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A simplified model of wheel/rail contact mechanics for non-Hertzian problems and its application in rail vehicle dynamic simulations
Abstract
The presented model assumes semi-elliptical normal pressure distribution in the direction of rolling. The contact area is found by virtual penetration of wheel and rail. The normal pressure is calculated by satisfying contact conditions at the geometrical point of contact. The calculation is non-iterative, fast and completely reliable. It may be carried out on-line in MultiBody Systems (MBS) computer codes. The tests using the programme CONTACT by Kalker and experience from application in MBS codes show that the model is suitable for technical applications. The creep forces have been calculated with the FASTSIM algorithm, adapted for a non-elliptical contact area. Some applications in rail vehicle dynamics and wear simulation have been outlined.
... Since the wheel-rail contact is likely to demonstrate a distinct manifestation of non-elliptic features when examining its contact patch and internal stress distributions, the Hertzian model may deviate sharply from reality, especially when the wheel-rail profiles are heavily worn or when wheel flange-rail gauge corner contact occurs. To this end, some researchers proposed a series of non-Hertzian models based on the 'virtual penetration' principle for on-line train-track interaction analysis so as to approximate CONTACT, such as Linder method [69], Kik-Piotrowski model [70], Ayasse-Chollet (STRIPES) model [71] and ANALYN model [72]. ...
Train-track coupled dynamics problems are particularly prominent in heavy-haul railways, due to the strong coupling effects of longitudinal , lateral, and vertical interactions between long heavy-haul trains and rail infrastructure. Poor train-track dynamic interaction performance could lead to a series of dynamics problems that significantly influence train operational safety and maintenance costs. This paper presents a comprehensive state-of-the-art review of train-track coupled dynamics theory and its applications in heavy-haul rail transportation. Several critical heavy-haul train-track coupled dynamics problems are discussed, including longitudinal coupler force induced safety issues, wheel-rail wear and rolling contact fatigue, and infrastructure degradation due to train-track interactions. The evolution of vehicle-track coupled dynamics theory and its extended development to heavy-haul train-track coupled dynamics is described briefly. A systematic review of the modelling methodology for key elements in heavy-haul train-track coupled dynamics is presented. Field test results focusing on the modelling validation are also briefly presented. The practical applications of heavy-haul train-track coupled dynamics theory are comprehensively discussed. Guidance is provided on further improving the heavy-haul train-track coupled dynamics and identifying future challenging research topics on better solving complicate engineering problems due to heavy-haul rail transportation.
... While there are various methods for detailed modelling of conformal contact [15], research is sparse on the effects of conformal contact modelling on the dynamics of a railway wheelset or vehicle. Piotrowski and Kik [19] applied a simplified approach to demonstrate the impact of the modelling of the curved contact area (conformal contact) using an idealised wheelset excluding the influence of the inertia forces. Pascal and Rismantab-Sany [20] explored the dynamics of an isolated wheelset, accounting for the effects of conformal contact with the multi-Hertzian approach. ...
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https://www.tandfonline.com/eprint/DDHR8IM23V8BRKYAVWYV/full?target=10.1080/00423114.2025.2480819
Low-friction surface conditions significantly contribute to the reduction of the wheel/rail adhesion capability and the occurrence of wheel/rail slipping behaviors, which may lead to the degradation of mechanical properties and frictional wear damage at the wheel/rail interface. To mitigate these undesirable consequences, modern railway locomotives are equipped with on-board anti-slip control systems. In this study, three different anti-slip controller models, comprising the traditional re-adhesion anti-slip controller and PID-based anti-slip controller with fixed threshold and with optimal threshold, are established. The wheel/rail rolling-slipping performances subjected to different anti-slip control algorithms under changing wheel/rail friction conditions are compared based on train-track interaction simulations. The results demonstrate that the PID-based anti-slip controller with an optimal threshold achieves the maximum utilization of wheel/rail adhesion in the presence of low-friction conditions, outperforming the other two types of anti-slip controllers. Additionally, the adoption of an anti-slip controller with a lower control threshold can effectively reduce the tread wear of locomotive wheels. This research can provide a deep going understanding of optimization design of anti-slip controller on railway vehicles.
There has been a growing interest in studying the impact of soil–structure interaction (SSI) on high-speed rail (HSR) bridges during near-fault earthquakes. A model of the HSR vehicle–track–bridge–soil–pile (HSR-VTBSP) system is created to compute its seismic response in both elastic and plastic states. The model is used to investigate how the SSI affects the elastic–plastic seismic response of bridges. The study reveals that SSI greatly affects the seismic response of bridges. Even when subjected to far-field earthquakes, SSI still significantly impacts the length of bridge girders. Consequently, the SSI effect has a notable influence on the safe operation of trains on bridges. This paper reports the following novel discoveries: The “derailment” condition is determined by the SSI effect, which is the safe limit amplitude of wheel–rail relative displacement with respect to the entire train derailment process in train track aggregation. r_max/disYw ≤ 70mm. The findings of this research may serve as a valuable reference for investigating the measurement of lateral displacement in the wheel–rail contact point.
