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.     

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.     

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.     

Development and validation of a wear model for the analysis of the wheel profile evolution in railway vehicles

The numerical wheel wear prediction in railway applications is of great importance for different aspects, such as the safety against vehicle instability and derailment, the planning of wheelset maintenance interventions and the design of an optimal wheel profile from the wear point of view. For these reasons, this paper presents a complete model aimed at the evaluation of the wheel wear and the wheel profile evolution by means of dynamic simulations, organised in two parts which interact with each other mutually: a vehicle's dynamic model and a model for the wear estimation. The first is a 3D multibody model of a railway vehicle implemented in SIMPACK?, a commercial software for the analysis of mechanical systems, where the wheel–rail interaction is entrusted to a C/C++user routine external to SIMPACK, in which the global contact model is implemented. In this regard, the research on the contact points between the wheel and the rail is based on an innovative algorithm developed by the authors in previous works, while normal and tangential forces in the contact patches are calculated according to Hertz's theory and Kalker's global theory, respectively. Due to the numerical efficiency of the global contact model, the multibody vehicle and the contact model interact directly online during the dynamic simulations.

The second is the wear model, written in the MATLAB® environment, mainly based on an experimental relationship between the frictional power developed at the wheel–rail interface and the amount of material removed by wear. Starting from a few outputs of the multibody simulations (position of contact points, contact forces and rigid creepages), it evaluates the local variables, such as the contact pressures and local creepages, using a local contact model (Kalker's FASTSIM algorithm). These data are then passed to another subsystem which evaluates, by means of the considered experimental relationship, both the material to be removed and its distribution along the wheel profile, obtaining the correspondent worn wheel geometry.

The wheel wear evolution is reproduced by dividing the overall chosen mileage to be simulated in discrete spatial steps: at each step, the dynamic simulations are performed by means of the 3D multibody model keeping the wheel profile constant, while the wheel geometry is updated through the wear model only at the end of the discrete step. Thus, the two parts of the whole model work alternately until the completion of the whole established mileage. Clearly, the choice of an appropriate step length is one of the most important aspects of the procedure and it directly affects the result accuracy and the required computational time to complete the analysis.

The whole model has been validated using experimental data relative to tests performed with the ALn 501 ‘Minuetto’ vehicle in service on the Aosta–Pre Saint Didier track; this work has been carried out thanks to a collaboration with Trenitalia S.p.A and Rete Ferroviaria Italiana, which have provided the necessary technical data and experimental results.     

A simplified model of wheel/rail contact mechanics for non-Hertzian problems and its application in rail vehicle dynamic simulations

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.     

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.     

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.     

Influence of AC system design on the realisation of tractive efforts by high adhesion locomotives

The main task for heavy haul railway operators is to reduce the cost of exported minerals and enhance the long-term viability of rail transport operations through increasing productivity by running longer and heavier trains. The common opinion is that this is achievable by means of implementation of high adhesion locomotives with advanced AC traction technologies. Modern AC high adhesion locomotives are very complex mechatronic systems and can be designed with two alternative traction topologies of either bogie or individual axle controls. This paper describes a modelling approach for these two types of AC traction systems with the application of an advanced co-simulation methodology, where an electrical system and a traction algorithm are modelled in Matlab/Simulink, and a mechanical system is modelled in a multibody software package. Although the paper concentrates on the analysis of the functioning for these two types of traction control systems, the choice of reference slip values also has an influence on the performance of both systems. All these design variations and issues have been simulated for various adhesion conditions at the wheel–rail interface and their influence on the high traction performance of a locomotive equipped with two three-axle bogies has been discussed.     

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.     

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.     

Simulation of curving at low speed under high traction for passive steering hauling locomotives

The traction control in modern electric and diesel electric locomotives has allowed rail operators to utilise high traction adhesion levels without undue risk of damage from uncontrolled wheel spin. At the same time, some locomotive manufacturers have developed passive steering locomotive bogies to reduce wheel rail wear and further improve locomotive adhesion performance on curves. High locomotive traction loads in curving are known to cause the loss of steering performance in passive steering bogies. At present there are few publications on the curving performance of locomotive steering with linkage bogies. The most extreme traction curving cases of low speed and high adhesion for hauling locomotives have not been fully investigated, with effects of coupler forces and cant excess being generally ignored. This paper presents a simulation study for three axle bogie locomotives in pusher and pulling train positions on tight curves. The simulation study uses moderate and high traction adhesion levels of 16.6% and 37% for various rail friction conditions. Curving performance is assessed, showing forced steering bogies to have considerable advantages over self steering bogies. Likewise it is shown that self steering bogies are significantly better than yaw relaxation bogies at improving steering under traction. As the required traction adhesion approaches the rail friction coefficient, steering performance of all bogies degrades and yaw of the bogie frame relative to the track increases. Operation with excess cant and tensile coupler forces are both found to be detrimental to the wear performance of all locomotive bogies, increasing the bogie frame yaw angles. Bogie frame pitching is also found to have significant effect on steering, causing increased performance differences between bogie designs.     

