Flexible-Rigid Wheelset Introduced Dynamic Effects due to Wheel Tread Flat

Abstract

The widespread faults that occur in railway wheels and can cause a massive dynamic impact are the wheel tread flat. The current work considered changes in vehicle speed or wheel radius deviation and studied the dynamic impact load. The modal technique for the impact evaluation induced by the wheel flat was proposed via the finite element analysis (FEA) software package ANSYS, integrated into a multibody dynamics model of the high-speed train CRH2A (EMU) through SIMPACK. The irregularity track line has developed and depends on the selected simulation data points. Additionally, a statistical approach is designed to analyze the dynamic impact load response and effect and consider different wheel flat lengths and vehicle speeds. The train speed influence on the flat size of the vertical wheel-rail impact response and the statistical approach are discussed based on flexible, rigid wheelsets. The results show that the rigid wheel flat has the highest vertical wheel impact load and is more significant than the flexible wheel flat force. The consequences suggest that the wheelset flexibility can significantly improve vertical acceleration comparably to the rigid wheel flats. In addition, the rendering of the statistical approach shows that the hazard rate, PDF, and CDF influence increase when the flat wheel length increases.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Wheel/rail interaction is one of the best multiple research subjects in railway engineering [1]. It considers the safety, reliability, performance, comfort, and ride indicators of railway vehicle dynamics systems. Due to the defects produced by wheel/rail interaction, the dynamic system behavior has an abnormal effect. Wheel flat is one type of railway train wheelset defect caused by wheel sliding, scraped-off parts of the wheel tread, and spots on the wheel rolling surface [2, 3]. Furthermore, this cause may be poorly adjusted; a brake block has frozen on the wheel after braking or defective breaks, or too high breaking forces concerning the wheel/rail adhesion [4]. Moreover, severe impact loads are generated on the wheel/rail contact [5]. A wheel defect is particularly significant at high speeds, affecting operational safety in a highly adverse manner [6]. It could also pose more significant derailment risks, especially with higher axle loads and high-speed operations, apart from the higher stresses in the infrastructure and railway system components [7, 8]. The result could be enormous subsequent wheel damage, and some illustration of wheel flat defects is shown in Figures 1(a)–1(c).

[figures omitted; refer to PDF]

Many researchers have concentrated their research on dynamic impact forces triggered by wheel/rail defects in the past few decades. The consequence of high impact forces can excessively damage railway vehicles and tracks, in addition to damaging axles, axle boxes, bearings, cracks that penetrate the wheels, and railroad breaks that may derail [9]. The analytical research of wheel/rail defects confirmed the fairly excellent agreement with the measured peak contact force in the presence of wheel defects [10, 11]. Sadeghi et al. [12] developed effect of unsupported sleepers on rail track dynamic behavior. The dynamic wheel/rail impact forces and damages owing to wheel flat have been the subjects of deliberate overanalytical studies [8, 13–23] and experimental cases [2, 24]. Field experiments [2, 6, 10, 25] measure wheel/rail impact forces in an instrumented rail [26, 27]. Bian et al. [13] established a three-dimensional (3D) finite element (FE) model for wheel flat impact analysis using ANSYS analysis software package. It is observed that the wheel flat extensively increases the dynamic force on each wheel and rail. Finite element simulation for wheel/rail interaction cases is proposed [28–30]. Multibody dynamics (MBD) methods are commonly used to simulate wheel/rail collisions caused by wheel flat [18, 31], and recently the newly developed finite element method (FEM) has been proposed [32, 33]. The wheel flat can be directly geometrically modeled or simulated by implementing the relative displacement excitation between the wheel and the rail [31]. Nielsen et al. [34] proposed a modal method to develop a nonrotating flexible wheelset model integrated into the vehicle-track dynamic model to investigate the influences of rail corrugation and wheel out-of-roundness on the vertical contact forces. The study concluded that considering the wheelset flexibility in the model yields lower magnitudes of wheel/rail contact forces than the rigid wheelset. Popp et al. [35] proposed a modal approach to developing a rotating flexible wheelset model to study the wheelset flexibility influences on the vehicle stationary wheel/rail interactions and hunting stability. The above research shows that wheelset flexibility dramatically contributes to the vehicle high-frequency dynamic response. Baeza et al. [36] developed a rotating flexible wheelset model using the Euler’s method to study the high-frequency vehicle response caused by wheel flat and track corrugations. Besides, the rigid wheelset model overestimates the wheel/rail contact force compared with the elastic wheelset [34]. The reported vehicle-track coupling model has also been used to predict the formation of railway corrugations [37, 38]. These show the more significant contribution of the wheelset to the third bending mode, resulting in relatively large regular and longitudinal forces compared to rigid wheelsets. Torstensson et al. [39] compared the wheel/rail contact force responses obtained from the rigid, nonrotating, and rotating flexible wheelset models at high speed. It is concluded that the rotating flexible wheelset model can more reliably estimate wheel/rail contact force at a high-frequency range.

Previous studies focused on the influence of forces in the track structure and the changing details in the time domain or distance domain. Some parametric studies have been conducted, but these studies are done using traditional methods. The most important thing is that the influence of wheels on dynamic impact has not been thoroughly studied based on actual conditions. Furthermore, a flexible and rigid wheel flat is introduced in multibody dynamics (MBS) to obtain vertical impact loads and statistical methods to evaluate dynamic performance and vehicle dynamic safety.

In the current research work, the 3D railway vehicle/track model is used to study the influence of wheel flat on dynamic vehicle response. The vehicle/track model is based on a typical Chinese high-speed train, CRH2A. The simulation is carried out with different types of wheel flats and their relative positions. The influence of flat on front/rear bogie wheels of the same or different wheelsets is also studied. The wheel flat may generate a higher impact load, which may cause wheel/rail interaction surface defects. The primary suspension and secondary suspension characteristics may affect the wheel flat induction response of the high-speed railway. Moreover, a statistical method has been developed to evaluate the vertical impact loads caused by wheel flat based on hazard rate (HR), probability (PDF), and cumulative function (CDF).

In this study, the flexible wheelset model was formulated and incorporated into the coupled vehicle-track dynamic model to explore wheelset flexibility in vehicle response when the wheel tread flat is present. The modal synthesis technique is implemented to evaluate the dynamic response of rotating flexible wheelsets affected by the wheel flat induction impact load, thus integrating the multibody dynamics model of high-speed trains into SIMPACK software. Hertz contact models are used in a couple of vehicles and track models. According to the impact force of wheel/rail and vertical acceleration, the dynamic vehicle response with flexible wheelsets is evaluated and discussed under the condition of a haversine method.

2. Materials and Methodology

This study selected the multibody dynamic simulation (MBS) method because it is a cost-effective technique for evaluating railway vehicle wheel flat analysis. MBS allows the use of experimental methods to analyze various economically unfeasible operating and fault conditions as obtainable in Figure 2. The process is separated into five stages. The SIMPACK commercial software package presents a typical CRH2A high-speed rail multivehicle model in the first stage. The second stage is provided by FEM ANSYS flexible wheel flat. The MATLAB code is provided to estimate the wheel flat size. The fourth stage model introduced four kinds of track irregularity. The final work provides self-programming software through statistical analysis to assess the hazard rate and failure phenomenon.

