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Research Papers

# Parallel Co-Simulation Method for Railway Vehicle-Track DynamicsOPEN ACCESS

[+] Author and Article Information
Qing Wu, Yan Sun, Maksym Spiryagin, Colin Cole

Centre for Railway Engineering,
Central Queensland University,
Rockhampton QLD4701, Australia;
Australasian Centre for Rail Innovation,
Canberra ACT2608, Australia

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 22, 2017; final manuscript received January 29, 2018; published online February 26, 2018. Assoc. Editor: Xiaobo Yang.

J. Comput. Nonlinear Dynam 13(4), 041004 (Feb 26, 2018) (9 pages) Paper No: CND-17-1467; doi: 10.1115/1.4039310 History: Received October 22, 2017; Revised January 29, 2018

## Abstract

This paper introduces a new parallel co-simulation method to study vehicle-track dynamic interactions. The new method uses the transmission control protocol/internet protocol (TCP/IP) to enable co-simulation between a detailed in-house track dynamics simulation package and a commercial vehicle system dynamics simulation package. The exchanged information are wheel-rail contact forces and rail kinematics. Then, the message passing interface (MPI) technique is used to enable the model to process track dynamics simulations and vehicle dynamics simulations in parallel. The parallel co-simulation technique has multiple advantages: (1) access to the advantages of both in-house and commercial simulation packages; (2) new model parts can be easily added in as new parallel processes; and (3) saving of computing time. The original track model used in this paper was significantly improved in terms of computing speed. The improved model is now more than ten times faster than the original model. Two simulations were conducted to model a locomotive negotiating a section of track with and without unsupported sleepers. The results show that the vertical rail deflections, wheel-rail contact forces and vehicle suspension forces are evidently larger when unsupported sleepers are present. The simulations have demonstrated the effectiveness of the proposed parallel co-simulation method for vehicle-track dynamic interaction studies.

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## Introduction

Railway track is usually the highest value part in a railway system. The track system has the functions of guiding trains and bearing various loads that are generated from the interactions of trains and track. Better understanding of track dynamics and responses under the excitations of passing trains can help to enhance train operational safety, to preserve the property value of the track system and to decrease the maintenance cost of both track and vehicle systems. Under the topic of track system dynamics, the track system is usually described by a combination of models for rails, fasteners, rail pads, sleepers, ballast, subballast, and subgrade [13]. In this paper, the track systems being discussed and studied are the ballast track systems; the ballastless or slab track systems [4,5] are not studied. However, it needs to be pointed out that the proposed parallel co-simulation method can also be used for slab tracks.

Track system dynamics is a highly nonlinear problem that is further complicated by the large number of degrees-of-freedom (DoFs); relevant studies have been reported worldwide and many models are developed [621]. Literature reviews regarding track system dynamics modeling can also be found in Refs. [2,2224]. These studies mainly use two different analysis approaches: frequency domain analyses [68] and time domain analyses [914]. Comparatively, frequency domain analyses are faster in computing and can directly reflect the vibration characteristics of the track system. However, frequency analyses often can only use linear models and simulate singular events. Time domain analyses can consider more nonlinearities and are more flexible with simulation scenarios. A major nonlinear component in the track system model is the rail model. Regarding the rail modeling, two continuous beam theories are commonly used. The first one is the Euler–Bernoulli beam theory while the second one is the Timoshenko beam theory [912]. The Euler–Bernoulli beam theory does not consider the shear deformation and the rotation of the cross section. And studies have shown that, compared with the Timoshenko beam models, Euler–Bernoulli beam models are simpler for modeling and faster for computing. However, they cannot accurately simulate the frequency response above 500 Hz while Timoshenko beam models have a capability of 1500 Hz for lateral and 2500 Hz for vertical motions of the rail. In terms of numerical solving methods for the track dynamics models, two methods are commonly seen, namely, the modal superposition method [912,25] and the finite element model [13,14]. Both methods are effective and it is hard to argue which one is faster for simulations. The finite element method needs to simulate a large number of elements while the superposition method needs to consider high order of vibration frequency.

Vehicle system dynamics is another area that has been vigorously studied all around the world. For general vehicle system dynamics, the most important model components are wheel-rail contact and suspension models. The wheel-rail contact problem can be further divided into the geometrical issue [26,27], the normal force issue [28], and the tangential force issue [2932]. For the geometrical issue, models that can handle single contact point and multiple contact points exist. Single contact point models are simpler and accurate enough for normal wheel-rail profiles with small lateral displacements and small yaw angles. Multiple contact points are better for simulation of worn profiles and flange contact. Regarding the normal force issue, besides the classic Hertz model, other semi-Hertz and non-Hertz models are also reported. Comparatively, semi-Hertz and non-Hertz models are more accurate for some special cases, but are more complicated to develop and implement. Regarding the tangential problem, three models are relatively more widely used: CONTACT [30], FASTSIM [31], and POLACH [32] models. They have a decreasing order of accuracy but an increasing order of computing speed.

Regarding rail vehicle suspension modeling, it is also a nonlinear problem which may entail modeling of dry friction, polymer, and hydraulic elements [33,34]. For freight wagons, friction dampers are commonly used and these can be modeled using different models as comprehensively reviewed in Ref. [35]. For passenger vehicles and locomotives, polymer and hydraulic dampers are commonly used. Modeling of these dampers has been comprehensively reviewed in Ref. [36].

The research of vehicle-track coupled dynamics is well developed. Both in-house [9,11] and commercial [37,38] software packages have been reported from literature. These coupled models integrate both track system dynamics and vehicle system dynamics via the simulation of wheel-rail contact. The wheel-rail contact has to account for considerations from both vehicle and track sides. Currently, most commercial software packages that can deal with railway vehicle system dynamics, such as simpack, universalmechanism, gensys, and nucars, have ready-to-use track models. However, these ready-to-use models usually consider rails, pads, and sleepers only. Ballast and structures underneath the ballast are not considered. Using commercial software packages, the simulation of full track models, which may include ballast, subballast, and subgrade, needs the import of external models (e.g., finite element model of the ballast and subgrade). In-house software packages in Refs. [9] and [11] can simulate both detailed track models and detailed vehicle models. However, these models cannot meet the requirements of some special needs due to, for example, the complexity of wheel-rail contact modeling. The wheel-rail contact models in these in-house software packages are not as advanced as those in commercial software packages. And the locomotive traction system models and adhesion control algorithms are either not available or are very basic.

