Abstract
Critical wind speed can play an important guiding role in developing an initial train operation schedule and knowledge of it may prevent safety risks for a train. Hence, the efficient and accurate calculation of the critical wind speeds of trains is critical. This study addresses this topic and focuses on the influence of different methods on the calculation of the critical wind speed. The result reveals that the five-mass and three-mass methods can both be used to determine the critical wind speed of a train more quickly with acceptable accuracy, but these two methods overestimate the crosswind safety risk of the train. With the increase of the train's operating speed, the nonlinearity of the vehicle system is further enhanced. In particular, the influence of the rolling motion between the car body and the bogie is more prominent, and the results of the five-mass method and the multi-body simulation method tend to be the closest. Last but not least, the damping parameters and inertial forces ignored by the quasi-static method will effectively reduce the wind forces transmitted to the track, resulting in a smaller overturning coefficient and higher critical wind speed.
1. Introduction
The crosswind safety of high-speed trains has become an increasingly popular research issue in the last 20 years, mainly motivated by crosswind accidents. The reports of high-speed train crosswind accidents can be traced from the previous century to the present. These accidents have occurred successively in Japan, the United Kingdom, Sweden, China and other countries [1,2]. A retrospective analysis of a series of typical wind-induced train accidents indicates that the most common type of crosswind accident for a train is overturning around one of the rails [3]. As a result, the overturning safety risk is the key index for assessing train operation safety under strong wind conditions. The investigation of train crosswind safety generally refers to calculating the critical wind speed based on the overturning coefficient, that is, the characteristic wind curve (CWC) [4].
In fact, the calculation of the CWC of high-speed trains involves two issues. One issue is the aerodynamic load acting on a train, and the other is the dynamic response of the vehicle's system. At present, the acquisition methods of aerodynamic loads are basically the same in various countries. These methods mainly involve wind tunnel tests or computational fluid dynamics (CFD) calculations. However, different methods have been developed for evaluating the dynamic responses of vehicles under crosswind conditions in different countries and regions. In Europe, the calculation of the CWC in crosswind conditions has been standardized. Either the aerodynamic forces obtained with CFD calculations or wind tunnel tests are then loaded onto a multi-body dynamic model or quasi-static methods are adopted, and then the critical wind speed can be obtained. In the above process, the evaluation of the dynamic response involves three different methods. The first method is the time-dependent multi-body simulation (MBS) model, which is also the most accurate. Moreover, the quasi-static methods of the three-mass and five-mass methods are also suggested to determine the crosswind risks of trains [5]. In Japan, a method that differs from the European method, the ‘Detailed Equation’ proposed by Hibino and Ishida, is used to determine the overturning risk of a train. In this method, the train's external force and the vibration of the car body are comprehensively taken into account [6]. At present, the Chinese method is similar to the European method and will not be repeated here. Both of the above methods are able to meet their current engineering needs.
With most of the high-speed railway lines under construction having to cross harsh environments with strong winds in China, an accurate and efficient engineering model that can predict the crosswind safety risk of a train is increasingly important for railway line design. The above-mentioned three-mass and five-mass methods provide good choices. However, the current research on train overturning safety mainly focuses on the CWC calculation of trains using the MBS method under different wind conditions, and there are few studies on the differences in the calculation results of the above different models (three-mass, five-mass and MBS methods), which is valuable and necessary information for engineering applications. It is worth mentioning that Dirk Thomas and Diedrichs Ben conducted a systematic study of the safety of train operations under different gust conditions. In reference [7], the accuracy of the train dynamic model was verified by a full-scale test, and then the CWC of a train passing through the curve under crosswind conditions was obtained. In reference [8], taking ICE2 and X2000 trains as the research objects to conduct numerical simulation research, it was found that when the train ran under crosswind conditions, the overturning moment of the carriage was about 100% higher than that of the middle carriage. Furthermore, the researchers studied the influence of the unsteady aerodynamic force on high-speed train crosswind safety by determining the dynamic responses of different gusts. The results showed that the change of the ramp distance of a simple artificial gust affected the wheel unloading of the front and rear bogies, resulting in a higher wheel unloading rate at a shorter ramp distance [9]. In particular, in reference [10], the crosswind safety of the high-speed train under gust conditions was studied with full-scale tests and simulations. The results indicated that the train's response to the gust was mainly an abrupt change in the overturning coefficient. The simulation results were in good agreement with the experimental results. However, because of the influence of the state of the air spring, the simulation results were larger than the experimental results in some cases, and the response of the train to the gust was slightly overestimated. In addition, some scholars have studied related issues, such as train operational safety under variable crosswind conditions. In reference [11], the influence of the Chinese hat gust on the turning dynamics of a train was studied using wind tunnel tests and dynamic simulations. It was found that the wind direction reversal had a greater impact on the safety of a train when overturning. Moreover, in our previous studies [4,12], the dynamic performance of a train under varying crosswind conditions and the correlation of the car body vibration and the overturning of a train under crosswind conditions were also investigated through simulations and full-scale experiments.