[figure omitted; refer to PDF]

Based on the above methodology, the influence of flexible and rigid wheel flat was studied. Furthermore,the vehicle impact load effects due to speed and wheel flat length are explored. Besides, the impact on the dynamic safety and performance of vehicle components is assumed. Previous studies reported in the literature are used to verify the vehicle-track model that caused wheel flat defects. After confirming the model, a series of simulations are performed, and the statistical method can cover various wheel flat lengths, vehicle speeds, and track irregularities. It can predict the vertical wheel/rail impact results, lateral wheel/rail forces, and wheelset vertical acceleration.

3. Railway Vehicle Dynamics

The railway vehicle dynamics model includes three basic submodels: vehicle, track, and wheel/rail interaction [40]. In the current work, a multibody 3D dynamic model was established and constructed for all three subgroups. Based on the finite element program and vehicle-track coupling [41], the dynamic equation of motion for the vehicle can be expressed in the form of a submatrix:$$\begin{array}{}\text{(1)}& M_V{\ddot{\mathbf{D}}}_V+C_V{\dot{D}}_V+K_V{D}_V=P_{vt},\end{array}$$where ${\ddot{\mathbf{D}}}_V$, ${\ddot{\mathbf{D}}}_V$, and $D_V$ are the vectors of acceleration, velocity, and displacement of the vehicle subsystem, respectively. The subscripts $V$ mean the vehicle dynamics and track of the subsystem, respectively; $M_V$, $C_V$, $K_V$, and $P_{vt}$ refer to the subsystem matrices of mass, stiffness, damping, and external force, respectively.

3.1. Vehicle Model

In China, there are several types of trains in operation. The CRH2A high-speed train having a speed of 200-350 km/h is selected. According to the structural characteristics of the CRH2A vehicle, a multibody dynamic model of the single-vehicle is established, which includes the car body, four wheelsets, and front/rear bogies, respectively, as shown in Figures 3(a) and 3(b). Additionally, most recent research has used 35 degrees of freedom (DOFs) [41, 42]. The train displacement vector and force vector are the continuous measures of the corresponding vectors of the vehicle.

[figures omitted; refer to PDF]

The displacement vector $D_V$can be written as$$\begin{array}{}\text{(2)}& D_v={{\mathbf{D}}_c^T,{\mathbf{D}}_{b_1}^T,{\mathbf{D}}_{b_2}^T,{\mathbf{D}}_{w_1}^T,{\mathbf{D}}_{w_2}^T,{\mathbf{D}}_{w_3}^T,{\mathbf{D}}_{w_4}^T}^T.\end{array}$$

The subscripts c, $b_1$, $b_2$, $w_1$, $w_2$, $w_3$, and $w_4$ denote the car body, the front frame, the rear frame, and the four wheelsets, respectively.

${\mathbf{D}}_c,{\mathbf{D}}_{b1},{\mathbf{D}}_{w1},{\mathbf{D}}_{w2},{\mathbf{D}}_{b2},{\mathbf{D}}_{w3},{\mathbf{d}}_{w4}$ denote car body, bogie, and wheel displacement, respectively. The sub-displacement vector of the car body, the bogie, and the wheelset is provided in the following equations, and the superscript T denotes the transpose matrix of the vehicle:$$\begin{array}{c}\text{(3)}& D_c={y_c,z_c,{\theta }_c{\phi }_c,{\psi }_c}^T,D_b={y_b,z_b,{\theta }_b{\phi }_b,{\psi }_b}^T,\\ D_w={y_w,z_w,{\theta }_w{\phi }_w,{\psi }_w}^T.\end{array}$$

$$\begin{array}{c}\text{(4)}& M_v=\text{diag}{M}_c{M}_{b1}{M}_{w1}{M}_{w2}{M}_{b2}{M}_{w3}{M}_{w4},M_c=\text{diag}{M}_c{J}_{cx}{J}_{cy}{M}_c{J}_{cz},\\ M_b=\text{diag}{M}_b{J}_{bx}{J}_{by}{M}_b{J}_{bz},\\ M_w=\text{diag}{M}_w{M}_w{M}_W{J}_{wx}{J}_{wz},\end{array}$$where $M_c$, $M_b,$ and $M_w$ are the submass matrix of car body, the bogie, and the wheelset of the vehicle, respectively. The damping and stiffness matrix of the train systems are symmetric and can be established by assembling the corresponding matrix of the vehicle body independent of the adjacent ones within the concerned degree of freedom$$\begin{array}{c}\text{(5)}& {\mathbf{K}}_v=\begin{array}{ccc}{\mathbf{K}}_c& & 0\\ & {\mathbf{K}}_b& \\ \text{sym}& & {\mathbf{K}}_w\end{array},{\mathbf{C}}_v=\begin{array}{ccc}{\mathbf{C}}_c& & 0\\ & {\mathbf{C}}_b& \\ \text{sym}& & {\mathbf{C}}_w\end{array},\\ {\mathbf{K}}_v=\begin{array}{ccccccc}{\mathbf{K}}_{cc}& {\mathbf{K}}_{cb1}& {\mathbf{K}}_{cb2}& 0& 0& 0& 0\\ & {\mathbf{K}}_{b1b1}& 0& {\mathbf{K}}_{b1w1}& {\mathbf{K}}_{b1w2}& 0& 0\\ & & {\mathbf{K}}_{b2b2}& 0& 0& {\mathbf{K}}_{b2w3}& {\mathbf{K}}_{b2w4}\\ & & & {\mathbf{K}}_{w1w1}& 0& 0& 0\\ & & & & {\mathbf{K}}_{w2w2}& 0& 0\\ & & & & & {\mathbf{K}}_{w3w3}& 0\\ \text{Sym}& & & & & & {\mathbf{K}}_{w4w4}\end{array},\\ {\mathbf{C}}_v=\begin{array}{ccccccc}{\mathbf{C}}_{cc}& {\mathbf{C}}_{cb1}& {\mathbf{C}}_{cb2}& 0& 0& 0& 0\\ & {\mathbf{C}}_{b1b1}& 0& {\mathbf{C}}_{b1w1}& {\mathbf{C}}_{b1w2}& 0& 0\\ & & {\mathbf{C}}_{b2b2}& 0& 0& {\mathbf{C}}_{b2w3}& {\mathbf{C}}_{b2w4}\\ & & & {\mathbf{C}}_{w1w1}& 0& 0& 0\\ & & & & {\mathbf{C}}_{w2w2}& 0& 0\\ & & & & & {\mathbf{C}}_{w3w3}& 0\\ \text{Sym}& & & & & & {\mathbf{C}}_{w4w4}\end{array},\end{array}$$where ${\mathbf{K}}_c$, ${\mathbf{K}}_b$, and ${\mathbf{K}}_w$ are substiffness matrix of the car body, bogie, and wheelset, respectively, also similar to the damping submatrix.

