A co-simulation method for the analysis of train running performance on a sea-crossing bridge in crosswind environment

Abstract

When a train crosses a bridge in a crosswind environment, the coupled vibration problem of the train-bridge system becomes prominent, and train safety and riding comfort are difficult to guarantee. Therefore, using the Pingtan Strait bridge in China as a case study, a co-simulation platform for the train-bridge system coupled vibration in crosswind environments was established based on computational fluid dynamics, finite element method, and the multi-body system dynamics. Based on this platform, dynamic response analysis of the train-bridge system was performed at different wind and train speeds. The results indicate that the dynamic response of the train and bridge under double-line conditions is greater than that under single-line conditions. With an increase in wind speed, the mid-span vertical displacement of the bridge changes little, while the lateral displacement increases significantly. Meanwhile, with increasing wind and train speeds, the train dynamic indexes obviously increase. Moreover, the dynamic index of the head car is the largest among all the train sections.

Keywords

sea-crossing bridge crosswind co-simulation dynamic response coupled vibration

Introduction

When a train runs in a crosswind environment, its aerodynamic characteristics undergo obvious changes (Paradot et al., 2015; Xiao et al., 2014; Yu et al., 2015). Consequently, the train aerodynamic load change substantially, potentially leading to lateral-swing overrun, train derailment, and even overturn and casualty accidents. In particular, if the train moves over a long-span bridge, the aerodynamic forces will change more significantly. Moreover, as a bridge structure with a long span is more flexible, the coupled vibration between bridge and vehicle is more severe, which further increases the possibility of derailment and overturning. The operational safety of trains on bridges under crosswind environments has received significant attention. Diana and Cheli (1989) first studied the additional dynamic effects of a moving vehicle under crosswind on the bridge structure. Cai and Chen (2004) presented a dynamic analysis framework of a coupled 3D vehicle-bridge system under strong winds, without considering the aerodynamic effects between the bridge and the vehicle, but only for road vehicles. Guo et al. (2007a, 2007b), Xia et al. (2008), Xu and Li (2016), Xu et al. (2003a, 2003b, 2004), and Zhang et al. (2016) calculated the vibration response of the Hong Kong Tsing Ma highway-railway bridge and Wuhan Tianxingzhou highway-railway bridge under wind and train loads. Based on the simulated fluctuating wind velocity field, the authors simulated the wind-induced vibration of the bridge and studied the influence of the vibration on the train-bridge coupled vibration characteristics. Li et al. (2003, 2005a, 2005b, 2013) comprehensively considered the wind-induced bridge vibration, coupled vibration between the train and bridge, and space fluctuation effects of wind on the train. They established a nonlinear, spatially coupled wind-train-bridge model and compiled a software BANSYS.

However, in existing studies on the coupled vibration in vehicle-bridge systems, the differential motion equations of position and the attitude coordinates are generally derived by means of the traditional kinetic analysis method, that is, using the second type of Lagrange equation or Newton-Euler equation, when the vehicle dynamics model is established. Most of these studies are confined to their own programs, which are not open to the public and have weak universality. With the development of railway engineering, vehicle models are gradually increasing and complicated. The demand for vehicle-bridge coupled vibration calculation is increasing, and the traditional calculation method has been difficult to meet the demand. However, the multi-body system (MBS) dynamics software SIMPACK has powerful wheel-rail analysis function, and can quickly build refined vehicle models. The finite element software ANASYS can build the bridge model efficiently (Li et al., 2018). Moreover, computational fluid dynamics (CFD) software FLUENT can conveniently simulate the fluid flow around the train and bridge, and obtain the aerodynamic load of the train and bridge (Chen et al., 2016).

Obviously, with the integration of numerical analysis and traditional dynamics, co-simulation is regarded as an important method for interdisciplinary research in different mechanical fields and can establish a platform for high-speed railway vehicle-bridge dynamic analysis under crosswind environments (He et al., 2017; Li et al., 2018; Xu and Li, 2016). This simulation study includes three domains, namely: wind, bridge, and train with complex wheel-rail relationships. The computational analysis of the wind load, bridge dynamic behavior, and vehicle running behavior can be conducted separately using CFD, FEM, and MBS software. Thereafter, in terms of the initial values for each sub-system obtained by the appropriate software, co-simulation can be performed by interconnecting these components.

