This paper presents a vehicle operation safety evaluation model; to this end, a nonlinear vehicletrack coupled dynamic system stochastic analysis model under random irregularity excitations based on probability density evolution method was developed. The nonlinear coupled vehicletrack dynamic system is used to accurately describe the wheelrail contact state. The stochastic functionspectral representation is used to simulate the random track irregularity in the time domain for the first time; consequently, the frequency components in the irregularity are preserved and random variables are reduced. In the process of evaluating the safety of train operation, the probability evolution, reliability of evaluation indices for different limit values, and evaluation indices for different probability limits are calculated for more accurate evaluation. The dynamic model and safety evaluation method was verified using the Zhaimodel and Monte Carlo method. The results show that, when the probability guarantee is increased, the running safety index of the vehicle increases more rapidly with running speed and the left/right wheelrail derailment coefficient increases rapidly at running speeds above 400 km/h. The computational model provides a novel direction for vehicle operation safety evaluation.
Vehicletrack coupled dynamic system is a nonlinear and stochastic system in nature as random track irregularity is one of its main random excitations. Meanwhile, the vehicle operation safety under random excitations is also the main transportation engineering concerns. In the past few decades, the popularization of highspeed railway and the continuous demand for high operating speed have brought many engineering challenges. In many countries, evaluation standards for train operation safety have been emplaced. Due to the complexity involved in the safety assessment of train operation, the problems generally need to be addressed through simulation methods. Several different traintrack coupled dynamics models or trainbridge coupled dynamics models have been established to analyze the dynamic response of the system. Wu et al. [
Due to the complexity of vehicletrack coupled dynamic system, Monte Carlo method (MCM) is a common analytical method in random analysis. Chen et al. [
The introduction of stochastic vibration analysis method makes it possible to use the probability theory to evaluate the safety of vehicletrack coupled power system or vehiclebridge coupled dynamic system. Rocha et al. presented a probabilistic methodology for the safety assessment of shortspan railway bridges for highspeed traffic [
In the existing random vibration analysis of vehicletrack coupled dynamics, due to the limitation of stochastic analysis methods or in order to improve the computational efficiency, linear or linearized models are often used. For the linearization of a model, the wheelrail relationship is often reduced to a linear model. The wheelrail relationship not only reflects the force state between wheel and rail, but also the excitation of track irregularity also needs to be input into the system through wheelrail relationship; this makes the wheelrail relationship one of the core issues in the modeling. The linearized model will not be able to accurately simulate the change of the size, direction, and action point of the wheelrail force, and it will also lead to a difference of the input track irregularity excitation input. Other than this, models with nonlinear wheelrail contact relationship allow for the evaluation of vehicle safety in a more realistic way than linear methods [
In this paper, a stochastic analysis model of nonlinear vehicletrack coupled dynamic system under random irregularity excitations was established based on probability density evolution method. The nonlinear wheelrail relationship has been considered to facilitate more convenient and accurate evaluation of vehicle operational safety. Generalized probability density evolution theory has been used, which has been widely applied to linear and nonlinear stochastic systems and have proven to obviously improve the efficiency the analysis [
Figure
Flow chart of the method for evaluating train operation safety.
Under random irregular excitations, the nonlinear vehicletrack coupled dynamic system is expressed as
Most of the randomness in a vehicletrack coupled dynamic system derives from the random track irregularities. A nonlinear vehicletrack dynamic system