The force vector applied to the vehicle is written as$$\begin{array}{}\text{(6)}& {\mathbf{P}}_v={{\mathbf{P}}_c{\mathbf{P}}_{b1}{\mathbf{P}}_{w1}{\mathbf{P}}_{w2}{\mathbf{P}}_{b2}{\mathbf{P}}_{w3}{\mathbf{P}}_{w4}}^T.\end{array}$$

If other external dynamic forces are not taken into account,$$\begin{array}{}\text{(7)}& {\mathbf{P}}_c={\mathbf{P}}_{b1}={\mathbf{P}}_{b2}=0,\end{array}$$where ${\mathbf{P}}_c$ and ${\mathbf{P}}_b$ are the subforce vectors acting into the car body and the bogie of this vehicle, respectively, and the submatrices $P_{wi}$ (i = 1, 2, 3, 4) denote the random load vector of the $i\text{th}$ wheelset. The complete information about a load of wheelset can be expressed in the wheel/rail interaction section.

3.2. Track Model

This kind of steel contains two parallel rails mounted on the pads by an elastic fastener, as illustrated in Figures 3(a) and 3(b). The connection between the guide rail and the sleeper (fastener) and the relationship between the sleeper and the roadbed include a longitudinal linear spring damper. The rail cant slope is 1 : 40, and this cant slop is available in SIMPACK rail properties. Detailed equations of motion of the rails and the sleepers can be found in [43]. The general equation of the track system can be expressed in matrix form:$$\begin{array}{}\text{(8)}& {\mathbf{M}}_t{\ddot{\mathbf{D}}}_t+{\mathbf{C}}_t{\dot{\mathbf{D}}}_t+{\mathbf{K}}_t{\mathbf{D}}_t=p_{tv},\end{array}$$where ${\ddot{\mathbf{D}}}_t$, ${\dot{\mathbf{D}}}_t$, and ${\mathbf{D}}_t$ are the vectors of displacement, velocity, and acceleration of the track subsystem, respectively. The displacement vector ${\mathbf{D}}_t$ can be written as$$\begin{array}{}\text{(9)}& {\mathbf{D}}_t={{\mathbf{D}}_r^L}^T,{{\mathbf{D}}_r^R}^T,{{\mathbf{D}}_s^T}^T,\end{array}$$

in which$$\begin{array}{c}\text{(10)}& {\mathbf{D}}_r^L={d_{y1}^L,\dots ,d{}_{ym}^L,d{}_{z1}^L,\dots ,d_{zm,}^L{d}_{t1,}^L,\dots ,d_{tm}^L}^T,{\mathbf{D}}_r^L={d_{y1}^R,\dots ,d_{ym}^R,d_{z1}^R,\dots ,d_{zm,}^R{d}_{t1,}^R,\dots ,d_{tm}^R}^T,\\ {\mathbf{D}}_s={y_{s1},\dots ,y_{sNs},z_{s1},\dots ,z_{sNs},{\varphi }_{s1},\dots ,{\varphi }_{sNs}}^T,\end{array}$$where $d_{y1}^L$, $d_{zk}^L,$ and $d_{tk}^L$ are the kth modal coordinates of the left rail and $d_{y1}^R$, $d_{zk}^R,$ and $d_{tk}^R$ are the modal coordinates of the right rail in lateral vertical bending and torsion directions, respectively; $m_1\sim m$, m is the number of modes considered for the rail beam; and $y_{sj}$, $y_{sj}$, and ${\varphi }_{sj}$ are the lateral, vertical, and roll angular displacements of the jth sleeper, respectively.

${\mathbf{M}}_t$, ${\mathbf{K}}_t$, ${\mathbf{C}}_t$, and ${\mathbf{M}}_s$ are the mass, stiffness, damping, and sleeper matrices of the track system, respectively, and can be expressed as$$\begin{array}{c}\text{(11)}& {\mathbf{M}}_t=\text{diag}{\mathbf{M}}_{rL}{\mathbf{M}}_{rR}{\mathbf{M}}_s,{\mathbf{C}}_r=\text{diag}{\mathbf{C}}_{rL}{\mathbf{C}}_{rR},\\ {\mathbf{K}}_r=\text{diag}{\mathbf{K}}_{rL}+{\displaystyle \sum _{k=1}^{N_w}{\mathbf{K}}_{\text{rail}}{{\mathbf{\Phi }}_k^z}^T{\mathbf{\Phi }}_k^z{\mathbf{K}}_{rR}}+{\displaystyle \sum _{k=1}^{N_w}{K}_{rail}{{\mathbf{\Phi }}_k^z}^T{\mathbf{\Phi }}_k^z},\\ {\mathbf{C}}_t=\begin{array}{ccc}{\mathbf{C}}_r^L& 0& {\mathbf{C}}_{rs}^L\\ 0& {\mathbf{C}}_{rr}^R& {\mathbf{C}}_{rs}^R\\ {\mathbf{C}}_{sr}^L& {\mathbf{C}}_{sr}^R& {\mathbf{C}}_{ss}\end{array},\\ {\mathbf{K}}_t=\begin{array}{ccc}{\mathbf{K}}_r^L& 0& {\mathbf{K}}_{rs}^L\\ 0& {\mathbf{K}}_{rr}^R& {\mathbf{K}}_{rs}^R\\ {\mathbf{K}}_{sr}^L& {\mathbf{K}}_{sr}^R& {\mathbf{K}}_{ss}\end{array},\end{array}$$where M_rL, M_rR, and ${\mathbf{M}}_s$ indicate the mass matrices of the left rail, right rail, and sleeper, respectively; K_rL and K_rR represent the stiffness matrices of the left and right rails, respectively; and C_rL and C_rR denote the damping matrices of the left and right rails, respectively.

