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.     

Creep force modelling for rail traction vehicles based on the Fastsim algorithm

The evaluation of creep forces is a complex task and their calculation is a time-consuming process for multibody simulation (MBS). A methodology of creep forces modelling at large traction creepages has been proposed by Polach [Creep forces in simulations of traction vehicles running on adhesion limit. Wear. 2005;258:992–1000; Influence of locomotive tractive effort on the forces between wheel and rail. Veh Syst Dyn. 2001(Suppl);35:7–22] adapting his previously published algorithm [Polach O. A fast wheel–rail forces calculation computer code. Veh Syst Dyn. 1999(Suppl);33:728–739]. The most common method for creep force modelling used by software packages for MBS of running dynamics is the Fastsim algorithm by Kalker [A fast algorithm for the simplified theory of rolling contact. Veh Syst Dyn. 1982;11:1–13]. However, the Fastsim code has some limitations which do not allow modelling the creep force – creep characteristic in agreement with measurements for locomotives and other high-power traction vehicles, mainly for large traction creep at low-adhesion conditions. This paper describes a newly developed methodology based on a variable contact flexibility increasing with the ratio of the slip area to the area of adhesion. This variable contact flexibility is introduced in a modification of Kalker's code Fastsim by replacing the constant Kalker's reduction factor, widely used in MBS, by a variable reduction factor together with a slip-velocity-dependent friction coefficient decreasing with increasing global creepage. The proposed methodology is presented in this work and compared with measurements for different locomotives. The modification allows use of the well recognised Fastsim code for simulation of creep forces at large creepages in agreement with measurements without modifying the proven modelling methodology at small creepages.     

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.     

The Kalker book of tables for non-Hertzian contact of wheel and rail

A new regularisation of non-elliptical contact patches has been introduced, which enables building the look-up table called by us the Kalker book of tables for non-Hertzian contact (KBTNH), which is a fast creep force generator that can be used by multibody dynamics system simulation programs. The non-elliptical contact patch is regularised by a simple double-elliptical contact region (SDEC). The SDEC region is especially suitable for regularisation of contact patches obtained with approximate non-Hertzian methods for solving the normal contact problem of wheel and rail. The new regularisation is suitable for wheels and rails with any profiles, including worn profiles.

The paper describes the new procedure of regularisation of the non-elliptical contact patch, the structure of the Kalker book of tables, and parameterisation of the independent variables of the tables and creep forces.

A moderate volume Kalker book of tables for SDEC region suitable for simulation of modern running gears has been computed in co-simulation of Matlab and program CONTACT.

To access the creep forces of the Kalker book of tables, the linear interpolation has been applied.

The creep forces obtained from KBTNH have been compared to those obtained by program CONTACT and FASTSIM algorithm. FASTSIM has been applied on both the contact ellipse and the SDEC contact patch. The comparison shows that KBTNH is in good agreement with CONTACT for a wide range of creepage condition and shapes of the contact patch, whereas the use of FASTSIM on the elliptical patch and SDEC may lead to significant deviations from the reference CONTACT solutions.

The computational cost of calling creep forces from KBTNH has been estimated by comparing CPU time of FASTSIM and KBTNH. The KBTNH is 7.8–51 times faster than FASTSIM working on 36–256 discretisation elements, respectively.

In the example of application, the KBTNH has been applied for curving simulations and results compared with those obtained with the creep force generator employing the elliptical regularisation. The results significantly differ, especially in predicted creepages, because the elliptical regularisation neglects generation of the longitudinal creep force by spin creepage.     

Modeling and Dynamics of Friction Wedge Dampers in Railroad Freight Trucks

A mechanical model for the vertical response of the bolster and friction-wedge suspension elements in a railroad freight truck is developed. The model incorporates linear stiffnesses and dry friction contact between rigid elements. Numerical bifurcation diagrams for sticking events, obtained for parameterizations of the amplitude and frequency of a harmonic track excitation, and for various friction laws, reveal distinct parameter-dependent dynamic responses: strongly nonlinear stick-slip oscillations, observed away from period-1 resonance; and weakly nonlinear, nearly harmonic responses, observed near the period-one resonance. The analytical method of harmonic balance is applied to develop a first order approximation for the period-1 response, and, as should be expected, is found to correspond well with the numerical results in the large amplitude limit and near the resonance location. Given that such weakly nonlinear, large amplitude, period-1 responses are not common for the speeds and system parameters of most freight vehicles, we conclude that friction-wedge models based on some form of equivalent linearization may be inadequate for simulating freight car dynamics in many cases of practical importance.     

Introduction of a friction coefficient dependent on the slip in the FastSim algorithm

One of the main limitations of algorithms relating forces and creepages at the wheel/rail contact is the use of a friction coefficient independent of the slip. This paper overcomes this limitation through a modification of the FastSim algorithm (based on the Simplified Theory of Kalker). A friction law based on the local value of the slip is established and the required formulation of the local slip elsewhere in the contact area is presented. Some difficulties of the method and the solutions adopted by the authors are also presented. Finally, the achieved improvements are shown through comparison of the results obtained both with the original and the modified FastSim algorithms.     

基于苏通大桥混凝土短期试验结果的徐变预测模型修正

为提高混凝土长期徐变预测精度,通过对不同加载龄期的苏通大桥用3组高强混凝土进行徐变试验,根据1个月内的试验结果,采用不同方法对ACI 209R,CEB-FIP 1990和B3徐变预测模型分别进行修正,通过比较变异系数及1年徐变试验结果,分析不同修正后模型的预测精度,结果表明,对于苏通大桥用高强混凝土,修正B3徐变模型具有最高的预测精度。     

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.     