Expressing the joint probability density function (
The generalized probability density evolves by the following equation [
In this formula,
Solving the above evolution equation of generalized probability density under the initial conditions, we obtain the joint probability density function of sample
For the probabilistic conservative vehicletrack coupled dynamic system under the random track irregularity excitation, the system dynamic response
In generalized probability density evolution theory, selecting the random variable samples is directly related to the computational efficiency of the random analysis. In this paper, the random variables are described by a lowdeviation sequence selected by number theory, which reduces the required number of random samples and accelerates the convergence of the calculation. However, as the lowdeviation sequence is uniformly distributed, it cannot be selected from a highdimensional space without forming void and cluster structures. Therefore, the uniformity will degrade and the calculation results will contain large errors.
Random irregularity is the superposition of irregular waves of different wavelengths, different phases, and different amplitudes. To accurately simulate the random track irregularities, more random frequencies and phase components (i.e., more selections) are needed. To avoid clustering in the random sample selection, the random track irregularities were simulated in the time domain by a stochastic functionspectral representation method, which requires just eight random variables. The generation method and validation of the sampling will be discussed in Section
The random samples in the probabilistic space Ω_{Θ} must then be divided. Probabilistic spaces are commonly divided into representative Voronoi regions with representative volumes
The assigned probability
This method obtains the initial probability of each random point sample.
Figure
Vehicletrack coupled dynamics model.
Figure
Vehicle dynamics model.
The antisnake shock absorber significantly affects the driving quality when the train is running at a high speed. Treating the force of the antisnake shock absorber on the vehicle and the antirolling moment of the vehicle’s central suspension as an additional force on the vehicle, the dynamic equation of the vehicle is given by
The freedom matrix of the vehicle is expressed as
The vehiclemass matrix is given by
The vehiclestiffness matrix is given by
The vehicledamping matrix
The additional moment on the vehicle is expressed as the matrix
In the above expressions,
The vehicleload matrix is expressed as
The track structure is divided into two main parts: the rail and the track plate. The rail is treated as an elastic support for an Euler beam, and the track plate and base plate are modeled by a plate element with high rigidity in the transverse direction, lateral displacement using beam element to simulate. Since the numerical integration solution step length is short and requires a lot of numerical calculation work, the method of the literature [
Structural model of the railtrack plate.
Rail force diagram.
The following expressions are shown for the right rail (the left rail is treated analogously). The fastener force matrix is expressed as
The torque of the wheelrail force, which induces rail torsion, is
The differential equations of the vertical, transverse, and torsional vibrations of the rail are, respectively given in the following [
Vertical:
Transverse:
Torsional:
The track and base plates are considered as horizontally rigid bodies and are modeled as fournode bending plate elements. They are connected by a fastener element, which is simulated by a linear spring damping element. Slabbase has been modeled by the principle of total potential energy with stationary value in elastic system dynamics [
The wheelrail relationship links the vehicle system to the track system. In the vehicletrack coupled dynamics, the wheelrail relationship mainly includes the wheelrail normalseeking solution, the wheelrail creep solution, and the wheelrail contact geometry. The wheelrail relationship is shown in Figure
Wheelrail contact model.
The pressure distribution on the contact surface is given by Hertz contact theory.
Simplified schematic of wheelrail contact.
The normal force on the wheel and rail is expressed as
And the maximum pressure between the wheel and rail is given by
In the above expressions,
Correction factors
Geometric wheelrail contact model.
Fitting parameters of the