The Pingtan Strait bridge, with a length of 11.15 km, is a critical control engineering project from the Fuzhou to Pingtan railway and Changle to Pingtan highway, and is the first actual highway-railway sea-crossing bridge in China. The design wind speed at the bridge site is 44.8 m/s, and it is difficult to guarantee safe train operation under strong winds. Taking Taking Yuanhongdao Bridge as an example, Fang et al. (2018) used the Newmark-β method to solve the governing equations of the wind-wave-vehicle-bridge model in the time domain to compute the dynamic response of bridges and vehicles.

In this study, a CRH3 train and steel truss girder cable-stayed bridge in the Pingtan Strait are used as case studies. The aerodynamic model of train-bridge system is established by using CFD software FLUENT, and the aerodynamic load is calculated. The finite element model of the bridge is established by ANSYS, and the train model is established by MBS software SIMPACK. Thus, through the co-simulation of CFD, FEM and MBS, the evaluation of train running performance in crosswind environment is realized. The flow chart of the co-simulation process is presented in Figure 1. Based on this method, the dynamic responses of the train and bridge are calculated when only one train passes through (hereinafter referred to as “single line condition”) or two trains pass through the bridge in reverse direction (hereafter referred to as the “double-line condition”).

Figure 1.

Co-simulation process.

Aerodynamic load of train-bridge system in crosswind environment

Calculation method

The Reynolds averaging method is used in the aerodynamic model. Because of the high-speed of the train, the Reynolds number is large, and the curved surface of the train head is complex, the RNG -model is adopted. The governing equation is discretized by the finite volume method (FVM), and the discretized control equation is solved by the separating solution. The pressure-velocity field is coupled by means of the SIMPLE method (Niu et al., 2018), and the pressure is simultaneously corrected with the iteration method.

Calculation model

The span of the highway-railway steel truss girder cable-stayed bridge between the Dalian and Xiaolian islands of the Pingtan Strait is 80 m+140 m+336 m+ 140 m + 80 m. The girder is an inverted trapezoid steel truss, and its size is depicted in Figure 2. A double-line railway is laid on the railway deck with a spacing of 4.4 m. The bridge tower height is 169.5 m, the upper part is 98.5 m, and the lower part is 71 m. There are 80 cables in the bridge; the cable spacing on the main beam is 14 m, while that on the bridge towers is 2.5 m. A CRH3 train was considered, the length of the head and tail cars is 25.85 m, and that of the middle car is 24.825 m. The train width is 3.265 m and the height is 3.89 m.

Figure 2.

Size of bridge (unit: m).

CFD has been applied successfully to the calculation of aerodynamic loads on trains and bridges (He et al., 2016; Wang et al., 2013; Yang et al., 2015). In this study, the aerodynamic models for the bridge and train were constructed using the CFD software FLUENT 16.0 (ANSYS, Inc., 2015a). Considering the calculation efficiency, the bridge aerodynamic model ignores the effects of deck attachment structures, track systems, cables, bridge piers, and bridge towers. The aerodynamic model of the train has three sections. The actual train surface contains many complex, detailed structures such as lights, windows, door handles, and bogies. These details are ignored in the calculation model, and the roughness height of the train surface is taken as 0.03 mm (García et al., 2017). Similarly, the track system of the railway bridge deck is ignored, and the roughness height of the bridge deck is taken as 2 mm (García et al., 2011, 2017) instead of the track system modeling. In addition, this study did not consider the effect of highway traffic vehicles. The simplified aerodynamic models of the train and bridge are illustrated in Figure 3.

Figure 3.

Aerodynamic model.

In order to simulate the train movement in real time, local dynamic layer grids were used for modeling, considering the advantages of dynamic and sliding grids. The grid partition, which includes fixed and moving regions, is illustrated in Figure 4. The moving region is composed of the stretching zone, compression zone, and moving train. The train and its surrounding grids together have rigid body motion, the front of the vehicle is the grid compression zone, and the rear is the grid stretching zone. Grid reconstruction in the stretching and compression zones is controlled by splitting and shrinkage factors. The static grid is used for the fixed region, and data are transferred between adjacent regions through the interface. The hexahedral grid and tetrahedral grid are combined to divide the mesh. The tetrahedral grid is used near the train and bridge, and the hexahedral grid is used for the external flow field. In addition, the boundary layer grids are divided on the surface of the train and the bridge, and the grids are densified. The height of the first layer grid on the surface of the train and bridge is 1 mm, the height growth ratio is 2.5, and the value of y+ is 30–60. The maximum grid size of the train surface and its computational domain is 0.4 m and 1 m, while the maximum grid size of the bridge surface and its computational domain is 1 m and 4 m. The total number of grids for single-line and double-line conditions is 7.6 million and 8.2 million respectively.

