Study of wind–vehicle–bridge system of suspended monorail during the meeting of two trains

Abstract

Based on the theories of aerodynamics, bridge dynamics, and vehicle dynamics, the aerodynamic performances and the vibration characteristics of the wind–vehicle–bridge coupling system of two suspended monorail trains passing each other are analyzed. First, a wind model is presented with spectral representation method, the aerodynamic coefficients of bridge and vehicles before and after meeting are obtained through computational fluid dynamic method, and wind tunnel tests are conducted to verify the aerodynamic coefficients. Then, a vehicle dynamic model and a bridge dynamic model are established with the multi-body dynamic method and finite element method. Finally, a three-dimensional wind–vehicle–bridge coupled vibration model is established in this article using the multi-body dynamic software SIMPACK. The effects of average wind and fluctuating wind on the wind–vehicle–bridge system are studied. It is shown that the aerodynamic coefficients vary greatly under different combinations of vehicle–bridge system. The responses of the leeward vehicle change abruptly at the beginning and the end of the meeting of the two trains. And the mean wind speed has a great negative contribution to the acceleration of leeward vehicle. The lateral responses of suspended monorail vehicle are sensitive to the fluctuating wind. The roll angle of vehicle is presented for describing the running safety of the suspended monorail vehicles.

Keywords

aerodynamic coefficients meeting of two trains suspended monorail transit system wind–vehicle–bridge coupling vibration

Introduction

In the natural wind environment, the wind loads usually lead to the deformation of static wind and buffeting response of bridge. Meanwhile, both wind loads and vibration of bridges will have significant influence on the dynamic responses of suspended monorail vehicles while running on a bridge under crosswind. Unlike railway vehicle, suspended monorail vehicle travels along the rail through the bogie frames above, meanwhile the vehicle body is suspended under the track beam, particularly, rubber tires are used in the transit system (Wang, 2003). A limited number of studies have been conducted for the dynamic behavior of the system. Meisinger (2006) studied the lateral gravity acceleration of monorail vehicle with periodic track irregularities by the numerical analysis method. Cai et al. (2019) evaluated the dynamic responses of suspended-type monorail system based on the multi-body dynamic method and finite element theory. Due to low lateral stiffness of the bridge structure, the wind load will cause a large roll angle of the moving vehicle. Though suspended monorail vehicle will not derail under the wind environment, it may affect the vehicle’s running stability and even safety.

If a sudden change of wind load occurs during the meeting of two trains, the interaction aerodynamic effect of the vehicle and the bridge changes significantly. Diana et al. (2002), Chenli et al. (2010), and Bocciolone et al. (2008) extended their work to study the aerodynamic behavior of railway vehicles under the action of crosswind, and the moving train tests are conducted to obtain the force coefficients. Baker (2010; Baker et al., 2013) investigated the simulation effect of crosswind on train dynamic systems. Xu et al. (2003, 2004) established a framework to perform the dynamic analysis of coupled train and bridge systems under crosswind; both buffeting and self-excited forces were considered as wind force acting on the bridge. He et al. (2018) proposed a rigid–flexible coupling method to study the dynamic behavior of train and bridge under wind loads. The above studies focused on the simulation of crosswind affecting the moving trains, and the effects of meeting of two trains have not been discussed.

Zhao et al. (2012) and Tian (2006) studied the flow field around trains passing by each other, and the pressure waves and aerodynamic forces were addressed. Li et al. (2015, 2005, 2013) presented a wind–vehicle–bridge (WVB) coupled vibration model for railway system, and the vibration characteristics of the wind, vehicle, and bridge during two trains meeting each other are studied. These researches (Wu et al., 2017; Xiang et al., 2015) showed that the relative position and different combinations of vehicle and bridge have obvious effect on the dynamic responses of vehicle and bridge. Particularly, the leeward vehicle will experience a sudden wind load reduction and load again during the meeting of two trains, which have negative effects on the running safety of leeward vehicle. However, there is few study on dynamic responses and running safety of the suspended monorail (Bao et al., 2016) WVB system (Cai and Chen, 2004; Chen and Cai, 2004) during two trains passing each other; the increase in the responses of the suspended monorail vehicle and the bridge may be a critical factor for the safety of the suspended monorail WVB system.

