A dynamic analysis scheme for the suspended monorail vehicle

Abstract

Based on the rigid–flexible coupling method, an original scheme for the dynamic analysis of the vehicle–bridge interaction between suspended monorail trains and horizontally curved bridges is proposed. Considering the compression deformation and contact model of walking tire and guiding tire, the geometric and mechanical coupling relationships between vehicle and bridge are studied, and the dynamic equations of suspended monorail vehicle–bridge interaction are derived. A vehicle–curved bridge coupling vibration system is established according to transformable relationship between the local coordinate system and the global coordinate system in SIMPACK. Considering a curved bridge under passage of suspended monorail vehicles as an example, the influences of critical system parameters, such as the superelevation, vehicle speed, and bridge curve radius, on the dynamic responses of vehicles and the curved bridge are explored. It is shown that the direction of the yawing moment of the front bogie is in accordance with the turning direction of the vehicle, while the yawing moment of the rear bogie is in the opposite direction. The superelevation has great influence on the lateral guiding force and vertical walking force of vehicle, and vehicle speed is a key factor to the running safety of suspended monorail vehicle. In addition, the curve negotiation ability of vehicle is better with the increase in bridge curve radius.

Keywords

curved bridge dynamic response suspended monorail transit system vehicle–bridge coupling vibration

Introduction

During the passage of suspended monorail vehicles over a curved bridge, the guiding tire provides lateral guidance force to make sure that vehicles will turn along the curved track direction. Meanwhile, the rubber walking tire exerts vertical, lateral, and longitudinal tire forces on the curved bridge, which is completely different from existing familiar transit system. Hence, it is necessary to figure out the interaction forces between vehicles and curved bridges, especially the lateral guiding force and walking force. Compared with traditional railways, the curved radius of suspended monorail line is much smaller, sometimes even can be less than 100 m, a dynamic analysis scheme should be proposed to investigate the driving safety of vehicles on the curved bridges.

Most studies on vehicle–bridge dynamic systems concentrate on straight bridges, still some research has been undertaken on the horizontally curved bridges. Li and Ren (2018) and Dimitrakopoulos and Zeng (2015) simplified and assumed the railway vehicles as a moving three-directional loads to investigate the vehicle-induced dynamic responses of curved bridges. Considering the suspension properties of vehicles, Mermertas (1998) established a four degrees of freedom (4-DOFs) vehicle model and studied the dynamic responses of a simply supported curved beam with the Newmark method. Senthilvasan et al. (2002) carried out a full-scale testing of truck driving on a curved concrete box girder bridge at different speeds, and the results of bridges in the experiment were compared with the analytical results to calibrate the analytical bridge–vehicle interaction model. With the growing computer technology and computational methods, research on the vehicle-induced vibrations of curved bridges became more precise. Xia et al. (2006, 2008) studied the lateral vibration responses of curved railway bridges with a coupled train–girder–pier dynamic system, in which the vehicle was modeled with 27-DOFs for a four-axle train, also the analysis results were verified with field experiments. Huang et al. (1998), Huang (2001) investigated the impact factors of curved box girder bridges under the truck loads and the road surface roughness was taken into account in the dynamic analysis. Yang et al. (2001, 2004) and Zeng et al. (2016) established a vehicle–bridge interaction (VBI) system to examine the dynamic responses of vehicles and curved bridges, and the resonance analysis was conducted. Results showed that the feedback effect of the curved bridge’s resonance to the vehicle response was noticeable. Cai et al. (2019) and He et al. (2019) investigated the dynamic responses of vehicle–bridge system, with emphasis on the suspended monorail vehicle, an experimental test was conducted to verify the numerical results, while the curve superelevation was not taken into account in this study. To the best knowledge of the authors, most of this research on curved bridges focuses on the railway or highway bridge (Li et al., 2015; Xu et al., 2003; Zhai et al., 2019). Research works on dynamic responses of suspended monorail curved bridges under the action of vehicles have rarely been studied. Moreover, vehicles are usually simplified as moving load, and the interaction between vehicles and bridges are ignored (Kim et al., 2005).

