The coupled vibration of train and bridge as high-speed trains meet in crosswind

Purpose – This paper aims to study the influence of aerodynamics force of trains passing each other on the dynamic response of vehicle bridge coupling system based on numerical simulation and multi-body dynamics and put forward the speed threshold for safe running of train under different crosswind speeds. Design/methodology/approach – The computational fluid dynamics method is adopted to simulate the aerodynamic force in the whole process of train passing each other by using dynamic grid technology. The dynamicmodelofvehicle-bridgecouplingsystemisestablishedconsideringtheeffectsofaerodynamicforceof trainpassingeachotherundercrosswind,thedynamicresponseoftrainintersectiononthebridgeundercrosswindiscomputedandtherunningsafetyofthetrainisevaluated. Findings – The aerodynamic force of trains ’ intersection has little effects on the derailment factor, lateral wheel-rail force and vertical acceleration of train, but it increases the offload factor of train and significantly increases the lateral acceleration of train. The crosswind has a significant effect on increasing the derailment factor, lateral wheel-rail force and offload factor of train. The offload factor of train is the key factor to control the threshold of train speed. The impact of the aerodynamic force of trains ’ intersection on running safety cannot be ignored. When the extreme values of crosswind wind speed are 15 m $ s (cid:1) 1 , 20 m $ s (cid:1) 1 and 25 m $ s (cid:1) 1 , respectively, the corresponding speed thresholds for safe running of train are 350 km $ h (cid:1) 1 , 275 km $ h (cid:1) 1 and 200 km $ h (cid:1) 1 , respectively. Originality/value – Theresearchcanprovideamoreprecisenumericalmethodtostudytherunningsafetyof high-speed trains under the aerodynamic effect of trains passing each other on bridge in crosswind.


Introduction
With bridge extensively adopted in China's high-speed railway as the substitute for the subgrade, the new approach ensures the safety, stability and riding comfort of high-speed RS 1,2 trains and minimizes the use of agricultural lands.With high-bridge ratio (in some lines even exceeding 90% (He et al., 2017)) as a distinctive feature, it is very likely to encounter the meeting of trains on a bridge.Due to the viscosity of air, the aerodynamic force of high-speed trains changes sharply as they meet (Huang, Li, & Yang, 2019;Lu, Zhang, Zheng, Lu, & Wu, 2018), and the rapid change increases train vibration.The train may derail or even rollover due to strong crosswind during running (Miao, Tian, & Gao, 2010;Hu & Guo, 2009).Under the combined action of strong crosswind and the aerodynamic force of train intersection, the riding comfort of passengers will be greatly compromised, and the possibility of derailment and overturning of trains will increase.
Scholars at home and abroad have conducted numerical simulations, wind tunnel tests, real train tests, etc. to study the aerodynamic force amid trains intersection and the operation safety of high-speed trains under crosswind.Li, Yang, Wu and Qiang (2015) studied the aerodynamic characteristics of trains passing each other on the bridge under crosswind through numerical simulation.Qiu, Li, Sha, Xiao and Wang (2018) carried out simulation when a train passes through a stopped train on single track in a wind tunnel test and analyzed the influence of train speed, wind speed, synthetic wind angle and other parameters on the aerodynamic coefficient amid train intersection.Zhai, Li, Zhang and Lv (2013) analyzed the influence of bridge height on the running safety of trains passing each other with the help of dynamic grid technology and numerical simulation.Chen, Zhang, He and Huang (2014) conducted onsite test to study the aerodynamic action of high-speed trains passing each other in tunnels.Tian, Xu, Liang and Liu (2006) and Xiong and Liang (2009) measured the air pressure wave as CRH 2 trains meet at a constant speed of 250 km$h À1 and obtained the conclusion that the track distance of 4.4 m will not affect the train's running safety.Guo and Zeng (2001) and Zhang, Xia, Guo and Xia (2009) analyzed the vehicle-bridge coupling dynamic response as a train passes through a large-span cable-stayed bridge of single track under the action of fluctuating wind, evaluated the train operation safety and proposed the speed limits of trains under different wind speeds.
In the above studies, only the influence of the train intersection on train aerodynamic force or the influence of crosswind on single-track train running safety was considered, while the influence of aerodynamic force of train intersection on train operation safety has been largely overlooked.In the current wind tunnel test, it is not possible to simulate train intersection for the study of the aerodynamic force in the process, which is generally calculated according to the aerodynamic coefficient of the relative positions of trains with the influence of train movement on the surrounding flow field ignored.
In this paper, the aerodynamic force of the meeting of two trains with three-car formation is simulated and calculated with the help of dynamic grid technology and numerical simulation.The time course is extended to that of eight-car formation trains with the help of a self-compiled program.The aerodynamic forces of the train and the bridge are applied to the 3D dynamic analysis model of wind-train-multi-span simply-supported beam jointly established by SIMPACK software and ANSYS software, and the dynamic response of train intersection on the multi-span simply-supported beam bridge under crosswind is studied.According to the safety index of train running, the speed threshold for safe running with consideration to the aerodynamic force of train intersection under crosswind is put forward.