The external force vector acting on the vehicle from the track $P_{tv}$ is written as$$\begin{array}{c}\text{(12)}& {\mathbf{P}}_{tv}={{{\mathbf{P}}_r^L}^T,{{\mathbf{P}}_r^R}^T,{{\mathbf{P}}_s^T}^T}^T,{\mathbf{P}}_r^L={\mathbf{P}}_r^{L1}+{\mathbf{P}}_r^{L2}+{\mathbf{P}}_r^{L3},\\ {\mathbf{P}}_r^R={\mathbf{P}}_r^{R1}+{\mathbf{P}}_r^{R2}+{\mathbf{P}}_r^{R3},\\ {\mathbf{P}}_r^{L1}=-{\displaystyle \sum _{i=1}^{Nw}{{\mathbf{P}}_{\mathbf{r}}^{\mathbf{Z}}{\lambda }^L{\mathbf{\Phi }}_i^Y{\gamma }_{wi}^{22L}{\dot{r}}^{YL}{\mathbf{\Phi }}_i^Y}^T},\\ {\mathbf{P}}_r^{R1}=-{\displaystyle \sum _{i=1}^{Nw}{{\mathbf{P}}_r^R{\lambda }^R{\mathbf{\Phi }}_i^Y+{\gamma }_{wi}^{22R}{\dot{r}}^{YR}{\mathbf{\Phi }}_i^Y}^T},\\ \begin{array}{c}{\mathbf{P}}_r^{L2}\\ {\mathbf{P}}_r^{R2}\end{array}=-{{\displaystyle \sum _{i=1}^{Nw}\begin{array}{c}{\mathbf{P}}_{wi}^L\\ {\mathbf{P}}_{wi}^R\end{array}\begin{array}{c}{\mathbf{\Phi }}_i^X\\ {\mathbf{\Phi }}_i^Y\\ {\mathbf{\Phi }}_i^Z\\ {\mathbf{\Phi }}_i^{\psi }\end{array},}}^T\\ {\mathbf{P}}_{rL}^3=-{\displaystyle \sum _{i=1}^{Nw}{k}_{1Z}{r}^{ZL}{\mathbf{\Phi }}_i^Z}+C_{1Z}{\dot{r}}^{ZL}{\mathbf{\Phi }}_i^Z,\\ {\mathbf{P}}_{rR}^3=-{\displaystyle \sum _{i=1}^{Nw}{k}_{1Z}{r}^{ZR}{\mathbf{\Phi }}_i^Z}+C_{1Z}{\dot{r}}^{ZR}{\mathbf{\Phi }}_i^Z,\end{array}$$where ${\mathbf{P}}_r^{L1}$ and ${\mathbf{P}}_r^{R1}$ denote the lateral random load vector induced by the normal wheel/rail contact force and the lateral creep force induced by random alignment track irregularity, respectively; ${\mathbf{P}}_r^{L2}$ and ${\mathbf{P}}_r^{R2}$ denote the random load vectors generated by the wheel/rail tangential contact forces. ${\mathbf{P}}_r^{L3}$ and ${\mathbf{P}}_r^{R3}$ represent the random load vectors generated by random vertical profile track irregularities. The subscripts L and R denote the left and right rail, respectively. $N_w$ is the number of the wheelsets. The force vectors of the wheelset of ${\mathbf{P}}_{wi}^L$ and ${\mathbf{P}}_{wi}^R$ are provided with detailed information in the wheel/rail interaction section.

3.2.1. Track Irregularity and Spectral Line Sectional Excitation

Rail irregularity can generally be described as a stationary and ergodic system in space due to its irregular nature [44]. The power spectrum density (PSD) is supported by the China rail spectrum presented in Figures 4(a) and 4(b), and the vertical and lateral rail irregularities are indicated [41].$$\begin{array}{c}\text{(13)}& Sf=\frac{A{f}^2+Bf+C}{f^4+D{f}^3+Ff+G},Sf=0.036f^{-3.15}.\end{array}$$

[figures omitted; refer to PDF]

A to G are specific parameters of the irregular nature of the lateral and vertical tracks.

Table 1 illustrates the values of these parameters for China’s high-speed rail lines (such as the Beijing–Shanghai line and the Beijing–Guangzhou line). The time-frequency transformation technique should be used to convert the power spectral density (PSD) function into the horizontal and vertical changes of the track geometry with the longitudinal distance of the track.

Table 1

Values of parameters of China rail mainline track spectrum.

Parameters	A	B	C	D	E	F	G
Left rail, vertical	0.127	−2.1531	1.5503	4.9835	1.3891	−0.0327	0.0018
Right rail, vertical	0.3326	−1.3757	0.5497	2.4907	0.4057	0.0858	−0.0014
Left rail, lateral	0.0627	−1.184	0.6773	2.1237	−0.0847	0.034	−0.0005
Right rail, lateral	0.1595	−1.3853	0.6671	2.3331	0.2561	0.0928	−0.0016

Table 2

Significant parameters for CRH2A of railway vehicles used in the present research.

Parameters	Notation	Value	Unit
Car body mass	mc	35067	kg
Car body roll moment of inertia	Jcx	119200	kgm2
Car body pitch moment of inertia	Jcy	1711800	kgm2
Car body yaw moment of inertia	Jcz	1615300	kgm2
Frame mass	mt	3630	kg
Frame roll moment of inertia	Jtx	2940	kgm2
Frame pitch moment of inertia	Jty	1990	kgm2
Frame yaw moment of inertia	Jtz	3630	kgm2
Wheelset mass	$m_w$	1794	kg
Wheelset roll moment of inertia	$J_{wx}$	900	kgm2
Wheelset pitch moment of inertia	$J_{wy}$	220	kgm2
Wheelset yaw moment of inertia	$J_{wz}$	950	kgm2
Vertical stiffness of primary suspension per axle side	kpz	980000	N/m
Lateral damping of primary suspension per axle side	cpy	5490	kNs/m
Vertical stiffness of primary suspension per axle side	kpz	1176000	N/m
Vertical damping of secondary suspension	cpz	20	kNs/m
Lateral stiffness of secondary suspension per bogie side	ksy	192	kN/m
Lateral damping of secondary suspension per bogie side	csy	60	kNs/m
Vertical stiffness of secondary suspension per bogie side	ksz	990.8	kN/m
Vertical damping of secondary suspension per bogie side	csz	9.8	kNs/m
Nominal wheel radius	ro	0.43	m
Half of the lateral distance between wheel/rail contact points	lt	1.25	m
Half of the lateral distance between primary suspensions of the two sides of the bogie	do		m

3.3. Wheel/Rail Interaction Model

The critical component of the vehicle-track coupled system is the wheel/rail interaction model. It primarily includes three submodels: the interaction geometry model, the normal force model, and the tangential creep force model.

The significant contact geometry model is used to describe the contact points positions of the wheel and rail interfaces and is the criterion for solving the wheel/rail contact forces. Figure 5 indicates the wheel/rail interaction model, the wheelset coordinates, and the rail force vectors definitions.

[figure omitted; refer to PDF]

Figure 6 illustrates the relation between the contact geometry of the wheel and rail. Furthermore, the specified equation for the contact geometry is presented in [45].

[figure omitted; refer to PDF]

In the conventional wheel/rail contact geometry computation method [46], the rails are supposed to be fixed without any movement. The wheel/rail contact geometry parameters are the nonlinear function of the wheelsets lateral displacement and yaw angle.

