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In the modeling of railway vehicle-track dynamics and wheel-rail damage, simplified tangential contact models based on ellipse assumption are usually used due to strict limitation of computational cost. Since most wheel-rail contact cases appear to be nonelliptic shapes, a fast and accurate tangential model for nonelliptic contact case is in demand. In this paper, two ellipse-based simplified tangential models (i.e., FASTSIM and FaStrip) using three alternative nonelliptic adaptation approaches, together with Kalker’s NORM algorithm, are applied to wheel-rail rolling contact cases. It aims at finding the best approach for dealing with nonelliptic rolling contact. Compared to previous studies, the nonelliptic normal contact solution in the present work is accurately solved rather than simplification. Therefore, it can avoid tangential modeling evaluation affected by inaccurate normal contact solution. By comparing with Kalker’s CONTACT code, it shows both FASTSIM-based and FaStrip-based models can provide accurate global creep force. With regard to local rolling contact solution, only the accuracy of FaStrip-based models is satisfactory. Moreover, Ayasse-Chollet’s local ellipse approach appears to be the best choice for nonelliptic adaptation.

Wheel-rail damage has been a main concern of railway industry for increasing maintenance cost and potential risk. It is of importance to develop a reasonable maintenance strategy. Therefore, an efficient contact model is needed for computation of both vehicle-track interaction and wheel-rail rolling contact. The term ‘efficient’ here means the contact model is capable of solving nonelliptic rolling contact accurately and fast, because wheel-rail damage simulation such as wear prediction always needs millions of contact cases to account for the actual operating distance. To this end, exact contact models based on finite element method [

The tangential wheel-rail interaction depends on rolling contact for low energy dissipation. Rolling contact problem was first studied by Carter [

The most widely used tangential contact model for fast calculation is Kalker’s simplified theory and its algorithm FASTSIM [

It is noted that both FASTSIM and FaStrip are developed based on the assumption of elliptic contact shape, thus triggering a struggle for many scholars to adapt them to nonelliptic cases. In the literature, three alternative nonelliptic adaptation approaches originally developed for FASTSIM are available [

However, it is worth mentioning in [

In this paper, both FASTSIM and FaStrip are adapted to nonelliptic contact condition using three available approaches proposed in [

The outline of this paper is as follows. Section

A methodology aiming at evaluating the performance of simplified tangential contact models on nonelliptic cases is presented in Figure

In previous comparative studies [

Both FASTSIM and FaStrip are employed to evaluate their performance on predicting global creep force and local contact solutions such as tangential stress distribution, stick-slip division, slip velocity, and wear distribution. Three alternative approaches proposed in [

Kalker’s CONTACT code is employed as a reference to evaluate the simplified contact models under the combination of creepage and spin.

Schematic diagram of the methodology to evaluate simplified models.

In Kalker’s NORM algorithm, the wheel-rail contact is locally approximated by elastic half-space. The potential contact zone is discretized by rectangular elements as shown in Figure _{c} and can be obtained from half-space equation, _{IJ} is the influence coefficient in normal direction, meaning the displacement in element

Discretized potential contact zone in NORM algorithm.

From (

Kalker’s FASTSIM is developed based on his simplified theory [_{1},_{2}, and_{3} in longitudinal, lateral, and normal directions as in (_{11},_{22}, and_{23} are Kalker’s creep coefficients depending on semiaxes ratio_{0}/_{0};_{x} and_{y}) in each strip can be estimated from the leading edge where the tangential stresses (_{x} and_{y}) are set to be zero, _{x},_{y}, and _{0} is the maximum pressure. Note that the parabolic traction bound is unrealistic but necessary in FASTSIM to ensure a sufficient exact creep force.

Schematic diagrams of (a) FASTSIM and (b) FaStrip.

If

In FaStrip algorithm, the contact patch is also discretized as strips that in FASTSIM; see Figure _{0} is the maximum pressure. The longitudinal and lateral shear stress in stick area can be obtained from _{t} (_{x}/_{y}/

In this part, it describes three nonelliptic adaptation approaches proposed by Kik-Piotrowski [

Schematic diagrams of approaches proposed by (a) Kik-Piotrowski, (b) Linder, and (c) Ayasse-Chollet.