Simulation of dynamic interaction between train and railway turnout

Dynamic train–track interaction is more complex in railway turnouts (switches and crossings) than that in ordinary tangent or curved tracks. Multiple contacts between wheel and rail are common, and severe impact loads with broad frequency contents are induced, when nominal wheel–rail contact conditions are disturbed because of the continuous variation in rail profiles and the discontinuities in the crossing panel. The absence of transition curves at the entry and exit of the turnout, and the cant deficiency, leads to large wheel–rail contact forces and passenger discomfort when the train is switching into the turnout track. Two alternative multibody system (MBS) models of dynamic interaction between train and a standard turnout design are developed. The first model is derived using a commercial MBS software. The second model is based on a multibody dynamics formulation, which may account for the structural flexibility of train and track components (based on finite element models and coordinate reduction methods). The variation in rail profile is accounted for by sampling the cross-section of each rail at several positions along the turnout. Contact between the back of the wheel flange and the check rail, when the wheelset is steered through the crossing, is considered. Good agreement in results from the two models is observed when the track model is taken as rigid.     

Non-steady state modelling of wheel–rail contact problem

Among all the algorithms to solve the wheel–rail contact problem, Kalker's FastSim has become the most useful computation tool since it combines a low computational cost and enough precision for most of the typical railway dynamics problems. However, some types of dynamic problems require the use of a non-steady state analysis. Alonso and Giménez developed a non-stationary method based on FastSim, which provides both, sufficiently accurate results and a low computational cost. However, it presents some limitations; the method is developed for one time-dependent creepage and its accuracy for varying normal forces has not been checked. This article presents the required changes in order to deal with both problems and compares its results with those given by Kalker's Variational Method for rolling contact.     

A study on high-speed rolling contact between a wheel and a contaminated rail

A 3-D explicit finite element model is developed to investigate the transient wheel–rail rolling contact in the presence of rail contamination or short low adhesion zones (LAZs). A transient analysis is required because the wheel passes by a short LAZ very quickly, especially at high speeds. A surface-to-surface contact algorithm (by the penalty method) is employed to solve the frictional rolling contact between the wheel and the rail meshed by solid elements. The LAZ is simulated by a varying coefficient of friction along the rail. Different traction efforts and action of the traction control system triggered by the LAZ are simulated by applying a time-dependent driving torque to the wheel axle. Structural flexibilities of the vehicle–track system are considered properly. Analysis focuses on the contact forces, creepage, contact stresses and the derived frictional work and plastic deformation. It is found that the longitudinal contact force and the maximum surface shear stress in the contact patch become obviously lower in the LAZ and much higher as the wheel re-enters the dry rail section. Consequently, a higher wear rate and larger plastic flow are expected at the location where the dry contact starts to be rebuilt. In other words, contact surface damages such as wheel flats and rail burns may come into being because of the LAZ. Length of the LAZ, the traction level, etc. are varied. The results also show that local contact surface damages may still occur as the traction control system acts.     

Locomotive dynamic performance under traction/braking conditions considering effect of gear transmissions

Traction or braking operations are usually applied to trains or locomotives for acceleration, speed adjustment, and stopping. During these operations, gear transmission equipment plays a very significant role in the delivery of traction or electrical braking power. Failures of the gear transmissions are likely to cause power loses and even threaten the operation safety of the train. Its dynamic performance is closely related to the normal operation and service safety of the entire train, especially under some emergency braking conditions. In this paper, a locomotive–track coupled vertical–longitudinal dynamics model is employed with considering the dynamic action from the gear transmissions. This dynamics model enables the detailed analysis and more practical simulation on the characteristics of power transmission path, namely motor–gear transmission–wheelset–longitudinal motion of locomotive, especially for traction or braking conditions. Multi-excitation sources, such as time-varying mesh stiffness and nonlinear wheel–rail contact excitations, are considered in this study. This dynamics model is then validated by comparing the simulated results with the experimental test results under braking conditions. The calculated results indicate that involvement of gear transmission could reveal the load reduction of the wheelset due to transmitted forces. Vibrations of the wheelset and the motor are dominated by variation of the gear dynamic mesh forces in the low speed range and by rail geometric irregularity in the higher speed range. Rail vertical geometric irregularity could also cause wheelset longitudinal vibrations, and do modulations to the gear dynamic mesh forces. Besides, the hauling weight has little effect on the locomotive vibrations and the dynamic mesh forces of the gear transmissions for both traction and braking conditions under the same running speed.     

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.