Figure 4.

Grid partition diagram.

The inlet boundary is set as the velocity inlet of exponential wind profile. The computational domain outlet boundary is set as the pressure outlet and the top surface is set as the symmetry boundary. The intersection of the moving and fixed regions, and the intersection between the two moving regions are all set as the interface. The surfaces of the train, bridge, and ground are all set as non-slip walls. The setting of computational domain and boundary conditions is shown in Figure 5. In CFD simulation, the trains run from one end of the girder to the other end, as shown in Figure 6.

Figure 5.

Computational domain and boundary conditions: (a) Computational domain, (b) Front view of the boundary conditions and (c) Side view of boundary conditions.

Figure 6.

Start and end positions of trains in CFD simulation.

Based on the aerodynamic model, the aerodynamic loads of train and bridge are calculated respectively. The train speed is 100, 200 and 300 km/h, respectively. The maximum wind speed taken during the calculation is 30 m/s and the wind speeds of 10 and 20 m/s are also calculated.

Fang et al. (2018) obtained the aerodynamic parameters of the section model of the bridge and vehicle through the wind tunnel test. The aerodynamic parameters calculated by the simulation model and the wind tunnel test are shown in Table 1. Comparison between the simulation and test shows that maximum difference of the aerodynamic parameters was less than 10%. This may be caused by insufficient grid density. In addition, the simplification of details on the surface of train and bridge may also be the cause of errors.

Table 1.

Comparison of aerodynamic parameters.

	Bridge			Vehicle
Aerodynamicparameters	Resistancecoefficient (C_d)	Liftcoefficient (C_l)	Momentcoefficient (C_m)	Resistancecoefficient (C_d)	Lift coefficient (C_l)	Momentcoefficient(C_m)
Test results	1.026	0.167	0.107	1.478	0.231	0.239
Simulation results	0.962	0.161	0.101	1.608	0.254	0.222
error (%)	−6.3	−3.5	−5.6	8.8	9.9	−7.0

The train aerodynamic loads include resistance (F_x), lateral force (F_y), lift force (F_z), roll moment (M_x), pitch moment (M_y), and yaw moment (M_z). The main girder aerodynamic loads include resistance (M_H), lift force (M_V), and moment (M_T). All of these aerodynamic loads of the train and bridge are composed of two components: the pressure and viscous force. The formula is:

F = F_{p} + F_{ν} = \sum_{i = 1}^{n} p \cdot A + \sum_{i = 1}^{n} f_{υ}

(1)

where F is the resultant force of the pressure and viscous force; F _p and F _υ are the pressure and viscous force vectors respectively; p is the grid cell central pressure; A is the grid cell area vector; n is the number of grid cells for the train and bridge surface; and f _υ is the grid cell viscous force. According to the torque center position, the aerodynamic moment can be calculated by the following equation:

M_{A} = r_{AB} \times F_{p} + r_{AB} \times F_{ν}

(2)

where M_A is the total moment; and r _AB is the moment vector from moment center A to acting point B of the force.

The train and bridge girder dynamic loads are respectively calculated under different train speeds and wind speeds. Figure 7 illustrates the middle section flow field characteristics of the 30–300 condition (the condition symbol represents a 30 m/s wind speed and 300 km/h train speed, with the same naming rule used in the following paragraphs) of the computational domain. In addition, Figure 8 shows the pressure distribution on the train surface under 30–300 condition. Owing to limited space, Figures 9 only indicate the maximum values of the aerodynamic loads of the single-line and double-line windward side head cars.

Figure 7.

Flow field characteristics of the 30–300 condition: (a) Single-line condition and (b) Double-line condition.

Figure 8.

Surface pressure distribution of train under 30–300 condition: (a) Single-line condition and (b) Double-line condition.

Figure 9.

