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.     

A fast-simplified wheel–rail contact model consistent with perfect plastic materials

A method is described which is an extension of rolling contact models with respect to plasticity. This new method, which is an extension of the STRIPES semi-Hertzian (SH) model, has been implemented in a multi-body-system (MBS) package and does not result in a longer execution time than the STRIPES SH model [J.B. Ayasse and H. Chollet, Determination of the wheel–rail contact patch in semi-Hertzian conditions, Veh. Syst. Dyn. 43(3) (2005), pp. 161–172]. High speed of computation is obtained by some hypotheses about the plastic law, the shape of stresses, the locus of the maximum stress and the slip. Plasticity does not change the vehicle behaviour but there is a need for an extension of rolling contact models with respect to plasticity as far as fatigue analysis of rail is concerned: rolling contact fatigue may be addressed via the finite element method (FEM) including material non-linearities, where loads are the contact stresses provided by the post-processing of MBS results [K. Dang Van, M.H. Maitournam, Z. Moumni, and F. Roger, A comprehensive approach for modeling fatigue and fracture of rails, Eng. Fract. Mech. 76 (2009), pp. 2626–2636]. In STRIPES, like in other MBS models, contact stresses may exceed the plastic yield criterion, leading to wrong results in the subsequent FEM analysis. With the proposed method, contact stresses are kept consistent with a perfect plastic law, avoiding these problems. The method is benchmarked versus non-linear FEM in Hertzian geometries. As a consequence of taking plasticity into account, contact patch area is bigger than the elastic one. In accordance with FEM results, a different ellipse aspect ratio than the one predicted by Hertz theory was also found and finally pressure does not exceed the threshold prescribed by the plastic law. The method also provides more exact results with non-Hertzian geometries. The new approach is finally compared with non-linear FEM in a tangent case with a unidirectional load and a complete slip: when plasticity is taken into account, and for large adhesion values, friction forces have an influence on the size of the contact patch. The proposed approach enables also to assess extensively the level of plasticity along a track through an indicator associated with a given yield stress.     

Kalker's algorithm Fastsim solves tangential contact problems with slip-dependent friction and friction anisotropy

This paper presents two extensions of Kalker's algorithm Fastsim of the simplified theory of rolling contact. The first extension is for solving tangential contact problems with the coefficient of friction depending on slip velocity. Two friction laws have been considered: with and without recuperation of the static friction. According to the tribological hypothesis for metallic bodies shear failure, the friction law without recuperation of static friction is more suitable for wheel and rail than the other one. Sample results present local quantities inside the contact area (division to slip and adhesion, traction) as well as global ones (creep forces as functions of creepages and rolling velocity). For the coefficient of friction diminishing with slip, the creep forces decay after reaching the maximum and they depend on the rolling velocity. The second extension is for solving tangential contact problems with friction anisotropy characterised by a convex set of the permissible tangential tractions. The effect of the anisotropy has been shown on examples of rolling without spin and in the presence of pure spin for the elliptical set. The friction anisotropy influences tangential tractions and creep forces. Sample results present local and global quantities. Both extensions have been described with the same language of formulation and they may be merged into one, joint 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.     

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.     

Determination of the wheel rail contact patch in semi-Hertzian conditions

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.     

Modification of the semi-Hertz wheel–rail contact method based on recalculating the virtual penetration value

ABSTRACT

Wheel–rail contact calculation is of vital importance in vehicle system dynamics. In the existing methods of wheel–rail contact calculation, the finite element method and Kalker’s CONTACT program, which are based on the complementary energy principle, are the two methods with accuracy recognised. However, because of its very slow calculation speed, it cannot meet the requirement of online calculation, so a variety of fast non-elliptic algorithms have been proposed. The semi-Hertz method, which is recognised for its great contributions to the fast wheel–rail contact calculation, is based on the concept of virtual penetration. The calculation of virtual penetration is crucial to evaluate the shape and normal pressure distribution of the contact patch. In practice, the virtual penetration is related to the curvature of the whole contact patch; however, the range of the contact patch is determined by the value of penetration. Such an interaction leads the calculation into a dead loop. In the semi-Hertz method, the penetration is calculated by the Hertz parameters of the initial contact point. Thus, the practical range of the method is limited. In this paper, a fast-iterative method for solving virtual penetration is proposed, and a reliable value of virtual penetration can be obtained under any lateral wheel–rail relative curvature variation with good stability and speed. The normal and tangential solutions are analysed with different methods in this paper.     