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.
This paper deals with the subject of the semi-Hertzian contact, which is a way to represent the wheel rail contact in railways or roller bearing applications. The method is based on the interpenetration of the two underformed bodies' profiles. The first step deals with the problem of the shape ratio; it is proposed to compensate the two main curvatures to obtain the good ratio in Hertzian conditions. Then, Hertz and Kalker's equations are used to establish the stresses at the level of a strip. These stresses expressions are used directly in a contact model discretized in strips and tabulated as a function of the lateral displacement between the wheel and the rail. The validation is made by comparison to the previous multi-Hertzian model of the VOCOLIN software. A first test shows identical results in Hertzian conditions; a second one shows only a small difference in semi-Hertzian conditions like S1002/UIC60 1:40.
A numerical method is presented for calculating the contact area and the normal stress distribution between two elastic bodies of revolution with parallel axes. For the discretization of the Fredhom integral equation the contact area is divided into strips perpendicular to the axes of revolution. The stress distribution in each strip is assumed to be semi-elliptical in the strip direction and constant in the other direction. The contact area is calculated iteratively by satisfying the contact condition and the stress condition. In addition for a given total normal force an additional iteration is required to improve the value obtained for the rigid-body approach. The contact area and the stress distribution between two ellipsoids are compared with Hertzian solutions. To investigate the non-Hertzian contact problem a number of cases of wheel-rail contact are considered.
The simulation results of a wheelset modelled with two different wheel-rail elements are compared to each other. The first model is a wheelset element together with a track element. The contact between wheel and rail is treated as constraint and the kinematics of the wheelset is solved in advance. The second model is a wheel-rail interconnection element. The contact between wheel and rail is flexible. The results of the two models are compared for three different simulations. In the first simulation the wheelset modelled with the flexible contact patch is oscillating around a position with two contact points on one wheel. The second simulation is hunting of the wheelset with flange contact and the third is a hunting movement with small amplitudes. The normal forces calculated with the two elements differ in all three simulations. Beside flexibility and two point contact the reason for differences is the variation of the curvature in the contact points. In simulations governed by two point contact or when the wheel is bouncing to the flange it is recommended to work with flexible wheel-rail contact. But the main advantage of the wheel-rail interconnection element is the flexibility gained in modelling. It is not restricted to a wheelset, it has the advantage to model wheel-rail contact in very different wheel-rail guiding mechanism.
1 The Rolling Contact Problem.- 2 Review.- 3 The Simplified Theory of Contact.- 4 Variational and Numerical Theory of Contact.- 5 Results.- 6 Conclusion.- Appendix A The basic equations of the linear theory of elasticity.- Appendix B Some notions of mathematical programming.- Appendix C Numerical calculation of the elastic field in a half-space.- Appendix D Three-dimensional viscoelastic bodies in steady state frictional rolling contact with generalisation to contact perturbations.- Appendix E Tables.
An algorithm “Fastsim” for the simplified theory of rolling contact is described which is 15-25 times as fast as the existing programs Simrol (Kalker), and 3 times as fast as Rolcon (Knothe). The relative total force computed with Fastsim differs at most 0.2 from that calculated with Simrol, Simcona (Goree & Law), Rolcon, and the “exact” program Duvorol (Kalker). Descriptions and lists of an Algol 60, and HP 67 program version are available upon request: the Fortran IV version is given in the paper.
In the present paper three problems in the simplified theory of rolling contact are investigated. As to the first problem, three benchmark loadings, derived by Kalker in 1973 for Hertzian rollingcontact, are in existence. Each of these loadings gives rise to a value of the flexibility parameter of the simplified theory. These values are combined to a single, creepage dependent value of the flexibility, which appears to have an error of at most 10 to 15%. Secondly, the law of Coulomb is generalised by introducing two values of the coefficient of friction. The FASTSIM algorithm is adapted to that, and it is found that the traction, and hence the displacement, show a discontinuity inside the contact area. The discontinuity in the displacement is removed by introducing damping in the constitutive relations of the simplified theory. The damping constant is determined experimentally. When the damping coefficient decreases, the damped solution tends to the solution obtained directly without damping. This establishes the correctness of the latter, but it does not compare well with the complete theory as implemented by the program CONTACT. Thirdly and finally, it is shown that inertial effects may be neglected at speeds of around 100 km/h and also for much higher rolling velocities.
In this paper the computer code Duvorol, dealing with the computation of three‐dimensional rolling contact with dry friction, is described. It is based on the variational principle of Duvaut and Lions for dry friction, which leads to an incremental theory. The relevant properties of Duvorol are: Generality. All half‐space steady‐state rolling contact problems with Hertzian normal contact can be treated.
Reliability. The total tangential force is always found with reasonable accuracy by a standard discretization.
Speed. On an IBM 370/158 the calculation of a case then takes only several seconds.