Maximum aerodynamic loads of head car: (a) Resistance (F_x), (b) Lateral force (F_y), (c) Lift force (F_z), (d) Roll moment (M_x), (e) Pitch moment (M_y) and (f) Yaw moment (M_z).

From the streamline distributions in Figure 7, when the train runs in crosswind environments, the airflow shunts, bypassing the train and bridge, simultaneously accompanied by the formation of vortices. These vortices are generated on the leeward side of the train, the top of the upper deck, and the bottom of the lower deck. From the pressure distributions in Figures 7 and 8, it can be seen that most of the windward area of the train and bridge is under positive pressure, while the leeward side is almost under negative pressure. The airflow on both sides of the train head is not symmetrical owing to the crosswind. The windward area is under positive pressure and the leeward area is under negative pressure. Moreover, an airflow stagnation point exists at the tip of the train, where the pressure reaches the maximum values of 4.48 kPa under the single-line condition and 4.75 kPa under the double-line condition. Owing to the influence of the wake, the train tail is subjected to “attraction” in the direction opposite to the crosswind. Thus, the pressure distribution on the train tail surface exhibits an unusual phenomenon whereby the leeward side pressure is greater than that of the windward side.

Figure 9 indicates that the aerodynamic components of the head car under the double-line condition are greater than those under the single-line condition at the same train and wind speeds, and the disparities become more obvious with an increase in train speed. Under both single-line and double-line conditions, the lateral force, rolling moment, and yaw moment of the head car all increase with increasing train and wind speeds. The lift force is more sensitive to wind speed than train speed and increases with higher wind speed. The resistance force of the head car increases with an increase in train speed, but decreases with an increase in wind speed. When the wind speed reaches 30 m/s, the resistance transforms into “traction” which acts in the same direction as that of the train’s motion. The pitch moment is caused by the resistance and lift forces together and is strongly influenced by the train speed, increasing with an increase in train speed.

Dynamic models of train and bridge

MBS dynamic model of train

The simulation model of the CRH3 train was established by the MBS software SIMPACK 9.9 (SIMPACK, GmbH, 2015). The train consists of 4 motor cars and 4 trailers. The motor car model consists of a car body, two bogies, four wheelsets, two motors and eight rotating arms for a total of 17 rigid bodies (see Figure 10), while the trailer has a total of 15 rigid bodies due to the absence of motors. These are connected by primary springs, secondary springs, shock absorbers, and antiroll torsions. The mass of the car body, bogie, wheelset, rotating arm and motor on the motor car are: 3.89, 2.27, 1.52, 0.12 and 1.93 tons, respectively. The mass of the car body, bogie, wheelset, and rotating arm on the trailer are: 4.41, 2.44, 1.71 and 0.12 tons, respectively. Weight of each train axle is 15 tons. The vertical stiffness of primary spring and secondary spring are 10.4 MN/m and 0.203 MN/m respectively. The motor car has 62° of freedom, including 54 independent joints and eight constraints among these. The trailer has 50° of freedom and includes 42 independent joints and eight constraints. When constructing this model, numerous nonlinear factors were taken into account, such as the anti-hunting damper, secondary damper and the secondary lateral stop.

Figure 10.

Schematic diagram of the vehicle mode.

The wheel-rail contact relationship mainly includes the geometrical compatibility condition and wheel-rail interaction force. The motion of wheelset is mainly constrained by the geometric shape of wheel and rail, and can be determined by cubic spline interpolation in SIMPACK. The S1002 wheel tread type and UIC60 rail type were used in the train model (Arrus et al., 2002). The wheel-rail interaction force includes the creep and normal forces, and the normal force can be calculated by the Hertz nonlinear elastic contact theory. Furthermore, the creep force can be solved by a simplified Kalker theory, namely the FASTSIM algorithm (Kalker, 1982). The German railway spectrum of low irregularity is used as the track excitation.

FE dynamic model of bridge

The numerical model of the bridge was constructed using the FEA software ANSYS 16.0 (ANSYS, Inc., 2015b). As shown in Figure 11, the beam elements (beam44) are used to simulate the bridge piers, the girder and the bridge towers, and the link elements (link10) are used to simulate the cables. The secondary permanent load was simulated by mass element (mass21). The secondary permanent load of the highway is 7 t/m, which is evenly distributed on two upper chords. The secondary permanent load of the railway is 9 t/m, which is evenly distributed on four longitudinal beams. The connections between the girder and the pier or bridge tower were simulated by the method of node coupling. Total number of nodes in the model is 8702, and the total number of elements is 12518. The bottom of the pier and the bridge tower were fixed and no consideration of soil-structure-interaction.