In this paper, a parallel co-simulation technique is developed to link an improved detailed in-house track model to a commercial vehicle system dynamics simulation software package (gensys). This method combines the previously discussed advantages of both in-house track models and commercial vehicle system dynamics software packages. Vehicle-track interaction simulations are conducted to prove the feasibility of the proposed method. Regarding the arrangement of this paper, Sec. 2 introduces the detailed track model and its improvement regarding the computational efficiency. Section 3 introduces the vehicle model (taking a locomotive model as an example) using the commercial software package gensys. Section 4 introduces the parallel co-simulation technique. Section 5 presents the results of vehicle-track interaction simulations. Section 6 concludes this paper with discussions and remarks.

## Track Model

The original track model was developed by Sun and coworkers [11,39] from the Centre for Railway Engineering. It is a model in the time domain and schematic diagrams of it are shown in Figs. 1 and 2. It can be seen that the track model is very detailed and has considered four layers of the track: rails, sleepers, ballast, and subballast. The fourth layer, i.e., the subballast, is supported by the subgrade.

In this model, the stiffness and damping coefficient of the subgrade are expressed by the connections between the subballast and subgrade. Therefore, the subgrade is modeled as a rigid boundary. Ballast and subballast are then modeled as equivalent pyramid blocks. Four blocks are used for each sleeper as shown in Figs. 1 and 2 (two in the vertical direction and two in the lateral direction). The pyramid blocks have one DoF (vertical) each and have the properties of mass, stiffness, and damping coefficient. Each block is connected to adjacent blocks that are in the same layer in the vertical direction.

Each sleeper has three DoFs: vertical translation (bounce), lateral translation, and rotation about the longitudinal axis (roll). The sleepers have the properties of mass, rotational (roll) inertia, stiffness, and damping coefficient. Fasteners and pads are modeled as linear springs and dampers that connect sleepers to rails. The rails are modeled as Timoshenko beams Display Formula

(1)$ρRAR∂2ωR∂t2−GARkR∂2ωR∂x2−∂φR∂x=∑i=1nsFpnδx−xsi−∑j=1nwFchδx−xwj$
Display Formula
(2)$ρRIRy∂2φR∂t2−EIRy∂2φR∂x2−GARkR∂ωR∂x−φR=−∑j=1nwMchδx−xwj$

where $ρR$ is the density of the rail, $AR$ is the cross-sectional area of the rail, $ωR$ is the deflection of the rail, $t$ is time, $G$ is the shear modulus of the rail, $kR$ is the Timoshenko shear coefficient, $x$ is the distance along the rail, $φR$ is rotation of the rail, $ns$ is the total number of sleepers, $i$ indicates the ith sleeper of the track, $Fpn$ is force between the rail and the sleeper, $δ$ indicates the Dirac delta function, $xsi$ is the longitudinal position of the sleeper under the rail, $nw$ is the total number of wheelsets on the rail, $j$ indicates the jth wheelset on the rail, $Fch$ is the wheel-rail contact force, $xwj$ is the longitudinal position of the wheelset on the rail, $IRy$ is the second moment of area of the rail section, $E$ is the Young's modulus, and $Mch$ is the creep moment of the wheel-rail contact.

The track model is numerically solved using the modal superposition method. For each mode of vibration, each rail has five motions: vertical translation (bounce), lateral translation, rotation about the lateral axis (pitch), rotation about the vertical axis (yaw), and rotation about the longitudinal axis (roll). In other words, each point of interest on the rail has five DoFs. Equations (1) and (2) can be used to describe the beam motions due to lateral and vertical loads: vertical translation, lateral translation, rotation about the lateral axis, and rotation about the vertical axis. The motion of rotation about longitudinal axis needs to be described using the following equation: Display Formula

(3)$ρRJOR∂2ϑR∂t2−GJRt∂2ϑR∂x2=∑i=1nsTpnδx−xsi−∑j=1nwTchδx−xwj$

where $JOR$ is the polar moment of inertia of the rail section, $ϑR$ is the rotation angle about the longitudinal axis, $JRt$ is a torsional parameter, $Tpn$ the torque from the ith sleeper, and $Tch$ is the torque from the wheel-rail contact. According to the modal superposition method, the following relationships exist: Display Formula

(4)$ωR=∑k=1nmapwktsinkπxL$
Display Formula
(5)$φR=∑k=1nmbpΦktcoskπxL$
Display Formula
(6)$ϑR=∑k=1nmcpΨktsinkπxL$

where $nm$ is the total number of vibration modes considered in the simulation, $k$ indicates the kth vibration mode of the rail, $ap,bp$, and $cp$ are constants that are related to the length and geometry of the rail, $L$ is the total length of the rail, $wk$, $Φk$, and $Ψk$ are time variables that need to be determined using Eqs. (1)(6). It has to be pointed out that the modal superimposition method assumes that the rail is infinitely long, so the rail model does not need constraints at the ends. However, this is not practical for numerical modeling. Actually, in the rail model, each rail has a certain length of $L$. The rail length has to be long enough to eliminate the implications of vehicle loads for the vibrations of rail ends. In other words, the deflections of the rail ends should always be zero or near zero. Therefore, the length of the simulated rail can be different for different simulation cases. Having determined the time variables, the motions of the rail can be determined Display Formula

(7)$Dy=∑k=1nmapsinkπxLwkyt;Dz=∑k=1nmapsinkπxLwkZt$
Display Formula
(8)$θx=∑k=1nmdpsinkπxLΨkt;θy=∑k=1nmepsinkπxLΦkyt;θz=∑k=1nmbpsinkπxLΦkzt$

where $Dy$ and $Dz$ are the lateral and vertical displacement of the rail, respectively, $θx$, $θy$, and $θz$ are rotation angle of the rail with respect to the longitudinal, lateral and vertical axis, respectively, $dp$ and $ep$ are constants that are related to the length and geometry of the rail, $wky$ and $Φky$ are time variables for the cases with lateral loads, and $wkZ$ and $Φkz$ are time variables for the cases with vertical loads.

Having reached this step, the track model has been developed and can be simulated. However, the original track model has a very low computational efficiency with a computing speed about 2300 times slower than real-time. For example, the simulation case listed in Table 1 (5 s of vehicle operation time) took the original track model about 3.2 h to finish. Under this situation, a source code profiling tool called Intel VTune Amplifier was used to profile the source code (in fortran language). It was found that most of the computing time (nearly 90%) was taken by the operations of sinusoidal functions in Eqs. (7) and (8). This is mainly due to (a) the sinusoidal function is a computationally expensive operation type compared with other basic operations; (b) the sinusoidal function has to be used for each rail, each DoF of the rail, each vibration mode of the rail; (c) Eqs. (7) and (8) have to be used for each point of the rail where a sleeper is attached so as to determine the motions of the sleepers, ballast, and subballast; (d) as the rails have very high stiffness and very low damping, a small time-step-size (0.01 ms) had to be used for the numerical solver, and the above operations have to be done for each time-step. For these reasons, the accumulated computing time for the sinusoidal functions is overwhelming.