Tyre rolling kinematics and prediction of tyre forces and moments: part I – theory and method

A new method to describe tyre rolling kinematics and how to calculate tyre forces and moments is presented. The Lagrange–Euler method is used to calculate the velocity and contact deformation of a tyre structure under large deformation. The calculation of structure deformation is based on the Lagrange method, while the Euler method is used to analyse the deformation and forces in the contact area. The method to predict tyre forces and moments is built using kinematic theory and nonlinear finite element analysis. A detailed analysis of the tyre tangential contact velocity and the relationships between contact forces, contact areas, lateral forces, and yaw and camber angles has been performed for specific tyres. Research on the parametric sensitivity of tyre lateral forces and self-aligning torque on tread stiffness and friction coefficients is carried out in the second part of this paper.     

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.     

Tangential problem solution for non-elliptical contact areas with the FastSim algorithm

This article deals with the application of the FastSim algorithm to the solution of the tangential contact problem for non-elliptical contact areas. At first, the causes creating problems for the solution of non-hertzian contact areas with this algorithm shall be analyzed. Then, different currently existing methods shall be studied, analyzing their accuracy characteristics and computational cost to determine whether or not they are appropriate to use in dynamic simulations. Finally, a new strategy shall be proposed that, in the opinion of the authors, offers good characteristics of precision and computational cost.     

Frequency-Dependent Creep Coefficients for Three-Dimensional Rolling Contact Problems

Kalker's creep coefficients for linear rolling contact problems are only valid in the steady state case. A method for the extension of linear contact mechanics into the high frequency range is presented. Frequency dependent creep coefficients are obtained, which can be interpreted as harmonic responses of the creep forces for harmonically varying creepages. The influence of different constant longitudinal creepages and of different shapes of the contact area on the harmonic responses are discussed. The analysis is based on Kalker [5]     

Tangential problem solution for non-elliptical contact areas with the FastSim algorithm

This article deals with the application of the FastSim algorithm to the solution of the tangential contact problem for non-elliptical contact areas. At first, the causes creating problems for the solution of non-hertzian contact areas with this algorithm shall be analyzed. Then, different currently existing methods shall be studied, analyzing their accuracy characteristics and computational cost to determine whether or not they are appropriate to use in dynamic simulations. Finally, a new strategy shall be proposed that, in the opinion of the authors, offers good characteristics of precision and computational cost.     

A numerical model of a HIL scaled roller rig for simulation of wheel–rail degraded adhesion condition

Scaled roller rigs used for railway applications play a fundamental role in the development of new technologies and new devices, combining the hardware in the loop (HIL) benefits with the reduction of the economic investments. The main problem of the scaled roller rig with respect to the full scale ones is the improved complexity due to the scaling factors. For this reason, before building the test rig, the development of a software model of the HIL system can be useful to analyse the system behaviour in different operative conditions. One has to consider the multi-body behaviour of the scaled roller rig, the controller and the model of the virtual vehicle, whose dynamics has to be reproduced on the rig. The main purpose of this work is the development of a complete model that satisfies the previous requirements and in particular the performance analysis of the controller and of the dynamical behaviour of the scaled roller rig when some disturbances are simulated with low adhesion conditions. Since the scaled roller rig will be used to simulate degraded adhesion conditions, accurate and realistic wheel–roller contact model also has to be included in the model. The contact model consists of two parts: the contact point detection and the adhesion model. The first part is based on a numerical method described in some previous studies for the wheel–rail case and modified to simulate the three-dimensional contact between revolute surfaces (wheel–roller). The second part consists in the evaluation of the contact forces by means of the Hertz theory for the normal problem and the Kalker theory for the tangential problem. Some numerical tests were performed, in particular low adhesion conditions were simulated, and bogie hunting and dynamical imbalance of the wheelsets were introduced. The tests were devoted to verify the robustness of control system with respect to some of the more frequent disturbances that may influence the roller rig dynamics. In particular we verified that the wheelset imbalance could significantly influence system performance, and to reduce the effect of this disturbance a multistate filter was designed.     