In summary, few studies have been conducted on the effect of different evaluation models on the CWC calculation of high-speed trains. Hence, based on the actual needs of the above-mentioned engineering problems, a numerical investigation is conducted to study this topic in this research. The remainder of the paper is organized as follows. In the second part of this study, the schematic of the CWC determination process and the methods to assess the crosswind safety risk of high-speed trains are introduced. Section 3 describes the numerical simulations and validation, and Section 4 is the result and discussion.
2. Methods to assess the crosswind safety risk of high-speed trains
2.1 Evaluation indicators
2.2 Crosswind safety risk assessment methods
To compare the influence of the assessment method on the critical wind speed of the train, this study selects a Chinese high-speed train as the investigation object, and determines the critical wind speed of the train according to the following different methods under the steady wind scenario; the simulation process is shown in Fig. 1. In addition, only some fundamental parameters of train dynamics can be described due to commercial secrets, as shown in Table 1.
Schematic of the CWC determination process.
Primary parameters for train model.
| Parameter | Value |
|---|---|
| Carbody mass | 33,766 kg |
| Wheelset mass | 1,850 kg |
| Bogie frame mass | 2,400 kg |
| Mass moment of inertia of the car body about the x-axis | 109,400 kg·m2 |
| Mass moment of inertia of the car body about the y-axis | 165,450 kg·m2 |
| Mass moment of inertia of the car body about the z-axis | 156,130 kg·m2 |
| Mass moment of inertia of the bogie frame about the x-axis | 19,44 kg·m2 |
| Mass moment of inertia of the bogie frame about the y-axis | 1,314 kg·m2 |
| Mass moment of inertia of the bogie frame about the z-axis | 2,400 kg·m2 |
| Mass moment of inertia of the wheelset about the x-axis | 967 kg·m2 |
| Mass moment of inertia of the wheelset about the y-axis | 123 kg·m2 |
| Mass moment of inertia of the wheelset about the I-axis | 967 kg·m2 |
| Gauge | 1,435 mm |
| Centre distance of bogie | 17,500 mm |
| Wheelbase | 2,500 mm |
| Wheel diameter | 860 mm |
| Lateral stiffness of the second suspension air spring | 165 kN/m |
| Vertical stiffness of the second suspension air spring | 33 kN/m |
| Lateral stiffness of the primary suspension steel spring | 980 kN/m |
| Vertical stiffness of the primary suspension steel spring | 1,176 kN/m |
| Parameter | Value |
|---|---|
| Carbody mass | 33,766 kg |
| Wheelset mass | 1,850 kg |
| Bogie frame mass | 2,400 kg |
| Mass moment of inertia of the car body about the x-axis | 109,400 kg·m2 |
| Mass moment of inertia of the car body about the y-axis | 165,450 kg·m2 |
| Mass moment of inertia of the car body about the z-axis | 156,130 kg·m2 |
| Mass moment of inertia of the bogie frame about the x-axis | 19,44 kg·m2 |
| Mass moment of inertia of the bogie frame about the y-axis | 1,314 kg·m2 |
| Mass moment of inertia of the bogie frame about the z-axis | 2,400 kg·m2 |
| Mass moment of inertia of the wheelset about the x-axis | 967 kg·m2 |
| Mass moment of inertia of the wheelset about the y-axis | 123 kg·m2 |
| Mass moment of inertia of the wheelset about the I-axis | 967 kg·m2 |
| Gauge | 1,435 mm |
| Centre distance of bogie | 17,500 mm |
| Wheelbase | 2,500 mm |
| Wheel diameter | 860 mm |
| Lateral stiffness of the second suspension air spring | 165 kN/m |
| Vertical stiffness of the second suspension air spring | 33 kN/m |
| Lateral stiffness of the primary suspension steel spring | 980 kN/m |
| Vertical stiffness of the primary suspension steel spring | 1,176 kN/m |
Primary parameters for train model.
