Traintrack interaction (TTI) is a classic research topic in railway engineering, which consists of three main parts, namely, train model, track model, and wheelrail interaction. To improve the computational accuracy and broaden the application range, an alternative calculation method to investigate TTI based on secondary development technology of the commercial software ANSYS through APDL language is introduced in this article. Primarily, the traintrack interaction theory is briefly presented. On this basis, TTI is programmed and implemented on the computing platform of ANSYS by fully taking the nonlinear wheelrail interaction into consideration. In this calculation method, the train model, which is established based on multibody dynamics theory and solved by an advanced explicit integration method, is programmed into ANSYS through APDL language, while the track part is simulated according to finite element theory. Then, the proposed calculation method is validated with field test results to verify the validity. Finally, a numerical demonstration is conducted employing the present method. Results show that the introduced method is effective and able to investigate TTI. Different complicated track systems can be accurately simulated employing this method. Moreover, this method is also adoptable to explore trainbridge interaction and traintrackbridge interaction.
Traintrack interaction (TTI) has been a traditional research topic for quite a long time [
So far, various calculation methods have been proposed to investigate TTI. These methods can be generally classified into the following categories:
Method A: dynamic equations of TTI are programmed on different compiled platforms employing different programming languages such as FORTRAN, MATLAB, and C, which are then solved with numerical integration methods. This is the most commonly used method. With this method, the train systems are usually established based on multibody dynamics [
Method B: train systems and track systems are all established in commercial finite element software such as ANSYS and ABAQUS [
Method C: different commercial software are combined to solve TTI. With this method, train systems are usually modeled by multibody software such as SIMPACK and UM, while the track structures are simulated using finite element software. Then, the modal properties of tracks are exported from finite element software into multibody software to make the two different systems coupled [
Method D: motion equations of trains are programmed with programming languages such as FORTRAN, while the track structures are modeled in finite element software. Then, mass matrix and stiffness matrix of tracks are exported to external files, which are further read in by the selfprogramed procedure to form the whole dynamic equations of the traintrack systems, and finally these equations can be calculated through different integration methods [
Through the above methods, several generalized conclusions are reached by the authors:
The train systems should be modeled based on multibody dynamics theory to improve the calculation speed
To conveniently establish various kinds of track structures and to accurately analyze the dynamic behaviors of tracks considering their nonlinear characteristics, finite element software is perfect tools to reach this goal
Track irregularity is hardly to be considered in finite element software
Comprehensively considering the computational efficiency and the computational accuracy, the train equations and track systems should be, respectively, solved by the explicit integration method and implicit integration method.
Based on the above investigations, to improve the computational accuracy and broaden the application range, an alternative method to solve TTI is proposed in this present work based on development of ANSYS. First, TTI is programmed and implemented on the computing platform of ANSYS in Section
Solutions of TTI adopting different technological means have been presented in previous works [
ANSYS is a worldfamous finite element software, which contains various kinds of elements and materials. The software has been widely used in railway engineering. Furthermore, APDL broadens the function and application range of ANSYS, resulting in convenient modelling and simulation. Adopting this language, the train system and track irregularity can be easily considered on the computing platform of ANSYS. Hence, the software ANSYS and APDL are adopted to investigate TTI.
Taking a 2D vehicleslab track system as an example, Figure
Typical traintrack interaction model (slab ballastless track).
Main notations adopted in this work.
Notation  Physical meaning  Unit 