 

Parameters  Error of fitting  Parameters  Error of fitting  

0.001227  SSE: 3.522e07 

0.002195  SSE: 3.156e07 

−0.05683  Rsquare: 1 

−0.07143  Rsquare: 1 

0.7315  Adjusted Rsquare: 1 

0.4331  Adjusted Rsquare: 1 

4.087  RMSE: 5.935e07 

4.816  RMSE: 5.618e07 

20.1 

41.2  

36.46 

21.94  

4.599 

5.123  

20.1 

41.2  

36.46 

21.94 
When calculating the normal contact force between the wheel and rail, the relative compression between the wheel and a rail with random irregularities is obtained by geometrically solving the wheelrail contact. The normal force between the wheel and rail is then obtained by solving the wheelrail normal contact model and determining the direction and magnitude variations in the normal wheelrail force at different contact points. Converting to the absolute coordinate system, the normal loads on the wheel and rail array are, respectively, expressed as
Here,
In the above expressions,
The calculated creep force and creep torque between the wheel and rail needs to be converted to the absolute coordinate system. The load array of the wheelrail creep force is expressed as
The random track random is regarded as the geometric displacement of the rail. Considering the transverse, vertical, and torsional degrees of freedom of the rail and the track irregularity, the discrete coordinate transformation of the rail is expressed as follows:
As shown in Figure
Comparison of track irregularity samples. (a) Vertical profiles of track irregularities, (b) track alignment irregularities, (c) track crosslevel irregularities, and (d) track gage irregularities.
Finally, the normal force between the wheel and rail is obtained by inserting the wheelrail normal compression into the wheelrail normal contact model.
In general, when simulating the random track irregularity timedomain samples, they often need dozens or hundreds of random variables. If the number theory method is used to select the random variables, it will inevitably produce certain aggregations, especially when highdimensional selection is performed. To accurately reflect the entire random variable space, it is necessary to distribute the random variable points in the random variable space as much as possible. The generation of the clustering phenomenon will increase the number of random samples required for the calculation to some extent. Therefore, the random track irregularities timedomain samples were simulated by the spectral representation randomfunction method, which reduces the required number of random variables in the simulation. In a onedimensional, univariate, stochastic process of the power spectral density function
Discretizing the above formula and introducing a set of standard orthogonal random variables {
The orthogonal random variables {
As an example, we consider the “TB/T33522014 highspeed railway ballastless track irregularity spectrum” implemented in 2015. Three hundred samples of random track irregularities were generated by the spectral representation randomfunction method. When generating the power spectrum of the track irregularities, the unit length and sampling interval were set to 1024 m and 0.25 m, respectively, with 4096 points in a single unit [
Track vertical profile irregularity:
Alignment irregularity:
Crosslevel irregularity:
Gage irregularity:
It should be noted that the track irregularity spectrum is averaged over many measured data. As the random error between a single sample and the track irregularity spectrum is 100%, the accuracy of the simulation method cannot be validated by comparing the simulated power spectrum of the track irregularity with that obtained from a single or finitelength sample, but validated by comparing the simulated power spectrum of the track irregularity with that obtained from a sample set.
Figure
The power spectral density and track irregularity power spectrum derived from a single sample did not accurately estimate the vertical profile, alignment, crosslevel, or gage irregularity of the track (Figure
The trackvehicle coupled dynamic model was solved by an explicitimplicit integration method, with time integral step less than 0.0001 seconds, and the mode number of rail is taken as half the number of rail support points. The displacement and speed of the train and rail at the (
The vehicle parameters are shown in Table
Vehicle dynamics parameters.
Item  Notation  Unit  Value  

Car body 


kg  33.2 × 10^{3} 
Mass moment of inertia about 

kg·m^{2}  1.07568 × 10^{5}  
Mass moment of inertia about 

Kg·m^{2}  1.6268 × 10^{6}  
Mass moment of inertia about 

kg·m^{2}  1.4027 × 10^{6}  
Bogie 


kg  2.6 × 10^{3} 
Mass moment of inertia about 

kg·m^{2}  2106  
Mass moment of inertia about 

kg·m^{2}  1424  
Mass moment of inertia about 

kg·m^{2}  2600  
Wheelset 


kg  1.97 × 10^{3} 
Mass moment of inertia about 

kg·m^{2}  623  
Mass moment of inertia about 

kg·m^{2}  78  
Mass moment of inertia about 

kg·m^{2}  623  
Primary suspension system  Longitudinal spring stiffness 

N/m  9.8 × 10^{6} 
Lateral spring stiffness 

N/m  9.8 × 10^{6}  
Vertical spring stiffness 

N/m  1.176 × 10^{6}  
Longitudinal damping coefficient 

N·s/m  0  
Lateral damping coefficient 

N·s/m  0  
Vertical damping coefficient 

N·s/m  0.196 × 10^{3}  
Second suspension system  Longitudinal spring stiffness 

N/m  1.784 × 10^{5} 
Lateral spring stiffness 

N/m  1.931 × 10^{5}  
Vertical spring stiffness 

N/m  1.931 × 10^{5}  
Longitudinal damping coefficient 

N·s/m  0  
Lateral damping coefficient 

N·s/m  5.88 × 10^{4}  
Vertical damping coefficient 

N·s/m  9.8 × 10^{4}  
Half the transverse distance between vertical primary suspension systems 

m  1.0  
Half the transverse distance between vertical primary suspension system absorber 

m  1.0  
Half the transverse distance between vertical second suspension systems 

m  1.25  
Half the transverse distance between vertical second suspension systems absorber 

m  1.25  
Vertical distance between center of gravity of car body and lateral secondary suspension system 

m  0.644  
Vertical distance between lateral secondary suspension system and center of gravity of bogie 

m  0.29  
Vertical distance between center of gravity of bogie and lateral primary suspension system 

m  0.08  
Half the longitudinal distance between center of rear bogie and center of front bogie 