The purpose of this study is to propose a dynamic analysis scheme on the vehicle-induced vibration of curved bridges for suspended monorail transit system. First, the critical parameters of curve line are designed, such as the transition curve length and curve superelevation, and a curved bridge model is established with the finite element method. Then, the geometric and mechanical coupling relationships between vehicle and bridge are studied by taking into account the compression deformation and contact model of walking tire and guiding tire, and the dynamic equations of suspended monorail VBI are derived. Next, according to transformable relationship between the local coordinate system and the global coordinate system, a detailed vehicle–curved bridge vibration system is established with the rigid–flexible coupling method. Finally, through a case study, the influences of superelevation, vehicle speed, and bridge curve radius on the dynamic characteristics of three-directional vehicle–curved bridge system are analyzed.

Case study: curved bridge structure

To investigate the dynamic responses of vehicles and the bridge when suspended monorail vehicles pass through the curved bridge, a six 25-m-span steel curved bridge of six spans is established, whose horizontal radius is 300 m. The curved bridge is idealized as a simply supported system, where the girder is a steel box beam with the bottom opening. The cross section of the steel box is 0.78 m wide and 1.256 m high, the pier height is 15 m, and the damping ratio of the bridge is 0.5%. The finite element model (FEM) of the curved bridge is established in ANSYS with the beam element, as shown in Figure 1.

Figure 1.

Finite element model of curved bridge structure.

Curve superelevation and limited speed

When suspended monorail vehicles pass through the curved bridge, vehicles and passengers will be under the action of centrifugal force, which has a negative influence on the ride safety and passenger comfort. The curve superelevation is usually set in curved line to balance the centrifugal force in railway systems. Referring to the straddle-type monorail line, the invariant line center height method is used to set the superelevation, namely, the inside part of the curved track lowers half of the superelevation and the outside part raises half of the superelevation. The curve superelevation can be defined by vehicle speed and curve radius, as follows

h = S \frac{v^{2}}{Rg}

(1)

i = \tan α = \frac{h}{S} \times 100 %

(2)

where $h$ is the superelevation of the curve line; S is the lateral distance between the center of walking tires; $v$ is the vehicle speed (m/s); $R$ is the curve radius (m); $g$ is the acceleration of gravity, 9.8 m/s²; $i$ is the superelevation ratio of the track.

When vehicles move through the curved bridge at a constant speed, the horizontal component of the gravity of vehicles and passengers equals to the centrifugal force, the superelevation turns into a balanced superelevation and the speed is called balanced speed. If vehicle speed exceeds the balanced speed, the deficient superelevation leads to irregular abrasions between inside and outside walking tires, thus the maximum vehicle speed passing through the curved bridge should be limited as

v_{max} = 3.6 \times \sqrt{(i + i_{q max}) Rg}

(3)

where $v_{max}$ denotes the limited speed (km/h); $i, i_{q max}$ denote the actual superelevation ratio and the allowable deficient superelevation ratio, respectively.

According to equations (1) to (3), the relationships between radius, curve superelevation, and limited speed are shown in Table 1.

Table 1.

Curve superelevation and limited speed.

Radius	Curve superelevation (mm)	Superelevation rates (%)	Limited speed (km/h)	Operating speed (km/h)
R400	45.0	10	87.3	65
R300	49.5	11	78.1	65
R200	54.0	12	65.7	65

Design of curve line

An actual operation curve line usually consists of the transition curve and the circular curve. The turning function of the suspended monorail vehicle is realized through the interaction between guiding tires and the guide rail installed on the side wall of the curved track beam. To realize the driving of suspended monorail vehicles along the curve line and avoid the needless vehicles vibration induced by the sudden change of line types between straight line and circular curve, line type of cubic parabola is employed to simulate the transition curve. The length of transition curve is determined by the curve radius, curve limited speed, and engineering condition. In this study, the length configurations of curve lines are shown in Table 2.

