A new solution method for wheel/rail rolling contact
© Yang et al. 2016
Received: 21 December 2015
Accepted: 7 April 2016
Published: 18 April 2016
To solve the problem of wheel/rail rolling contact of nonlinear steady-state curving, a three-dimensional transient finite element (FE) model is developed by the explicit software ANSYS/LS-DYNA. To improve the solving speed and efficiency, an explicit–explicit order solution method is put forward based on analysis of the features of implicit and explicit algorithm. The solution method was first applied to calculate the pre-loading of wheel/rail rolling contact with explicit algorithm, and then the results became the initial conditions in solving the dynamic process of wheel/rail rolling contact with explicit algorithm as well. Simultaneously, the common implicit–explicit order solution method is used to solve the FE model. Results show that the explicit–explicit order solution method has faster operation speed and higher efficiency than the implicit–explicit order solution method while the solution accuracy is almost the same. Hence, the explicit–explicit order solution method is more suitable for the wheel/rail rolling contact model with large scale and high nonlinearity.
With constant upgrades in operating speed and axle load of trains, the interaction between wheel and rail becomes more intense, especially for the curved track which contains larger contact normal force and tangential force as well as larger creep force and creep force moment in the contact zone. Two of the most important problems in railway system are rolling contact fatigue and corrugation caused by the long-term wheel/rail rolling contact process which seriously influence the transport safety (Li et al. 2009). Hence, the wheel/rail rolling contact has been a hot issue.
For numeric simulation, the transient wheel/rail rolling contact in railway system is a very complex and time-consuming process. Reasonable solution method would reduce the computational effort and improve the solution efficiency in a certain extent. At present, there are several related solution methods that are applied to study the wheel/rail rolling contact problem (Lu et al. 2009; Zhao et al. 2013; Zhai and Huang 1991; Zhao et al. 2014; Telliskivi and Olofsson 2001; Arias-C et al. 2010; Zhai 2007; Lian 2004; Chang et al. 2010). The representative solution methods are as follows. Lu et al. (2009) applied implicit algorithm to solve the stress of wheel/rail rolling contact, but the implicit algorithm took much time when solving the problem with large scale. Zhao et al. (2013) analyzed the transient behavior of wheel/rail rolling contact at high speed by explicit algorithm, yet did not take into account the structure pre-loading. Zhuo et al. performed the simulation of larger dynamic problems of train on PCs with self-procedure programming by the Newmark explicit and predictor–corrector integration method, which greatly improved the solving speed. However, the application range of self-procedure was relatively limited for all kinds of restrictions (Zhai and Huang 1991). Zhao et al. (2014) studied the problem of wheel/rail rolling contact by the implicit–explicit order solution method, and found that the implicit analysis results would have a certain influence on the later explicit computation and the influence effect would increase as the train speed increases for the difference between implicit and explicit algorithm.
The solution methods for wheel/rail rolling contact that are based on ANSYS/LS-DYNA mainly include the explicit solution method and the implicit–explicit order solution method. The independent explicit solution method without considering the pre-loading does not tally with the actual situation, so it is not applicable for the wheel/rail rolling contact problem. The implicit–explicit order solution method is widely used to solve the rolling contact problem of railway system which needs to consider the pre-loading. However, the implicit–explicit order solution method applies implicit algorithm to define the initial pre-loading, which requires much solution time. Moreover, the solution efficiency and convergence decrease as the model nonlinearity and DOFs (degrees of freedom) increase.
To improve the solving speed and efficiency, an explicit–explicit order solution method is put forward based on analysis of the features of implicit and explicit algorithm. Results show that the explicit–explicit order solution method has faster operation speed and higher efficiency than the implicit–explicit order solution method while the accuracy of the calculations are almost the same. Hence, the explicit–explicit order solution method is more suitable for solving the wheel/rail rolling contact problem with large scale.
The FE model of wheel/rail rolling contact
Mechanical properties of wheel and rail
7.83 × 103
Poisson’s ratio (ν)
Elastic modulus (E)
2.06 × 105
Parameter values for lateral and vertical spring
Stiffness of lateral spring (k l )
Damping of lateral spring (C l )
Stiffness of lateral spring (k v )
Damping of lateral spring (C v )
A typical solving process of wheel/rail rolling contact
A typical wheel/rail rolling contact solving process includes two parts. Part 1: calculate the initial pre-loading (the static deformation and stress of equilibrium state) of wheel/rail rolling contact under gravity. Part 2: initialize the initial pre-loading including the deformation and stress (calculated in Part 1) and then explicitly solve the transient wheel/rail rolling contact process. Note that the initial pre-loading calculated in Part 1 is to ensure the wheelset achieves a steady-state rolling.