Vertical displacements of car body, frame, and wheelset  m 

Pitch angles of car body, frame, and wheelset  ° 

Masses of car body, frame, and wheelset  kg 

Damping of primary suspension and secondary suspension  N·s/m 

Stiffness of primary suspension and secondary suspension  N/m 

Wheel base  m 

Length between bogie centers  m 

Contact constant of wheel and rail  m/N^{2/3} 

Elastic compression deformation  m 

Wheel displacement  m 

Rail displacement  m 

Track irregularity  m 

Vertical, lateral and torsional displacements of the rail  m 

Rail mass per unit length  kg/m 

Rail density  kg/m^{3} 

Locations of 
m 

Numbers of fasteners and wheelsets  — 

Dirac delta function  — 
The train system contains a series of vehicles, which are evenly placed on the track. Each vehicle is modeled as a massspringdamper system consisting of a car body, two bogie frames, four wheelsets, and twostage suspensions, as shown in Figure
Vertical motion of car body:
Pitch motion of car body:
Vertical motion of front frame:
Pitch motion of front frame:
Vertical motion of rear frame:
Pitch motion of rear frame:
Vertical motion of 1^{st} wheelset:
Vertical motion of 2^{nd} wheelset:
Vertical motion of 3^{rd} wheelset:
Vertical motion of 4^{th} wheelset:
Furthermore, the dynamic equations of the train system can be given in matrix form as follows:
The detailed expressions of the mass, the stiffness, and the damping matrices of the train system can be referred to literature [
These matrices of the train system cannot be calculated by ANSYS automatically, which should be written into the software manually. The relevant program code is given as follows.
In the above code,
The track system consists of many parts with different mechanical characteristics. ANSYS provides different kinds of elements and materials, which are enough to accurately simulate different components in track structures. According to the modelling principle in research [
In the above code,
The train system and the track system are coupled by wheelrail interaction relationship, which can be described by the nonlinear Hertz contact theory [
Furthermore,
It should be underlined that the wheelrail contact relationship is programmed into ANSYS rather than employing the default contact in this software.
The rail is modeled by the finite element method; thus, all the wheels will not be at the node locations exactly in each integration step. In most cases, the wheels are located between the two adjacent nodes. Therefore, it is necessary to allocate the wheelrail force to the two adjacent nodes, as shown in Figure
Allocation of wheelrail force on the beam element of rail.
As shown in Figure
Also, in the finite element model, the vibration displacements of all the nodes are calculated. In order to obtain the rail displacement at the wheelset location between the adjacent two nodes, the following equation is employed based on the interpolation theory:
As a result, the rail displacement at any point can be obtained combining equations (
The allocation of wheelrail force in APDL language is written as
In the above program,
Moreover, for other finite element software such as ADINA and ABAQUS, it is rather difficult to consider track random irregularity in investigating TTI, leading to the calculation processes of train system and track system being usually separate. For instance, precalculated wheelrail forces, which are calculated with other tools, are applied on the track FE model to investigate the track vibrations.
As for ANSYS, the APDL provides a convenient program language to implement secondary development for ANSYS. After the displacements of train and track systems are determined, the wheelrail forces can be further calculated by considering track irregularity and the relevant program is given as follows. In the program,
The model and written code for 2D traintrack dynamic system are introduced in Section
The train submodel is built by considering seven rigid bodies of a car body, two bogies, and four wheelsets. Five DOFs are taken into consideration for each rigid body, describing bounce, sway, roll, yaw, and pitch motions. The detailed equations of motion of all the seven parts can be referred to literature [
The rails are modeled as Euler beams and discretely supported by fasteners which are simulated as linear springdamping elements. Three DOFs of each rail are taken into account, describing vertical motion, lateral motion, and torsional motion, and the equations of motion of the rail are given as follows:
Vertical motion:
Lateral motion:
Torsional motion:
In equations (
In the 3D traintrack dynamic model, the nonlinear Hertzian elastic contact theory is used to calculate the wheelrail normal contact forces:
Obviously, when
Based on Kalker’s linear creep theory, the wheelrail longitudinal creep force
Since Kalker’s linear creep theory is only suited to the cases with small creepages, the nonlinear modification should be made, for example, by the Shen–Hedrick–Elkins model, for the situations of large creepages. Similar to the 2D traintrack model, the 3D wheelrail relationship is also programmed into ANSYS rather than employing the default contact in the software.
The wheelrail interaction forces mainly include wheelrail normal contact forces derived by nonlinear Hertzian elastic contact theory according to the elastic compression deformation of wheels and rails at contact points in the normal directions and tangential wheelrail creep forces, which are calculated first by Kalker’s linear creep theory and then modified with the ShenHedrick–Elkins nonlinear model [
Due to the space limitation, the code for the 3D traintrack dynamic model is not listed below, which can be referred in the first author’s doctor degree dissertation [
Through the above explanation, the whole dynamic equations of the traintrack system are established, which can be solved by the stepbystep integration method. In this present calculation model, the track is modeled by FEM, while the vehicle system and the wheelrail relationship are programmed into ANSYS by secondary development with APDL language. The Newmark implicit integration method is selected to calculate the FEM, which ensures the stability of the FEM calculation. In order to improve the computational efficiency, the Zhai method [
The frame of Zhai method is implemented in ANSYS as
As seen from the code, in the beginning of each integration step, the displacement and velocity of the train system are calculated, while the acceleration of the train is determined in the end of each integration step.
The methodology of investigating TTI with ANSYS is shown in Figure
Procedure of investigating TTI with ANSYS adopting APDL language.
The calculation method is introduced in detail, and on this basis, two kinds of validations are conducted to verify its validity.
Based on vehicletrack dynamics [
Parameters of the adopted vehicle.
Item  Vehicle  Unit 