Examining Eqs. (7) and (8), one can see that all sinusoidal functions are the same and the functions are only related to three variables: the vibration mode of the rail, the position of the sleeper, and the total length of the rail. Actually, all variables can be determined before the time integration, so the sinusoidal functions can be replaced by a two-dimensional look-up table with the indexes of sleeper sequence and vibration mode. The look-up table can be setup before the time integration, so the sinusoidal functions are only needed once. Using this method and some other programing techniques such as pointer arrays, the computing speed of the track model was significantly improved. The improved model is now more than ten times faster than the original code depending on different simulation cases.

Modal superposition is a widely used solving method for track modeling as reviewed in Sec. 1. Therefore, computing speed issues regarding the sinusoidal functions could also exist in other models. The look-up table method described in this section can also be beneficial for other track models.

## Vehicle Model

A locomotive model was used as an example to demonstrate the parallel co-simulation method. Note that the method does not differentiate between different vehicle models as the connection between the track model and the vehicle model is the wheel-rail contact which is a common part for all types of railway vehicles. The locomotive model is based on a heavy haul diesel-electric locomotive [40] and was developed using the commercial software package gensys. Key parameters of the locomotive model are listed in Table 2. The model simulates a six axle locomotive with two bogies; the bogie spacing is 14.32 m. The locomotive has the maximum power of 3100 kW and an axle load of 22.3 tons. The suspension system is shown in Fig. 3. Each primary suspension has two vertical coil springs and a longitudinal traction rod. A pair of vertical dampers is also used for the first and third wheelsets of each bogie. Regarding the secondary suspension, the simple solution of rubber springs is used. The rubber springs provide vertical stiffness and small vertical damping; they also provide shear stiffness for the longitudinal and lateral directions. Longitudinal forces are mainly transferred via the traction rod and yoke.

The wheel-rail contact model used in this paper has a multiple point contact geometrical model. It allows up to three simultaneous points of contact. The wheel and rail profiles are shown in Fig. 4. The normal forces and contact patch geometries are determined using linear springs (with damping) and the Hertz theory. Specifically, an equivalent penetration is determined by the geometrical model for each contact point. Then, the magnitude of the normal force is determined by using a linear spring model with a small amount of damping. Finally, the contact patch geometries are determined using the Hertz model. The FASTSIM algorithm is used to determine the tangential forces. To demonstrate the feasibility of the proposed parallel co-simulation method and demonstrate the interactions of the locomotive model and track model, a track defect will be used for simulations. Track irregularities are not used in this paper so the patterns in the results can be better seen. Note that this is for the case of locomotive simulations. If wagon models with friction suspensions are simulated, track irregularities are recommended to be used to excite the model so as to avoid jamming in the friction suspension.

## Parallel Co-Simulation Technique

As the track model and locomotive model were developed in two separate software packages, a co-simulation technique is needed to combine these models. In addition, as the track and vehicle models can run mostly independently, a parallel computing technique is applied to improve the simulation speed as well as to simplify the model integration process.

###### Co-Simulation Based on Transmission Control Protocol/Internet Protocol.

Various co-simulation techniques are available for dynamics simulations [4143]; the one used in this paper is based on the transmission control protocol/internet protocol (TCP/IP) as explained in Refs. [41] and [42]. The TCP/IP technique allows the in-house model, which is written in fortran language, to be connected without recompilation of the commercial software package. For gensys, using in-house fortran codes without co-simulations often involves recompilation of the main gensys program, as demonstrated in Ref. [44]. Note that most commercial software packages including gensys have ready-to-use co-simulation interface with matlab. But the connections to in-house models written in lower levels of languages, such as fortran, c/c++, are still limited.

In this paper, gensys is assigned as the host while the track model is assigned as the client as shown in Fig. 5. Reprogramming was mainly conducted in the track model. For the track model, the communication process is as follows:

1. (1)establish a communication socket and then connect to the host;
2. (2)ask addresses for the data that need to be sent to gensys and for that need to be received from gensys;
3. (3)send required data to the predefined addresses;
4. (4)send a message to gensys and ask gensys to proceed to the next time step;
5. (5)transfer the data in the receiving addresses into the track model;
6. (6)close the socket when the simulation is finished.

Steps (3)–(5) are executed repeatedly for each communication cycle. Note that the communication cycle (or the communication time-step-size) does not need to be the same as the computing time-step-size. The communication step-size can be larger than the computing step-size for most simulations.

As shown in Fig. 5, during the communication process, rail positions and velocities are sent to gensys first. The positons and velocities include those of vertical translation, lateral translation, and rotation about the longitudinal axis. Having received these information, gensys will then update the rail positions and velocities accordingly. Using the adjusted rail positions and velocities, vehicle dynamics and wheel-rail contact forces are calculated. The wheel-rail contact forces are then sent to the track model. These wheel-rail contact forces include normal contact forces, lateral creep forces, and longitudinal creep forces. The positions of the wheel-rail contact points are also sent back to the track model.

Having received wheel-rail contact forces and contact positons from gensys, the track model will attach all contact forces to the corresponding contact positons. Then, a new computing time-step can be proceeded with. In this paper, the computing step-sizes for the track model and the locomotive model were 0.01 ms and 1 ms, respectively; the communication step-size was set to be 1 ms. The step-sizes were selected according to the authors' many years of simulation experience. And the selection has the following technical considerations. First, the simulated rails have very high stiffness, and they require small step-sizes to achieve accurate and stable simulations. For the specific rail model used in this paper, a maximum step-size about 0.03 ms is required when using the Newmark-Beta numerical solver (this solver is used for this paper). Sensitivity analyses regarding different step-sizes of the rail model can be found in Ref. [40]. Therefore, in this paper, the step-size of 0.01 ms was selected for the simulation of the track. Note that in this specific track model, the 0.01 ms step-size was used in for all track model components including sleepers, ballast, and subballast. Further computing speed improvement can be achieved by using different (larger) step-sizes for sleepers, ballast, and subballast. This concept has been demonstrated by D'Adamio et al. [45]. Regarding the step-size of the locomotive model, 1 ms is commonly used for the gensys software package and has been tested for many simulations. Having set the step-size of the locomotive model to be 1 ms, then it is justified to use the same step-size for the communication process.

