J. Cent. South Univ. Technol. (2008) 15: 121-126
DOI: 10.1007/s11771-008-0024-9
Analysis theory of spatial vibration of high-speed train and slab 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：The motor and trailer cars of a high-speed train were modeled as a multi-rigid body system with two suspensions. According to structural characteristic of a slab track, a new spatial vibration model of track segment element of the slab track was put forward. The spatial vibration equation set of the high-speed train and slab track system was then established on the basis of 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 equation set was solved by the Wilson-θ direct integration method. The contents mentioned above constitute the analysis theory of spatial vibration of high-speed train and slab track system. The theory was then verified by the high-speed running experiment carried out on the slab track in the Qinghuangdao-Shenyang passenger transport line. The results show that the calculated results agree well with the measured results, such as the calculated lateral and vertical rail displacements are 0.82 mm and 0.9 mm and the measured ones 0.75 mm and 0.93 mm, respectively; the calculated lateral and vertical wheel-rail forces are 8.9 kN and 102.3 kN and the measured ones 8.6 kN and 80.2 kN, respectively. The interpolation method, that is, the lateral finite strip and slab segment element, for slab deformation proposed is of simplification and applicability compared with the traditional plate element method. All of these demonstrate the reliability of the theory proposed.
Key words: slab track; track segment element; high-speed train; spatial vibration
1 Introduction
With wide development of railway construction of passenger transport line in China, it is urgent to understand the dynamic performance of ballastless track and high-speed train when it runs on the ballastless track[1]. So spatial vibration analysis for the high-speed train and ballastless track system is demanded. For a long time, many researchers studied the ballastless track by static method. But the related dynamics aspect researches on the ballastless track were few. ZHAI et al[2] studied the dynamic performance of the slab track by a dynamic model of vertical interaction between vehicle and slab track. CAI et al[3] simplified the slab as an elastic thin plate and established a dynamic coupling model of vehicle and slab track by general plate elements. These elements are with many degrees of freedom and complex in calculation. ZHANG[4] established three dynamic models of ballastless track, such as long sleeper buried type track, elastically bearing block type track and slab track by superposition beam model, and then analyzed the interaction between high-speed vehicle and ballastless track. Researchers in other countries chiefly studied vertically coupling dynamics of vehicle and ballastless track system[5-8]. But related research on spatial vibration of high-speed train and ballastless track system has not been reported.
In this work, taking example for “Xianfeng” high-speed train in China and slab track in Japan, the analysis theory of spatial vibration of the high-speed train and slab track system was proposed and then verified by the high-speed running experiment carried out on the slab track in the Qinghuangdao-Shenyang passenger transport line.
2 Analysis model of spatial vibration of high-speed train
Each car of the high-speed train is modeled as a multi-rigid body system with two suspensions. The car body and bogie of a car have 6 degrees of freedom (DOFs) including longitudinal, lateral, vertical, rolling, pitching and yawing vibrations. For each wheel set, lateral and vertical vibrations are considered. This results in a total of 26 DOFs for each car. When the displacement modes for each car are confirmed, the potential energy of vibration of each car Пvi can be derived, in which i denotes the car number of the train[9-13].
3 Analysis model of spatial vibration of slab track structure
3.1 Track segment element of spatial vibration of slab track
According to structural characteristic of the slab track in ballastless track types, a new spatial vibration model of track segment element with 32 DOFs of the slab track was established as shown in Fig.1.