In order to study the dynamic responses of the bridge and the running safety of the vehicle during the meeting of two trains, an analytical model of WVB coupled vibration system is presented in the time domain. A suitable bridge model of double-line 30-m-span simple beam is established and selected as a numerical example. Both the static wind load and the buffeting force on the bridge and the moving vehicle are generated in the time domain using the aerodynamic coefficients and spectral representation method. The suspended monorail vehicle has a new-type structure, which is totally different from the existing vehicle model. The multi-body software SIMPACK is a commercial software, and it can simulate the dynamic vehicle structure conveniently. Thus, the dynamic responses of the vehicle and the bridge for suspended monorail transit system are calculated by SIMPACK. The influences of stochastic winds and wind speed are exploded in detail.

Modeling of WVB system

Dynamic model of bridge

A double-line 30-m-span simple beam for suspended monorail transit under construction is adopted in this study, which is a steel structure consisting of track beams and piers. The bridge structure is modeled as a three-dimensional system using finite element method. The equation of motion for the whole bridge can be expressed as

\begin{matrix} [M_{b}] {{\overset{\cdot\cdot}{u}}_{b}} + [C_{b}] {{\overset{\cdot}{u}}_{b}} + [K_{b}] {u_{b}} \\ = {P_{v}^{b}} + {P_{st}^{b}} + {P_{bu}^{b}} + {P_{se}^{b}} \end{matrix}

(1)

where ${u_{b}}$ , ${{\overset{\cdot}{u}}_{b}}$ , and ${{\overset{\cdot\cdot}{u}}_{b}}$ are, respectively, the displacement, velocity, and acceleration vectors of the bridge; $[M_{b}]$ , $[C_{b}]$ , and $[K_{b}]$ are, respectively, the mass, damping, and stiffness of the bridge; ${P_{st}^{b}}$ , ${P_{bu}^{b}}$ , and ${P_{se}^{b}}$ are, respectively, the static wind load, buffeting wind load, and self-excited aeroelastic load acting on the bridge; and ${P_{v}^{b}}$ is the wheel–rail interaction which is exerted on the running surface by vehicles through wheel–rail interactions.

Different from railway vehicle, the suspended monorail vehicle and bridge are coupled through contact relation between the rubber and the bridge. Referring to the tires used in the road vehicle system, Pacejka’s method is adopted to model the tire. The vertical contact force between tire and the bridge is obtained as

F_{Z} = {\begin{matrix} c_{z} (R_{0} - R_{H}) - F_{z N} - d_{z} {[v_{B_{k}, B_{l}}]}_{z} & for & R_{H} \leq R_{0} \\ 0 & for & R_{H} > R_{0} \end{matrix}

(2)

where $c_{z}$ and $d_{z}$ are the vertical damper stiffness and vertical damping of the tire, respectively; $R_{0}$ and $R_{H}$ are the radius of the unload tire and the current height of the center of the tire above the road, respectively; $F_{zN}$ is the nominal vertical force; and $[v_{B_{k}, B_{l}}]_{z}$ is the vertical component of translation speed of the center point of the tire relative to the road.

The finite element model of the double-line bridge is established using the software ANSYS, and beam element is used. The finite element bridge model is shown in Figure 1. After the analysis of natural vibration characteristics, the fundamental frequencies of the bridge of transversal and vertical bending are 3.430 and 4.689 Hz, respectively.

Figure 1.

Finite element model of double-line bridge structure.

Dynamic model of train

A suspended monorail vehicle is composed of a vehicle body, bogies, wheel-sets, and suspension system, which is different from traditional railway vehicle. Each bogie consists of a frame, four walking tires, and four guiding tires. The walking tires carry the load of the vehicle and drive inside the bridge, and the guiding tires travel along the guiding track and provide lateral guidance; the schematic diagram of a suspended monorail is shown in Figure 2. To better understand the structure of monorail vehicles, a topology graph of the dynamic vehicle model is applied to express the relationship and relative movement between each vehicle component. As shown in Figure 3, the vehicle body, bogies, wheel-sets, and electric motors are regarded as rigid components with no elasticity. The suspension system is modeled as force element, such as primary spring, tire force, and guiding force. Also, the relationship and relative movement between each component is clearly identified.