| Parameter | Value |
|---|---|
| Carbody mass | 33,766 kg |
| Wheelset mass | 1,850 kg |
| Bogie frame mass | 2,400 kg |
| Mass moment of inertia of the car body about the x-axis | 109,400 kg·m2 |
| Mass moment of inertia of the car body about the y-axis | 165,450 kg·m2 |
| Mass moment of inertia of the car body about the z-axis | 156,130 kg·m2 |
| Mass moment of inertia of the bogie frame about the x-axis | 19,44 kg·m2 |
| Mass moment of inertia of the bogie frame about the y-axis | 1,314 kg·m2 |
| Mass moment of inertia of the bogie frame about the z-axis | 2,400 kg·m2 |
| Mass moment of inertia of the wheelset about the x-axis | 967 kg·m2 |
| Mass moment of inertia of the wheelset about the y-axis | 123 kg·m2 |
| Mass moment of inertia of the wheelset about the I-axis | 967 kg·m2 |
| Gauge | 1,435 mm |
| Centre distance of bogie | 17,500 mm |
| Wheelbase | 2,500 mm |
| Wheel diameter | 860 mm |
| Lateral stiffness of the second suspension air spring | 165 kN/m |
| Vertical stiffness of the second suspension air spring | 33 kN/m |
| Lateral stiffness of the primary suspension steel spring | 980 kN/m |
| Vertical stiffness of the primary suspension steel spring | 1,176 kN/m |
| Parameter | Value |
|---|---|
| Carbody mass | 33,766 kg |
| Wheelset mass | 1,850 kg |
| Bogie frame mass | 2,400 kg |
| Mass moment of inertia of the car body about the x-axis | 109,400 kg·m2 |
| Mass moment of inertia of the car body about the y-axis | 165,450 kg·m2 |
| Mass moment of inertia of the car body about the z-axis | 156,130 kg·m2 |
| Mass moment of inertia of the bogie frame about the x-axis | 19,44 kg·m2 |
| Mass moment of inertia of the bogie frame about the y-axis | 1,314 kg·m2 |
| Mass moment of inertia of the bogie frame about the z-axis | 2,400 kg·m2 |
| Mass moment of inertia of the wheelset about the x-axis | 967 kg·m2 |
| Mass moment of inertia of the wheelset about the y-axis | 123 kg·m2 |
| Mass moment of inertia of the wheelset about the I-axis | 967 kg·m2 |
| Gauge | 1,435 mm |
| Centre distance of bogie | 17,500 mm |
| Wheelbase | 2,500 mm |
| Wheel diameter | 860 mm |
| Lateral stiffness of the second suspension air spring | 165 kN/m |
| Vertical stiffness of the second suspension air spring | 33 kN/m |
| Lateral stiffness of the primary suspension steel spring | 980 kN/m |
| Vertical stiffness of the primary suspension steel spring | 1,176 kN/m |
2.2.1 MBS method
In crosswind conditions, according to EN 14067–6, modelling the MBS model must take into account the most critical vehicle of the train, and that vehicle must be empty. Additionally, the influence of the track irregularity can be ignored in this procedure. Hence, a Chinese high-speed train tailer is selected as the simulation object in this study. The specific modeling process is shown in Section 3.2.
2.2.2 Three-mass method
![Analytical model of the three-mass method [5].](https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/tse/4/3/10.1093_tse_tdac029/3/m_tdac029fig2.jpeg?Expires=1747930892&Signature=1dfQIK4lvTlB90SMZKzK5S9F4fJGhqgGuTULKpEe6NYe35YDIvqATQe1lYDaUyp~EfoZZ6WQQGEpnQ7Ghs8AgHjwjLST6JNEnE2yFW2qYAp12qxhsTMgZ~nWAD66MZnO3YfEINtNfzGybbU~3P2sjveslgng75yQp0ntgLtkvYcMn7ANzAN4EtBEk2NfKvhUo9JVFQfn-AuHC5ph68lasaAESj~5qWACwxIA4FMJ7UPCxf3Zf9WOXD0l40wiooMdE9aStK19X4t7syakJKNELcJjzsEm6l1UooMnPF6ip3NaITyec6447bWmQijJowffIrRxSCBOaIP3KaY01D8N0A__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
Analytical model of the three-mass method [5].
Definition of the relative wind speed Vrel and yaw angleβ.