A track segment element is taken between two adjacent fasteners. In fact, it is a short track. For each element, the rail is regarded as an Euler beam supported by discrete viscoelastic supports, and the nodal displacement parameter takes in the rail both sides. The fastener is replaced by a linear spring and damp, and Kul represents the lateral spring factor, Cul the lateral damping factor, Kuv the vertical spring factor, Cuv the vertical damping factor. The slab is regarded as an elastic thin plate and the nodal displacement parameter takes in the plate element four vertex. The slab is connected with the concrete base by the cement asphalt mortar (CAM) layer, and the vibrations of the concrete base and its foundation are ignored. The CAM layer is regarded as a continuous plane spring and damp, and Kdl represents lateral spring factor, Cdl the lateral damping factor, Kdv the vertical spring factor, Cdv the vertical damping factor. The nodal displacement parameter of the track segment element is described as follows:
(1)
where
(2)
(3)
the superscripts R and S denote the displacements of rail and slab, respectively; the subscripts 1 and 2 denote the left and right nodes of the track segment element in direction X, respectively; the subscripts L and R denote the left and right hand of the track segment element in direction Y, respectively; the subscripts X, Y and Z denote longitudinal, lateral and vertical directions, respectively; V and W denote linear displacement in directions Y and Z, respectively; and θ denotes rotation displacement.
3.2 Displacement mode
When the nodal displacement of the track segment element is chosen, the displacements of the rail and slab can be gained with the nodal displacement by an interpolation method. The vertical and lateral rail displacements as well as the lateral slab displacement at an arbitrary point are determined by Hermitian cubic interpolation function. As to the vertical displacement of the slab, the following method was applied.
Fig.1 Model of track segment element of spatial vibration of slab track: (a) Three dimensional view; (b) Cross-section view
As shown in Fig.1, for an unit finite strip in breadth abcd, it can be modeled as a beam ij. Then the deformation of the beam ij at any point (its coordinate in direction Y is y) can be obtained by using displacement in the both sides of beam ij and through Hermitian cubic interpolation function. Furthermore, the displacement on the both sides of beam ij can also be gained by nodal displacement of the slab through interpolation method, in which the linear displacement is obtained by the Hermitian cubic interpolation function and the rotation displacement by the Hermitian linear interpolation function. Finally, the vertical displacement at any point of the slab can be expressed as
(4)
where
(5)
The number above the element in matrix denotes the column order and the other elements not labeled are zero, and
,
,
,
,
,
,
,
,
,
.
The above interpolation method for slab deformation is called as the method of lateral finite strip and slab segment element. This method is of simplification and applicability compared with the traditional plate element method.
3.3 Potential energy of spatial vibration of slab track structure ΠT
When the displacement modes of all components of each track element are defined, the elastic strain energy and potential energy of inertial force of the rail and slab of track segment element can be deduced, and so as the elastic strain energy and potential energy of damping force of the CAM and fastener. Combining these energy items, ΠTi, the potential energy of vibration of the ith track segment element is obtained. If there are N track segment elements in the calculation length of the slab track, the potential energy of spatial vibration of the slab track structure Π T can be written as
(6)
where
(7)
in which, 
and 
are the elastic strain energy of theⅠth and Ⅱth rails, respectively, 
and 
are the potential energy of inertial force of theⅠth and Ⅱth rails, respectively, US is the elastic strain energy of the slab,
is the potential energy of inertial force of the slab,
is the elastic strain energy of the CAM, 
is the potential energy of damping force of the CAM, 
is the elastic strain energy of the fastener, and 
is the potential energy of damping force of the fastener.
4 Establishment and solution of equation set of spatial vibration of system
If there are M cars on the slab track at time t, the potential energy of vibration of the train at time t can be written as
(8)
So the total potential energy of spatial vibration of the train and slab track system at time t, considering the preceding subsection 3.3, can be expressed as
(9)
By using the principle of total potential energy with stationary value in elastic system dynamics[14] and the “set-in-right-position” rule for formulating system matrices[15], the equations of the spatial vibration of the high-speed train and slab track system at time t can be written as
(10)
where the matrices [M], [C] and [K] are the global mass, damping and stiffness matrices of the high-speed train and slab track system, respectively, and the vectors 
, 
and 
denote the acceleration, velocity and displacement vectors of the system, and 
is the global load vector of the system. Eqn.(10) can be then solved by direct time integration such as Wilson-θ method.
Thus, sections 2, 3 and 4 form the analysis theory of spatial vibration of high-speed train and slab track system.