In Figure _{e} and_{e} are semiaxes length of the equivalent ellipse;_{L} and_{R} are left and right boundary of the contact patch;_{c} and

The idea of Linder’s approach is to apply the tangential model on every strip of contact patch as shown in Figure _{e} as defined in (_{e} and longitudinal axis keeps the same as the original one,

This approach also regards the nonelliptic contact as several strips along the rolling direction. Each strip is assumed to locate in the center of respective virtual ellipse as shown in Figure _{0} is characteristic length;_{f} are original and filtered lateral curvature. In each

For wear calculation, the wear model proposed by KTH Royal Institute of Technology [

Wear chart of the wear rate applied in the wear model (

In order to evaluate the performance of aforementioned six simplified models, it will present a comparative study of wheel-rail rolling contact under the combination of creepage and spin. Standard wheel-rail profile of S1002CN-CN60 is considered, with prescribed lateral displacement, yaw angle, friction coefficient, and rail cant. A further study for worn profiles will be investigated in the next work. Note that before performing a contact simulation; it should carry out a contact geometry calculation [

Main input parameters for computation of contact geometry and rolling contact.

Name | Parameter | Value |
---|---|---|

Wheel-rail contact geometrical parameters | Flange back separation | 1353 mm |

Wheel’s nominal rolling radius | 460 mm | |

Track gauge | 1435 mm | |

Rail cant | 1:40 | |

| ||

Wheel-rail’s material property | Friction coefficient between wheel-rail interface | 0.3 |

Poisson’s ratio | 0.28 | |

Elastic modulus | 205.9 GPa |

In every lateral displacement case, the input penetration is obtained using Hertz theory under the prescribed load of 83.3 kN [

Variation of contact shape and pressure on rail surface under different lateral displacements ranging from -3 mm to 6 mm at 1 mm increment.

Location of contact patch

Contact patch and pressure distribution

For including lateral creep behavior, specified yaw angles are applied to the ten shift cases. It means the yaw angle (being defined as 1.2 times the lateral displacement) increases from -3.6 mrad to 7.2 mrad in 1.2 mrad increments. Since the yaw angle is not enough large to change contact shape obviously, we only consider its effect on creepage. For a detailed investigation of yaw angle’s role on contact shape and creepage, readers are referred to [_{x} and_{y}) and spin (_{0} are rolling radius and nominal radius, respectively.

The variation of longitudinal, lateral creepage, and spin with lateral shift are illustrated in Figure _{x}). For lateral creepage, it changes linearly since lateral creepage can be considered as yaw angle when it is small. The variation of spin is similar to that of longitudinal creepage because the determination of spin mainly depends on contact angle whose variation is close to rolling radius difference.

Creepage and spin in Figure

Lateral shift (mm) | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

Long. creepage, _{x} (%) | 0.072 | 0.055 | -0.037 | -0.058 | -0.084 | -0.114 | -0.147 | -0.193 | -0.251 | -0.732 |

Lat. creepage, _{y} (%) | 0.360 | 0.240 | 0.120 | 0 | -0.120 | -0.239 | -0.358 | -0.477 | -0.596 | -0.691 |

Spin, | -0.056 | -0.068 | -0.132 | -0.146 | -0.162 | -0.181 | -0.200 | -0.229 | -0.265 | -0.605 |

Variation of creepage and spin with lateral shift.

For the sake of simplicity, three typical nonelliptic contact cases under different shifts (0, -2 and 6 mm), creepage, and spin are selected to evaluate detailed tangential solutions including ‘

Tangential stress distribution and stick-slip division for different shift cases.

Δ

Δ

Δ

Figure

Comparison of tangential stress and stick-slip division for Δ

Longitudinal, lateral, and total tangential stress (at

Compared to the reference, it is found FASTSIM-based models result in different tangential stress distribution. This is because the tangential stress in FASTSIM increases linearly in the adhesion area until it saturates in the slip area and follows a parabolic distribution. However, in FaStrip-based models, the stress distributions in the adhesion area are nonlinear and follow the elliptic traction bound in the slip area, which are similar to that in CONTACT. Among them, AC’s approach performs the best.