2.2.3 Five-mass method
The five-mass model method is a quasi-static method that accounts for the kinematics of the main components of the vehicle but excludes the dynamic effects caused by dampers, etc. This method uses coupling elements to connect the concentrated masses, which include five mass blocks in total, namely, the car body (CB), the front bogie frame (BG-1), the rear bogie frame (BG-2), and the front and rear wheelsets (WS-1, WS-2). Moreover, the primary and secondary suspensions are located between these mass blocks and are constructed as spring units with a certain stiffness, as shown in Fig. 4. The car body has 5 degrees of freedom, translation along the y and z directions and rotation degrees of freedom around all axes. The bogie frame has 3 degrees of freedom and the wheelset has 0 degrees of freedom. As a consequence, the system as a whole has 11 degrees of freedom.
![Analytical model of the five-mass method [6].](https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/tse/4/3/10.1093_tse_tdac029/3/m_tdac029fig4.jpeg?Expires=1747930892&Signature=VxT~34Lt~cOcFpQbAapXYozNyUCCUpXIyjjDlr4Q-LN8VIGZGZBMO80jhFuqnY6sRiqSGxDNaz3KeA9dvr9kzT5D8HetrF5QY61gU6r0KYoIkyHdzNEoLKcW8t-qzqo9DQe7S-m70kPlhZnRjrIWWdgEQsp6WfQZzuZcrlENX3wtqvVPiotMJY6zza3~3gcB0H8NBngXhuVhySmfbs5eRn1EiVjNcWbA~M6arEKoOOVpFp1Dj8O2W0kptTy5VuofesPx1wbcq~GAlhKddMr4-xV9RanrACi7ONEb26vXmW-0nWMid4caNojSdr65BpiFLKoWUxF2DQeE71qTQFi1aQ__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
Analytical model of the five-mass method [6].
2.2.4 Comparison of parameters involved in different methods
As can be seen from the above description, the system complexity of the above three methods is different and the parameters involved are also different. In order to further compare the characteristics of the above methods, the factors involved are listed and compared, as shown in Table 2. It can be seen that the MBS model is the most complex and involves the most factors. The five-mass method is less complex. Compared with the MBS method, lateral force, lift force, rolling moment, pitch moment and yaw moment are also considered simultaneously, but the five-mass method ignores the effect of the shock absorber and axle boxes. Furthermore, the three-mass method is the simplest and requires much less information. Aerodynamic forces only involve lateral force, lift force and overturning moments due to the fact that these three loads contribute the most to the crosswind safety risk of the train, and the effects of shock absorbers, anti-roll bars, buffers and axle boxes are ignored.
Comparison of three different methods.
| Type | Three-mass | Five-mass | MBS |
|---|---|---|---|
| Number of blocks | 3 | 5 | 5+8 (axle box) |
| Primary suspensions | √ | √ | √ |
| Secondary suspensions | √ | √ | √ |
| Buffers | / | √ | √ |
| Anti-roll bar | / | √ | √ |
| Primary and secondary shock absorbers | / | / | √ |
| Aerodynamic forces | Fy, Fz, Mx | Fy, Fz, Mx, My, Mz | |
| Overturning coefficient | Average of the front and rear bogies | Critical bogie |
| Type | Three-mass | Five-mass | MBS |
|---|---|---|---|
| Number of blocks | 3 | 5 | 5+8 (axle box) |
| Primary suspensions | √ | √ | √ |
| Secondary suspensions | √ | √ | √ |
| Buffers | / | √ | √ |
| Anti-roll bar | / | √ | √ |
| Primary and secondary shock absorbers | / | / | √ |
| Aerodynamic forces | Fy, Fz, Mx | Fy, Fz, Mx, My, Mz | |
| Overturning coefficient | Average of the front and rear bogies | Critical bogie |
Comparison of three different methods.
