Dynamics of an isolated railway wheelset with conformal wheel–rail interactions

ABSTRACT

In multi-body dynamics, we model a non-conformal wheel/rail contact at one centre point since this contact is flat and Hertzian. However, the quasi-conformal contact requires more points since the contact is curved and non-Hertzian. The methodology in solving these points during dynamic simulations is the basis of this development. In this online effort, first, we present a general contact joint and the gutter search method of Pascal and Jourdan [The rigid-multi-Hertzian method as applied to conformal contacts. USA: ASME; 2007] in the context of a multi-body approach. Next, by adopting the non-iterative approach, a subset of these points with positive profile interpenetrations is selected to idealise one curved contact by a set of multi-Hertzian patches for which the Hertz normal contact solution is available. Finally, the feasibility of this multibody approach together with its implementation in two different codes is evaluated by simulating the motion of an isolated wheelset with realistic inertia. On tangent tracks, the wheelset with non-conformal pairing displays self-excited unstable oscillations while it displays a stable behaviour below a critical speed with conformal profiles. A further study reveals the net friction losses at multiple patches within the curved contact being the reason for the stable behavior.     

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 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.     

The Elastic Cross-Influence between Two Quasi-Hertzian Contact Zones

In this paper a type of contact between two bodies is considered, which leads to the formation of two separate contact zones. The contact zones considered are Hertzian ellipses if the distance between them is large. When the distance between the zones is finite it is necessary to take into account the elastic cross-influence of the two zones. In this paper an approximate method is proposed which allows the determination of the actual contact zones as Hertz's ellipses but without disregarding the cross-influence. Numerical results are presented for two, two-humped bodies pressed against each other and for steady state rolling of a flanged wheel along a steel rail.

It is shown that neglecting the cross influence for a flanged wheel leads to a negligible error in the contact forces but the forces are transmitted between wheel and rail through contact patches which are generally more slender than with the cross-influence neglected.     

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.     

The Elastic Cross-Influence between Two Quasi-Hertzian Contact Zones

SUMMARY

In this paper a type of contact between two bodies is considered, which leads to the formation of two separate contact zones. The contact zones considered are Hertzian ellipses if the distance between them is large. When the distance between the zones is finite it is necessary to take into account the elastic cross-influence of the two zones. In this paper an approximate method is proposed which allows the determination of the actual contact zones as Hertz's ellipses but without disregarding the cross-influence. Numerical results are presented for two, two-humped bodies pressed against each other and for steady state rolling of a flanged wheel along a steel rail.

It is shown that neglecting the cross influence for a flanged wheel leads to a negligible error in the contact forces but the forces are transmitted between wheel and rail through contact patches which are generally more slender than with the cross-influence neglected.     

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 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 optimal pole-matching observer design for estimating tyre–road friction force

In this paper, considering the dynamical model of tyre–road contacts, we design a nonlinear observer for the on-line estimation of tyre–road friction force using the average lumped LuGre model without any simplification. The design is the extension of a previously offered observer to allow a muchmore realistic estimation by considering the effect of the rolling resistance and a term related to the relative velocity in the observer. Our aim is not to introduce a new friction model, but to present a more accurate nonlinear observer for the assumed model. We derive linear matrix equality conditions to obtain an observer gain with minimum pole mismatch for the desired observer error dynamic system. We prove the convergence of the observer for the non-simplified model. Finally, we compare the performance of the proposed observer with that of the previously mentioned nonlinear observer, which shows significant improvement in the accuracy of estimation.     

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.     

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.     

Optimisation of railway crossing geometry considering a representative set of wheel profiles

A numerical method for robust geometry optimisation of railway crossings is presented. The robustness is achieved by optimising the crossing geometry for a representative set of wheel profiles. As a basis for the optimisation, a crossing geometry is created where rail cross-sectional profiles and longitudinal height profiles of both wing rails and crossing nose are parameterised. Based on the approximation that the two problems are decoupled, separate optimisations are performed for the cross-sectional rail profiles and the longitudinal height profiles. The rail cross sections are optimised to minimise the maximum Hertzian wheel–rail contact pressure. The longitudinal height profiles are optimised to minimise the accumulated damage in the wing rail to crossing nose transition zone. The accumulated damage is approximated using an objective criterion that accounts for the angle of the wheel trajectory reversal during the transition from the wing rail to the crossing nose as well as the distribution of transition points for the utilised wheel profile set. It is found that small nonlinear height deviations from a linear longitudinal wing rail profile in the transition zone can reduce the objective compared to the nominal design. It is further demonstrated that the variation in wheel profile shapes, lateral wheel displacements and the feasible transition zone length of the crossing will determine the longitudinal height profiles of the wing rail and crossing nose if all wheel profiles are to make their transition within the transition zone.     