Figure 2.

Mass–spring–damper model of vehicle.

Figure 3.

Topological structure of vehicle.

A four-axle suspended monorail vehicle can be modeled into a mass–spring–damper system, and the degree of freedom (DOF) along the longitudinal direction is assumed to be neglected. Each vehicle body or bogie has 5 DOFs: vertical displacement $Z$ , lateral displacement $Y$ , roll displacement $R_{x}$ , yaw displacement $R_{z}$ , and pitch displacement $R_{y}$ . The DOFs of other rigid bodies such as suspension frame, center pin, and bolster have also been simplified. As a result, a vehicle with a total of 31 DOFs is shown in Table 1. The model of the vehicle–bridge system was introduced in the previous study (Bao et al., 2016).

Table 1.

Degree of freedom of suspended monorail vehicle.

Vehicle parts	Lateral	Vertical	Rolling	Yawing	Pitching
Vehicle body	Y_c	Z_c	φ_c	ψ_c	θ_c
Bogie (i = 1, 2)	Y_ti	Z_ti	φ_ti	ψ_ti	θ_ti
Walking wheel (i = 1–4)	–	–	–	–	θ_zi
Guide wheel (i = 1–8)	–	–	–	Ψ_di	–
Suspension frame (i = 1, 2)	–	–	Φ_yi	–	–
Center pin (i = 1, 2)	–	–	Ψ_si	–	–

Wind loads of vehicle and bridge

Since the width of the suspended monorail vehicle is relatively small and the cross-section is blunt, aeroelastic coupling effects between vehicles and wind are weak, and the self-excited forces on the vehicle are ignored. In this study, static wind loads and buffeting loads are calculated and considered in the vehicle–bridge system in SIMPACK. The wind forces were applied to the vehicle–bridge system with force element and moved markers. Finally, the WVB coupling vibration system for the suspended monorail system is established, and the model of WVB system is shown in Figure 4.

Figure 4.

Wind–vehicle–bridge (WVB) coupling model.

Static wind loads

Static wind loads are determined by the mean part of wind flow and the static aerodynamic coefficients. The expressions of static wind loads acting on the vehicle and bridge deck are similar; the formulations of the drag force, the lift force, and the moment are shown, respectively, as follows,

F_{H} = \frac{1}{2} ρ U^{2} C_{H} (α_{0}) D

(3)

F_{V} = \frac{1}{2} ρ U^{2} C_{V} (α_{0}) B

(4)

F_{M} = \frac{1}{2} ρ U^{2} C_{M} (α_{0}) B^{2}

(5)

where $ρ$ is the air density; $U$ is the mean wind speed; $D$ and $B$ are the height and width of the vehicle or bridge deck, respectively; and $C_{H} (α_{0})$ , $C_{V} (α_{0})$ , and $C_{M} (α_{0})$ are the drag coefficient, lift coefficient, and moment coefficient at the wind attack angle of $α_{0}$ , respectively.

Buffeting wind loads

The buffeting loads on the vehicle and bridge deck are induced by the fluctuating winds, consisting of the along-wind part and the vertical part. According to the quasi-steady theory, the buffeting loads acting on the vehicle and bridge can be expressed as follows

D_{bu} (x, t) = \frac{1}{2} ρ U^{2} B [2 \frac{A}{B} C_{D} (α_{0}) \frac{u (x, t)}{U} γ_{1} (t)]

(6)

\begin{matrix} L_{bu} (x, t) = - \frac{1}{2} ρ U^{2} B {2 C_{L} (α_{0}) \frac{u (x, t)}{U} γ_{2} (t) \\ + [{C'}_{L} (α_{0}) + \frac{A}{B} C_{D} (α_{0})] \frac{w (x, t)}{U} γ_{3} (t)} \end{matrix}