| Type | Three-mass | Five-mass | MBS |
|---|---|---|---|
| Number of blocks | 3 | 5 | 5+8 (axle box) |
| Primary suspensions | √ | √ | √ |
| Secondary suspensions | √ | √ | √ |
| Buffers | / | √ | √ |
| Anti-roll bar | / | √ | √ |
| Primary and secondary shock absorbers | / | / | √ |
| Aerodynamic forces | Fy, Fz, Mx | Fy, Fz, Mx, My, Mz | |
| Overturning coefficient | Average of the front and rear bogies | Critical bogie |
| Type | Three-mass | Five-mass | MBS |
|---|---|---|---|
| Number of blocks | 3 | 5 | 5+8 (axle box) |
| Primary suspensions | √ | √ | √ |
| Secondary suspensions | √ | √ | √ |
| Buffers | / | √ | √ |
| Anti-roll bar | / | √ | √ |
| Primary and secondary shock absorbers | / | / | √ |
| Aerodynamic forces | Fy, Fz, Mx | Fy, Fz, Mx, My, Mz | |
| Overturning coefficient | Average of the front and rear bogies | Critical bogie |
3. Numerical simulation
3.1 Computational fluid dynamics simulations
3.1.1 Train geometric model
The simulated object in this study is a simplified version of a Chinese CRH high-speed train, consisting of three carriages (designated C-1, C-2 and C-3, respectively). The full-scale train model dimensions are 78 m (length) × 3.38 m (width) × 3.7 m (height). In addition, the shape of high-speed trains is complex and the slenderness ratio is very large. Hence, the bogies and windshields are simplified in accordance with EN 14067–6 to increase the grid quality, as shown in Fig. 5.
Train geometry model.
3.1.2 Computational domain and boundary conditions
The computational domain and boundary conditions for the numerical simulations are shown in Fig. 6, with the domain dimension specified as a function of the train's height H. The train is located 10H from the velocity inlet, where a constant wind with a low turbulence intensity of 1% is established to replicate the train operation speed, and the train aerodynamic forces and moments are observed [14]. The computational domain's side face is 10H away from the train model area and is also specified as the velocity inlet boundary condition. The computational domain's leeward side face is 20H away from the train model. Additionally, the domain's height is set to 10H with a symmetry plane boundary condition. The rear face is positioned 30H away from the train tail to allow the flow wake to fully emerge [15, 16]. In addition, the domain's bottom is adjusted to no-slip conditions, and the movement speed is designed to match the incoming wind speed, as shown in Fig. 6.
Computational domain and boundary conditions: (a) side-view; (b) front-view.
3.1.3 Numerical solver and mesh strategy
In this study, the train aerodynamic forces are simulated using detached-eddy simulation with the shear-stress transport k-w turbulence model, which has been widely used to predict the aerodynamic performances of trains. Moreover, the semi-implicit method for the pressure-linked equation technique is used to process the pressure and velocity coupling and the flow field is solved with the commercial application STAR-CCM+ [17].
The quality of CFD simulation is highly dependent on the mesh resolution; hence, a hexahedral-dominated mesh that has been widely used for numerical predictions of the train aerodynamics is generated in STAR-CCM+ in this study. Three refinement boxes are set to better predict the flow separation around the train, as shown in Fig. 7. The first refinement box aims to refine the mesh resolution around the high-speed train to the same mesh size as the train surface. The second refinement box is used to refine the mesh around the high-speed train again and the mesh size is doubled at the interface between the refinement regions, and the third larger grid size refinement box is designed for refining the meshes around the high-speed train again. In addition, to simulate the flow field close to the train surface, 10 prism layers were added to the boundary layer of the train and an adaptive mesh technique was employed to adjust the meshes around the train based on preliminary calculations to ensure that the averaged Y+ was approximately within an acceptable range.
Medium grid distribution of simulation: (a) top- and side-view of refinement zone; (b) head zone; (c) bogie zone.
3.1.4 Numerical validation and mesh sensitivity test
Comparison of simulation and wind tunnel results and pressure under different grid strategies: (a) wind tunnel test; (b) head car force coefficients; (c) pressure on the windward side and (d) pressure on the leeward side.
Different grid information.
| Grids | |${n^ + }$| | |${l^ + }$| | |${s^ + }$| | Roof and door | Surface | Total number |
|---|---|---|---|---|---|---|
| Coarse | 1.5 | 1,000 | 1,000 | 0.02H | 0.01H–0.02H | 1.5×107 |
| Medium | 1.5 | 500 | 500 | 0.015H | 0.01H–0.02H | 3.0×107 |
| Fine | 1.5 | 250 | 250 | 0.01H | 0.005H–0.02H | 6.0×107 |
| Grids | |${n^ + }$| | |${l^ + }$| | |${s^ + }$| | Roof and door | Surface | Total number |
|---|---|---|---|---|---|---|
| Coarse | 1.5 | 1,000 | 1,000 | 0.02H | 0.01H–0.02H | 1.5×107 |
| Medium | 1.5 | 500 | 500 | 0.015H | 0.01H–0.02H | 3.0×107 |
| Fine | 1.5 | 250 | 250 | 0.01H | 0.005H–0.02H | 6.0×107 |
Different grid information.