Derivation of Frequency Dependent Creep Coefficients Based on an Elastic Half-Space Model

Kalker's creep coefficients for linear rolling contact problems are only valid in the steady state case. A method for the extension of linear contact mechanics into the high frequency range is presented. Frequency dependent creep coefficients are obtained, which can be interpreted as harmonic responses of the creep forces for harmonically varying creepages. For linear investigations of contact phenomena in the high frequency range simple laws governing the Kalker coefficients C₁₁and C₂₂ are given.     

A new method for the solution of the normal contact problem in the dynamic simulation of railway vehicles

Due to requirements related to computational efficiency, in the majority of railway dynamic simulators the Hertz theory is used for solving the normal problem in wheel/rail contact. This theory is based on a large number of assumptions. Particularly noteworthy is the assumed simplification that the undeformed distance between the bodies in contact can be assimilated by a quadratic function. There are many situations in which the undeformed distance cannot be represented by this kind of function. As such, the results obtained with Hertz theory in these cases are not accurate.

In this paper, a new method for solving the normal problem that overcomes the above-mentioned limitation is presented. First, the exactness of the new method is tested with Hertzian cases. The results obtained are almost exact. Second, the results calculated with the new method in more general cases are compared with the ones obtained with the variational method of Kalker (more exact but computationally less efficient).     

About Multi-Hertzian-Contact Hypothesis and Equivalent Conicity in the Case of S1002 and UIC60 Analytical Wheel/Rail Profiles

In a previous paper [1], a new method using 'multi-Hertzian' elasticity for identifying equivalent parameters of railway wheelsets rolling on defined tracks was presented. Another method also using Hertzian elasticity [2] has produced different results. Without knowing the origin of these discrepancies, the question was asked whether any of these methods could adequately describe actual non-Hertzian contacts. This paper, using as a basis Kalker's 'Contact' software which solves the non-Hertzian case for normal and tangential forces, answers this question, demonstrating that the assumption of summing Hertzian contacts is realistic provided that enough ellipses are considered. On this basis, the 'equivalent conicity' of the standard wheelset equipped with S1002 profiles rolling on UIC60 (1/40) rails is analysed in the range of small lateral displacements: around 2 mm it amounts to between 0.13 and 0.16. It is demonstrated that, due to actual wheel and rail elasticity, linearization of rigid profiles conicity, whatever method being used, is not realistic in practice.     

Simulations of roughness growth on rails - results from a 2D non-Hertzian, non-steady contact model

A new model for simulating rail roughness growth on tangent track is presented in this paper. The model consists of three relatively independent components: (1) a time-domain vehicle/track interaction model; (2) a 2D non-Hertzian and non-steady wheel/rail contact model; and (3) a wear model. Wheel/rail contact forces for a given initial roughness obtained from the vehicle/track interaction model are used by the contact model to calculate the contact patch size, normal pressure and tangential stresses with material removal assumed to be linearly proportional to the friction work in the contact patch. The roughness profile is updated and fed back into vehicle/track interaction model. The 2D contact model is initially compared with a 3D model for various wavelength of initial sinusoidal roughness. Long term roughness growth is then simulated with the 2D contact model. Simulation shows that all initial sinusoidal roughness of wavelengths between 20-100 mm are levelled out. The wavelength-fixing mechanism, that has previously been used to explain the cause of corrugation, is not found in the present investigations.     

Wheel life prediction model – an alternative to the FASTSIM algorithm for RCF

In this article, a wheel life prediction model considering wear and rolling contact fatigue (RCF) is developed and applied to a heavy-haul locomotive. For wear calculations, a methodology based on Archard's wear calculation theory is used. The simulated wear depth is compared with profile measurements within 100,000?km. For RCF, a shakedown-based theory is applied locally, using the FaStrip algorithm to estimate the tangential stresses instead of FASTSIM. The differences between the two algorithms on damage prediction models are studied. The running distance between the two reprofiling due to RCF is estimated based on a Wöhler-like relationship developed from laboratory test results from the literature and the Palmgren-Miner rule. The simulated crack locations and their angles are compared with a five-year field study. Calculations to study the effects of electro-dynamic braking, track gauge, harder wheel material and the increase of axle load on the wheel life are also carried out.