(7)

\begin{matrix} M_{bu} (x, t) = \frac{1}{2} ρ U^{2} B^{2} [2 C_{M} (α_{0}) \frac{u (x, t)}{U} γ_{4} (t) \\ + {C'}_{M} (α_{0}) \frac{w (x, t)}{U} γ_{5} (t)] \end{matrix}

(8)

where $C'_{L} (α_{0})$ and $C'_{M} (α_{0})$ are the slopes of $C_{L}$ and $C_{M}$ at the wind attack angle of $α_{0}$ , respectively; $γ_{1} (t), γ_{2} (t), \dots, γ_{5} (t)$ are the aerodynamic admittance functions, whose values are taken as 1.0 in this study; $A$ and $B$ are the height and width of the bridge or the vehicle, respectively; and $u (x, t)$ and $w (x, t)$ are the lateral and vertical fluctuation wind speed at the position of $x$ , respectively.

Aerodynamic coefficients during the meeting of two trains

When vehicles are moving on a suspended monorail bridge, the existence of vehicles under the track beam will alter the ambient flow significantly. Similarly, the aerodynamic forces on the vehicles are also affected by the track beam. Particularly, when the windward train and leeward train pass through the bridge from the opposite directions, the aerodynamic coefficients of vehicles and bridge vary greatly in the process of meeting. Furthermore, the double-line bridge structure is composed of two track beams: the windward beam and the leeward beam, thus the flow field around trains will be more complicated.

In order to investigate the variation of the aerodynamic characteristics of vehicles passing by each other, the computational fluid dynamic (CFD) method was adopted to analyze the aerodynamic coefficients, and the trains and bridge are named train A, train B, beam A, and beam B, respectively. The local mesh generation is shown in Figure 5. The aerodynamic coefficients at the attack angle of 0° are shown in Table 2.

Table 2.

Aerodynamic coefficients of bridge and vehicle.

Combination of vehicle and bridge		Drag coefficients, C_D	Lift coefficients, C_L	Moment coefficients, C_M
One train: train A	Train A	1.6875	–0.0377	0.3211
	Beam A	1.8969	–0.6474	–0.2002
	Beam B	0.3120	0.2040	–0.0083
One train: train B	Train B	1.8965	–0.2381	0.2328
	Beam A	1.2247	0.0101	–0.0294
	Beam B	2.4905	–0.2366	–0.1567
Meeting of two trains	Train A	1.9679	–0.2159	0.3711
	Train B	–0.0946	–0.1219	0.0919
	Beam A	1.9535	–0.6487	–0.2063
	Beam B	0.5673	–0.0255	–0.0068
No trains	Beam A	2.0993	0.0829	–0.0574
No trains	Beam B	0.8388	0.1877	–0.0375

Figure 5.

Train position and schematic diagram of bridge and vehicle.

The flow line and pressure nephograms of different combinations of vehicle and bridge at the attack angle of 0° are shown in Figure 6. It can be further seen that during the meeting of two trains, the wind pressure at the front of train B (leeward train) dramatically reduces to a negative value due to the shielding effect of train A (windward train), which leads to an increase in the drag coefficients of the windward vehicles, and the drag coefficients of the leeward vehicles decrease suddenly at the same time. As shown in Table 2, even the drag coefficient of train B in the cases of two trains turns to a negative value compared to one train on the same track. Thus, the local wind field of the bridge and vehicle is obviously different during the meeting of two suspended monorail trains. The real aerodynamic forces on the vehicle and bridge are determined by the local wind filed.

Figure 6.

Flow line and pressure nephogram of vehicle–bridge system: (a) no trains, (b) meeting of two trains, (c) one train: train A, and (d) one train: train B.

To verify the accuracy of the numerical analysis of CFD method, the aerodynamic forces on the vehicle and bridge are obtained by the wind tunnel test with the crossed slot system (Li et al., 2013) designed to separate the wind loads on the bridge and vehicle. The test model of bridge and vehicle are shown in Figure 7, and the geometrical scale was 1:25. The width and height of the vehicle model are 0.085 and 0.126 m, respectively. The tested and numerical drag coefficients at the attack angle of 0° are shown in Table 3.