The wheelset rolling angle is corrected iteratively until left and right smallest vertical distances between the wheel and rail surfaces are equal. For the track vibration, especially the rail motions were proposed by Chen and Zhai and used to solve the wheel/rail contact geometry relation [40]. The equation of the wheel force vector can be expressed as$$\begin{array}{c}\text{(14)}& {\mathbf{P}}_{wi}^L={\begin{array}{c}{p}_{xL}^{cr}\cos {\psi }_{wi}+p_{yL}^{cr}\cos {\alpha }_{Li}+{\theta }_{wi}\sin {\psi }_{wi}\\ p_{xL}^{cr}\cos {\psi }_{wi}+p_{yL}^{cr}\cos {\alpha }_{Li}+{\theta }_{wi}cos{\psi }_{wi}-p_{yL}^{cr}\sin {\alpha }_{Li}+{\theta }_{wi}cos{\psi }_{wi}\\ \begin{array}{c}-p_{xL}^{cr}\sin {\psi }_{wi}-p_{yL}^{cr}\cos {\alpha }_{Li}+{\theta }_{wi}\cos {\psi }_{wi}{R}_{wi}\\ -p_{yL}^{cr}\sin {\alpha }_{Li}+{\theta }_{wi}cos{\psi }_{wi}{d}_o-p_{xL}^{cr}\cos {\psi }_{wi}+p_{yL}^{cr}\cos {\alpha }_{Li}+{\theta }_{wi}\sin {\psi }_{wi}{d}_o\\ +M_{zL}^{cr}\cos {\alpha }_{Li}+{\theta }_{wi}\end{array}\end{array}}^T,{\mathbf{P}}_{wi}^R={\begin{array}{c}-p_{xR}^{cr}\cos {\psi }_{wi}+p_{yR}^{cr}\cos {\alpha }_{Ri}-{\theta }_{wi}\sin {\psi }_{wi}\\ -p_{xR}^{cr}\cos {\psi }_{wi}+p_{yR}^{cr}\cos {\alpha }_{Ri}-{\theta }_{wi}cos{\psi }_{wi}\\ p_{yR}^{cr}\sin {\alpha }_{Ri}-{\theta }_{wi}cos{\psi }_{wi}\\ p_{xR}^{cr}\sin {\psi }_{wi}-p_{yR}^{cr}\cos {\alpha }_{Ri}-{\theta }_{wi}cos{\psi }_{wi}{R}_{wi}-p_{yR}^{cr}\sin {\alpha }_{Ri}-{\theta }_{wi}cos{\psi }_{wi}{d}_o\\ -p_{xR}^{cr}\cos {\psi }_{wi}-p_{yR}^{cr}\cos {\alpha }_{Ri}-{\theta }_{wi}\sin {\psi }_{wi}{d}_o+M_{zR}^{cr}\cos {\alpha }_{Ri}-{\theta }_{wi}\end{array}}^T,\\ {\mathbf{P}}_s=0,\end{array}$$where $\dot{r}$ is the first derivative of the track irregularity and L and R indicate the left and right rails. ${\lambda }_i^L$ and ${\lambda }_i^R$denote the wheel/rail contact angle slope of the left and right rails, respectively; W_a is the axle weight concerning the self-weight of all components of the vehicle distributed on each wheelset; and ${\mathbf{P}}_s$ is the sleeper force. $R_{wiL}$ and $R_{wiR}$ denote the instantaneous rolling radius of the left and right sides of the wheels, respectively, at the ith wheelset of the vehicle. The tangential contact forces $p_{xL}^{cr}$, $p_{yL}^{cr}$, and $M_{xL}^{cr}$ in the left wheel/rail pair and $p_{xR}^{cr}$, $p_{yR}^{cr}$, and $M_{xR}^{cr}$ in the right wheel/rail pair can be calculated using the Kalker linear rolling contact theory [47]. The tangential contact forces of the wheel/rail interaction can be written as$$\begin{array}{c}\text{(15)}& p_{xL}^{cr}=-{\gamma }_{wi}^{11L}{\Psi }_{xwi}^L,p_{yL}^{cr}=-{\gamma }_{wi}^{22L}{\Psi }_{ywi}^L-{\gamma }_{wi}^{23L}{\Psi }_{\psi wi}^L,\\ M_{zL}^{cr}={\gamma }_{wi}^{23L}{\Psi }_{ywi}^L-{\gamma }_{wi}^{33L}{\Psi }_{\psi wi}^L,\\ p_{xR}^{cr}=-{\gamma }_{wi}^{11R}{\Psi }_{xwi}^R,\\ p_{yR}^{cr}=-{\gamma }_{wi}^{22R}{\Psi }_{ywi}^R-{\gamma }_{wi}^{23R}{\Psi }_{\psi wi}^R,\\ M_{zR}^{cr}={\gamma }_{wi}^{23R}{\Psi }_{ywi}^R-{\gamma }_{wi}^{33R}{\Psi }_{\psi wi}^L,\end{array}$$where ${\gamma }_{wi}^{aL}$ and ${\gamma }_{wi}^{aR}$ (a = 11, 22, 23, 33) characterize the creep coefficient between the ith wheelset and the left and right rails, respectively, at the consistent position of the vehicle; ${\mathrm{\Psi }}_{xwi}^L$, ${\mathrm{\Psi }}_{ywi}^L$ and ${\mathrm{\Psi }}_{\psi wi}^L$ denote the longitudinal creep rate, the lateral creep rate, and the spin creep rate of the wheel/rail contact point of the ith wheelset on the vehicle acting on the left rail at the consistent position, respectively; and the corresponding values for the right rail are ${\mathrm{\Psi }}_{xwi}^R$, ${\mathrm{\Psi }}_{ywi}^R$, and ${\mathrm{\Psi }}_{\psi wi}^R,$respectively. The creep coefficients ${\gamma }_{wi}^{aL}$ and ${\gamma }_{wi}^{aR}$ (m = 11, 22, 23, 33) can be obtained by [47].

3.3.1. Wheel Flat Model

Wheel/rail interaction is different from other forms of transportation. In this research, the interaction is established by the nonlinear Hertzian contact [11, 19, 25, 48–50]. According to the Hertz method, the wheel/rail contact force is nonlinearly related to the rail deflection. Figures 7(a) and 7(b) show the wheel flat geometry representation and wheel radius variation of the LMA type wheel profile.$$\begin{array}{}\text{(16)}& F_{c}t=C_{H\Delta Z{t}^{3/2}},\end{array}$$where z(t) is the vertical direction of the wheel/rail overlap and C_H is the Hertz contact stiffness coefficient. In the presence of wheel defects, the overlapping is defined by the relative movement of the wheels concerning the rail.$$\begin{array}{}\text{(17)}& \Delta zt=Z_{w}t-Z_{r}x,t,\end{array}$$where $Z_{w}t$ and $Z_{r}x,t$ are vertical wheel and rail deflections. In the occurrence of wheel defects, the instantaneous overlap between the wheel and the rail z(t) is represented as follows:$$\begin{array}{}\text{(18)}& \Delta zt=Z_{w}t-Z_{r}x,t-rt,\end{array}$$where $rt$ is the wheel flat function, in the case of the haversine flat, as shown in Figure 7(a); the role of the wheel flat geometry is represented as follows:$$\begin{array}{c}\text{(19)}& rt=0.5D_{f}1-\cos \frac{2\pi X_f}{L_f},D_f=\frac{L_f^2}{16R_w}.\end{array}$$

[figures omitted; refer to PDF]