| Grids | |${n^ + }$| | |${l^ + }$| | |${s^ + }$| | Roof and door | Surface | Total number |
|---|---|---|---|---|---|---|
| Coarse | 1.5 | 1,000 | 1,000 | 0.02H | 0.01H–0.02H | 1.5×107 |
| Medium | 1.5 | 500 | 500 | 0.015H | 0.01H–0.02H | 3.0×107 |
| Fine | 1.5 | 250 | 250 | 0.01H | 0.005H–0.02H | 6.0×107 |
| Grids | |${n^ + }$| | |${l^ + }$| | |${s^ + }$| | Roof and door | Surface | Total number |
|---|---|---|---|---|---|---|
| Coarse | 1.5 | 1,000 | 1,000 | 0.02H | 0.01H–0.02H | 1.5×107 |
| Medium | 1.5 | 500 | 500 | 0.015H | 0.01H–0.02H | 3.0×107 |
| Fine | 1.5 | 250 | 250 | 0.01H | 0.005H–0.02H | 6.0×107 |
Fig. 8(b) compares the wind tunnel test results to the simulation results of the head car lateral force and lift force coefficients for different mesh configurations at a yaw angle of 15 ° The results show that the aerodynamic force coefficient simulation results correspond well with the test results. Moreover, the medium and fine mesh simulation results are more accurate than the coarse mesh simulation results and the medium mesh simulation results are close to the fine mesh simulation results. Fig.s 8(c)–(d) further compares the results of the pressure on the windward and leeward sides for different grid conditions. It can be seen that the pressure difference between the medium grid and fine grid on the windward side is small, and the pressure for the coarse grid is lower than those for the simulation results for the above two meshes conditions. Similarly, the simulation results for the medium grid and fine grid on the leeward side are almost the same, but the value of the coarse grid is significantly different. Therefore, the medium grid can meet the simulating requirements and the medium grid strategy is adopted in this study.
Fig. 9 shows the comparison of the wind tunnel test and the simulation results of the head car aerodynamic forces at different yaw angles of 5 °, 10 ° and 15 °. The results reveal that the lateral force coefficient and lift force coefficient simulation results agree well with the test results and the maximum relative errors are less than 10%. Hence, the numerical simulation method can meet the requirements of this study.
Comparison of the lateral and lift force coefficients of the head car at different yaw angles.
3.2 Multi-body simulations
Following EN 14067–6, a trailer is chosen as the simulated object in this study, and a full-scale multi-body vehicle model is developed using the commercial software SIMPACK [18], as shown in Fig. 10. The MBS model consists of 15 rigid bodies with 50 degrees of freedom, including one car body, two bogie frames, four wheelsets and eight axle boxes. In addition, the wheel-rail contact force and the primary and secondary suspensions are modelled with force element No. 5 and the external aerodynamic load is modelled with force element No. 93 (the position of the aerodynamic load is consistent with the reference point of the CFD simulation). In addition, the model does not consider the influence of mass changes during the service, such as anti-roll torsion bars, springs and shock absorbers. Furthermore, according to EN 14067–6, in this study, the influence of track irregularities is not considered and the focus is on the situation when the train is running on a straight track.
Train multi-body dynamics model: (a) Simpack model; (b) front-view of the vehicle dynamics topology; (c) side-view of the vehicle dynamics topology.
3.3 Aerodynamic forces
It is well known that the aerodynamic forces of the head car are much greater than that of the middle and the trailing cars under strong crosswind; hence, the crosswind safety risk of the head car is the greatest. Therefore, the focus is usually on the crosswind safety risk of the head car. As shown in Fig. 11(a), the lateral force coefficient of the head car is unsteady initially and stabilizes after the 1.0 s, which implies that the train aerodynamic forces fluctuate steadily around an average value under a constant wind speed. In other words, this average value can be considered as the aerodynamic force on the train. Furthermore, this study focuses on the influence of assessment methods on determining the critical wind speed and many different cases are involved; hence, the aerodynamic loads are calculated with Equations (13) and (14). First, the aerodynamic force coefficients Ci at typical yaw angles are obtained with CFD simulation or a wind tunnel test and then a polynomial is obtained by fitting the aerodynamic force coefficient and the yaw angle. Finally, the aerodynamic force coefficients at any yaw angles can be obtained with the polynomial. Fig. 11(b) shows the lateral force, lift force, overturning moment, pitch moment and yawing moment coefficients of the head car for different yaw angle conditions. It can be seen that the variation trends of the lateral force coefficient and the overturning moment coefficient with the yaw angle are essentially the same. The values both first increase and then decrease with the increase of the yaw angle and maximums are at a yaw angle of 70 °. The changing trend of the lift force and yaw moment coefficients is similar to that of the lateral force coefficient, but it is a maximum at a yaw angle of 50 °. In summary, the variation of the yaw angle has an obvious influence on the aerodynamic forces of the train.