Table 3.

Drag coefficients of bridge and vehicle.

Cases	Train A	Train B	Beam A	Beam B
Test values	1.6621	–0.0371	2.3558	0.6311
Numerical values	1.9679	–0.0946	1.9535	0.5673

Figure 7.

Wind tunnel test of aerodynamic forces.

As seen in Table 3, despite the existence of installation error and the influence of model scale ratio, the results demonstrate that the numerical drag coefficients of the vehicle and bridge (except train B) coincide well with the test values in general. That is because the influence of flow field has a significant effect on the aerodynamic forces of train B and the drag coefficient of train B is very small. It can be drawn that it is practicable to obtain the aerodynamic coefficients with the CFD method. Hence, the various numerical aerodynamic coefficients of vehicles and bridges in Table 2 can be used to express the shielding effect. Before meeting, the aerodynamic coefficients of single train were adopted. During meeting, the aerodynamic coefficients of two trains are applied. Using the method, the time-variable process of meeting of two trains can be simulated more accurately in the WVB system.

Dynamic analysis of the meeting of two trains

To investigate the running stability and safety of vehicles during the meeting of two trains, as well as the dynamic behavior of the bridge, a suitable double-line 30-m-span simple beam bridge was established as a numerical example as shown in Figure 1.

A spectral representation method is adopted here for the simulation of stochastic wind velocity field. It is assumed that the track beam is horizontal at the same elevation, and the mean wind speed and the wind spectrum do not vary along the track beam. Simu spectrum and Panofsky spectrum were adopted to simulate the horizontal and the vertical fluctuating wind velocity fields. To eliminate the effect of the loading of wind forces, a 300-m line containing 100 wind speed simulation points with an interval of 3 m is set just before and after the track beam. For track beam, the simulation points are chosen corresponding to the nodes’ location of finite element model with a total of 112 points.

A typical train formation of suspended monorail transit is adopted with a composition of M_c + M + M_c (here M_c is the abbreviation of vehicle with cab and M is the abbreviation of vehicle without cab), where the vehicle speed is 65 km/h. The track irregularities are not taken into account in this study. The damping ratio of the bridge is 0.5%. The procedure of two suspended monorail trains passing by each other in SIMPACK is shown in Figure 8.

Figure 8.

Schematic diagram of two trains passing by each other.

Effect of fluctuating wind

In order to evaluate the influence of fluctuating wind on the WVB model of suspended monorail system, the wind forces on the bridge and train are simulated with the spectral representation method introduced above. The average height of the track beam and the mean wind speed at the track level are set as 15 m and 15 m/s, respectively. The time histories of fluctuating wind on trains are spatial and temporal distribution curves, which are associated with the vehicle speed, thus the lateral wind forces on the train during the meeting of two trains are shown in Figure 9.

Figure 9.

Time histories of lateral wind forces on trains.

The two suspended monorail trains driving into the bridge at the same time in the opposite directions at the vehicle speed of 65 km/h were analyzed at the wind speed of 15 m/s, and the static wind forces and the lateral wind forces (both the static wind forces and the buffeting forces, as shown in Figure 9) are considered, respectively, to show the effects of buffeting forces. The lateral acceleration responses of middle vehicle are shown in Figure 10. It shows that in the lateral acceleration responses, two peaks appear at the beginning (26.04 s) and the end (27.15 s) due to the effect of the sudden change of wind loads when two trains pass by each other. As shown in Figure 10(a), the mean wind speed has negative contributions to the lateral responses of leeward vehicle compared with that of windward vehicle during the meeting of two trains. The lateral acceleration of windward vehicle increases significantly due to the action of fluctuating wind as shown in Figure 10(b), which shows that the lateral responses of suspended monorail vehicle are sensitive to the fluctuating wind.

Figure 10.

Time histories of lateral acceleration of middle vehicle: (a) static wind forces and (b) lateral wind forces.

Effect of mean wind speed

By considering various wind speeds, the effect of wind speed on the running safety of vehicle and dynamic responses of bridge are analyzed at the vehicle speed of 65 km/h. The buffeting loads are ignored here to eliminate the influence of fluctuating wind.