3.3.2. Finite Element Result Input to SIMPACK

The master node is used to connect the flexibility to the surrounding multibody system. This subset of the master node is defined during the reduction in the FE code by explicitly selecting the nodes of the reduced finite element model. Moreover, the master node of the reduced model in the finite element code will generate the markers in the flexible body. The rigid region body element is defined by an independent degree of freedom (DOF) at one node and a dependent degree of freedom at the other nodes, as shown in Figure 8(c). Figure 8(c) indicates that, at the center of the wheelset axle, the single node is generated independently and at the specified surface of the wheelset axle has more than two degrees of freedom Additionally, to create the master node and rigid region, as shown in Figure 8(c), the following parameters are provided: element type (solid 185, mass 21, target 170, and contact 173) and material properties (modulus of elasticity 2.06e¹¹, Poisson’s ratio 0.3, and density 7830 kg/m³). From the finite element solution, apply the analysis type and then select the substructure. From the analysis options, give the stiffness + mass conditions. In addition, select the master degrees of freedom (DOF), define the master nodes, select the provided wheelset circumference line from the wheel/rail contact, apply all degrees of freedom, archive model, and finally solve the finite element model. From the solution output, we will get .substr and .cdb. Moreover, see ANSYS Mechanical APDL multibody analysis [51–53] for detailed information about integrating FE and multibody dynamics.

[figures omitted; refer to PDF]

3.3.3. Generated Flexible Body in SIMPACK

The .Fbi file contains the physical and structural data from third-party finite element software that is required for importing the flexible body to SIMPACK. The .Fbi file generator is used to convert FE-output data to flexible body input (.Fbi) files needed for defining a flexible body in the body properties. The following important information should be considered to generate the flexible body inputs (.Fbi).

(1) Define the interface nodes of your initial model.

(i) The MBS interacts with the FE substructure on these nodes.

(ii) The markers should be located at the interface nodes.

(2) Connect the interface nodes with the structure. You can connect the interface nodes with the FE model using the following element types:

(i) Rigid Body Element.

(ii) Interface node has independent degrees of freedom.

(iii) Coupling nodes on the FE structure have dependent degrees of freedom.

They are launching the SIMPACK. Fbi file generator by selecting Utilities - Fbi files - Generation. Moreover, for detailed information, refer to the SIMPACK documentation.

3.3.4. Flexible Wheelset Model

The complete model of the high-speed vehicle-track system is articulated in three combinations of the component models. These models include flexible wheelsets, tracks, and vehicles. Figure 8 demonstrates the simulation system concerning finite element (FE) models of the flexible wheelset and the track coupled with the vehicle dynamic model in the SIMPACK platform. In ANSYS, the input file for the dynamic reduction can be run either in the ANSYS-GUI or in the ANSYS-Batch mode. Accordingly, the modal vector is integrated into the vehicle model using the finite element multibody system (FEMBS) interface available in SIMPACK as explained in Sections 3.3.2 and 3.3.3.

P_s, F_y(L, R), and F_z(L, R) represent the primary suspension and the lateral and vertical wheel/rail contact force for the left and right positions. The position of any point M on the wheelset can be demonstrated as$$\begin{array}{}\text{(20)}& xt=X_{0}t+P_{xM}+X_f{X}_M,t,\end{array}$$

where X₀(t) is the movement of the wheelset center as a rigid body, P = P(φ) is the transformation matrix due to the rotation as a rigid body $\stackrel{⟶}{\phi }$ = [φ(t), x(t), ψ(t)], and $\stackrel{⟶}{x_M}$ is the vector of the limited (local) coordinates of a general independent variable on the wheelset. Due to the flexibility of the system, the movement of ${\stackrel{⟶}{x}}_f$ ($\stackrel{⟶}{x_M}$, t) point ${\stackrel{⟶}{x}}_M$ is considered, so $\stackrel{⟶}{x_f}$ is adjusted according to the different flexible modes of the wheelset:$$\begin{array}{}\text{(21)}& x_f{X}_M,t=P{\displaystyle \sum {\varnothing }_k{X}_M{\eta }_{k}t}.\end{array}$$

The shape of the position mode is ${\varnothing }_k$(X_M); X_M and η_k(t) are the amplitudes of the instantaneous t of vibration k mode. Thus, when presenting the elasticity of wheelsets, the structure of each wheelset contains new equations. Use equations 20 and 21 to indicate a more effective way to do the governing modal matrix.$$\begin{array}{}\text{(22)}& Xt=X_{0}t+P_{xM}+P{\displaystyle \sum {\varnothing }_k{X}_M{\eta }_{k}t}.\end{array}$$

The model was created on the ANSYS software package platform using 185 solid elements indicated in Figures 8(a)–8(c). The model vector of the wheelset comes from Eigen analysis. Considering that the high impact load will excite several maximum-frequency modes of the wheelset, 46 vibration modes were examined at frequencies up to 2467 Hz to determine the wheelset response. The modal matrix is combined into the model of the SIMPACK platform through the FEMBS.

4. Numerical Model

There are several types of trains in operation in China, among which the CRH2A high-speed train that can travel at 300 km/h is selected. The railway vehicle dynamics model includes three basic submodels: vehicle submodel, track submodel, and wheel/rail interaction model [40]. Moreover, the railway vehicle model and its SIMPACK implementation were verified using the wheel defect manual [54], flexible wheelset and wheel flat [55], and 3D wheel flat model [23]. In simulated results of a similar study reported by Zhai [56], the author puts forward assessing vertical impact load influences due to vehicle speed change and the wheel flat length to evaluate the vehicle systems dynamic performance, safety, and stability. Furthermore, Table 2 provides the main parameters implemented in the present work; the data are presented from CRRC Qingdao Sifang Co., Ltd. The vehicle-track coupled dynamic model established in SIMPACK is shown in Figure 9.

[figure omitted; refer to PDF]

4.1. Simulation Results Validation Comparison

Table 3 describes the speed limits of simulated railway vehicles. The speed limit shown in the wheel defect manual is revealed in [54]. It specifies that the simulation results match the speed limit access to the wheel defect manual [54]. For example, in the current model and the two methods, the allowable length of the infinite wheel defect is 25 mm. The speed limit available in the present paper is 105 km/h and the value given [23, 54] is 110,115 km/h, respectively, as the flat length is 40 mm. Besides, when the wheel length is near 60 mm, the existing operating speed limit is approximately 45 km/h, in the wheel defect manual [54] is 40 km/h, and in [23] is 50 km/h, respectively. The current work considers flexible, rigid wheel flat and track irregularities, so the speed limit is consistent with the two existing methods.

Table 3

Comparison of vehicle speed limit due to wheel flat length increment.

Flat length	Vehicle speed limit
	Wheel defect manual [54]	Ren Zunsong [23]	Present paper
l ≤ 25 mm	No speed limit	No speed limit	No speed limit
25 mm < l ≤ 40 mm	115 km/h	110 km/h (L = 40 mm)	105 km/h (L = 40 mm)
40 mm < l ≤ 60 mm	40 km/h	50 km/h (L = 60 mm)	45 km/h (L = 60 mm)
60 mm < l ≤ 100 mm	25 km/h	No calculation	No calculation

In summary, the techniques and models presented in the current article are reasonable and available.