###### Parallel Computing and Parallel Co-Simulation Architecture.

Having achieved the co-simulation between the track model and the locomotive model, a deeper look into Fig. 5 reveals that the simulations of the track and the locomotive are mostly independent. Specifically, within each communication step, the simulations of the track and the locomotive do not need the exchange of data. And the simulations within the communication step take a much longer time than the data exchange. Therefore, the co-simulation case is suitable for parallel computing [46].

Parallel computing is a computational method that uses multiple computing cores to process multiple computing tasks simultaneously. The primary purpose of parallel computing is to save computing time. In the case of this paper, both track simulation and locomotive simulation are time consuming. Therefore, parallel computing can be a good option to further improve the computing speed of the model. To conduct parallel computing, multiple computer cores are needed. And usually one computer core serves as the master core while all other cores serve as slave cores. The master core is usually assigned to coordinate the program and to distribute computing tasks. The slave cores usually receive computing tasks and then report the results to the master core. Slave cores work in parallel so as to save computing time. After the computing task distribution, the master core can also join the slave cores and process some computing tasks in parallel with the slave cores. To facilitate parallel computing, some enabling techniques such as the message passing interface (MPI) [47,48] and open multiprocessing (OpenMP) [42] are needed. These techniques define the rules regarding communications between different computer cores.

Specifically, in this paper, the parallel co-simulation architecture is shown in Fig. 6. The MPI technique is used. An initiation and coordination module was added to:

1. (1)initialize the program;
2. (2)initialize MPI and multiple computer cores;
3. (3)define some global parameters such as the total operation time and communication time step-size.

Two computer cores are used: core 0 as the master core and core 1 as the salve core. In this specific case, the master core and the slave core will process computing tasks in parallel: core 0 undertakes track simulation while core 1 undertakes locomotive simulation.

As the TCP/IP co-simulation interface was coded using c/c++ language while the track model was developed using fortran language, another module (data collection in Fig. 6) was developed using fortran to make the parallel programming easier. Hence, the coordination process for the parallel computing is only related to the track model and the data collection module. The TCP/IP co-simulation interface and gensys are linked to the data collection module. In other words, the co-simulation interface and gensys participate in the parallel computing indirectly via the data collection module.

There are multiple advantages using this parallel computing architecture. First, the architecture offers flexibility for further extension of the simulation model. For example, an ongoing project is to add a longitudinal train dynamics model [49] and a traction control model [50] into the simulation. In this case, no changes are need for the current architecture, and the add-on models can be “plugged” in easily. Second, nearly 90% of computing time that was taken by the locomotive simulation can be saved. Note that the computing speed of the track model is significantly slower than that of the locomotive model, being about 200 times and 5 times slower than real time for the former and the latter. The final computing speed while using parallel computing is mainly limited by the computing speed of the track model. This is due to the uneven load balance between different computer cores as explained in Ref. [51]. Third, the parallel computing architecture only requires two computer cores which can be provided by all modern personal computers (no super computer is needed).

Regarding the disadvantages or limitations of using this method, the following has been observed. First, two layers of message passing are used, i.e., MPI and TCP/IP. Understandably, there is time overhead for message passing between individual computing cores as discussed in Ref. [50]. Second, also due to the utilization of MPI and TCP/IP techniques, the implementation is more complicated than other applications which uses only one technique.

## Vehicle-Track Interactions

Vehicle-track interaction simulations are conducted for the locomotive model described in Sec. 3 negotiating a section of track with three consecutive unsupported sleepers. Unsupported sleepers [5257] are common track faults which can be caused by track settlement, poor drainage, etc. In this paper, the unsupported sleepers are modeled as shown in Fig. 7. The ballast blocks underneath the unsupported sleepers are completely removed. This is achieved by setting the stiffness and damping coefficient that connect the unsupported sleepers and the removed ballast blocks as zero.

Two simulations were conducted for the cases with and without the unsupported sleepers. Key parameters of the track model are listed in Table 3. In the simulations, the locomotive was set to run at a constant speed of 80 km/h. The initial positon of the locomotive was set at 18.02 m on the track section. The unsupported sleepers were set at the middle point of the track.

Three demonstrative results were selected and presented in this paper. (1) Vertical deflections of the right rail (in the running direction of the vehicle) at the central point over the three unsupported sleepers (Fig. 8). This can be regarded as a demonstrative result for the track model. (2) Normal forces of the wheel-rail contact point of the right wheel of the leading wheelset (Fig. 9). The wheel-rail contact force is a demonstrative result of the wheel-rail interface which is also the interface of the track model and the vehicle model. (3) Vertical forces in the right primary suspension on the leading wheelset (Fig. 10), which can be regarded as a demonstrative result of the vehicle model.

Figure 8 shows clear deflection changes when two bogies are passing the middle point of the rail. These are represented by the two dips in the figure. Figure 8 also shows that the vertical deflection of the rail is significantly larger when the sleepers are unsupported. The maximum deflection of the rail under the normal condition was about 2.8 mm while that under the unsupported condition reached 13.4 mm.

Figure 9 shows that, at the initial stage of the simulations, there were shock forces in the model due to gravity. This is commonly seen in many dynamics models and can be avoided by performing static equilibrium calculations. Note that the initial shocks do not affect the consequent simulations and the final conclusions of this paper. Figure 9 shows that the normal forces in the wheel-rail contact patches have evident fluctuations when sleepers are unsupported. The biggest force differences during the fluctuations were about 30 kN and about 28% of the static normal forces.

Similar to Fig. 9, the initial shock forces due to gravity can also be seen in Fig. 10. Figure 10 shows that, with unsupported sleepers, due to the force fluctuations in the wheel-rail contact, the vertical forces in the primary suspensions also show clear fluctuations. The biggest force differences during the fluctuations were about 17 kN and 18% of the static vertical forces.

## Discussion and Conclusion

This paper developed a parallel co-simulation method to study vehicle-track dynamic interactions. The parallel co-simulation method uses the TCP/IP protocol to enable the co-simulation between a detailed in-house track dynamics simulation package and a commercial vehicle system dynamics simulation package (gensys). Then, the MPI technique is used to enable the model to process track dynamics simulations and vehicle dynamics simulations in parallel.

The co-simulation technique offers access to the advantages of both in-house track simulation packages and commercial vehicle simulation packages. The parallel computing technique offers flexibility for further extension of the simulation model. New parts can be easily added in as new parallel processes. Computing time can also be saved using the parallel computing technique. Note that the computing time that can be saved depends on the distribution of computing loads among the parallel processes.