m  8.75  
Half the bogie axle base 

m  1.25 
The vehicletrack coupled dynamic model proposed by Zhai and Chen [
Comparison of calculation results for a single sample. (a) Reduction rate of wheel load, (b) derailment coefficient of wheel load, (c) acceleration of car body, and (d) lateral wheelset force.
Random variables were selected by the MCM, and the random track irregularity was solved by the trigonometric series method. Five thousand random track irregularity samples were selected for computing the dynamic response of the vehicletrack coupled power system, and the statistical characteristics of each evaluation index were calculated. The probabilistic evolution processes of the indices were then calculated by the developed model, varying the number of samples as 100, 500, and 1000. Figure
Comparison of cumulative probabilities of the major vehiclesafety evaluation indices. Reduction rate of left wheel load (a) and right wheel load (b). Derailment coefficient of left wheel load (c) and right wheel load (d). Vertical acceleration of car body (e). Lateral acceleration of car body (f). Lateral wheelset force (g).
The ballastless track structure in China offers a comfortable riding experience. When a single vehicle is traveling at 350 km/h, its safety evaluation index (calculated by the developed method) meets the requirements of common specifications. As limiting values in the safety evaluation, the wheel load reduction rate was taken as 0.3, the derailment coefficient as 0.15, the lateral and vertical accelerations of the car body as 1.0 m/s^{2} and 0.5 m/s^{2}, respectively, and the lateral force as 13 kN. Figure
Equalprobability and reliability curves of the train safety evaluation indices. Reduction rates of left wheel load (a) and right wheel load (b). Derailment coefficients of left wheel load (c) and right wheel load (d). Vertical acceleration of car body (e). Lateral acceleration of car body (f). Lateral wheelset force (g).
Probabilities of limit values of the vehiclesafety evaluation indices.
Evaluating indicator  Reduction rate  Derailment coefficient  Vertical acceleration of car body  Lateral acceleration of car body  Lateral wheelset force  

Left  Right  Left  Right  
Min  0.9787  0.9857  0.9874  0.9821  0.9885  0.9853  0.9725 
Max  0.9989  0.9988  0.9989  0.9984  0.9997  0.9987  0.9948 
Given a vehicle and track structure with random track irregularities, the calculation results of a single sample cannot obtain the limits of the evaluation indices. In the presence of random irregularities, the limit values can be obtained only with a certain probability. For this purpose, the probabilistic index was introduced, providing a new way of computing the limit values of the indices of vehicletrack coupled dynamics.
Figure
Speed dependences of the probabilistic train safety evaluation indices. Reduction rates of left wheel load (a) and right wheel load (b). Derailment coefficients of left wheel load (c) and right wheel load (d). Vertical acceleration of car body (e). Lateral acceleration of car body (f). Lateral wheelset force (g).
This paper presents an evaluation model of safe railway operation based on probability theory and a nonlinear vehicletrack coupled power system. The model more accurately captures the force state between the wheel and rail than the linearized vehicletrack coupling dynamic system. The random track irregularities were sampled in the time domain by a stochastic functionspectral representation method, which reduces the required number of samplings. The developed simulation method was verified in comparisons with an established method. The safety of the running vehicles was probabilistically determined by the evaluation indices (derailment coefficient, acceleration of the car body, and the lateral force of the wheel axle). The model results were compared with those of the Zhaimodel and the MCM. Next, the vehiclesafety evaluation indices were computed for different probabilities of their limit values. At the lower probability (68.3%), the limiting evaluation indices were little affected by the vehicle speed, but at the highest probability (99.7%), they were significantly increased at higher vehicle speeds. The derailment coefficient deserves special attention when driving at high speed. The developed computational model provides a new research direction for vehicle operation safety evaluation. In future work, the model will be applied to safe railway operation under different conditions, such as different structures, different track or foundation diseases, and different track irregularities.
The analysis result data used to support the findings of this study are included within the article. The calculation data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
The research work described in this paper was supported by the Highspeed Railway Joint Fund of National Natural Science Foundation of China (grant U1734208), the Major Program of National Natural Science Foundation of China (grant 11790283), the Hunan Provincial Natural Science Foundation of China (grant 2019JJ40384), and the Hunan Provincial Innovation Foundation for Postgraduate (CX2017B058).