Head-car aerodynamic force coefficients of the high-speed train at different yaw angles: (a) time-history of the lateral force coefficient; (b) force coefficients at different yaw angles.
4. Results and discussion
4.1 Fundamental characteristics of train crosswind safety
Before assessing the critical wind speed of the train, it is necessary to first understand the fundamental characteristics of the crosswind safety of high-speed trains. Fig. 12 shows the maximum value of the train overturning coefficient, derailment coefficient, wheel unloading ratio and lateral wheelset force for different operating conditions. It can be seen from Fig. 12 that the maximum overturning coefficient, derailment coefficient, wheel unloading ratio and lateral wheelset force all significantly increase with the increase of the train operating speed and the crosswind speed. Moreover, the influence of the increase in the wind speed on the safety indexes is more significant than that of the increase in the train running speed. In other words, the influence of the wind speed on the crosswind safety of train operation is greater than that of the running speed. In addition, an interesting phenomenon is also found in that when the overturning coefficient and wheel unloading ratio exceeds the safety limit value, the derailment coefficient and the lateral wheelset force are still within the safe value range, and there is a significant margin. In other words, in this case, the overturning coefficient and the wheel unloading ratio are more likely to exceed the critical values. Additionally, it can be seen that the changing trend of the wheel unloading rate and the overturning coefficient are similar, but the value of the wheel unloading ratio is slightly larger than the overturning coefficient. This is because of the similar definitions of the wheel unloading ratio and the overturning coefficient. However, the derailment coefficient and the wheel unloading ratio are both derived from the necessary condition that the wheelset must climb the rail. From the point of view of the process of the wheelset climbing the rail, the wheelset must be attached to the rail when climbing the rail. Additionally, there should be a certain angle of attack between the wheelset and the rail. The rail climbing process takes a certain amount of time. However, the full-scale test reveals that the wheel unloading ratio cannot directly reflect the contact between the wheel flange and the rail, which implies that the wheel unloading ratio may cause the train crosswind safety risk to be overestimated. Therefore, it is reasonable to select the overturning coefficient as the safety index to determine the critical wind speed of a high-speed train.
Head-car maximum value of the train safety indicators under different conditions: (a) overturning coefficient; (b) derailment coefficient; (c) wheel unloading ratio and (d) lateral wheelset force.
4.2 Critical wind speed from the three-mass method
The multi-body dynamics analysis described above indicates that the overturning coefficient is the most likely safety index to exceed the critical value when the train operates in crosswind conditions. Hence, this section further compares the critical wind speed of the train under different operating conditions using the three-mass method.
Fig. 13 shows the critical wind speed at different operating speeds. The train speed is increased from 40 km/h to 380 km/h. It can be seen that overall, with the increase in the train speed, the critical wind speed decreases significantly. When the train operating speed is in the range of 40 km/h–140 km/h, the train speed is increased by 100 km/h and the critical wind speed is reduced by about 12 m/s. However, when the train speed is in the 140 km/h–380 km/h interval, the train speed is increased by 240 km/h, but the critical wind speed decreases approximately linearly, by only about 11 m/s. This indicates that the reduction rate of the critical wind speed when the train speed is less than 140 km/h is significantly greater than that when the train speed is greater than 140 km/h. In addition, it can also be seen from Fig. 13(b) that the unbalanced lateral acceleration of the car body has a significant impact on the critical wind speed of the train. With the increase of the lateral acceleration of the car body, the critical wind speed also decreases approximately linearly. Hence, it can be concluded that when a high-speed train encounters a strong crosswind, the speed reduction operation can effectively improve the safety of the train.
Critical wind speed under different conditions: (a) different train speeds and (b) different car body unbalanced lateral accelerations.