Figure 11.

FE model of bridge.

Analysis of train-bridge system coupled vibration in crosswind environments

Co-simulation of MBS and FEM

FEA element software has been widely used in building bridge models, while MBS software can build fine vehicle models. The coupling vibration of vehicle and bridge can be solved by FEA and MBS co-simulation. The initial value problems for the vehicle behavior and bridge dynamic behavior are solved separately by the MBS dynamics and FEM. Thereafter, the interaction between the bridge and the vehicle is realized by the data exchange of the force element. The applicability of the SIMPACK and ANSYS co-simulation in the vehicle and bridge coupled vibration has been validated (Cui, 2009; Li et al., 2018).

Firstly, the substructure analysis of the finite element model is needed to obtain the *.cdb and *.sub files containing the information of mass matrix, stiffness matrix, node coordinates and modal shape of the bridge. Then the pre-processing program of the MBS software SIMPACK calls these data and generates a flexible body input file (*.fbi), so that the bridge is integrated into the MBS as a flexible body. In the MBS software, the moving markers on the bridge that follow the corresponding wheelset are constructed, and the flexible bridge and vehicle are connected by rail-track force elements between these moving markers and the rails (illustrated in Figure 12). The vertical and lateral stiffness and damping of these force elements are 70 kN/m and 70 kN•s/m, respectively. Moreover, aerodynamic loads calculated by CFD software are applied to the bridge and train by means of time excitation in MBS. Data exchange between bridge and vehicle is carried out by the force elements established between rails and moving markers. The deformation compatibility and force equilibrium conditions between the vehicle and bridge can be expressed as follows:

u_{r} (t) = u_{b} (t, s)

(3)

{\begin{matrix} Y (t) = F_{y} (t) \\ Q (t) = F_{z} (t) \end{matrix}

(4)

Figure 12.

Connection between wheel, rail, and bridge.

where ur(t) is the track displacement; u_b(t,s) is the displacement at the “s” position in the bridge axial direction; Y(t) and Q(t) are the lateral and vertical wheel-track forces, respectively; and F_y(t) and F_z(t) are the lateral and vertical forces of the force element, respectively.

According to GB 5599-85 (1985) and TB10002-2017 (2017), the evaluation criteria for the bridge and vehicle in this study are listed in Table 2.

Table 2.

Evaluation criteria.

Evaluation index of bridge and vehicle		Limited standard
Bridge	Lateral deflection to span ratio		1/1200
	Vertical deflection to span ratio		1/600
	Lateral acceleration (m/s²)		1.4
	Vertical acceleration (m/s²)		3.5
Vehicle	Safety	Derailment coefficient	0.8
		Wheel unloading ratio	0.6
		Wheel-rail lateral force (kN)	80.0
	Comfort	Car-body lateral vibrationacceleration (m/s²)	1.0
		Car-body vertical vibrationacceleration (m/s²)	1.3
		Lateral Sperling index (Dumitriuand Mihai, 2018)	≤2.5 (Excellent), ≤2.75 (good), ≤3.0(Qualified)
		Vertical Sperling index	≤2.5 (Excellent), ≤2.75 (good), ≤3.0(Qualified)

Bridge dynamic response

The maximum displacement and acceleration of the mid-span when the train passes over the bridge are illustrated in Figure 13.

Figure 13.

Bridge dynamic response: (a) Lateral displacement, (b) Vertical displacement, (c) Lateral acceleration and (d) Vertical acceleration.

