1 Introduction
2 Wheel–rail contact formulation
2.1 Geometric contact problem
2.1.1 Convex contact search
2.1.2 Concave contact search
2.2 Normal contact problem
2.3 Tangential contact problem
3 Vehicle–structure coupling system
3.1 Equilibrium equations
3.2 Constraint equations
3.3 Hybrid system of equations to make the couple between vehicle and structure
4 Numerical validation of the wheel–rail contact model
4.1 Initial considerations
4.2 Manchester Benchmarks
4.2.1 Benchmark description
Software | Normal contact formulation | Tangential contact formulation |
---|---|---|
CONPOL | Hertzian | FASTSIM |
CONTACT PC92 | CONTACT | CONTACT |
DYNARAIL | Hertzian and Multi-Hertzian | USETAB |
GENSYS | Non-Hertzian (equivalent contact ellipses) | FASTSIM |
LaGer | CONTACT | CONTACT |
OCREC | Multi-Hertzian | FASTSIM |
NUCARS | Multi-Hertzian | Lookup tables based on DUVOROL |
TDS CONTACT | Hertzian | FASTSIM |
VAMPIRE | Hertzian | Lookup tables based on DUVOROL |
VOCOLIN | Semi-Hertzian | FASTSIM |
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Case A1.1: The wheelset is subjected to a prescribed lateral displacement from 0 to 10 mm with 0.5 mm increments. A static analysis is performed in each position, and the normal contact is evaluated.
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Case A1.2: The wheelset is subjected to the previously described lateral displacements combined with a yaw rotation from 0 to 24 mrad with 1.2 mrad increments. A static analysis is performed in each position, and the normal contact is evaluated.
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Case A2.1: Forward speed of 2 m·s-1 is given to the wheelset on straight track while it is subjected to the previously described lateral displacements. A dynamic analysis is performed, and both the normal and tangential contacts are evaluated.
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Case A2.2: The wheelset is subjected to the combinations of lateral displacements and yaw rotations described in the case A1.2. The dynamic conditions are the same as for the case A2.1, and both the normal and tangential contacts are evaluated.
4.2.2 Contact point positions
4.2.3 Rolling radius difference
4.2.4 Contact angles
4.2.5 Longitudinal creepages
4.2.6 Lateral creepages
4.2.7 Spin creepages
4.2.8 Final remarks
4.3 Hunting stability analysis of a suspended wheelset
4.3.1 Hunting phenomenon
4.3.2 Numerical models
Model | Variable | Description | Value |
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A, B, C | mws | Wheelset mass | 1568 kg |
Ix,ws | Roll mass moment of wheelset | 656 kg·m2 | |
Iy,ws | Pitch mass moment of wheelset | 168 kg·m2 | |
Iz,ws | Yaw mass moment of wheelset | 656 kg·m2 | |
\(a\) | Contact ellipse longitudinal semi-axis | 5.667 mm | |
\(b\) | Contact ellipse lateral semi-axis | 4.284 mm | |
\(c_{11}\) | Longitudinal creepage coefficient | 4.523 | |
\(c_{22}\) | Lateral creepage coefficient | 4.121 | |
\(R_{0}\) | Initial rolling radius | 456.6 mm | |
\(\gamma_{0}\) | Conicity | 0.025 | |
2Lcp | Lateral distance between initial contact points | 1435 mm | |
l | Distance between longitudinal suspensions | 1800 mm | |
\(c_{1,x}\) | Damping of the longitudinal primary suspensions | 0 kN·s/m | |
\(c_{1,y}\) | Damping of the lateral primary suspensions | 0 kN·s/m | |
A | \(k_{1,x}\) | Stiffness of the longitudinal primary suspensions | 0 kN/m |
\(k_{1,y}\) | Stiffness of the lateral primary suspensions | 0 kN/m | |
B, C | \(k_{1,x}\) | Stiffness of the longitudinal primary suspensions | 135 kN/m |
\(k_{1,y}\) | Stiffness of the lateral primary suspensions | 250 kN/m |
4.3.3 Governing equations of motion of the semi-analytical model
4.3.4 Bases of the analysis
4.3.5 Validation results
5 Experimental validation of the vehicle–structure interaction model
5.1 Initial considerations
5.2 Numerical models
5.2.1 Track
Parameter | Unit | Value | |
---|---|---|---|
Vertical | Lateral | ||
kp | kN/m/m | 9.42 \(\times\) 104 | 47.1 \(\times\) 104 |
Ir | cm4 | 3032 | 509 |
Lt | m | 3 | 3 |
E | GPa | 210 | 210 |
5.2.2 Vehicle
5.3 Validation results
6 Conclusions
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In the first application, the Manchester Benchmark is revisited to validate the concave contact search proposed in this article. The benchmark comprised a series of tests that consisted of prescribing, both statically and dynamically, lateral displacements and yaw rotations to a single wheelset to analyze its behavior. Several contact characteristics were analyzed during the benchmark, namely, the contact point positions on both wheels of the wheelset, the rolling radius difference between wheels, the contact angles, and the creepages. The results obtained with the proposed model for all the analyzed quantities showed an excellent agreement with those obtained with other railway vehicle dynamics multibody software, such as GENSYS, NUCARS, and VAMPIRE. The few discrepancies observed are mainly justified by limitations of the contact models adopted by some of the tested software, especially CONPOL and VOCOLIN, rather than by limitations of the proposed model.
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The second validation example consists of evaluating the lateral stability of a single wheelset running at several speeds. The dynamic response of the wheelset calculated with the proposed model is compared with that obtained using a semi-analytical model with two degrees of freedom available in the literature. A good agreement between the responses obtained with the proposed model and those obtained by the integration of the equations of motion of the semi-analytical model is observed. As expected, for speeds below the critical limit, both the lateral displacement and the yaw rotation of the wheelset tend to damp out after being driven away by a lateral disturbance. This is due to the energy dissipation provided by the creep forces and to the stability provided by the suspensions. However, when the speed exceeds the critical value, the behavior of the wheelset becomes unstable, leading to a hunting motion that grows indefinitely. The critical speed predicted by the proposed formulation using a logarithmic decrement factor is also in a perfect agreement with the theoretical value determined from a stability study described in the literature. Moreover, the two-point contact scenario (flange and tread contact points) was also validated through an example in which a flange was introduced to the wheel profile. The results showed that, exactly after the moment when the gap between flange and rail closes, the wheelset lateral movement became limited by the flange, and an abrupt increment in lateral contact forces occurred, demonstrating the capabilities of the model on capturing the flange impacts with the rail. Finally, differences were detected in the magnitude of the contact forces and in the moment when the first flange impact occurs between the simplified model presented in one of the authors’ previous publications and the model presented here that considers the concave region of the wheel profile.
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Finally, an experimental test conducted in a test rig from the RTRI, in which a full-scale railway vehicle runs over a track subjected to vertical and lateral deviations, is reproduced numerically. Unlike in one of the authors’ previous publication, where only a single example has been used to validate the model (lateral accelerations measured for a scenario where the train runs at 300 km/h), a much more comprehensive validation is presented in this paper, consisting of the comparison between the experimental and numerical responses of the vehicle in the vertical and lateral directions caused by track deviations imposed by the actuators on both directions and for train speeds ranging between 100 and 400 km/h. The results showed a good agreement between all the examples, proving the capacity of the model to deal with generic railway dynamic applications. Regarding the differences between the simplified and the realistic model presented here, they are only notorious during the occurrence of lateral flange impacts, because it is in these moments that the contact points may be located in the concave transition zone.