Implicit and explicit algorithm
In Eq. (3), the element mass matrix M is always symmetrical or centralized, and the node acceleration is absolutely determined by the element mass and the element force. Hence, when solving the model acceleration equations, the solution procedure is greatly simplified, which improves the whole calculation efficiency to a large extent.
Comparison and analysis
The solution time of explicit algorithm is proportional to the number of nodes and inversely proportional to the minimum size of elements. Meanwhile, more experiences show that the solution time of implicit algorithm is roughly proportional to the number of nodes squared. So for the same model, the same element size and the same number of nodes, the explicit algorithm would take less solution time and is more efficient. Moreover, the superiority of solution time would increase quickly as the number of nodes increases relative to the implicit algorithm.
Although the implicit algorithm is provided with relatively higher solution accuracy than the explicit algorithm for the static problems (i.e. the initial pre-loading of transient wheel/rail rolling contact), the superiority of solution time of explicit algorithm would obviously exceed its shortage of solution accuracy when solving the problems with large scale and high nonlinearity. So, at the expense of a certain solution accuracy, the explicit algorithm can be used to solve the initial pre-loading of wheel/rail rolling contact instead of the implicit algorithm to reduce the solution time. Meanwhile, a large number of simulation results show that the influence of relatively small solution error of initial pre-loading can be neglected. Therefore, a new explicit–explicit order solution method is put forward to solve the transient wheel/rail rolling contact problem in this paper.
The explicit–explicit order solution method
Key steps of explicit–explicit order solution method
The explicit–explicit order solution method consists of three key steps. Step 1: apply the explicit algorithm to solve the initial pre-loading (the static deformation and stress of equilibrium state) of wheel/rail rolling contact only under gravity which is based on ANSYS/LS-DYNA. Step 2: import the results of deformation and stress (calculated in step 1) into the FE model of wheel/rail rolling contact to initialize the equilibrium state. Step 3: apply the explicit algorithm of ANSYS/LS-DYNA to solve the dynamic process of wheel/rail rolling contact by setting the initial velocity of wheelset and related boundary conditions. Note that the settings of boundary conditions are as follows (see Fig. 1): apply symmetric boundary conditions to the rail ends and the axle ends of wheelset, and fix all the outer ends of lateral and vertical springs and dampers.
Time step of explicit–explicit order solution method
Error analysis of explicit–explicit order solution method
In addition, the total solution time of the explicit–explicit order solution method is about one-third of the implicit–explicit order solution method (Baed on Intel(R) Core(TM) i7-3930 K CPU @ 3.20 GHz 16 G RAM), which proves that the explicit–explicit order solution method has higher calculation efficiency.
According to the actual parameters, the three-dimensional transient FE model of wheel/rail rolling contact is created by ANSYS/LS-DYNA, and the single-layer track dynamic model is taken into account. Based on analysis of the features of implicit and explicit algorithm, the explicit–explicit order solution method is proposed to improve the solving speed and efficiency.
Results show that the solution accuracy of the explicit–explicit order solution method and implicit–explicit order solution method is almost the same. And the total solution time of the explicit–explicit order solution method is about one-third of the implicit–explicit order solution method. Hence, the explicit–explicit order solution method is more efficient and more suitable for solving the wheel/rail rolling contact model with large scale and high nonlinearity.
JY and HS conceived and designed the study. JY, HS and LF performed the simulation experiments. WL took charge of the parameter testing of the FE model. JY wrote the paper. HS, LF, MW and WL reviewed and edited the manuscript. All authors read and approved the final manuscript.
My deepest gratitude goes first and foremost to Professor Hua Song, my supervisor, for his constant encouragement and guidance. He has walked me through all the stages of the writing of this thesis. Without his consistent and illuminating instruction, this thesis could not have reached its present form. Second, I would like to express my heartfelt gratitude to Professor Lihua Fu, who led me into the world of translation. Last my thanks would go to my beloved family for their loving considerations and great confidence in me all through these years. I also owe my sincere gratitude to my friends and my fellow classmates who gave me their help and time in listening to me and helping me work out my problems during the difficult course of the thesis.
The authors declare that they have no competing interests.
This work is supported by the National Natural Science Foundation of China (51374127).
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