J. Cent. South Univ. Technol. (2009) 16: 0520-0524

DOI: 10.1007/s11771-009-0086-3

Effect of cross-wind on spatial vibration responses of train and track system

XIANG Jun(向 俊), HE Dan(赫 丹), ZENG Qing-yuan(曾庆元)

(School of Civil and Architectural Engineering, Central South University, Changsha 410075, China)

Abstract: By taking cross-wind forces acting on trains into consideration, a dynamic analysis method of the cross-wind and high-speed train and slab track system was proposed on the basis of the analysis theory of spatial vibration of high-speed train and slab track system. The corresponding computer program was written by FORTRAN language. The dynamic responses of the high-speed train and slab track under cross-wind action were calculated. Meanwhile, the effects of the cross-wind on the dynamic responses of the system were also analyzed. The results show that the cross-wind has a significant influence on the lateral and vertical displacement responses of the car body, load reduction factor and overturning factor. For example, the maximum lateral displacement responses of the car body of the first trailer with and without cross-wind forces are 32.10 and 1.60 mm, respectively. The maximum vertical displacement responses of the car body of the first trailer with and without cross-wind forces are 6.60 and 3.29 mm, respectively. The maximum wheel load reduction factors of the first trailer with and without cross-wind forces are 0.43 and 0.22, respectively. The maximum overturning factors of the first trailer with and without cross-wind forces are 0.28 and 0.08, respectively. The cross-wind affects the derailment factor and lateral Sperling factor of the moving train to a certain extent. However, the lateral and vertical displacement responses of rails with the cross-wind are almost the same as those without the cross-wind. The method presented and the corresponding computer program can be used to calculate the interaction between trains and track in cross-wind.

Key words: slab track; high-speed train; cross-wind; spatial vibration; displacement; dynamic responses

1 Introduction

Dynamic response of trains running on a railway track has been a subject of great interest of vehicle designers, maintenance engineers, as well as track designers for many years. This interest is motivated by the desire to improve ride quality, to reduce wear to vehicle and track components, to prevent vehicle hunting and, most important of all, to ensure safe operation. In particular, the aerodynamic forces on a train moving through a cross-wind may be sufficiently large to overturn the train. Many incidents caused by cross-wind have been observed in the world [1-3]. To guarantee comfort and safety for a train in cross-wind, one has to understand the effect of cross-wind on the dynamic interaction between the train and track.

However, compared with studies on vehicle/bridge interaction in cross-wind, researchers on the interaction of train and track in cross-wind are relatively few [4-5]. SUZUKI et al [6] made three kinds of wind tunnel tests to evaluate the aerodynamic characteristics of typical configurations of the vehicles that are on typical configurations of infrastructures such as bridges and embankments. CAI and CHEN [7] proposed a framework of dynamic analysis of the coupled three-dimensional vehicle-bridge system under strong wind. And the effects of driving speed on the dynamic performance of the vehicles as well as the bridge were discussed. It is found that the driving speed mainly affects the vehicle’s vertical relative response while it has an insignificant effect on the rolling response of vehicles. Vehicle’s absolute response is dominated by the bridge response when wind speed is high, while it is dominated by road roughness when the wind speed is low. XU and GUO [8] presented an evaluation of the ride comfort of road vehicles running on a long span cable-stayed bridge under cross-wind. And the ride comfort of the high-sided road vehicle under various conditions of road roughness, vehicle speed, and cross-wind speed was investigated for the vehicle running on either the ground or the cable-stayed bridge with and without cross-wind. XU and DING [5] focused on the effect of cross-wind on the dynamic responses of railway vehicles running on a tangent ballasted track. The vehicle subsystem and the track subsystem were coupled through contacts between wheels and rails based on the contact theory. The dynamic responses of the vehicle and track in cross-wind were calculated by taking track irregularities and cross-wind forces as the exciting sources.

Beijing-shanghai high-speed railway line traverses 8 large cities: Beijing, Tianjing, Cangzhou, Dezhou, Jinan, Bengbu, Nanjing and Shanghai. The maximum wind speed may exceed 30 m/s in these cities. The running safety of the high-speed train subjected to cross-wind action may suffer great challenge. The 1 196 km ballastless track with 91% of the total line will be paved in the Beijing-Shanghai high-speed line, and the main track type is the slab track.

Therefore, the analysis method of spatial vibration of cross-wind and high-speed train and slab track system was established on the basis of the analysis theory of spatial vibration of high-speed train and slab track system presented by XIANG et al [9]. And the effect of cross-wind on the dynamic performances of the wind-train-track system was studied.