The original track model used in this paper was significantly improved in terms of computing speed. The improved model is now more than ten times faster than the original model. The improvement was mainly achieved by tabulating the sinusoidal operations which are needed for the modal superposition method. Modal superposition is a widely used solving method for track modeling. The look-up table method described in this paper can also be beneficial for other track models.

Two simulations were conducted to model a locomotive negotiating a section of track with and without unsupported sleepers. The results show that the vertical rail deflections, wheel-rail contact forces, and vehicle suspension forces are evidently larger when unsupported sleepers are present. The simulations have demonstrated the effectiveness of the proposed parallel co-simulation method for vehicle-track dynamic interaction studies.

## Acknowledgements

The editing contribution of Mr. Tim McSweeney (Adjunct Research Fellow, Centre for Railway Engineering) is gratefully acknowledged.

## Funding Data

• Australasian Centre for Rail Innovation (ACRI) (HH3).

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Yang, S. , 2009, “ Enhancement of the Finite-Element Method for the Analysis of Vertical Train–Track Interactions,” Proc. Inst. Mech. Eng., Part F, 223(6), pp. 609–620.
Zhang, J. , Gao, Q. , Tan, S. , and Zhong, W. , 2012, “ A Precise Integration Method for Solving Coupled Vehicle—Track Dynamics With Nonlinear Wheel—Rail Contact,” J. Sound Vib., 331(21), pp. 4763–4773.
Martínez-Casas, J. , Giner-Navarro, J. , Baeza, L. , and Denia, F. , 2017, “ Improved Railway Wheelset-Track Interaction Model in the High-Frequency Domain,” J. Comput. Appl. Math., 309(1), pp. 642–653.
Ferrara, R. , Leonardi, G. , and Jourdan, F. , 2013, “ A Contact-Area Model for Rail-Pads Connections in 2-D Simulations: Sensitivity Analysis of Train-Induced Vibrations,” Veh. Syst. Dyn., 51(9), pp. 1342–1362.
Nilsson, C. , Jones, C. , Thompson, D. , and Ryue, J. , 2009, “ A Waveguide Finite Element and Boundary Element Approach to Calculating the Sound Radiated by Railway and Tram Rails,” J. Sound Vib., 321(3–5), pp. 813–836.
Gómez, J. , Vadillo, E. , and Santamaría, J. , 2006, “ A Comprehensive Track Model for the Improvement of Corrugation Models,” J. Sound Vib., 293(3–5), pp. 522–534.
Koro, K. , Abe, K. , Ishida, M. , and Suzuki, T. , 2004, “ Timoshenko Beam Finite Element for Vehicle-Track Vibration Analysis and Its Application to Jointed Railway Track,” Proc. Inst. Mech. Eng., Part F, 218(2), pp. 159–172.
Gry, L. , 1996, “ Dynamic Modelling of Railway Track Based on Wave Propagation,” J. Sound Vib., 195(3), pp. 477–505.
Dong, R. , Sankar, S. , and Dukkipati, R. , 1994, “ A Finite Element Model of Railway Track and Its Application to the Wheel Flat Problem,” Proc. Inst. Mech. Eng., Part F, 208(1), pp. 61–72.
Knothe, K. , and Grassie, S. , 1993, “ Modelling of Railway Track and Vehicle/Track Interaction at High Frequencies,” Veh. Syst. Dyn., 22(3–4), pp. 209–262.
Popp, K. , Kruse, H. , and Kaiser, I. , 1999, “ Vehicle-Track Dynamics in the Mid-Frequency Range,” Veh. Syst. Dyn., 31(5–6), pp. 423–464.
Blanco, B. , 2017, “Railway Track Dynamic Modelling,” Licentiate thesis, KTH, Stockholm, Sweden.
Meli, E. , and Pugi, L. , 2013, “ Preliminary Development, Simulation and Validation of a Weigh in Motion System for Railway Vehicles,” Meccanica, 48(10), pp. 2541–2565.
Sugiyama, H. , and Suda, Y. , 2009, “ On the Contact Search Algorithms for Wheel/Rail Contact Problems,” ASME J. Comput. Nonlinear Dyn., 4(4), p. 041001.
Sugiyama, H. , and Suda, Y. , 2008, “ Wheel/Rail Two-Point Contact Geometry With Back-of-Flange Contact,” ASME J. Comput. Nonlinear Dyn., 4(1), p. 011010.
Piotrowski, J. , and Kik, W. , 2008, “ A Simplified Model of Wheel/Rail Contact Mechanics for Non-Hertzian Problems and Its Application in Rail Vehicle Dynamic Simulations,” Veh. Syst. Dyn., 46(1–2), pp. 27–48.
Afshari, A. , and Shabana, A. , 2010, “ Directions of the Tangential Creep Forces in Railroad Vehicle Dynamics,” ASME J. Comput. Nonlinear Dyn., 5(2), p. 021006.
Kalker, J. , 1991, “ Wheel-Rail Rolling Contact Theory,” Wear, 144(1–2), pp. 243–261.
Kalker, J. , 1982, “ A Fast Algorithm for the Simplified Theory of Rolling Contact,” Veh. Syst. Dyn., 11(1), pp. 1–13.
Polach, O. , 2005, “ Creep Forces in Simulations of Traction Vehicles Running on Adhesion Limit,” Wear, 258(7–8), pp. 992–1000.
Ju, S. , 2015, “ Study of Train Derailments Caused by Damage to Suspension Systems,” ASME J. Comput. Nonlinear Dyn., 11(3), p. 031008.