Six components of aerodynamic force on train
In the natural wind field, the aerodynamic force acting on the high speed train is shown in Figure 1, including lateral force F x , resistance F y , lift force F z , pitching moment M x , roll moment M y and lateral deflection moment M z .

The coupled vibration of train and bridge
2.2 Three components of aerodynamic force on bridge When natural wind flows through the bridge, the bridge is subject to the aerodynamic force formed by aerodynamic (Han & Cai, 2009), including aerodynamic drag force F D , lift force F L and moment F M , as shown in Figure 2.

Calculation model of aerodynamic force of train intersection
Using FLUENT software as the calculation platform, the paper adopts dynamic grid technology to establish a 1:1 full-scale 3D model for the intersection of two trains on a bridge and calculate the aerodynamic force of the trains under crosswind during the intersection on bridge.The flow field as the trains meet on the bridge is considered a viscous, incompressible and inconstant one, and the turbulent movement caused by the high-speed movement of the train is studied by the reynolds average navier-stokes method.Given the calculation capacity of the computer and the similar aerodynamic characteristics of the intermediate cars, the train of three car formation (head car, intermediate car and tail car, 25.25 m long, 3.27 m wide and 3.82 m high each) is used for numerical simulation.The bridge in simulation is a 32 m-long double-track simply-supported beam, which is commonly used in the China high-speed railway, with a total of seven spans.On the premise of calculation accuracy and speed, the influence of train windows, pantographs, bridge guardrails and tracks are waived.The calculation area and the overall coordinate system are shown in Figure 3 (Hong, 2020), which registers 300, 240 and 60 m in length, width and height.The flow field is divided with the help of unstructured tetrahedral meshes which are radioactive.The meshes near the train and the bridge are comparatively small, and the meshes far away from these components are appropriately enlarged so as to ensure the calculation accuracy within the calculation workload.The minimum mesh size of the whole area is 0.2 m, and the total number of meshes is about 3,994,500.Train movement is achieved by editing the user define function program.
Train movement causes mesh deformation.When the mesh quality is lower than a certain value, the local low-quality meshes are updated by the local remeshing method and smoothing method to ensure the mesh quality and the results accuracy amid train movement.At the beginning, the distance between the nose tips of the head cars of the two trains is 72 m, which is recorded as 72 m.During the analysis, the trains are considered to pass each other at a constant speed, and the crosswind direction is perpendicular to the train operation direction.(3) In the whole process of intersection, the head cars of the train are subjected to drag force opposite to the forward direction of trains.The drag forces of the head cars increase first, then decrease and finally increase to the size before intersection.Due to the shielding effect of the head car, the drag forces of the intermediate cars are almost equal to zero, which is far less than that of the head car.3. Mechanical model of wind-train-bridge coupling system

Vehicle dynamics model
The vehicle model is of secondary suspension and four axles, which consists of seven rigid bodies, including one carbody, two bogies and four wheelsets.Each rigid body has six and ψ c are the degrees of freedom of the carbody in x, y, z, θ, w and ψ directions, respectively; x t , y t , z t , θ t , w t and ψ t are the degrees of freedom of bogie in x, y, z, θ, w and ψ directions, respectively; x w , y w , z w , θ w , w w and ψ w are the degrees of freedom of wheelsets in x, y, z, θ, w The coupled vibration of train and bridge and ψ directions, respectively.2d 1 and 2d 2 are the wheelset spacing and bogie spacing, respectively; 2a 1 and 2b 1 are the transverse distances of secondary suspension and primary suspension, respectively; h 1 ,h 2 and h 3 are the heights of car centroid, secondary suspension vertex and secondary suspension vertex to bogie centroid and bogie centroid to wheelset centroid, respectively; c x1 , c y1 , c z1 , k x1 , k y1 and k z1 are the damping and stiffness of primary suspension in x, y and z directions, respectively; c x2 , c y2 , c z2 , k x2 , k y2 and k z2 are the damping and stiffness of secondary suspension in x, y and z directions, respectively (Guo, Hong, & Wang, 2020).