The peak values of lateral and vertical acceleration of the middle vehicle and the displacements of mid-span bridge at different mean wind speeds are shown in Table 4. It can be observed that the lateral and vertical acceleration of leeward vehicle increase greatly with the increase in the mean wind speed, while the acceleration responses of windward vehicle show a slight change, which indicates that the meeting of two trains has limited effect on the windward vehicle. Figure 11 shows the time history of lateral displacements of mid-span bridge. Also, it can be found that the lateral displacement of bridge increases significantly with the increase in the mean wind speed, but the vertical displacement almost remains the same. The static wind forces will not enlarge the vibration of vehicle, and under different wind speeds, the difference between the acceleration responses of vehicle is caused by the effects of the sudden change of static wind forces. Therefore, the mean wind speed has a great negative contribution to the acceleration of leeward vehicle during the meeting of two suspended monorail trains.

Table 4.

Peak values of dynamic responses of vehicle and mid-span bridge.

Mean wind speed (m/s)		0	10	15	20
Windward vehicle	a_l (m/s²)	0.54	0.62	0.69	0.79
Windward vehicle	a_v (m/s²)	0.60	0.62	0.75	0.96
Leeward vehicle	a_l (m/s²)	0.54	0.76	1.57	2.72
Leeward vehicle	a_v (m/s²)	0.62	0.71	1.02	1.68
Windward track beam	d_l (mm)	3.14	3.69	4.51	6.24
Windward track beam	d_v (mm)	27.27	27.42	27.53	27.76
Leeward track beam	d_l (mm)	2.97	2.83	3.20	4.23
Leeward track beam	d_v (mm)	27.25	27.29	27.39	27.32

Figure 11.

Time histories of lateral displacement responses of mid-span of bridge: (a) windward track beam and (b) leeward track beam.

Figure 12 shows the time histories of roll angle responses of the middle vehicle. It can be seen that the peak values of windward vehicle and leeward vehicle both increase greatly with the increase in the mean wind speed, and the roll angle responses show significant variation when two trains pass by each other (26.04–27.15 s). The reason is that the car body of the suspended monorail vehicle is suspended under the track beam, and the motion path of the car body is in some ways similar to the simple pendulum movement under the action of wind load jump. When the deviation value of roll angle from the equilibrium position reaches a peak, the lateral acceleration of the vehicle reaches a peak value at the same time. Correspondingly, the lateral acceleration value of the vehicle approaches zero when the roll angle is located in the equilibrium position.

Figure 12.

Time histories of roll angle responses of middle vehicle: (a) windward vehicle and (b) leeward vehicle.

Conclusion

In order to evaluate the effect of two trains meeting under wind forces for the safety of suspended monorail transit system, CFD method is used to analyze the aerodynamic characteristics of the vehicle and bridge, and it was verified through a wind tunnel test. A detailed numerical study is conducted to investigate the driving safety of the vehicle and the dynamic behavior of the bridge during the meeting of two trains. An analytical model of WVB coupled vibration for suspended monorail system is proposed, which can consider the opposite-direction driving of two suspended monorail trains. Several conclusions can be drawn as follows:

The aerodynamic characteristics of the vehicle and bridge vary widely during the meeting of two trains, and the drag coefficients of leeward vehicle change significantly at the beginning and end of the meeting. Therefore, the chosen aerodynamic coefficients of the vehicle and bridge in the WVB system should depend on the procedure of two trains meeting each other.

During the meeting of two trains, the sudden change of static wind loads has obvious impact on the lateral responses of leeward vehicle. The lateral responses of suspended monorail vehicle are sensitive to the fluctuating wind. The lateral responses of leeward vehicle are critical factors during the meeting of the two trains.

The mean wind speed has a great negative contribution to the acceleration of leeward vehicle. The acceleration of leeward vehicle and lateral displacements of bridge increase greatly with the increase in the mean wind speed.

The roll angle of vehicle is presented for describing the running safety of suspended monorail vehicles.