4.3 Critical wind speed based on the five-mass method
This section further compares the critical wind speed of the train under different operating conditions with the five-mass method. According to EN 14067–6, it is first necessary to check the accuracy of the five-mass calculation procedure. In this study, the established five-mass calculation procedure is verified by the examples given in the standard, as shown in Fig. 14. It can be seen from the figure that the maximum error between the critical wind speed obtained with the five-mass calculation procedure established in this study and the result given in the standard is less than 0.5 m/s, which indicates that this procedure is correct.
Comparison of the result of the established five-mass calculation procedure and the standard: (a) EN14067-Ex1; (b) EN14067-Ex2.
Fig. 15 shows the critical wind speed calculated with the five-mass method at different operating speeds. The train speed range is from 40 km/h to 380 km/h. It can be seen that overall, the variation trend of the critical wind speed with the operating speed is similar to the three-mass calculation result, but the amplitude is slightly larger than that of the three-mass method. In addition, it can also be seen that the inflection point of the critical wind speed reduction ratio is 120 km/h, which is less than 140 km/h for the three-mass method. Similarly, the critical wind speed decreases approximately linearly with the increase of the lateral acceleration of the car body. Additionally, the magnitude of the critical wind speed for different unbalanced lateral accelerations is slightly larger than the value of the three-mass method. This indicates that the critical wind speed of high-speed trains obtained with the three-mass method is more conservative than that of the five-mass method.
Critical wind speed under different conditions: (a) different train speeds and (b) different car body unbalanced lateral accelerations.
4.4 Comparison of critical wind speeds based on different assessment methods
According to the EN14067-6, a method factor that considers uncertainties is defined in the three-mass method and the suggested value for passenger vehicles is 1.2. Therefore, this study uses the three-mass method to determine the critical wind speed divided into two cases: with the method factor and without the method factor, as shown in Fig. 16. The results indicate that the variation trend of the critical wind speeds calculated by different methods is similar, the critical wind speed calculated by the three-mass method without the method factor is the largest. Moreover, the critical wind speed calculated by the three-mass method with the method factor is the smallest, and the critical wind speed values calculated by the five-mass method are between the three-mass method with the method factor and the MBS method. In addition, the Error of the three-mass method with the method factor is less than 15% and is significantly larger than that of the five-mass method. This indicates that both the five-mass method and the three-mass method can be used to determine the critical wind speed of the train at any steady winds. However, these two methods will overestimate the crosswind safety risk of the train and the critical wind speed obtained for the train operation is conservative. The three-mass method with the method factor is the most conservative. This implies that all damping parameters and inertial forces ignored by the quasi-static method will effectively reduce the wind forces transmitted to the track, resulting in a smaller overturning coefficient and higher critical wind speed.
Simulation results with different methods: (a) critical wind speed and (b) Error.
In addition, it can also be seen from Fig. 16(a) that when the train speed is less than 250 km/h, the variation trends of the critical wind speed with the change of the train speed obtained with the different methods are approximately parallel. However, when the train speed exceeds 250 km/h, the calculation results of the five-mass method and the MBS method are closer. In particular, at the train speed of 350 km/h, the Error is less than 5%, while the changing trend of the value of the three-mass method with the method factor is similar to that when the train speed is less than 250 km/h. This indicates that with the increase in the operating speed, the nonlinearity of the vehicle system is further enhanced. Specifically, the influence of the rolling motion between the car body and the bogie is more prominent, which can be seen from the fact that the five-mass method accounts for the effects of the buffers and anti-roll bars.
5. Conclusions
In this study, the effects of different calculation methods on determining the critical wind speed of a high-speed train and application scenarios are numerically investigated in the same wind scenario. The following conclusions can be drawn:
When a high-speed train is completely exposed to a crosswind, the safety risk of the train overturning increases significantly, and the risk of the train overturning first is greater than the risk of derailment. The influence of the wind speed on the crosswind safety of train operation is more significant than that of the running speed. When a high-speed train encounters a crosswind, reducing the speed can improve the safety of the train.
The MBS model is the method closest to a real situation, and it is also the most accurate model for studying the dynamic performance of trains. Both the five-mass and three-mass methods can be used to determine the critical wind speed of a train. However, these two methods will overestimate the crosswind safety risk of a train. The critical wind speed obtained for train operation is conservative. Specifically, the three-mass method is the most conservative.
Acknowledgements
The work presented here was supported by the China National Postdoctoral Program for Innovative Talents (Grant No. BX2021379). We are grateful for resources from the High-performance Computing Center of Central South University.
Conflict of interest statement
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