Table 2.

Length configurations of curved line (unit: m).

Radius	Straight line	Transition curve	Circle curve	Transition curve	Straight line	Total line
R400	100	60	150	60	100	460
R300	100	60	150	60	100	460
R200	100	60	150	60	100	460

A rigid track FEM of straight line track and transition curve track is established at the front and the end of the curved bridge model, as shown in Figure 2. The dynamic responses of the curved bridge are the main research objectives, and the rigid track is only established to ensure vehicles turn along the curve line smoothly. After the analysis of vibration characteristics, the nature fundamental frequencies of the curved bridge of transversal and vertical bending are 4.52 and 5.96 Hz, respectively.

Figure 2.

Finite element model of the curve line.

Dynamic analysis of the vehicle–curved bridge interaction

The dynamic system consists of the bridge subsystem and the vehicle subsystem. The two subsystems are coupled through the interactions between vehicles and bridges. This study simulates the curved bridge with FEM and the dynamic vehicles as multi-rigidity-body systems. First, the substructure analysis should be conducted in ANSYS to get the essential information files of bridge (*_struct.sub flie, *_eigen.rst file, *_cad.cdb file). Then, the standard input data (SID) files are generated through the interface program FEMBS in SIMPACK, which realizes the transform procedure of the FE modeling of the bridge to the flexible body modeling. To realize the co-simulation of vehicle and bridge coupling system, the rigid–flexible coupling method is used to unite the vehicle subsystem and bridge subsystem, each walking wheel should link to the moved marker of the bridge, and the guiding wheel are contacted to the guiding rail through the spring–damper force element. So far, the proposed vehicle–curved bridge dynamic system is realized with ANSYS and SIMPACK as shown in Figure 3.

Figure 3.

Procedure of the interaction for suspended monorail vehicle–bridge coupling system.

Vehicle modeling

The suspended monorail vehicle is modeled as a multibody system, which consists of one car body and two bogies, which are all assumed as rigid bodies (Bao et al., 2019). Each bogie is comprised of four walking tires and four guiding tires, the walking tires located on the walking surface carry the vertical load of the whole vehicle and passengers. Besides, the walking tires also provide lateral and longitudinal forces according to the cornering and sliding characteristics. The guiding tires provide the lateral guide force through contacting with the guiding rails installed on the side wall of the bridge. Figure 4 presents a typical three-dimensional dynamic vehicle model with 34-DOFs established in this study. The car body and two bogies are assigned 6-DOFs each: the longitudinal displacement, the vertical displacement, the lateral displacement, the rolling rotation, the pitching rotation, and yawing rotation. In addition, the other parts such as walking tires and guiding tires have its own certain DOFs.

Figure 4.

Dynamic model of suspended monorail vehicle.

VBI modeling

The contact model of suspended monorail vehicle is different from railway vehicle, rubber tires are employed to drive and guide the vehicle. Referring to the rubber tires used in the road vehicle system, the similarity method of Pacejka is adopted to model the walking tire, the lateral and longitudinal slip forces and the lateral force of the guiding tire are also taken into consideration in this work.

Vertical force of the walking tire

With the measurement of the relative movement between the tire and road, the vertical force of the tire is obtained as

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

(4)

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

Lateral and longitudinal forces of the walking tire

With the similarity method, the lateral and longitudinal forces of the tire can be calculated through the existing creep and frictional coefficients as the following equations

F_{y} = - \frac{σ_{y}}{σ_{theo}} \frac{F_{z}}{F_{z_{0}}} \frac{μ_{y}}{μ_{y_{0}}} F_{0} (σ_{e q_{y}})

(5)

F_{x} = - \frac{σ_{x}}{σ_{theo}} \frac{F_{z}}{F_{z_{0}}} \frac{μ_{x}}{μ_{y_{0}}} F_{0} (σ_{e q_{x}})