2 Analysis method

The dynamic model of train and track system in cross-wind includes vehicle model, track model and wind force model. The high-speed train and slab track have been modeled by the authors in Refs.[9-10]. The wind force model was introduced as follows.

Wind forces acting on a train in cross-wind can be divided into two parts: the steady and unsteady aerodynamic forces. The steady wind forces are due to the mean wind speed component and the unsteady wind forces are caused by the fluctuating wind speed components of natural wind. Only the former was considered in this work.

The mean wind speed was assumed to be horizontal and normal to the direction of motion of the train in this work. Only wind forces acting on the car body of the train were taken into account. Wind forces acting on the car body of the train refer mainly to side force, lift force and rolling moment as shown in Fig.1.

Fig.1 Sketch of aerodynamic forces of vehicle

According to aerodynamics theory [11-13], the coefficients of the side force, lift force and rolling moment acting on the mass center of a car body can be defined as

F_S=0.5C_Sρu²hl                               (1)

F_L=0.5C_Lρu²bl                               (2)

M_R=0.5C_Rρu²b²l                              (3)

where  F_S, F_L and M_R are the side force, lift force, and rolling moment acting on the mass center of the car body, respectively; C_S, C_L and C_R are the side force, lift force, and rolling moment coefficients, respectively; ρ is the air density; u is the cross-wind speed; h is the reference height, which is normally taken as the height of the car body; b is the width of the car body; and l is the length of the car body.

And now, Y_c, Z_c and θ_c are taken as the lateral, vertical and rolling displacements of the car body of the trailer or motor car in a high-speed train, respectively. The cross-wind forces are expressed in Eqns.(1)-(3). Therefore, the potential energy of cross-wind forces, that is, the negative numerical values of work done by the cross-wind forces, of the ith vehicle in the train, can be expressed as

               (4)

If the train has m vehicles, the total potential energy of the cross-wind forces acting on the whole train can be written as

                              (5)

When Π_V denotes the potential energy of the train vibration and Π_T denotes the potential energy of the slab track vibration [9-10], the total potential energy of spatial vibration of the cross-wind and high-speed train and slab track system can be presented as

Π=Π_V+Π_T +Π_W                               (6)

According to the principle of total potential energy with stationary value in elastic system dynamics and the “set-in-right-position” rule for formulating system matrices [14], the matrix equation of the spatial vibration of the system at time t is as follows:

                (7)

where matrices [M], [C] and [K] are the global mass, damping and stiffness matrices of the system, respectively, and vectors ,  and {δ} are the acceleration, velocity and displacement vectors of the system, and {P} denotes the global load vector of the system.

Load vector {P} consists of wheel-set load, cross-wind load and vertical track irregularities. The random vertical track irregularities simulated from low disturbed track spectrum of Germany were taken as the exciting source of random vertical vibration of the system. And the artificial hunting wave of bogie frame of the high-speed train was taken as the exciting source of random lateral vibration of the system [14]. Eqn.(7) can be solved by direct time integration such as Wilson-θ method. Meanwhile, a computer program was written by FORTRAN language.

3 Results and discussion

The case study concerned a high-speed train ICE3 running over a 200 m long slab track at a constant speed of 200 km/h with cross-wind forces (u=20 m/s) and without cross-wind forces (u=0 m/s). The high-speed train concerned in this work consists of one motor car and 7 trailers. In the simulation of cross-wind forces on the high-speed train ICE3, the side force, lift force, and rolling moment coefficients of the train were taken according to Table 1 corresponding to wind speed u=20 m/s and train speed v=200 km/h [15]. The cross-wind forces acting on the train were calculated according to Eqns.(1)-(3). And then the cross-speed forces, vertical track irregularities and artificial hunting waves of bogie frame of high-speed train were input into Eqn.(7). Finally, the time histories of spatial vibration responses of high-speed train and slab track system with cross-wind forces can be computed. The comparison of the maximum values of spatial vibration of the system with and without cross-wind forces is shown in Table 2.

Table 1 Aerodynamic force coefficients of high-speed train ICE3 (u=20 m/s)

Table 2 Comparison of maximum values of vibration response with and without cross-wind forces

Fig.2(a) depicts the time histories of lateral displacement response of the car body of the first trailer with and without cross-wind forces, while Fig.2(b) displays the time histories of vertical displacement response of the car body of the first trailer with and without cross-wind forces. The first trailer selected here has the maximum displacement response among all the trailers. It is seen that both lateral and vertical displacement responses of the car body of the first trailer increase significantly due to cross-wind forces, especially in the lateral direction where the lateral displacement of the car body is very small without cross-wind forces but increases significantly under cross-wind forces. The maximum lateral displacement response of the car body of the first trailer is 32.10 mm with cross-wind forces and 1.60 mm without cross-wind forces, and the former is about 20 times as much as the latter. The maximum vertical displacement response of the car body of the first trailer is 6.60 mm with cross-wind forces and 3.29 mm without cross-wind forces, and the former is about 2 times as much as the latter.