5 Results and discussion
The spatial vibration responses of the “Xianfeng” high-speed train formed with 5 trailers hauled by a motor car running at 200 km/h on the 200 m slab track are calculated and then verified by the high-speed running experiment carried out on the slab track in the Qinghuangdao-Shenyang passenger transport line[16]. The primary parameters of the “Xianfeng” high-speed train are as follows: the length between truck centers of the motor car is 18 m, the rigid wheelbase of the motor car 2.5 m, the length between truck centers of the trailer 18 m, the rigid wheelbase of the trailer 2.5 m, the axle-load of the motor car 145 kN, and the axle-load of the trailer 137.5 kN. The primary parameters of the slab track are listed as follows: the rail mass is 60 kg/m, the stiffness of the fastener 6×107 N/m, the length, width and thick of the slab are 4.93 m, 2.4 m and 0.19 m, respectively, and the elastic modulus of the CAM 300 MPa.
The comparison between maximum calculated and measured results is listed in Table 1. It is shown that the calculated results agree well with the measured results, which demonstrates the reliability of the theory proposed in this work.
Table 1 Comparison between maximum calculated and mea- sured results
Furthermore, the waveforms of vibration of the high-speed train and slab track system can also given to prove the rationality of the calculated results. So some typically calculated waveforms of vibration of the system are shown in Figs.2-9. From these figures we can see that the calculated waveforms and values are in accord with the physic concept. For example, from Figs.2-5, the following phenomena can be found: when wheels of the train pass the fixed point of the rail, the lateral and vertical rail displacements are larger, otherwise they are smaller or even zero. In addition, we can also find how many cars pass the fixed point of the rail. All of these demonstrate the reliability of the theory presented in the work.
Fig.2 Lateral rail displacement wave
Fig.3 Vertical rail displacement wave
Fig.4 Lateral slab displacement wave
Fig.5 Vertical slab displacement wave
Fig.6 Lateral rail acceleration wave
Fig.7 Lateral slab acceleration wave
Fig.8 Lateral force wave of third wheel set of motor car
Fig.9 Derail factor wave of third wheel set of motor car
6 Conclusions
1) An analysis theory of spatial vibration of high-speed train and slab track system is proposed. The main contents of this theory are as follows: The high-speed train is modeled as a multi-rigid body system and the potential energy of spatial vibration of the train is deduced. According to structural characteristic of the slab track, a new spatial vibration model of track segment element of the slab track is put forward and the potential energy of spatial vibration of the slab track is derived. Combining these energy items, the total potential energy of spatial vibration of the high-speed train and slab track system is obtained. According to 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 equations of spatial vibration of the system are established, and the equations are solved by the Wilson-θ direct integration method.
2) An interpolation method for slab deformation is proposed, that is, the method of lateral finite strip and slab segment element. This method has simplification and applicability compared with the traditional plate element method.
3) The maximum calculated value of lateral rail displacement is 0.82 mm, the vertical rail displacement 0.90 mm, the lateral wheel-rail force 8.9 kN, the vertical wheel-rail force 102.3 kN, the derail factor 0.2, and the wheel load reduction rate 0.32. The maximum measured value of lateral rail displacement is 0.75 mm, the vertical rail displacement 0.93 mm, the lateral wheel-rail force 8.6 kN, the vertical wheel-rail force 80.2 kN, the derail factor 0.08, and the wheel load reduction rate 0.14. It is shown that the calculated results agree well with the measured results. All of these demonstrate the reliability of the theory proposed.
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(Edited by YANG Hua)
Foundation item: Project(2007CB714706) supported by the National Basic Research Program of China; Project (50678176) supported by the National Natural Science Foundation of China; Project(NCET-07-0866) supported by the Program for New Century Excellent Talents in University
Received date: 2007-04-22; Accepted date: 2007-06-18
Corresponding author: XIANG Jun, Professor, PhD; Tel: +86-731-2656728; E-mail: jxiang@mail.csu.edu.cn
