularities will strongly underestimate the contact forces as it does not describe occurring mechanisms correctly. Further, it is shown that in principle it is not possible to describe the second type of contact fully correct with a lumped wheel model. Both wheel models are formulated mathematically for some basic contact cases. Afterwards, results are applied to a linear track model. Analytical closed-form solutions are found in the frequency domain for arbitrary type of contact and numerically transformed to the time domain. Finally, the necessity is shown to avoid situations where transient multiple-point-contact may occur (like rail joints) in practice.     

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.     

Mathematical modelling of train–turnout interaction

The paper proposes a mathematical model of train–turnout interaction in the mid-frequency range (0–500 Hz). The model accounts for the effects of rail profile variation along the track and of local variation of track flexibility. The proposed approach is able to represent the condition of one wheel being simultaneously in contact with more than one rail, allowing the accurate prediction of the effect of wheels being transferred from one rail to another when passing over the switch toe and the crossing nose. Comprehensive results of train–turnout interaction during the negotiation of the main and the branch lines are presented, including the effect of wear of wheel/rail profiles and presence of track misalignment. In the final part of the paper, comparisons are performed between the results of numerical simulations and line measurements performed on two different turnouts for urban railway lines, showing a good agreement between experimental and numerical results.