Dynamic models of bridge and track structure
The bridges, tracks and piers are simulated by spatial beam elements in the finite element analysis.The track, considered the Euler beam supported by discrete elastic points, is connected with the main girder by fasteners, therefore seen as spring-damping The stiffness of the pier foundation is superimposed on the corresponding nodes.The damping of the bridge system is considered as Rayleigh damping, and the damping ratio at low frequency is 2-5% generally.Secondary dead load of the bridge is considered part of the mass of the main girder.

Implementation of wind-train-bridge coupling simulation
The coupling vibration of train-bridge system involves two sub-systems of train and bridge.
The finite element model of bridge is established on ANSYS and imported into SIMPACK to form the a vehicle/train-bridge coupling vibration dynamic analysis model with the vehicle model and track model.The track irregularity is used as the internal excitation of coupling system, while the aerodynamic forces of train and bridge are used as the external excitation of vehicle/train-bridge coupling system and are applied to the coupling system via No. 93 force element in SIMPACK.On the wheel-rail contact surface, discrete information points are used for the exchange of data on displacement, speed and force to realize the simulation analysis of coupling vibration (Zhu, Qi, & Jia, 2014).In the vibration calculation, the train is regarded as a rigid body and the bridge as a flexible one.Different algorithms are used to solve their kinematic equations, backward differential formula for the train and modal superposition method for the bridge.The two are solved in turn and in iteration within time step, as shown in Figure 7.The wheel-rail normal force is calculated using the Hertz nonlinear contact theory, and the wheel-rail tangential creep force is calculated using the simplified Kalker creep assumption (Du, Xu, & Xia, 2012).

Calculation examples 4.1 Basic parameters of models
The 32 m-long seven-span double-track simply-supported bridge, commonly used in highspeed railway, is taken as the object of study with the track distance being 5.0 m.The main girder is of box-shaped section and C50 concrete.The net width of the bridge deck is 13.4 m, the beam height is 3.05 m, the beam length is 32.6 m and the net span is 31.5 m.Secondary dead load of 184 kN$m À1 is considered to be equivalent to the mass of the main girder.The pier adopts C30 concrete pier of hollow rectangular section, with a longitudinal length of 3.3 m, a transverse length of 6.8 m, a wall thickness of 0.5 m and a pier height of 16 m.The pier bottom support is considered to be consolidated.The elastic modulus E and Poisson's ratio μ of the material are determined according to the current bridge regulations, and the model damping ratio is 0.02.The train adopts German ICE3 high speed train.Refer to reference (Chen, 2019) for vehicle parameters.The train formation is as follows: 1 motor car þ 6 trailer cars þ 1 motor car.The track structure adopts bi-block ballastless track.The rail adopts RS 1,2 standard 60 rail, with a track length and gauge being 660 m and 1.5 m.The set spacing of fasteners is 0.6 m, the lateral stiffness of fasteners is 60 MN$m À1 , the vertical stiffness is 120 MN$m À1 , the lateral damping is 0.12 MN$s$m À1 , and the vertical damping is 0.15 MN$s$m À1 .According to the suggestions in the Technical Conditions for High Speed Test Trains in China, the simulation of track irregularity is conducted with the help of the German track low interference spectrum, in which the spatial step length is 0.2 m.

Influence of aerodynamic force of train intersection without crosswind on the dynamic response of train
The paper aims to unveil the aerodynamic force amid train intersection with no crosswind and its impact on the dynamic response of the vehicle/train-bridge coupling system.In this sense, it calculates the vibration response of train both intersecting and not intersecting another train at 350 km$h À1 in a no crosswind environment.The dynamic response of the head car of the train is shown in Figure 8.It can be seen from Figure 8 that the dynamic response of the train in the non-crosswind environment is mainly caused by track irregularity, and the vibration response is closely related to track irregularity; in both cases with and without the aerodynamic force of train intersection, the time course of the derailment coefficient, the lateral wheel-rail force as well as the time course of vertical acceleration generally follows the same trends.With consideration of the aerodynamic force of train intersection, the wheel load reduction rate of the train increases; the aerodynamic force of train intersection significantly increases the lateral acceleration of the train, which increases significantly near 3.5 s and 4.5 s, and it is caused by the sudden change of the aerodynamic force on the train at the beginning and the ending of intersection.