Figures 10(a) and 10(b) show the vertical impact of the wheel/rail obtained by the method in this article, and the forces obtained using the procedure provided by Zhai [21, 56] at 50 mm wheel flat length are presented. Figure 10(a) shows that the recent flexible and rigid wheelset vertical impact is 275 and 320 kN, respectively. Besides, the vertical impact loads for the two methods are 255.275 kN, as shown in Figure 10(b). From the above results, it can be concluded that the impact load of the flexible wheelset is similar to that of the two methods, but the rigid wheel flat impact force is more advanced than that of both approaches. Facts have proved that this method is more reasonable for realizing the wheel/rail dynamic effect due to increased wheel flat length and different vehicle speeds.

[figures omitted; refer to PDF]

5. Results and Discussion

5.1. Simulation Result

The wheel/rail vertical and lateral force and wheel acceleration in the time domain are shown in Figures 11 and 12. Moreover, the impact load and the acceleration are specified from the 0–400 km/h sample of speed increment, and the wheel flat length L = 50 mm. This shows that the wheel/rail impact occurs relatively fast and has high-frequency vibration. Furthermore, the vertical impact load drops rapidly after the speed reaches 150 km/h. Due to wheel radius deviation and track irregularities, the maximum peak wheel/rail vibration results during each wheel transition period are different. As shown in Figure 11(a), view parts of Figures 11(b) and 11(c) are enlarged. Figures 11(b) and 11(c) specify that the vertical force of the flexible wheel level is less than the vertical force indicated by the rigid wheel level.

[figures omitted; refer to PDF]

[figures omitted; refer to PDF]

Figure 12(a) and the partially enlarged view in Figure 12(b) show that the lateral force generated by the flexible wheel flat (first wheelset) model and the rigid wheel flat (third wheelset) model is presented. The results clearly show that the flexible wheelset lateral force change is much more significant than that of the rigid wheelset, attributed to the wheelset elastic deformation.

Figure 12(c) and the partially enlarged view in Figure 12(d) show the vertical acceleration values generated by the flexible wheel flat (first wheelset) model and the rigid wheel flat (third wheelset) model. Furthermore, the vehicles traveling at 300 km/h on a straight-line track with a wheel flat length of 50 mm are presented. The two-wheel flat model results are generally consistent, producing different peak acceleration values and acceleration pulse event durations. It is shown that the vertical acceleration of a flexible wheelset has a significant influence on the rigid wheelset, and the irregular effect of the track also has a considerable impact on dynamic performance.

5.2. Influence of Vehicle Speed

For examining the effect of train speed on the vertical wheel/track impulse response, seven speeds (namely, 100, 150, 200, 250, 300, 350, and 400 km/h) were proposed. Figures 13(a) and 13(b) show a set of vertical wheel/rail forces versus vehicle speed caused by a flat of 15, 20, 30, 40, 50, 60, 70, and 80 mm. It is observed that the historical curve of the vertical force response varies at seven different train speeds. The maximum vertical impact force changes nonmonotonously with the train speed. Therefore, when the train speed is less than or equal to 150 km/h, the leading vertical impact force increases with the increase in the train speed, reaching a peak value of 678 kN, and it slowly decreases with speed up to 400 km/h.

[figures omitted; refer to PDF]

Figure 13(c) shows the vertical wheel/rail impact and time curve caused by 50 mm flat and a vehicle speed of 300 km/h. The results demonstrate that the vertical force history curves have different effects on flexible and rigid wheel flat models. Rigid wheels have a higher dynamic response than those considering flexibility. The nonmonotonic participation between the extreme vertical impact force of all flat lengths and the train speed is proposed. When the train speed is less than or equal to 60 mm, the maximum vertical impact peak reaches 150 km/h, while the peak produced by the flat wheel appears at an 80 mm flat point, and the speed is about 190 km/h. This nonmonotonic correlation concerning the maximum vertical force and train speed could also be related to the contact loss at different speeds. The maximum contact force at 80 mm wheel flat is higher than that of other units, which is as good as the group’s contact effect with the guide rail. The results are revealed in Figures 14(a) and 14(b). Figures 14(a) and 14(b) indicate that for rigid and flexible wheel flat, the wheel/rail vertical forces are 60 to 678 kN and 60 to 634 kN, respectively. Moreover, they show the influence of wheel/rail vertical force concerning flat length and vehicle speed; furthermore, the rigidity and flexible impact force increase nonmonotonously.

[figures omitted; refer to PDF]

5.3. Influence of Flat Length

Eight different flat lengths (namely, 15, 20, 30, 40, 50, 60, 70, and 80 mm) were designated to study the effect of flat size on the wheel/rail vertical impact response. For the first right wheelset (flexible) and the third right wheelset (rigid), with a flat length of 50 mm, the wheel/rail vertical impact is caused by a train speed of 300 km/h.

The force history curve is plotted in Figure 15(c). Clearly, for a wheel with a flat length of 80 mm, the maximum vertical impact force generated by a flat wheel increases with the increase of the flat size, while the peak values of the peak vertical impact force of the flexible and rigid flat wheel are 634 and 678 KN, respectively. Figures 15(a) and 15(b) show the maximum wheel/rail vertical impact force as a flat length function and different train speeds.

[figures omitted; refer to PDF]

The highest wheel/rail vertical impact force increases nonlinearly with the wheel flat length, significantly impacting the peak vertical force in the speed range of 150 to 250 km/h. Besides, it will produce average peaks at different flat lengths and different vehicle speeds. The outcomes are revealed in Figures 16(a) and 16(b). Figures 16(a) and 16(b) indicate that for rigid and flexible wheel flat, the wheel/rail vertical forces are 60 to 678 kN and 60 to 634 kN, respectively. Moreover, they show that as the flat length and vehicle speed increase, the rigidity and flexible impact force increase monotonously.

[figures omitted; refer to PDF]

6. The Statistical Model for Wheel Flat Data

6.1. Approach

The authors proposed a method to determine an appropriate statistical model of fatigue life in the former approach in [57]. To reasonably describe the current wheel/rail vertical impact load caused by the eight-wheel flat data, the successive sizes should be considered to determine the appropriate statistical model of 15, 20, 30, 40, 50, 60, 70 80 mm, and the following notes matter:

(1) Select appropriate eight-wheel flat data. Additionally, the comparison between the seven commonly used statistical distributions is consistent with the description of the vertical impact load physical characteristics, the predicted risk in the left tail area of the model application, and the total effect of the model fitting record, namely, three-parameter (3-P) Weibull, two-parameter (2-P) Weibull, normal, lognormal, extreme minimum (extr.mini.) value, extreme maximum (extr.maxi.) value, and exponential distribution.

(2) Considering the agreement on the physics of vertical impact loads, the appropriate statistical models should reflect the critical characteristics of increased risk predictions. This type of flat wheel damage is triggered by the accumulation of irregular plastic deformation and occasional friction and wear, which is affected by the slip between the wheel/rail contact point and the carved marks of foreign objects. Therefore, the exponential distribution should not be applied to the current data because it has a constant risk rate prediction.