Pasquale, G. , Somà, A. , and Zampieri, N. , 2012, “ Design, Simulation, and Testing of Energy Harvesters With Magnetic Suspensions for the Generation of Electricity From Freight Train Vibrations,” ASME J. Comput. Nonlinear Dyn., 7(4), p. 041011.
Wu, Q. , Cole, C. , Spiryagin, M. , and Sun, Y. Q. , 2014, “ A Review of Dynamics Modelling of Friction Wedge Suspensions,” Veh. Syst. Dyn., 52(11), pp. 1389–1415.
Bruni, S. , Vinolas, J. , Berg, M. , Polach, O. , and Stichel, S. , 2011, “ Modelling of Suspension Components in a Rail Vehicle Dynamics Context,” Veh. Syst. Dyn., 49(7), pp. 1021–1072.
Rodikov, A. , Pogorelov, D. , Mikheev, G. , Kovalev, R. , Lei, Q. , and Wang, Y. , 2016, “ Computer Simulation of Train-Track-Bridge Interaction,” Conference on Railway Excellence, Melbourne, Australia, May 16–18, pp. 1–7.
Li, Y. , Xu, X. , Zhou, Y. , Cai, C. , and Qin, J. , 2016, “ An Interactive Method for the Analysis of the Simulation of Vehicle–Bridge Coupling Vibration Using ANSYS and SIMPACK,” Proc. Inst. Mech. Eng., Part F, epub.
Sun, Y. , Dhanasekar, M. , and Roach, D. , 2003, “ A Three-Dimensional Model for the Lateral and Vertical Dynamics of Wagon-Track Systems,” Proc. Inst. Mech. Eng., Part F, 217(1), pp. 31–45.
Spiryagin, M. , Wolfs, P. , Cole, C. , Spiryagin, V. , Sun, Y. Q. , and McSweeney, T. , 2016, Design and Simulation of Heavy Haul Locomotives and Trains, CRC Press, Boca Raton, FL.
Spiryagin, M. , Simson, S. , Cole, C. , and Persson, I. , 2012, “ Co-Simulation of a Mechatronic System Using Gensys and Simulink,” Veh. Syst. Dyn., 50(3), pp. 495–507.
Wu, Q. , Spiryagin, M. , Cole, C. , and Sun, Y. Q. , 2017, “ Introducing Wheel-Rail Adhesion Control Into Longitudinal Train Dynamics,” Int. J. Heavy Veh. Syst., accepted.
Burgelman, N. , Sichani, M. , Enblom, R. , Berg, M. , Li, Z. , and Dollevoet, R. , 2015, “ Influence of Wheel–Rail Contact Modelling on Vehicle Dynamic Simulation,” Veh. Syst. Dyn., 53(8), pp. 1190–1203.
Spiryagin, M. , Wu, Q. , Duan, K. , Cole, K. , Sun, Y. , and Persson, I. , 2017, “ Implementation of a Wheel–Rail Temperature Model for Locomotive Traction Studies,” Int. J. Rail Transp., 5(1), pp. 1–15.
D'Adamio, P. , Escalona, J. , Galardi, E. , Meli, E. , Pugi, L. , and Rindi, A. , 2016, “ Real Time Modelling of a Railway Multibody Vehicle: Application and Validation on a Scaled Railway Vehicle,” Third International Conference on Railway Technology: Research, Development and Maintenance, Sardinia, Italy, Apr. 5–8, Paper No. 259.
Negrut, D. , Serban, R. , Mazhar, H. , and Heyn, T. , 2014, “ Parallel Computing in Multibody System Dynamics: Why, When and How,” ASME J. Comput. Nonlinear Dyn., 9(4), p. 041007.
Wu, Q. , Cole, C. , and Spiryagin, M. , 2016, “ Parallel Computing Enables Whole-Trip Train Dynamics Optimizations,” ASME J. Comput. Nonlinear Dyn., 11(4), p. 044503.
Wu, Q. , Spiryagin, M. , and Cole, C. , 2017, “ Parallel Computing Scheme for Three-Dimensional Long Train System Dynamics,” ASME J. Comput. Nonlinear Dyn., 12(4), p. 044502.
Wu, Q. , Spiryagin, M. , and Cole, C. , 2016, “ Longitudinal Train Dynamics: An Overview,” Veh. Syst. Dyn., 54(12), pp. 1688–1714.
Spiryagin, M. , Wolfs, P. , Szanto, F. , and Cole, C. , 2015, “ Simplified and Advanced Modelling of Traction Control Systems of Heavy-Haul Locomotives,” Veh. Syst. Dyn., 53(5), pp. 672–691.
Wu, Q. , and Cole, C. , 2015, “ Computing Schemes for Longitudinal Train Dynamics: Sequential, Parallel and Hybrid,” ASME J. Comput. Nonlinear Dyn., 10(6), p. 064502.
Sun, Y. , Spiryagin, M. , and Cole, C. , 2017, “ Simulation of Wheel-Rail Dynamics Due to Track Ballast Voids,” 25th International Symposium on Dynamics of Vehicles on Roads and Tracks, Rockhampton, Australia, Aug. 14–18, pp. 1–6.
Zhang, S. , Xiao, X. , Wen, Z. , and Jin, X. , 2008, “ Effect of Unsupported Sleepers on Wheel/Rail Normal Load,” Soil Dyn. Earthquake Eng., 28(8), pp. 662–673.
Zakeri, J. , Fattahi, M. , and Ghanimoghadam, M. , 2015, “ Influence of Unsupported and Partially Supported Sleepers on Dynamic Responses of Train-Track Interaction,” J. Mech. Sci. Technol., 29(6), pp. 2289–2295.
Lundqvist, A. , and Dahlberg, T. , 2005, “ Load Impact on Railway Track Due to Unsupported Sleepers,” Proc. Inst. Mech. Eng., Part F, 219(2), pp. 67–77.
Recuero, A. , Escalona, J. , and Shabana, A. , 2011, “ Finite-Element Analysis of Unsupported Sleepers Using Three-Dimensional Wheel–Rail Contact Formulation,” Proc. Inst. Mech. Eng., Part K, 225(2), pp. 153–165.
Zhu, J. , Thompson, D. , and Jones, C. , 2011, “Effect Unsupported Sleepers Dynamic Behaviour a Railway Track,” Veh. Syst. Dyn., 49(9), pp. 1389–1408.
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## References