Distance between bogie centers  17.375  m 
Bogie wheelbase  2.5  m 
Wheel rolling circle diameter  0.92  m 
Carbody mass  38.884  t 
Bogie frame mass  2953  kg 
Wheelset mass  1517  kg 
Inertia moment of carbody  1905.3  t·m^{2} 
Inertia moment of bogie frame  1293  kg·m^{2} 
Inertia moment of wheelset  118  kg·m^{2} 
Primary suspension stiffness  0.8865  MN/m 
Secondary suspension stiffness  0.195  MN/m 
Subject to impact loads, such as wheel flat and welding joint, the dynamic responses of the traintrack system are illustrated in Figure
Dynamic responses of the traintrack system subject to impact loads: (a) wheelrail force and (b) rail displacement at the impact location.
As seen from this result, the wheelrail forces and rail displacements determined through the two tools are almost the same, indicating the proposed calculation method is effective and able to investigate TTI. It should be noted that the periodic fluctuation of the wheelrail force from ANSYS is caused by the length of each rail element.
UM (Universal Mechanism) is developed by the Laboratory of Computational Mechanics in Bryansk State Technical University, which is designed to automate the analysis of mechanical objects representing as a multibody system. By far, the software has been widely adopted in the field of railway engineering worldwide.
Excited by lateral and vertical harmonic irregularities, the dynamic responses of the train, whose dynamic parameters can be seen in Table
Calculated responses of CRH3 by UM and the proposed method.
Indicator  Calculated result  

By UM  By the proposed method  
Vertical acceleration of carbody (m/s^{2})  ±0.133  ±0.134 
Vertical wheelrail force (kN)  60.89∼64.11  60.76∼63.97 
Lateral acceleration of carbody (m/s^{2})  ±0.142  ±0.161 
Lateral wheelrail force (kN)  −1.35∼3.83  −1.39∼4.43 
A 40day field experimental test of the traintrack interaction dynamic was carried out in a certain highspeed railway in 2013 (shown in Figure
Field test on a highspeed railway with the slab ballastless track: (a) measurement of the displacement and acceleration of the rail, (b) measurement of the acceleration of the slab, (c) measurement of the acceleration of the base, and (d) the data acquisition system.
The details of the field test, e.g. the track irregularity, the parameters of the traintrack system, and the operation speed, can be referred to the authors’ published paper [
The validation is illustrated in Figure
Comparison between test results and simulated results: (a) calculated rail acceleration, (b) measured rail acceleration, (c) comparison of rail displacement, and (d) comparison of slab acceleration.
Every calculation method certainly has advantages and disadvantages, which will be discussed in this section.
The advantages are given as follows:
The track structures can be easily established, even if the structure is abnormal and peculiar
With the help of finite element software, the nonlinear materials and mechanical behaviors can be easily simulated
The train and track systems are coupled and calculated simultaneously
Track irregularity is easily considered
Wind loads and earthquake loads are convenient to be considered in investigating TTI with ANSYS
The calculation method can also be used in investigating trainbridge interaction, traintrackbridge interaction, traintunnel interaction, and so on
On the contrary, the disadvantages of the method are listed below:
The computational efficiency is lower than that programmed with FORTRAN, C and so on
The train model is simpler than the established models in SIMPACK and UM
Boundary condition is very important in solving the FE model, which should be seriously and accurately considered and applied
Compared to the proposed calculation method with the classified 4 methods (methods AD) in Section
Evaluations of computational accuracy and calculation time of different methods.
Method  Computational accuracy  Calculation time  