The safety of a railway vehicle concerns mainly

Fig.2 Time histories of (a) lateral displacement and (b) vertical displacement of car body of the first trailer at its center

derailment. There are two important factors that should be considered in the evaluation of the safety of a railway vehicle. One is the derailment factor, the other factor is the wheel load reduction factor.

Fig.3 shows the time histories of derailment factor of the first wheel-set in the first trailer with and without cross-wind forces, while Fig.4 displays time histories of the left wheel load reduction factor of the first wheel-set in the first trailer with and without cross-wind forces. It is seen that cross-wind forces certainly affect the derailment factor and wheel load reduction factor. In particular, the wheel load reduction factor with cross-wind forces is much larger than that without cross-wind forces. The maximum derailment factor is 0.25 with cross-wind forces and 0.15 without cross-wind forces, and the former is 1.67 times as much as the latter. The maximum wheel load reduction factor is 0.43 with cross-wind forces and 0.22 without cross-wind forces, and the former is 1.95 times as much as the latter. It can be seen that the safety of the train in cross-wind is controlled by the wheel load reduction factor rather than by the derailment factor.

Fig.3 Time histories of derailment factor the first wheel-set in the first trailer

Fig.4 Time histories of left wheel load reduction factor of the first wheel-set in the first trailer

Fig.5 shows the time histories of overturning factor of the first trailer with and without cross-wind forces. It is seen that the overturning factor of the first trailer increases significantly due to cross-wind forces. The maximum overturning factor of the first trailer is 0.28 with cross-wind forces and 0.08 without cross-wind forces, and the former is 3.5 times as much as the latter. These results show that the safety of the train in strong cross-wind is controlled by the overturning factor.

Fig.5 Time histories of overturning factor of the first trailer

To assess the effects of cross-wind forces on the ride comfort of the high-speed train running on the slab track, the Sperling factor with cross-wind forces was compared with that without cross-wind forces. Fig.6 shows the time histories of lateral Sperling factor of the first trailer with and without cross-wind forces. It is seen that the maximum lateral Sperling factor is 3.33 with cross-wind forces and 3.16 without cross-wind forces, and the former is 1.05 times as much as the latter. These results indicate that the cross-wind forces only slightly affect the lateral Sperling factor.

To know if the cross-wind forces affect the slab track, the responses of the track vibration were also computed with the program. The displacement responses

Fig.6 Time histories of lateral Sperling factor of the first trailer

of the rail are plotted in Fig.7 for both lateral and vertical directions with and without cross-wind forces. It is seen that both the lateral and vertical displacement responses of the rail with cross-wind forces are similar to those without cross-wind forces.

Fig.7 Time histories of (a) lateral rail displacement and (b) vertical rail displacement

4 Conclusions

(1) The cross-wind and high-speed train and slab track are considered as a whole system. The potential energy of cross-wind forces acting on the train is derived. And then, by using the principle of total potential energy with stationary value in elastic system dynamics and the rule of “set-in-right-position” for formulating system matrices, the matrix equation of spatial vibration of high-speed train and slab track system in cross-wind is established on the basis of the analysis theory of spatial vibration of the high-speed train and slab track system. The corresponding computer program is also written.

(2) The dynamic responses of the high-speed train and slab track system in cross-wind are calculated corresponding to wind speed u=20 m/s and train speed v=200 km/h. The dynamic responses with cross-wind forces are compared with those without cross-wind forces. The results show that the cross-wind has a significant influence on the lateral and vertical displacement responses of the car body, wheel load reduction factor and overturning factor. And the cross-wind certainly affects the derailment factor and Sperling factor of the moving train. The results also show that the lateral and vertical displacement responses of the rail with cross-wind forces are almost the same as those of the rail without cross-wind forces.

(3) The analysis method proposed may be further used to study critical wind speeds and train speeds of high-speed train running on the track in cross-wind.

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 (Edited by CHEN Wei-ping)

Foundation item: Project (2007CB714706) supported by the Major State Basic Research and Development Program of China; Project (50678176) supported by the National Natural Science Foundation of China; Project (NCET-07-0866) supported by the New Century Excellent Talents in University

Received date: 2008-09-20; Accepted date: 2008-11-26

Corresponding author: XIANG Jun, Professor, PhD; Tel: +86-731-2656728; E-mail: jxiang@mail.csu.edu.cn