Zhai, W. , Wang, K. , and Cai, C. , 2009, “ Fundamentals of Vehicle–Track Coupled Dynamics,” Veh. Syst. Dyn., 47(11), pp. 1349–1376.
Dahlberg, T. , 2006, “ Track Issue,” Handbook of Railway Vehicle Dynamics, Iwnicki, S. , ed., Taylor & Francis, London, Chap. 9.
Tanabe, M. , Sogabe, M. , Wakui, H. , Matsumoto, M. , and Tanabe, Y. , 2016, “ Exact Time Integration for Dynamic Interaction of High-Speed Train and Railway Structure Including Derailment During an Earthquake,” ASME J. Comput. Nonlinear Dyn., 11(3), p. 031004.
Zhai, W. , Xia, H. , Cai, C. , Gao, M. , Li, X. , Guo, X. , Zhang, N. , and Wang, K. , 2013, “ High-Speed Train–Track–Bridge Dynamic Interactions—Part I: Theoretical Model and Numerical Simulation,” Int. J. Rail Transp., 1(1–2), pp. 3–24.
Zhai, W. , Wang, S. , Zhang, N. , Gao, M. , Xia, H. , Cai, C. , and Zhao, C. , 2013, “ High-Speed Train–Track–Bridge Dynamic Interactions—Part II: Experimental Validation and Engineering Application,” Int. J. Rail Transp., 1(1–2), pp. 25–41.
Grassie, S. , Gregory, R. , Harrison, S. , and Johnson, K. , 1982, “ The Dynamic Response of Railway Track to High Frequency Vertical Excitation,” J. Mech. Eng. Sci., 24(2), pp. 77–90.
Thompson, D. , and Jones, C. , 2000, “ A Review of the Modelling of Wheel/Rail Noise Generation,” J. Sound Vib., 231(3), pp. 519–536.
Heckl, M. , 2002, “ Coupled Waves on a Periodically Supported Timoshenko Beam,” J. Sound Vib., 252(5), pp. 849–882.
Zhai, W. , and Wang, K. , 2010, “ Lateral Hunting Stability of Railway Vehicles Running on Elastic Track Structures,” ASME J. Comput. Nonlinear Dyn., 5(4), p. 041009.
Nielsen, J. , and Igelan, A. , 1995, “ Vertical Dynamic Interaction Between Train and Track—Influence of Wheel and Track Imperfections,” J. Sound Vib., 187(5), pp. 825–839.
Sun, Y. , and Dhanasekar, M. , 2002, “ A Dynamic Model for the Vertical Interaction of the Rail Track and Wagon System,” Int. J. Solids Struct., 39(5), pp. 1337–1359.
Baeza, L. , and Ouyang, H. , 2011, “ A Railway Track Dynamics Model Based on Modal Substructuring and a Cyclic Boundary Condition,” J. Sound Vib., 330(1), pp. 75–86.
Yang, S. , 2009, “ Enhancement of the Finite-Element Method for the Analysis of Vertical Train–Track Interactions,” Proc. Inst. Mech. Eng., Part F, 223(6), pp. 609–620.
Zhang, J. , Gao, Q. , Tan, S. , and Zhong, W. , 2012, “ A Precise Integration Method for Solving Coupled Vehicle—Track Dynamics With Nonlinear Wheel—Rail Contact,” J. Sound Vib., 331(21), pp. 4763–4773.
Martínez-Casas, J. , Giner-Navarro, J. , Baeza, L. , and Denia, F. , 2017, “ Improved Railway Wheelset-Track Interaction Model in the High-Frequency Domain,” J. Comput. Appl. Math., 309(1), pp. 642–653.
Ferrara, R. , Leonardi, G. , and Jourdan, F. , 2013, “ A Contact-Area Model for Rail-Pads Connections in 2-D Simulations: Sensitivity Analysis of Train-Induced Vibrations,” Veh. Syst. Dyn., 51(9), pp. 1342–1362.
Nilsson, C. , Jones, C. , Thompson, D. , and Ryue, J. , 2009, “ A Waveguide Finite Element and Boundary Element Approach to Calculating the Sound Radiated by Railway and Tram Rails,” J. Sound Vib., 321(3–5), pp. 813–836.
Gómez, J. , Vadillo, E. , and Santamaría, J. , 2006, “ A Comprehensive Track Model for the Improvement of Corrugation Models,” J. Sound Vib., 293(3–5), pp. 522–534.
Koro, K. , Abe, K. , Ishida, M. , and Suzuki, T. , 2004, “ Timoshenko Beam Finite Element for Vehicle-Track Vibration Analysis and Its Application to Jointed Railway Track,” Proc. Inst. Mech. Eng., Part F, 218(2), pp. 159–172.
Gry, L. , 1996, “ Dynamic Modelling of Railway Track Based on Wave Propagation,” J. Sound Vib., 195(3), pp. 477–505.
Dong, R. , Sankar, S. , and Dukkipati, R. , 1994, “ A Finite Element Model of Railway Track and Its Application to the Wheel Flat Problem,” Proc. Inst. Mech. Eng., Part F, 208(1), pp. 61–72.
Knothe, K. , and Grassie, S. , 1993, “ Modelling of Railway Track and Vehicle/Track Interaction at High Frequencies,” Veh. Syst. Dyn., 22(3–4), pp. 209–262.
Popp, K. , Kruse, H. , and Kaiser, I. , 1999, “ Vehicle-Track Dynamics in the Mid-Frequency Range,” Veh. Syst. Dyn., 31(5–6), pp. 423–464.
Blanco, B. , 2017, “Railway Track Dynamic Modelling,” Licentiate thesis, KTH, Stockholm, Sweden.
Meli, E. , and Pugi, L. , 2013, “ Preliminary Development, Simulation and Validation of a Weigh in Motion System for Railway Vehicles,” Meccanica, 48(10), pp. 2541–2565.
Sugiyama, H. , and Suda, Y. , 2009, “ On the Contact Search Algorithms for Wheel/Rail Contact Problems,” ASME J. Comput. Nonlinear Dyn., 4(4), p. 041001.
Sugiyama, H. , and Suda, Y. , 2008, “ Wheel/Rail Two-Point Contact Geometry With Back-of-Flange Contact,” ASME J. Comput. Nonlinear Dyn., 4(1), p. 011010.
Piotrowski, J. , and Kik, W. , 2008, “ A Simplified Model of Wheel/Rail Contact Mechanics for Non-Hertzian Problems and Its Application in Rail Vehicle Dynamic Simulations,” Veh. Syst. Dyn., 46(1–2), pp. 27–48.
Afshari, A. , and Shabana, A. , 2010, “ Directions of the Tangential Creep Forces in Railroad Vehicle Dynamics,” ASME J. Comput. Nonlinear Dyn., 5(2), p. 021006.
Kalker, J. , 1991, “ Wheel-Rail Rolling Contact Theory,” Wear, 144(1–2), pp. 243–261.
Kalker, J. , 1982, “ A Fast Algorithm for the Simplified Theory of Rolling Contact,” Veh. Syst. Dyn., 11(1), pp. 1–13.
Polach, O. , 2005, “ Creep Forces in Simulations of Traction Vehicles Running on Adhesion Limit,” Wear, 258(7–8), pp. 992–1000.