The VOCO multi-body software in the context of real-time simulation

The railway multi-body software developed for more than 20 years by INRETS (now IFSTTAR), under the name VOCO^®, has been dedicated from the origin to highly nonlinear elements, such as the dry friction dampers of freight bogies and the wheel–rail contact based on measured profiles. A second important step has been the discovery of a particular method in order to simulate on sinuous tracks. In the aim of industrial applications, the specification has always been to reach the goal of real time. Although it is not possible in all the cases, the recent non-Hertzian contact development is allowing real-time simulation to be achieved.     

Emission-robust operation of diesel HEV considering transient emissions

In this paper, we consider a method to create an engine emission simulation model for cycle and customer driving of a vehicle. The emission model results from an empiric approach, also taking into account the effects of engine dynamics on emissions. We analysed transient engine emissions in driving cycles and during representative customer driving profiles and created emission meta models. The analysis showed a significantly higher correlation in emissions when simulating realistic customer driving profiles using the created verified meta models (< 1 % model error) compared to static approaches, which are commonly used for vehicle simulation. Therefore, a transient modelling approach is conducted, which shows a great increase in accuracy in customer driving operation.     

常温施工式填缝料低温性能评价体系的研究

结合东北地区的气候特点．分析了规范中关于常温施工式填缝料低温性能评价方法的不足．补充了用于评定填缝料与水泥混凝土粘结性能的低温拉伸试验．并结合-10℃的弹性试验,通过对7种常温施工式填缝料的室内研究,建立了常温施工式填缝料的低温性能评价体系．提出了相应的指标要求。采用此评价体系和指标对7种填缝料进行了综合评定和分级．结果与实际工程的使用效果一致。     

基于环模型的轮胎有限元建模与仿真研究

为了提高仿真模型的计算效率,基于环模型理论系统研究了轮胎二维有限元模型的建模技术、参数确定方法和轮胎包容特性分析技术.从试验与仿真结果对比分析可知,利用有限元方法基于轮胎REF模型建模,在对轮胎胎侧弹性进行非线性模型修正后,得到的轮胎低速滚动仿真结果与试验结果基本吻合,验证了轮胎模型的有效性,同时为车辆一地面系统虚拟试验提供了一种实时高精度的轮胎面内特性仿真建模方法.     

弧齿锥齿轮的数字化滚检

提出了一种弧齿锥齿轮数字化滚检的方法,该方法反映了实际齿面的印痕与传动误差以及实际齿面相对于理论齿面在啮合性能上的差别,解决了由加工误差造成的弧齿锥齿轮设计与实际检验的评价基准不统一的问题.以某弧齿锥齿轮为例进行了数字化滚检,并与该对弧齿锥齿轮实际滚检试验结果进行对比,验证了该方法的正确性及可行性.     

An inverse design method for rail grinding profiles

Inspired by the optimisation design method for restoration of worn wheel profiles, an inverse design method based on optimal rail grinding profiles is presented in this paper. To improve grinding quality, vehicle dynamic performance is chosen as the main criteria and rolling radii difference function is selected as the key factor (also main target function) determining dynamic performance. Grinding material to be removed is chosen as the auxiliary target aimed at extending rail service life. Besides that, wheel–rail contact distribution is also taken into consideration as an auxiliary target preventing stress concentration and fatigue growth. By introducing certain presuppositions, all the design targets will form an inverse design problem. This problem can be solved using hybrid discrete numerical methods. Considering different grinding requirements, two examples of grinding profile design for straight and curved track will be discussed. Results show that the presented method is efficient and effective. Practical implementation has been carried out at several grinding sites in China.     

Some New Results in Rolling Contact

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.