From Figure 13, both the displacement and acceleration of the bridge under the double-line condition are greater than those under the single-line condition. With increasing wind speed, the variation in the mid-span vertical displacement is very small, with a maximum increase of only 6.6% under the single-line condition and 3.5% under the double-line condition. This is owing to the absolutely dominant role of the train dead load, leading to the bridge vertical displacement. The mid-span lateral displacement increases significantly with an increase in wind speed, by up to 6.55 times under the single-line condition and up to 5.56 times under double line condition. With increasing wind speed, the vertical and lateral mid-span vibration accelerations also increase obviously, by up to 24.7% and 19.3% for vertical and lateral acceleration, respectively, under the double-line condition, and 32.9% and 25.2%, respectively, under the single-line condition. For the studied conditions of different wind and train speeds, the maximum vertical and lateral deflection-to-span ratios are calculated as 1/3482 and 1/17909, respectively. Thereby, the maximum vertical and lateral deflection-to-span ratios, and vertical and lateral vibration acceleration of the mid-span all meet the requirements of Table 2.

Train dynamic response

The dynamic response of the train is shown in Table 3, where “S” stands for single line, “D” stands for double line, “H” stands for head car, “M” stands for middle car, “T” stands for tail car, “W” stands for windward side, “L” stands for leeward side. Taking the 30–300 condition under single line as an example, the derailment coefficient, wheel-rail lateral force, wheel unloading ratio, car body vibration acceleration, and comfort index of the head car are all higher than the relative values of the middle and tail cars. The other conditions have the same rules, but the data is not listed in the table due to the limitation of space. The values in Table 3 that exceed the thresholds described in Table 2 are shown in bold font.

Table 3.

Dynamic response of train.

Conditions		Derailmentcoefficient	Wheel-rail lateralforce (kN)	Wheelunloadingratio	Car- body Vertical vibrationacceleration (m/s²)	Car- bodyLateralvibrationacceleration(m/s²)	VerticalSperling index	Lateral Sperlingindex
0-300	S-H	0.095	11.212	0.400	0.815	0.627	2.428	2.351
	D-H-W	0.118	21.638	0.494	0.914	0.637	2.496	2.347
	D-H-L	0.122	20.697	0.506	0.896	0.623	2.491	2.355
10-100	S-H	0.134	13.4	0.297	0.635	0.543	1.839	2.261
	D-H-W	0.253	28.2	0.382	0.774	0.676	1.935	2.249
	D-H-L	0.218	38.8	0.376	0.839	0.534	1.927	2.301
10-200	S-H	0.156	16.5	0.391	0.966	0.610	2.320	2.404
	D-H-W	0.178	33.7	0.473	0.996	1.247	2.419	2.607
	D-H-L	0.184	31.4	0.471	0.901	1.103	2.454	2.463
10-300	S-H	0.207	18.6	0.574	1.030	0.803	2.521	2.474
	D-H-W	0.352	29.1	0.581	1.310	2.411	2.674	2.583
	D-H-L	0.342	42.0	0.575	1.298	1.420	2.675	2.745
20-100	S-H	0.211	19.2	0.356	0.529	0.789	1.981	2.446
	D-H-W	0.244	31.6	0.482	0.868	0.795	1.924	2.465
	D-H-L	0.197	33.5	0.477	0.853	0.685	2.034	2.388
20-200	S-H	0.234	26.3	0.484	0.948	0.862	2.351	2.577
	D-H-W	0.239	31.2	0.595	0.957	1.417	2.425	2.639
	D-H-L	0.237	34.6	0.588	1.213	1.126	2.439	2.685
20-300	S-H	0.285	23.6	0.681	1.014	1.013	2.561	2.582
	D-H-W	0.585	30.4	0.698	1.463	2.555	2.684	2.587
	D-H-L	0.348	47.5	0.688	1.425	1.608	2.684	2.745
30-100	S-H	0.264	31.1	0.549	0.810	0.818	2.315	2.480
	D-H-W	0.269	32.2	0.582	0.826	0.948	2.280	2.479
	D-H-L	0.305	38.2	0.539	1.155	0.886	2.519	2.437
30-200	S-H	0.498	41.7	0.635	0.996	1.581	2.347	2.684
	D-H-W	0.653	41.7	0.712	1.146	2.319	2.482	2.674
	D-H-L	0.362	53.9	0.675	1.265	1.478	2.548	2.700
30-300	S-H	0.736	45.5	0.737	1.306	1.923	2.652	2.725
	S-M	0.228	23.8	0.586	0.920	1.016	2.435	2.451
	S-T	0.232	29.2	0.691	1.002	0.957	2.510	2.388
	D-H-W	0.764	46.0	0.749	1.556	3.171	2.726	2.760
	D-H-L	0.514	66.3	0.677	1.589	1.853	2.727	3.024

Note. Bold font indicates that the values in Table 3 exceed the thresholds in Table 2.