(6)

where $σ_{x}$ and $σ_{y}$ represent the longitudinal slip and lateral slip; $σ_{theo} = \sqrt{σ_{x}^{2} + σ_{y}^{2}}$ is the theoretical overall slip at the tire; $F_{z_{0}}$ , $μ_{y_{0}}$ are the vertical load and frictional constant for the graph of lateral force against the lateral slip; $μ_{x}$ , $μ_{y}$ are the current frictional coefficients for the longitudinal and lateral force calculation; $σ_{e q_{x}}$ and $σ_{e q_{y}}$ represent the differing frictional behavior of the tire in the X-axle and Y-axle direction, respectively; $F_{0} (σ_{e q_{x}})$ and $F_{0} (σ_{e q_{y}})$ represent the initial characteristic of the tire under the nominal conditions in the longitudinal and lateral directions.

Lateral force of the guiding tire

The guiding tire provides the lateral guiding force to help vehicles pass the curved bridge smoothly, which also plays an important role in holding the stability of bogies. The linear spring–damping element is adopted to model the guiding force between the guiding tire and the guiding rail, the lateral contact force can be expressed as

F_{D Y} = {\begin{matrix} K_{y} (Y_{r} - Y_{0}) - C_{y} {[v_{D_{k}, D_{l}}]}_{y} + F_{D y 0} & f o r & Δ r > 0 \\ 0 & f o r & Δ r \leq 0 \end{matrix}

(7)

where $Δ r$ is the amount of radial compression, $K_{y}$ and $C_{y}$ are the lateral stiffness and damping coefficient of guiding tire, respectively, $Y_{r}$ and $Y_{0}$ are the radial compression of guiding tire at the initial and running moment, respectively, $F_{D y 0}$ is the initial guiding force of tire, $[v_{D_{k}, D_{l}}]_{y}$ is the lateral component of translation speed of the tire center point relative to the guiding rail.

Curve coordinate transformation

In the study of vehicles moving over the curved bridge, there are two coordinate systems to express the motions and responses of the vehicle–bridge system in SIMPACK: an inertia system O-XYZ and a body-fixed system O_r-X_rY_rZ_r. The dynamic responses of vehicle and bridge obtained from the post processor in SIMPACK are based on the inertia system, to get the tangential (longitudinal) and normal (lateral) dynamic responses of vehicles and the bridge in the local coordinate system fixed on the trajectory of vehicles, formulas should be established to transfer the responses from the inertia system to the body-fixed system.

Lateral responses of the bridge

Assumed that the FEM of curved bridge is established according to the inertia coordinate system in SIMPACK, more specially, the direction to the right is X+ direction, the upward direction is Y+ direction and the direction toward the front is the Z+ direction. The tangent of each node on the curved bridge has a constant angle γ relative to X-axis, and clockwise direction around the Z-axis is positive. Then, the lateral responses of the bridge node can be obtained according to the equation

[\begin{matrix} X_{b} & Y_{b} \end{matrix}] = [\begin{matrix} X & Y \end{matrix}] [\begin{matrix} \cos γ & \sin γ \\ - \sin γ & \cos γ \end{matrix}]

(8)

where $X$ and $Y$ are the responses based on the X-axis and Y-axis in the inertia coordinate system, $X = [\begin{matrix} x_{1} & x_{2} & \dots & x_{n} \end{matrix}]^{T}$ , $Y = [\begin{matrix} y_{1} & y_{2} & \dots & y_{n} \end{matrix}]^{T}$ ; $X_{b}$ and $Y_{b}$ are the tangential and normal responses of the bridge, respectively.