Influence of aerodynamic force of train intersection under crosswind on dynamic response of train
In order to study the impact of the aerodynamic force of train intersection under crosswind on the vibration response of the vehicle-bridge coupling system, the paper calculates the vibration response of trains passing on the bridge at 350 km$h À1 both with and without aerodynamic force of train intersection under the crosswind speed at 10 m$s À1 .Please refer to Figure 9 for the dynamic response of the head car.The coupled vibration of train and bridge The comparison of Figures 8 and 9 shows that the crosswind significantly increases the derailment coefficient, the wheel load reduction rate and the lateral wheel-rail force of trains, and the effects of crosswind on the lateral and vertical accelerations of trains are relatively small; the aerodynamic force of train intersection under crosswind has a significant effect on the rise of the lateral acceleration of the train, and at the same time, it pushes up the wheel load reduction rate by a large margin.

Speed threshold for trains intersection in crosswind
In order to ensure the running safety in crosswind, the high speed railway authorities set corresponding threshold of running speed under different levels of crosswind.The dynamic response of trains' intersection at different speeds under crosswind is calculated.The crosswind speed increases with a gradient of 5 m$s À1 in the range of 0-25 m$s À1 , and the vehicle speed changes with a gradient of 25 km$h À1 in the range of 200-350 km$h À1 .A total of 42 working conditions are calculated.The train dynamic response amid the intersection of trains running at 300 km$h À1 under 20 m$s À1 crosswind is shown in Figure 10.It can be seen that the derailment coefficient, lateral wheel-rail force and wheel load reduction rate of the trains tend to increase amid intersection, while the derailment coefficient, lateral wheel-rail force as well as the wheel load reduction rate of the head car largely exceed those of the intermediate and the tail cars.Figure 10a show that during the whole process of train intersection, the wheel load reduction rate of the head car exceeds the specification limit at the time when the two head cars meet and the two tail cars leave each other, while the wheel load reduction rate of the intermediate car and the tail car are less than the specification limit.
Figure 11 shows the dynamic response of the trains passing each other at a speed of 200-350 km$h À1 at a crosswind speed of 0-25 m$s À1 .The dotted horizontal line in the figure is the safety limits of the dynamic response of trains specified in the relevant technical specification.It can be seen from Figure 10 that the derailment coefficient, lateral wheel-rail force and wheel load reduction rate of the train increase with the rise of wind speed and train speed; under each working condition, the derailment coefficient and lateral wheel-rail force of the train meet the requirements of relevant specifications.However, in some cases, the wheel load reduction rate of the train exceeds the specified limits, which makes the rate the control index for the safe intersection of trains on the bridge.

The coupled vibration of train and bridge
It can also be seen from Figure 11b that when the crosswind speed is less than 15 m$s À1 and the trains pass each other at a speed of 350 km$h À1 , the maximum wheel load reduction rate of the train is 0.548, which is less than the safety limit of 0.6 specified in relevant specifications.Therefore, when the crosswind speed is less than 15 m$s À1 , the speed threshold shall be 350 km$h À1 .When the crosswind speed is between [15, 20] m$s À1 and the trains pass each other at 300 km$h À1 , the maximum wheel load reduction rate is 0.618; when the trains pass each other at 275 km$h À1 , the maximum wheel load reduction rate is 0.578; when the crosswind speed is [15, 20] m$s À1 , the speed threshold is 275 km$h À1 ; when the crosswind speed is between [20, 25] m$s À1 and the trains pass each other at 225 km$h À1 , the maximum wheel load reduction rate of the train is 0.631; when the trains pass each other at 200 km$h À1 , the maximum wheel load reduction rate of the train is 0.589; when the crosswind speed is [20,25] m$s À1 , the train speed threshold shall be 200 km$h À1 .

Conclusion
(1) Given the similarity of the aerodynamics characteristics of intermediate cars, the time course of the aerodynamic force of the intersection of trains of three-car formation can be extended to one of eight-car formation passing.
(2) The crosswind plays a significant role in the increase of the wheel load reduction rate, the lateral wheel-rail force as well as the derailment coefficient and has no obvious impact on the train acceleration.
(3) The aerodynamic force of train intersection has a significant impact on the lateral acceleration of the train and has, to some extent, push up the wheel load reduction ratio of the train but has no obvious effect on other dynamic responses of the train.The coupled vibration of train and bridge Figure 1.Schematic diagram of six components of aerodynamic force on train

Figure 4 .
Figure 4. Time course curve of aerodynamic force of the head car and the intermediate car of two intersected trains of three-car formation Figure 5.Time course curve of lateral force of head cars of two trains with eight-car formation passing each other Figure 7. Iteration diagram of train-bridge system's solution Figure 8.Time course of dynamic response of head car with no crosswind Figure 9. Head car dynamic response time course Figure 10.Time course of dynamic response of trains' intersection at speed of 300 km$h À1 under 20 m$s À1 crosswind