Description  Level  Description  Level  
A  To write vibration equations, complicated structures are simplified in large extent; moreover, inapparent mistakes in the written codes are difficult to find out  Middle  Compiled programs are highly efficient in solving TTI  Fast 
B  Rail random irregularity is hard to be accurately considered  Low  Computational efficiency is low due to the implicit integration algorithm in FE software  Slow 
C  Nonlinear characteristics of tracks and foundations cannot be considered  Middle  Adopting the algorithm of multibody dynamics and mode superposition method or modal synthesis method, computational efficiency can be ensured  Middle 
D  Due to the changes of mass stiffness matrices, nonlinear properties of substructures cannot be considered  Middle  Compiled programs are highly efficient in solving TTI  Fast 
Proposed method  Disadvantages of the above methods are settled  High  Hybrid explicitimplicit integration method is adopted to efficiently solve TTI  Middle 
As seen from the above table, the computational accuracy of the proposed method can be improved due to (a) nonlinear characteristics of tracks and foundations can be accurately considered, (b) complicate substructures can be accurately modeled, and (c) track irregularity is easily and accurately considered.
To demonstrate the wide application range of the proposed calculation method, the dynamic responses of a highspeed train running through a bridgesubgrade transition section are investigated using this present method. In previous studies, to explore such a complex problem, the most common method is to simplify the transition section as an additional track irregularity [
A widely used bridgesubgrade transition section with a doubleblock ballastless track (Figure
Bridgesubgrade transition section with the doubleblock ballastless track.
Key parameters of the bridgesubgrade transition section.
Item  Value  Unit 

Stiffness of fastener  30  MN/m 
Damping of fastener  3 × 10^{4}  N·s/m 
Elasticity modulus of slab  3.4 × 10^{4}  MPa 
Elasticity modulus of support base  0.5 × 10^{4}  MPa 
Elasticity modulus of subgrade top layer  180  MPa 
Elasticity modulus of subgrade bottom layer  110  MPa 
Elasticity modulus of embankment  30  MPa 
Adopting the above parameters, a 2D finite element model of this transition section is displayed in Figure
FE model of bridgesubgrade transition section in ANSYS.
Deformation of rail caused by settlement.
Structural dynamic stress of the infrastructure at a certain time.
Dynamic responses of the traintransition section system: (a) wheelrail force and (b) dynamic stress in subgrade bottom layer.
The rail deformation caused by settlement is displayed in Figure
As seen from Figure
Figure
Based on secondary development of ANSYS adopting APDL, this paper has proposed an alternative calculation method for investigating traintrack interaction (TTI). In this method, the train system and wheelrail relationship have been programmed into ANSYS based on multibody dynamics, while the track system is established in ANSYS with the help of different kinds of elements. An explicitimplicit hybrid integration method has been employed to solve the large complex dynamic system. Finally, the proposed calculation method has been validated. From this present research, the following conclusions can be reached:
The proposed calculation method is effective to investigate TTI. The whole traintrack system is coupled and calculated synchronously.
This paper also provides a guidance to calculate a coupled mechanical system with different integration methods in ANSYS.
This calculation method has a wide application range, which can also be used to study traintrackbridge interaction, traintunnel interaction, and so on.
The computational efficiency of this present method is lower than that of programming with FORTRAN, C, and so on, indicating that new calculation methods to speed the calculation in ANSYS, such as parallel computing, should be further explored.
No data were used to support this study.
The authors declare that there are no conflicts of interest in preparing this article.
This work was supported by the Basic Natural Science and Frontier Technology Research Program of the Chongqing Municipal Science and Technology Commission (Grant numbers: cstc2018jcyjAX0271 and cstc2019jcyjmsxmX0777), the National Science Fund for Distinguished Young Scholars of China (Grant number: 51425801), the Open Research Fund of State Key Laboratory of Traction Power (Grant number: TPL1901), the China Postdoctoral Science Foundation (Grant number: 2019M650236), the Open Research Fund of MOE Key Laboratory of Highspeed Railway Engineering (Grant number: 2017HRE01), the Open Research Fund of Chongqing Key Laboratory of Railway Vehicle System Integration and Control (Grant number: CKLURTSICKFKT201804), and the Project of Fund Cultivation of Chongqing Jiaotong University (Grant number: 2018PY14).