It can be seen from Table 3 that the dynamic indexes of trains in cross wind environment are significantly larger than those in no wind environment. Compared with the no wind environment, at 10 m/s wind speed and 300 km/h vehicle speed, the derailment coefficient increased by 1.18 times under single-line conditions, the wheel unloading ratio increased by 43.5%, and the car body lateral vibration acceleration increased increased by 28.1%; under double line conditions, the derailment coefficient, wheel unloading ratio and the car body lateral vibration acceleration increased by up to 1.98 times, 17.6% and 2.78 times, respectively.

Tables 3 also indicate that, with an increase in wind and train speed, the safety indexes (derailment coefficient, wheel-rail lateral force, and wheel unloading ratio) and comfort indexes (car body vibration acceleration and comfort index) all exhibit an increasing trend under both the single-line and double-line conditions. Compared to the dynamic indexes under the single-line condition at the same wind and train speeds, the derailment coefficient increases by a maximum of 51.28%; the wheel-rail lateral force increases by a maximum of 65.52%; the wheel unloading ratio increases by a maximum of 26.14%; the car body vertical vibration acceleration increases by a maximum of 39.06%; and the car body lateral vibration acceleration increases by a maximum of 66.69% under the double-line condition.

When the wind and train speeds are within 30 m/s and 300 km/h, respectively, the train derailment coefficient and wheel-rail lateral force can meet all the requirements listed in Table 2, but the wheel unloading ratio, car-body acceleration, and comfort index do not fully meet the requirements. Figures 14 and 15 illustrate the wheel unloading ratio and car body lateral vibration acceleration of the head car on the windward side at a wind speed of 30 m/s and train speed of 300 km/h. It can be observed that the wheel unloading ratio reaches its maximum when the train runs across the bridge mid-span. Moreover, the car body lateral vibration acceleration under the double line condition achieves a maximum that is almost twice as high as that under the single line condition, which is caused by its mutation when two trains pass by one another.

Figure 14.

Time-history curve of wheel unloading ratio under 30–300 condition.

Figure 15.

Time-history curve of lateral acceleration under 30–300 condition.

Conclusions

Taking full advantage of the advantages of CFD software FLUENT, FEA software ANSYS and MBS software SIMPACK, this study presented a co-simulation method for calculating train-bridge coupled vibration in crosswind environment. Then, using the crossing sea bridge between the Dalian and Xiaolian islands of the Pingtan Strait and the CRH3 train for a case study, a co-simulation model was established, and the dynamic responses of vehicles and the dynamic response of the train-bridge coupled system was calculated under different wind and train speeds. The main conclusions can be summarized as follows.

The co-simulation method based on CFD, MBS and FEM software provides an effective means for the study of coupled vibration of train-bridge system in crosswind environment.

In crosswind environments, the bridge dynamic response under the double-line condition is greater than that under the single-line condition. With an increase in wind speed, the variation in the mid-span vertical displacement is very small. However, the lateral displacement of the mid-span increases significantly with an increase in wind speed, while the vertical and lateral vibration accelerations of the mid-span also increase obviously. For the studied conditions of different wind and train speeds, the mid-span deflection and the acceleration meet all of the requirements.

When the train runs in crosswind environments, the dynamic index of the head car is the largest among all train sections; thus, this can be used to evaluate the safety and comfort of the entire train. The head car dynamic characteristics under the double-line condition are more unfavorable than those under the single-line condition, and the dynamic indexes are larger under the double-line condition. With increasing wind and train speed, the safety and comfort indexes all exhibit an increasing trend under the single-line and double-line conditions. The wheel unloading ratio reaches its the maximum when the train runs across the bridge mid-span, and the car body lateral vibration acceleration under the double line condition achieves its maximum when two trains pass by one another.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: We acknowledge the National Natural Science Foundation of China-China Railway Corporation High-speed Railway Basic Research Joint Fund (under Grant No. U1834207), the National Natural Science Foundation of China (under Grant No. 51678492), and the Sichuan Province Science and Technology Project (under Grant No. 2019YFG0001 and No. 2019YFG0258).

ORCID iD

Shengai Cui

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