Lateral responses of the vehicle

The body-fixed system follows the longitudinal and tangential responses to the trajectory of vehicles, with its origin fixed to the center mass of the car body. The motion of a car body in the body-fixed system is described with a time-dependent coordinate relative to the inertia system: the yawing rotation about the Z-axis. The lateral responses of the vehicle can be expressed as

[\begin{matrix} X_{v} & Y_{v} \end{matrix}] = [\begin{matrix} X & Y \end{matrix}] [\begin{matrix} \cos ϒ & \sin ϒ \\ - \sin ϒ & \cos ϒ \end{matrix}]

(9)

where $X$ and $Y$ are the vehicle responses in the X-axis and Y-axis in the inertia coordinate system, $X = [\begin{matrix} x_{1} & x_{2} & \dots & x_{t} \end{matrix}]$ , $Y = [\begin{matrix} y_{1} & y_{2} & \dots & y_{t} \end{matrix}]$ ; $X_{v}$ and $Y_{v}$ are the longitudinal and tangential responses of the vehicle; $ϒ$ is the rotation angle around the Z-axis, $ϒ = [\begin{matrix} γ_{1} & γ_{2} & \dots & γ_{t} \end{matrix}]$ .

Parametric analysis and results

To investigate the running stability and safety of vehicles during suspended monorail vehicles passing through the curved bridge, as well as dynamic behaviors of the curved bridge, a case with 470 m (100 m + 60 m + 150 m + 60 m + 100 m) length track route where a six 25-m-span simple beam curved bridge located on the circular curve established in section “Case study: curved bridge structure” is used as a numerical example as shown in Figure 2.

In this section, a typical train formation of a composition of (M_c + M + M + M_c; M_c is the abbreviation of vehicle with cab; M is the abbreviation of vehicle without cab) is adopted. The track irregularities are not taken into account in this study. The procedure of suspended monorail vehicles passing through the curved bridge in SIMPACK is shown in Figure 5.

Figure 5.

Schematic of vehicles passing through a curved bridge.

Effect of superelevation

Curve superelevation is usually setup in the curved line to balance the centrifugal force of vehicles during passing through the curved bridge. To investigate the influences of curve superelevation on dynamic responses of vehicles and the bridge, various curve superelevations (0, 25, 49.5, 75 mm) are set in the study. The vehicle speed is 65 km/h, and the pre-guiding force of the guiding tire is set as 5000 N. As we can see in Table 1, the balanced superelevation is 49.5 mm at the speed.

Table 3 shows the dynamic responses of the curved bridge at midspan under different superelevation conditions, it shows that the effect of superelevation on the dynamic responses of the bridge is not obvious. Figure 6 presents the acceleration time histories of the front vehicle under different superelevation conditions. As shown in Figure 6, the curve superelevation has little influences on the lateral or vertical accelerations of vehicles, when vehicles move over the curved bridge, the lateral accelerations of vehicles are much larger than the vertical accelerations.

Table 3.

Dynamic responses of curved bridge at middle-span.

Superelevation (mm)	Displacement of bridge (mm)		Acceleration of bridge (m/s²)		Torsion angle (rad/1000)
Superelevation (mm)	Vertical	Lateral	Vertical	Lateral	Torsion angle (rad/1000)
0	14.71	3.15	1.13	0.45	6.34
25	14.71	3.12	0.99	0.50	6.72
49.5	14.76	3.11	1.03	0.52	6.51
75	14.81	3.06	1.09	0.54	6.59

Figure 6.

The lateral and vertical accelerations of vehicles.

Figures 7 and 8 show the lateral force of guiding tires on the front and rear bogie, respectively, when vehicles pass through the curved bridge (9.4–17.7 s). As shown in Figure 7, when the actual superelevation (H) equals the balanced superelevation, the centrifugal force would be greatly offset by the horizontal component of the load of vehicles, thus the guiding force of inside guiding tires is similar to that of outside guiding tires. In addition, the variation of guiding force compared with the pre-guiding force is smallest.

Figure 7.

The lateral forces of guiding tires (front bogie): (a) H = 0 mm, (b) H = 25 mm, (c) H = 49.5 mm, and (d) H = 75 mm.

Figure 8.

The lateral forces of guiding tires (rear bogie): (a) H = 0 mm and (b) H = 49.5 mm.