Footnotes

Acknowledgements

The content of this paper reflects the views of the authors, who are responsible for the facts and the accuracy of the information presented.

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: This study was financially supported by the Sichuan Science and Technology Program (no. 2017GZ0083), the National Natural Science Foundation of China (nos. 51778544 and 51525804), and the Sichuan Province Youth Science and Technology Innovation Team (no. 15CXTD0004).

ORCID iD

Huo-yue Xiang

References

Baker

(2010) The simulation of unsteady aerodynamic crosswind forces on trains. Journal of Wind Engineering and Industrial Aerodynamics 98(2): 88–99.

Baker

Quinn

Sima

et al . (2013) Full-scale measurement and analysis of train slipstreams and wakes. Part 1: ensemble averages. Institution of Mechanical Engineering 228(5): 451–467.

Bao

Ding

(2016) A case study of dynamic response analysis and safety assessment for a suspended monorail system. International Journal of Environmental Research and Public Health 13(11): 1121–1138.

Bocciolone

Chenli

Corradi

et al . (2008) Crosswind action on rail vehicles: wind tunnel experimental analyses. Journal of Wind Engineering and Industrial Aerodynamics 96(5): 584–610.

Cai

Zhu

et al . (2019) Dynamic interaction of suspended-type monorail vehicle and bridge: numerical simulation and experiment. Mechanical Systems and Signal Processing 118: 388–407.

Cai

Chen

(2004) Framework of vehicle–bridge–wind dynamic analysis. Journal of Wind Engineering and Industrial Aerodynamics 92(7–8): 579–607.

Chen

Cai

(2004) Accident assessment of vehicles on long-span bridges in windy environments. Journal of Wind Engineering and Industrial Aerodynamics 92: 991–1024.

Chenli

Ripamonti

Rocchi

et al . (2010) Aerodynamic behaviour investigation of the new EMUV250 train to cross wind. Journal of Wind Engineering and Industrial Aerodynamics 98(4–5): 189–201.

Diana

Cheli

Collina

et al . (2002) The development of a numerical model for railway vehicles comfort assessment through comparison with experimental measurements. Vehicle System Dynamics 38(3): 165–183.

10.

Gai

(2018) Simulation of train-bridge interaction under wind loads: a rigid-flexible coupling approach. International Journal of Rail Transportation 6(3): 163–182.

11.

Cai

(2015) Special issue on wind loads and wind-induced responses of vehicle-bridge systems. Wind and Structures 20(2): I–I.

12.

Qiang

Liao

et al . (2005) Dynamics of wind-rail vehicle-bridge systems. Journal of Wind Engineering and Industrial Aerodynamics 93(6): 483–507.

13.

Xiang

Wang

et al . (2013) Dynamic analysis of wind-vehicle-bridge coupling system during the meeting of two trains. Advances in Structural Engineering 16(10): 1663–1670.

14.

Meisinger

(2006) Dynamic analysis of the Dortmund University campus sky train. In: Second international conference on dynamics vibration and control, Beijing, P.R. China, 23–26 August.

15.

Tian

(2006) Study development of train aerodynamics in China. Journal of Traffic and Transportation Engineering 6(1): 1–9 (in Chinese).

16.

Wang

(2003) Development orientation of urban transit in China. Journal of Railway Engineering Society 3: 43–47 (in Chinese).

17.

Zhang

(2017) Impacts of wind shielding effects of bridge tower on railway vehicle running performance. Wind and Structures 25(1): 63–77.

18.

Xiang

Wang

et al . (2015) Numerical simulation of the protective effect of railway wind barriers under crosswinds. International Journal of Rail Transportation 3(3): 151–163.

19.

Xia

Yan

(2003) Dynamic response of suspension bridge to high wind and running train. Journal of Bridge Engineering 8(1): 46–55.

20.

Zhang

Xia

(2004) Vibration of coupled train and cable-stayed bridge systems in cross winds. Engineering Structures 26: 1389–1406.

21.

Zhao

Zhang

et al . (2012) Aerodynamic performances and vehicle dynamic response of high-speed trains passing each other. Journal of Modern Transportation 20(1): 36–43.