高强混凝土收缩徐变试验及预测模型研究

通过苏通大桥连续刚构所用高强混凝土的收缩徐变试验,以及其他几组不同强度等级的高强混凝土收缩徐变试验,探讨了目前常用收缩徐变模型对高强混凝土收缩徐变的适用性。试验结果表明,高强混凝土的徐变系数一般低于常用的徐变模型预测值;而现桥规采用的CEB-FIP90收缩模型有低估高强混凝土收缩发展的危险,并且,随着混凝土抗压强度的提高,预测精度有降低的趋势。针对高强混凝土收缩徐变的特点,提出了考虑混凝土强度因素的修正收缩、徐变模型。最后运用B3变异系数法比较了这几种模型预测高强混凝土收缩徐变的精度,比较结果表明,修正收缩、徐变模型对高强混凝土收缩徐变预测的精度相对于现有模型有较大提高。     

A linear complementarity method for the solution of vertical vehicle–track interaction

A new method is proposed for the solution of the vertical vehicle–track interaction including a separation between wheel and rail. The vehicle is modelled as a multi-body system using rigid bodies, and the track is treated as a three-layer beam model in which the rail is considered as an Euler-Bernoulli beam and both the sleepers and the ballast are represented by lumped masses. A linear complementarity formulation is directly established using a combination of the wheel–rail normal contact condition and the generalised-α method. This linear complementarity problem is solved using the Lemke algorithm, and the wheel–rail contact force can be obtained. Then the dynamic responses of the vehicle and the track are solved without iteration based on the generalised-α method. The same equations of motion for the vehicle and track are adopted at the different wheel–rail contact situations. This method can remove some restrictions, that is, time-dependent mass, damping and stiffness matrices of the coupled system, multiple equations of motion for the different contact situations and the effect of the contact stiffness. Numerical results demonstrate that the proposed method is effective for simulating the vehicle–track interaction including a separation between wheel and rail.     

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.     

Direct Covariance Analysis for the Calculation of Creepages and Creep-Forces for Various Bogies on Straight Track with Random Irregularities

In this paper creep-forces and creepages for an ICE express train coach and an electric locomotive BR 120 of the DB on straight track with random irregularities are investigated. The calculations are made with the multibody program MEDYNA using direct covariance analysis. One of the aims of this article is to emphasize the advantages of the direct covariance method (time domain) compared with the power spectral density method (frequency domain).     

大吨位徐变仪球铰非线性接触分析

因研究桥用高强度混凝土材料徐变性能的需要,自主研制了大吨位徐变仪。对于处于接触受力状态的徐变仪球铰区域,先用Hertz接触理论估算球铰区域尺寸范围,然后利用ANSYS通用有限元软件对该区域分别取2种垫板凹槽球径的模型进行接触分析,通过对比球铰区域的最大接触压力、mises应力、第3主应力来进一步确定垫板凹槽的合理球径。用该徐变仪对2座大桥成功进行了高强度混凝土徐变试验,并实际验证了试验荷载下徐变仪球铰区域的受力性能,以及有限元接触分析结果的正确性。     

Re-adhesion control for a railway single wheelset test rig based on the behaviour of the traction motor

A method for detecting wheel slip/slide and re-adhesion control of AC traction motors in railway applications is presented in this paper. This enables a better utilisation of available adhesion and could also reduce wheel wear by reducing high creep values. With this method, the wheel–rail (roller) creepage, creep force and friction coefficient can be indirectly detected and estimated by measuring the voltage, current and speed of the AC traction motor and using an extended Kalman filter. The re-adhesion controller is designed to regulate the motor torque command according to the maximum available adhesion based on the estimated results. Simulations under different friction coefficients are carried out to test the proposed method.     

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.     

强迫应变法在混凝土桥梁徐变分析中的应用

根据积分中值定理,利用时间段中点时间徐变系数推导了有效弹性模量计算公式,以替代传统的老化系数方法;并视徐变应变为强迫应变,推导了节段施工桥梁徐变分析的有限元方法。有限元列式直接用应力的方式给出,以适应桥梁的三维空间有限元的需要。最后通过一座逐跨施工的混凝土连续梁桥算例验证了方法的可行性。     

High Speed Stability for Rail Vehicles Considering Varying Conicity and Creep Coefficients

The critical or hunting speed of solid axle rail vehicles is known to be a strong function of primary suspension stiffness, wheel/rail profile geometry (conicity and gravitational stiffness), wheel/rail friction forces (creep coefficients), bogie/carbody inertia properties, and secondary suspension design. This paper deals with the problem of maximizing the critical speed through design of the primary and secondary suspension but with control only over the range of wheel/rail geometry and friction characteristics. For example, the conicity may varie from .05 to .3 and the linear creep coefficients from 25% to 100% of the predicted Kalker values.

It is shown that the maximum critical speed is greatly limited by the wheel/rail geometry and friction variations. It is also shown that, when lateral curving and ride quality are considered, the best design approach is to select an intermediate primary longitudinal stiffness, to limit the lowest value of conicity (e.g. to .1 or .2) by wheel profile redesign, increasing the secondary yaw damping value (yaw relaxation) and optimizing the primary and secondary lateral stiffness.     

Direct Covariance Analysis for the Calculation of Creepages and Creep-Forces for Various Bogies on Straight Track with Random Irregularities

SUMMARY

In this paper creep-forces and creepages for an ICE express train coach and an electric locomotive BR 120 of the DB on straight track with random irregularities are investigated. The calculations are made with the multibody program MEDYNA using direct covariance analysis. One of the aims of this article is to emphasize the advantages of the direct covariance method (time domain) compared with the power spectral density method (frequency domain).