For the case Δ

Comparison of tangential stress and stick-slip division for Δ

Longitudinal, lateral, and total tangential stress (at

Full slip is observed in the case Δ

Comparison of tangential stress and stick-slip division for Δ

Longitudinal, lateral, and total tangential stress (at

Figure

Relative slip (at

Δ

Δ

Δ

It is mentioned that the rigid slip contained in the relative slip for all models are identical while the elastic slip contributes to the difference. The discrepancy of elastic slip is influenced by different flexibility parameters. This influence is evident for case Δ

Then, the capability of simplified models on simulating wear distribution is explored. From (

The wear distributions under 200 km/h predicted by different models are illustrated in Figure

Wear depth for different shift cases.

Δ

Δ

Δ

The difference of wear depth using different models is reduced in the case Δ

In this part, we evaluate the performance of simplified models on predicting creep force, which is very important for vehicle dynamics simulation such as vehicle negotiating a curved track [

Figure

Predicted longitudinal, lateral, and total creep force.

In order to quantify the performance of these models for the calculation of creep force, we define an average error in (_{ijs} and_{ijc} are predicted normalized creep force by simplified models and CONTACT for every shift case.

The average errors caused by the six simplified models are compared in Figure

Errors of predicted creep force by six simplified models.

The present work is an extension of our previous study [

For instance, in Figures

Comparison of contact patch prediction by CONTACT and Hertz theory.

Δ

Δ

On the other hand, in the case Δy = -2 mm, the predicted results using three nonelliptic adaptation approaches are similar. It is due to the fact that, in the case Δy = -2 mm, the contact patch is similar to an ellipse as shown in Figure

By contrast, FaStrip is not as sensitive to nonelliptic adaptation approaches as FASTSIM. This is because in FaStrip, the expressions of stick zone and tangential stress are nonlinear, and they cannot significantly be affected by nonelliptic adaptation approaches as shown in (

It is also found though the total tangential stress using FaStrip is very accurate, the component of it in x- and y-direction deviates a bit from the reference as presented in the second rows of Figures

Even so, the simplified models in this paper are considered as promising solutions to be used in vehicle-track dynamics and wheel-rail damage simulation, because they are both computationally efficient. FaStrip is a bit slower than FASTSIM since it uses FASTSIM to identify the component of analytical total tangential stress in slip zone. In addition, the nonelliptic adaptation approaches will not markedly increase their computational cost since they can be solved explicitly. In our DELL Precision T7910 workstation with the processor of Intel Xeon E5-2630 v3 @ 2.4 GHz, the average computational time is about 2.4 ms and 1.7 ms for FaStrip and FASTSIM, which are about 1000 times faster than CONTACT.

In this paper, two simplified tangential models and three nonelliptic adaptation approaches are firstly recalled. They are then combined with Kalker’s NORM to carry out a comparative study of wheel-rail rolling contact and evaluated by Kalker’s CONTACT.

All simplified models can provide satisfactory global creep force, meaning all of them are applicable to vehicle-track dynamics modeling. This is while only FaStrip-based models can capture local nonlinear distribution of tangential stress and slip velocity, contributing to more exact prediction of wear distribution.

By using Ayasse-Chollet’s local ellipse approach, it results in the best prediction compared to the reference. It implies varying flexibility parameters throughout the contact patch are more close to the nature of contact rather than constant flexibility parameter employed in Kik-Piotrowski’s approach. This approach can be further applied to the extension of other ellipse-based method, such as thermal wheel-rail contact analysis [

FaStrip is found not very sensitive to the selection of nonelliptic approach. This characteristic shows Kik-Piotrowski’s and Linder’s approaches are also alternative choice and they are expected to be more suitable for worn wheel-rail profile cases, in which the nonsmoothing curvature may affect the accuracy of Ayasse-Chollet’s approach [

All data of the current study are available from the corresponding author on reasonable request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The present work is supported by National Key R&D Plan of China (grant no. 2016YFB1102600), National Natural Science Foundation of China (51605318, 51608459, 51778542, and U1734207), Fundamental Research Funds for the Central Universities (2682018CX01), and Jiangsu Provincial Natural Fund project (16KJB580008).