(3) Considering the predicted risk of applying the left tail region, if the probability values of the first data and the second data are P_e1 and P_e2, the predicted values of the model are P_P1 and P_P2, respectively, and the following differences (d₁ and d₂) should be applied to make judgments:$$\begin{array}{c}\text{(23)}& d_1=p_{e1}-p_{p1},d_2=p_{e2}-p_{p2},\end{array}$$

  when d₁ is positive and less than or equal to d₂, it indicates a safety prediction trend; further, once d₁ is less than zero, it specifies that the safety prediction can be attained for the wheel flat value W_X; it is less than W₁.

(4) Considering the overall effect of the model suitable for the data, the representative of the relationship coefficient R_ep based on the intermediate empirical value and the predicted value must be used for judgment to define the representative R_ep.$$\begin{array}{}\text{(24)}& R_{ep}=\sqrt{1-\frac{{\displaystyle {\sum }_{i=1}^{n_s}{p}_{ei}-1/n_s{\displaystyle {\sum }_{i=1}^{n_s}{p}_{ei}}}{\displaystyle {\sum }_{i=1}^{n_s}{p}_{pi}-1/n_s{\displaystyle {\sum }_{i=1}^{n_s}{p}_{pi}}}}{{\displaystyle {\sum }_{i=1}^{n_s}{p}_{pi}-1/n_s{\displaystyle {\sum }_{i=1}^{n_s}{p}_{pi}}}}.}\end{array}$$

P_ei and P_Pi, i = 1, 2, …, n_s, are midrank-based experience values, and the probabilities predicted values corresponding to the ordered data W_i, i = 1, 2, …, n_s. The closer the R_ep value to 1, the better the modeling fitting data. Additionally, n_s is the number of samples for vertical impact force due to wheel flat for statistical analysis.

6.2. Statistical Tests

The current statistical test is similar to that in [57], in terms of the linear regression method, a statistical test parameter described by equation Y = A + BX linear correlation coefficient R_xy. is defined [58]:$$\begin{array}{}\text{(25)}& R_{xy}=\frac{n_s{\displaystyle {\sum }_{i=1}^{n_s}{X}_i{Y}_i-{\displaystyle {\sum }_{i=1}^{n_s}{X}_i{\displaystyle {\sum }_{i=1}^{n_s}{Y}_i}}}}{n_s{{\displaystyle {\sum }_{i=1}^{n_s}{X}_i^2-{\displaystyle {\sum }_{i=1}^{n_s}{X}_i}}}^2{n}_s{\displaystyle {\sum }_{i=1}^{n_s}{Y}_i^2{{\displaystyle {\sum }_{i=1}^{n_s}{Y}_i}}^2}},\end{array}$$and the critical value R_XYc for accepting the linear description is formulated as$$\begin{array}{}\text{(26)}& R_{xyc}=\frac{t_a{n}_s-2}{\sqrt{n_s-2+t_a^2{n}_s-2}},\end{array}$$where t_α(n_s − 2) is the Student t-distribution function with n_s − 2 degrees of freedom at a significance level of α. The linearization models of seven commonly used distributions fitting the present wheel/rail vertical impact load data of three parameters are constructed to make the R_xy. and R_XYc evaluations.

6.3. Appropriate Models for the Present Data

In the above method, the parameters d₁, d₂, R_ep, and R_xy are given by the wheel/rail vertical impact load results in the last part, of which seven distributions describe eight-wheel flat data. Since the left tail region has a complete safety prediction, the final parameter values can be obtained for subsequent annotations. Moreover, the 2-P Weibull distribution is best for the following analysis. Describe the hazard rate (HR) of the 2-P Weibull model of the current vertical impact data W_X in the k^th wheel flat; the probability distribution function (PDF) and cumulative distribution function (CDF) can be expressed as scale (β_k) and shape (β_x) of the 2-P Weibull model fitting the vertical impact load data in the k^th wheel flat studied.$$\begin{array}{c}\text{(27)}& \lambda w_x=\frac{f{w}_x}{1-F{w}_x}=\frac{{\beta }_x}{w_{xsdk}}{\frac{w_x+w_{xo}}{w_{xsdk}}}^{{\beta }_k-1},f{w}_x=\frac{{\beta }_k}{w_{xsdk}}{\frac{w_x+w_{xo}}{w_{xsdk}}}^{{\beta }_k-1}\exp -{\frac{w_x+w_{xo}}{w_{xsdk}}}^{{\beta }_k},\\ F{w}_x=1-\exp -{\frac{w_x+w_{xo}}{w_{xsdk}}}^{{\beta }_k}.\end{array}$$

Figures 17(a)–17(f) show the hazard rate, PDF, and CDF curves of 2-P Weibull model for reasonable distribution of the high-speed train CRH2A eight-wheel flat data. This work compares the impact loads of rigid and flexible wheels and obtains excellent results. When considering the statistical method shown in Figures 17(a)–17(c) and Figures 17(d) and 17(e), the wheel/rail vertical impact will be affected due to the length of the wheel flat. The outcomes demonstrate that the flexible wheelset vertical contact force impact as shown in Figures 17(a)–17(c) is lower than that of the rigid wheelset in Figures 17(d) and 17(e). The hazard rate (HR), PDF, and CDF curve show that the maximum wheel flat length has a more significant impact on the vertical wheel/rail interaction. Using the current statistical methods, vehicle performance, safety, comfort, and reliability can be evaluated.

[figures omitted; refer to PDF]

7. Conclusion

The SIMPACK platform and ANSYS have been established. The multibody system model of coupled high-speed vehicles/track integrated with FEM is used to study flexible and rigid wheelsets and introduce dynamic effects when various wheel flat lengths and different speeds appear. Moreover, a statistical method has been established to evaluate the influence of wheel flat on dynamic performance. The necessary conclusions are as follows:

(1) By comparing the dynamic response of rigid wheelsets, the contribution of the flexible wheel flat is demonstrated. Compared with the rigid wheel flat, the results show that the flexible wheel flat significantly affects the vertical acceleration.

(2) The relationships between wheel/rail impact force magnitude and flat wheel length were nonlinear and monotonically increasing with flexible and rigid wheel flat.

(3) The speed results show that the historical curves of the vertical force response of the seven different train speeds are different. The maximum vertical impact force varies with the train speed (nonmonotonous); that is, the train reaches a maximum speed of 150 km/h or below and slowly decreases to 400 km/h.

(4) By considering statistical methods, the wheel/rail vertical force effect due to the wheel flat length is evaluated. The results show that the flexible wheelset contact force is smaller than that of the rigid wheelset. The hazard rate (HR), PDF, and CDF curves indicate that the maximum wheel flat length affects the vertical wheel/rail interaction. Furthermore, vehicle performance, safety, comfort, and stability can be evaluated according to the current statistical methods.

Acknowledgments

The present research was supported by the China 973 Program (2015CB654801) and China Railway Science and Technology Development Program by China Railway Co., Ltd. (2015J007-E).

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