When the actual superelevation (H) is less than the balanced superelevation, with an increasing deficient superelevation, the force of outside guiding tires increases accordingly to resist the extraversion movement of the car body, which reaches the maxima when H is zero. Meanwhile, the force of inside guiding tires decreases accordingly, and it even can reduce to negative values, which means inside guiding tires may run off guiding rails at some certain conditions. When the actual superelevation (H) is larger than the balanced superelevation, the inside guiding force increases with the increasing surplus superelevation to resist the introversion movement of car body. It can also be seen that the difference between outside and inside guiding tires increases with the increasing deficient superelevation or surplus superelevation, leading to an aggravation abrasion of guiding tires.

Comparing the guiding force of the front bogie with the rear bogie (Figure 8), it can be concluded that the lateral forces of front left and rear right guiding tires (Dwheel_F_L and Dwheel_R_R) are larger than that of front right and rear left guiding tires (Dwheel_F_R and Dwheel_R_L) for the front bogie. It is because the lateral forces generate a yawing torque in the same direction as vehicles move along the curved bridge. While, the rear bogie generates a yawing torque in the opposite direction to the front bogie, as shown in Figure 9, these two yawing torques guarantee a favorable curve negotiation performance of vehicles.

Figure 9.

The force state of bogies passing by curved bridge.

Figure 10 shows the time histories of vertical forces of walking tires. Obviously, there is a difference between the vertical forces of left and right walking tires because of the action of centrifugal force, the vertical forces of left (outside) tires are larger than that of right (inside) tires. The difference between left and right walking tires increases with the increasing superelevation, as well as the extremum of vertical force of walking tires, which is disadvantageous to the driving safety of vehicles.

Figure 10.

The vertical forces of walking tires: (a) H = 0 mm, (b) H = 25 mm, (c) H = 49.5 mm, and (d) H = 75 mm.

Effect of vehicle speed

When suspended monorail vehicles pass through the curved bridge at a certain speed, different speeds have a corresponding balanced superelevation. Assuming that the actual superelevation of curve is 49.5 mm (H = 49.5 mm), the pre-guiding force is 5000 N, and the vehicle speed ranges from 45 to 70 km/h at an interval of 5 km/h. The relationship between vehicle speed and surplus superelevation can be calculated according to equation (1), as shown in Table 4.

Table 4.

Surplus superelevations at different vehicle speeds.

Vehicle speed (km/h)	40	45	50	55	60	65	70
Balanced superelevation (mm)	18.9	23.9	29.5	35.7	42.5	49.9	57.9
Surplus superelevation (mm)	31.0	26.0	20.4	14.2	7.4	0.4	−8.0

Figures 11 and 12 show the dynamic responses of the bridge and the vehicle at different speeds. As we can see, there is no obvious law between the vehicle speed and dynamic responses of the bridge. The vertical acceleration of vehicle and the lateral displacement of track beam increase significantly with the increase of the vehicle speed. As for the roll angle of vehicle, it has the same variable law with the surplus superelevations, the tilt direction of car body switches from introversion to extraversion when vehicles pass through the curved bridge with the increase of vehicle speed, which shows the increase of the centrifugal force.

Figure 11.

The displacement and acceleration of the bridge.

Figure 12.

The vibration acceleration and the roll angle of the vehicle.

Effect of bridge curve radius

The curved radius is an important factor to the running safety of suspended monorail vehicles and dynamic responses of curved bridge. The bridge curve radius of R = 200, R = 300, and R = 400 are taken into account in this study, and the corresponding analysis model of vehicle–bridge system are built respectively. The actual curve superelevations are set as shown in Table 1, the pre-guiding force is 2500 N, and the vehicle speed is 65 km/h.

Table 5 shows the dynamic responses under different curve radius conditions, the dynamic vertical and lateral displacements of the bridge decrease with the increase in radius, especially for the torsion angle of the midspan. It can be drawn that the lateral responses of the bridge are closely associated with the curve radius.

Table 5.

Dynamic responses of curved bridge at middle-span of different radii.