Ju, S. , 2015, “ Study of Train Derailments Caused by Damage to Suspension Systems,” ASME J. Comput. Nonlinear Dyn., 11(3), p. 031008.
Pasquale, G. , Somà, A. , and Zampieri, N. , 2012, “ Design, Simulation, and Testing of Energy Harvesters With Magnetic Suspensions for the Generation of Electricity From Freight Train Vibrations,” ASME J. Comput. Nonlinear Dyn., 7(4), p. 041011.
Wu, Q. , Cole, C. , Spiryagin, M. , and Sun, Y. Q. , 2014, “ A Review of Dynamics Modelling of Friction Wedge Suspensions,” Veh. Syst. Dyn., 52(11), pp. 1389–1415.
Bruni, S. , Vinolas, J. , Berg, M. , Polach, O. , and Stichel, S. , 2011, “ Modelling of Suspension Components in a Rail Vehicle Dynamics Context,” Veh. Syst. Dyn., 49(7), pp. 1021–1072.
Rodikov, A. , Pogorelov, D. , Mikheev, G. , Kovalev, R. , Lei, Q. , and Wang, Y. , 2016, “ Computer Simulation of Train-Track-Bridge Interaction,” Conference on Railway Excellence, Melbourne, Australia, May 16–18, pp. 1–7.
Li, Y. , Xu, X. , Zhou, Y. , Cai, C. , and Qin, J. , 2016, “ An Interactive Method for the Analysis of the Simulation of Vehicle–Bridge Coupling Vibration Using ANSYS and SIMPACK,” Proc. Inst. Mech. Eng., Part F, epub.
Sun, Y. , Dhanasekar, M. , and Roach, D. , 2003, “ A Three-Dimensional Model for the Lateral and Vertical Dynamics of Wagon-Track Systems,” Proc. Inst. Mech. Eng., Part F, 217(1), pp. 31–45.
Spiryagin, M. , Wolfs, P. , Cole, C. , Spiryagin, V. , Sun, Y. Q. , and McSweeney, T. , 2016, Design and Simulation of Heavy Haul Locomotives and Trains, CRC Press, Boca Raton, FL.
Spiryagin, M. , Simson, S. , Cole, C. , and Persson, I. , 2012, “ Co-Simulation of a Mechatronic System Using Gensys and Simulink,” Veh. Syst. Dyn., 50(3), pp. 495–507.
Wu, Q. , Spiryagin, M. , Cole, C. , and Sun, Y. Q. , 2017, “ Introducing Wheel-Rail Adhesion Control Into Longitudinal Train Dynamics,” Int. J. Heavy Veh. Syst., accepted.
Burgelman, N. , Sichani, M. , Enblom, R. , Berg, M. , Li, Z. , and Dollevoet, R. , 2015, “ Influence of Wheel–Rail Contact Modelling on Vehicle Dynamic Simulation,” Veh. Syst. Dyn., 53(8), pp. 1190–1203.
Spiryagin, M. , Wu, Q. , Duan, K. , Cole, K. , Sun, Y. , and Persson, I. , 2017, “ Implementation of a Wheel–Rail Temperature Model for Locomotive Traction Studies,” Int. J. Rail Transp., 5(1), pp. 1–15.
D'Adamio, P. , Escalona, J. , Galardi, E. , Meli, E. , Pugi, L. , and Rindi, A. , 2016, “ Real Time Modelling of a Railway Multibody Vehicle: Application and Validation on a Scaled Railway Vehicle,” Third International Conference on Railway Technology: Research, Development and Maintenance, Sardinia, Italy, Apr. 5–8, Paper No. 259.
Negrut, D. , Serban, R. , Mazhar, H. , and Heyn, T. , 2014, “ Parallel Computing in Multibody System Dynamics: Why, When and How,” ASME J. Comput. Nonlinear Dyn., 9(4), p. 041007.
Wu, Q. , Cole, C. , and Spiryagin, M. , 2016, “ Parallel Computing Enables Whole-Trip Train Dynamics Optimizations,” ASME J. Comput. Nonlinear Dyn., 11(4), p. 044503.
Wu, Q. , Spiryagin, M. , and Cole, C. , 2017, “ Parallel Computing Scheme for Three-Dimensional Long Train System Dynamics,” ASME J. Comput. Nonlinear Dyn., 12(4), p. 044502.
Wu, Q. , Spiryagin, M. , and Cole, C. , 2016, “ Longitudinal Train Dynamics: An Overview,” Veh. Syst. Dyn., 54(12), pp. 1688–1714.
Spiryagin, M. , Wolfs, P. , Szanto, F. , and Cole, C. , 2015, “ Simplified and Advanced Modelling of Traction Control Systems of Heavy-Haul Locomotives,” Veh. Syst. Dyn., 53(5), pp. 672–691.
Wu, Q. , and Cole, C. , 2015, “ Computing Schemes for Longitudinal Train Dynamics: Sequential, Parallel and Hybrid,” ASME J. Comput. Nonlinear Dyn., 10(6), p. 064502.
Sun, Y. , Spiryagin, M. , and Cole, C. , 2017, “ Simulation of Wheel-Rail Dynamics Due to Track Ballast Voids,” 25th International Symposium on Dynamics of Vehicles on Roads and Tracks, Rockhampton, Australia, Aug. 14–18, pp. 1–6.
Zhang, S. , Xiao, X. , Wen, Z. , and Jin, X. , 2008, “ Effect of Unsupported Sleepers on Wheel/Rail Normal Load,” Soil Dyn. Earthquake Eng., 28(8), pp. 662–673.
Zakeri, J. , Fattahi, M. , and Ghanimoghadam, M. , 2015, “ Influence of Unsupported and Partially Supported Sleepers on Dynamic Responses of Train-Track Interaction,” J. Mech. Sci. Technol., 29(6), pp. 2289–2295.
Lundqvist, A. , and Dahlberg, T. , 2005, “ Load Impact on Railway Track Due to Unsupported Sleepers,” Proc. Inst. Mech. Eng., Part F, 219(2), pp. 67–77.
Recuero, A. , Escalona, J. , and Shabana, A. , 2011, “ Finite-Element Analysis of Unsupported Sleepers Using Three-Dimensional Wheel–Rail Contact Formulation,” Proc. Inst. Mech. Eng., Part K, 225(2), pp. 153–165.
Zhu, J. , Thompson, D. , and Jones, C. , 2011, “Effect Unsupported Sleepers Dynamic Behaviour a Railway Track,” Veh. Syst. Dyn., 49(9), pp. 1389–1408.

## Figures

Fig. 2

Cross-sectional view of the track model

Fig. 1

Longitudinal view of the track model

Fig. 3

Locomotive suspension: (a) primary suspension and (b) secondary suspension

Fig. 4

Wheel and rail profiles

Fig. 5

Co-simulation architecture

Fig. 6

Parallel co-simulation architecture

Fig. 7

Track model with an unsupported sleeper

Fig. 8

Vertical deflection of the rail

Fig. 9

Vertical wheel-rail contact force

Fig. 10

Vertical force of primary suspension

## Tables

Table 1 Improvement of computing speed
Table 2 Key locomotive model parameters
Table 3 Key track model parameters

## Discussions

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