Radius (m)	Displacement of bridge (mm)		Acceleration of bridge (m/s²)		Torsion angle (rad/1000)
Radius (m)	Vertical	Lateral	Vertical	Lateral	Torsion angle (rad/1000)
R = 200	15.58	3.99	1.17	0.62	8.43
R = 300	14.76	3.06	1.02	0.52	6.85
R = 400	14.20	2.66	1.13	0.51	4.78

Figure 13 shows the vibration accelerations of vehicles. As it can be seen, the vertical stiffness of curved bridge under the three radii are basically the same, thus the vertical accelerations of the vehicle are consistent with each other. The lateral vibration accelerations of vehicle decrease significantly with the increase of the curve radius, it is because that when vehicles pass through the bridge with the car body suspended under the track beam, the new vehicle structure is sensitive to the centrifugal force caused by the radius. The lateral force time histories of front right guiding tire (Dwheel_F_R) and rear right guiding tire (Dwheel_R_R) are shown in Figure 14, and the sudden changes in the curve are caused by the tolerance of curve line. We can draw conclusion that the guiding force time history of front right tire has a negative value in the condition with the pre-guiding force of 2500 N, which indicates that the guiding tire runs off the guiding rail. Thus, the inside guiding tires are more likely to appear to derail when the curve radius is smaller, it is suggested to increase the pre-guiding force appropriately.

Figure 13.

The lateral and vertical accelerations of vehicles of different radii.

Figure 14.

The lateral guiding forces of guiding tires: (a) Dwheel_F_R and (b) Dwheel_R_R.

Conclusion

To investigate the impact of suspended monorail VBI on the dynamic responses of vehicles and curved bridges, this article proposes an analysis scheme with the rigid–flexible coupling method. To this end, dynamic equations of suspended monorail VBI are derived according to the geometric and mechanical coupling relationships between vehicle and bridge, and a detailed vehicle–curved bridge coupling vibration system is established. Some interesting conclusions can be obtained through this study, including:

When suspended monorail vehicle passes through the curved bridge, it is shown that the direction of the yawing moment of the front bogie is in accordance with the turning direction of the vehicle, while the yawing moment of the rear bogie is in the opposite direction, which provides a favorable curve negotiation performance with the vehicle.

The curve superelevation has little influences on the lateral and vertical accelerations of the vehicle, while it has great influence on the guiding force and walking force of the vehicle. The guiding force of inside guiding tires is similar to that of outside guiding tires, and the variation of guiding force compared with the pre-guiding force is smallest when the actual superelevation is equal to the balanced superelevation.

The difference between left and right walking tires increases with the increasing superelevation, as well as the extremum of vertical force of walking tires, which is disadvantageous to the driving safety of vehicles.

There is no obvious law between the vehicle speed and dynamic responses of the bridge. With the increase of vehicle speed, the vertical acceleration of vehicle and the lateral displacement of track beam increase significantly and the tilt direction of car body switches from introversion to extraversion.

The lateral responses of the bridge are closely associated with the curve radius. When the curve radius is small, the inside guiding tires is more likely to derail, it is suggested to increase the pre-guiding force appropriately.

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: The authors are grateful for the financial supports from the National Natural Science Foundation of China (51778544 and 51525804).

ORCID iD

Huoyue Xiang

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, et al. (2019) Train–track–bridge dynamic interaction: a state-of the-art review. Vehicle System Dynamics 57(7): 984–1027.

A dynamic analysis scheme for the suspended monorail vehicle–curved bridge coupling system

Abstract

Keywords

Introduction

Case study: curved bridge structure

Curve superelevation and limited speed

Design of curve line

Dynamic analysis of the vehicle–curved bridge interaction

Vehicle modeling

VBI modeling

Vertical force of the walking tire

Lateral and longitudinal forces of the walking tire

Lateral force of the guiding tire

Curve coordinate transformation

Lateral responses of the bridge

Lateral responses of the vehicle

Parametric analysis and results

Effect of superelevation

Effect of vehicle speed

Effect of bridge curve radius

Conclusion

Footnotes

Acknowledgements

Declaration of Conflicting Interests

Funding

ORCID iD

References