A Parameter-Driven Methodology of Wheel Flat Modeling for Wheel–Rail Impact Dynamics

Abstract

A wheel flat is a typical wheel defect that significantly impacts the wheel–rail system, posing substantial challenges to vehicle operation safety. In the existing literature, the wheel flat plane model does not account for the contribution of the width direction to the impact response and thus cannot accurately reveal the wheel–rail contact state with a flat. This paper systematically proposes a three-dimensional analytical model that considers multiple worn stages and constructs a spatial complex surface reconstruction model for flats based on NURBS technology. A vehicle–track coupled dynamics model, considering the geometry of the flat, is established to investigate the effects of flat geometry on the wheel–rail impact response and contact relationship in detail. The results show that in the subcritical regime, the wear degree of the flat predominantly affects the impact force, while in the transcritical regime, both the wear degree and velocity together determine the magnitude of the wheel–rail impact force. As the wear degree increases, the moment of wheel lateral jump occurs earlier. The spatial modeling method for flats proposed in this paper offers a novel technical approach for accurately simulating the dynamic behavior of wheel–rail contact when a flat is present.

1. Introduction

A wheel flat is a common wheel defect that typically occurs when a wheel slides along a rail during braking, as illustrated in Figure 1.

This sliding causes severe wear on the wheel tread where it contacts the rail, forming a nearly circular or oval damage area. Once a wheel flat develops, it generates significant impact forces on both the vehicle and the rail, as well as periodic impact noise. These effects pose substantial challenges to vehicle operation safety and riding comfort.

The effect of wheel flats on vehicles and track systems has attracted considerable attention from researchers.

Uzzal et al. [1] investigated the dynamic contact loads at the wheel–rail contact points and found that the nonlinear rail pad and ballast model could better predict the wheel–rail impact force. On this basis, they developed a railway vehicle–track model containing multiple flats [2] and analyzed in detail the effect of multiple flats on the peak acceleration of wheels of different sizes and relative positions. To further simulate the high-frequency response during impacts involving wheel flats, a rigid–flexible coupling model was considered. Xu et al. [3] established a rigid–flexible vehicle-turnout dynamic coupling model and investigated the most unfavorable phase of flats on the wheels of a vehicle crossing a turnout. The studies [4,5] established a joint simulation model considering the combination of an electric drive subsystem and a coupled locomotive–track dynamics model to analyze the dynamic response of a locomotive under the impact of flats of different sizes. Wu et al. [6] developed a numerical model for predicting radiated noise from wheel tracks in the presence of flats. Newton et al. [7] conducted a field test in which unevenness was introduced at the rail head to simulate a series of vehicle flats, monitoring the loads and rail stresses. Steenbergen et al. [8] classified different stages of wheel flats and concluded that the circumferential curvature of the wheel tread is the key parameter controlling the dynamic interaction between the wheel and the rail when the wheel flat is worn. Dukkipati et al. [9] developed a vehicle–track finite element model and investigated the factors affecting the impact loads on wheel flats. Hossein et al. [10] performed a three-dimensional elasto-plastic finite element analysis of undamaged wheels and wheels containing flats, considering the effect of impact loads on the parameters of the contact area that influence stresses. Bian et al. [11] developed a three-dimensional finite element model to analyze the effect of flat impact on railroad track performance. The authors of [12] analyzed the dynamic effects of wheel flats on axle box bearings using rigid–flexible coupling modeling.

To accurately predict the impact response of a wheel flat, most complex programs use the Hertzian contact model to evaluate the elastic properties of the wheel–rail contact. However, when the wheel–rail contact occurs in the flat region, the basic assumptions of the Hertzian contact model are not satisfied, leading to deviations between the model’s calculations and field test results. Consequently, detailed studies have been conducted on wheel–rail contact modeling in the presence of flats. Baeza et al. [13] compared the Hertzian contact model with a non-Hertzian contact model in simulating wheel flat impacts and found that the peak contact force calculated using the non-Hertzian contact model was significantly lower. Pieringer et al. [14] analyzed the effect of contact modeling on simulated wheel flat impacts and investigated wheel–rail dynamic interactions using a three-dimensional non-Hertzian contact time-domain model based on Kalker’s variational method [15]. They determined the errors introduced by the simplified method. Their results showed that accurately modeling the longitudinal geometry of the wheel flat actually has a greater impact on the maximum impact force than the choice of the contact model. Yang et al. [16] developed a detailed time-domain wheel–rail interaction model with non-Hertzian contact values and used the field-measured roughness of different cross-sections as input. They predicted the vibration impact response and noise caused by the wheel flat using a numerical model. Additionally, the use of a detailed contact model provides significant improvements at high frequencies compared with Hertzian theory.

In the existing research, multi-body dynamics (MBD) or finite element methods (FEMs) are primarily used to study the dynamic effects of wheel flats. Regardless of the method used, the accuracy of the flat geometry modeling significantly impacts the final simulation results. This paper summarizes the common methods for describing the geometry of the wheel flat, as shown in Table 1.

According to Table 1, all existing modeling methods have limitations.

Firstly, the most common methods—wheel center trajectory description and radius deviation description—only describe the geometry of the wheel in the longitudinal direction and do not account for the width of the wheel flat. The authors of [17] found that for a fixed flat length, a wider flat increases the wheel–rail vertical impact, and a wide flat results in a jump at the wheel–rail contact point compared with a narrower flat.

Secondly, the wear process in each stage of the wheel flat should be continuous. The authors of [8] classified the wear process of the wheel flat in detail and found significant differences in the dynamic response at different wear stages. However, current methods only control the shape of the wheel flat in the complete wear stage by changing the length and do not propose control parameters for the shape in the edge-worn stage.

Additionally, reconstruction methods for complex 3D surfaces in flats have not yet been developed for modeling based on measured roughness data in the wheel flat region. The limited number of measured rolling circle sections and simple linear interpolation can lead to geometric discontinuities within the flat region.

Therefore, this paper contributes to addressing these issues as follows:

(1)

A three-dimensional wheel flat modeling method considering a continuous wear process is proposed. This method uses uniform control parameters and analytical forms to describe wheel flats at different wear stages, with the constraints between the control parameters determined by numerical iteration.

(2)

A complex 3D wheel flat surface reconstruction method is proposed based on NURBS technology. Three different sizes of wheel flats were machined on the wheel tread, and roughness tests were conducted to demonstrate the effectiveness of the reconstruction method using the measured data.

(3)

A vehicle–track coupled dynamics model is established to compare the variability in the wheel–rail impact response under different wheel flat morphologies. Based on the flat geometry modeling method proposed in this paper, the pattern of wheel–rail impact under different wear states is investigated, as well as the effect of the wear state on wheel–rail contact geometry.

Table 1. The main methods for describing the geometry of a wheel flat.

Method	Formula	2D/3D	Consider Wear?	Simulation Type	Citation
Center trajectory	$z(x)=\left\{\begin{array}{ll}{x}^2/2R_w& 0\le x\le l/2\\ {(l-x)}^2/2R_w& l/2<x\le l\end{array}\right.$ (new flat)	2D	Consider	MBD	[5,6]
	$z(x)=\left\{\begin{array}{ll}4d/{(x/l)}^2& 0\le x\le l/2\\ 4d/{[(l-x)/l]}^2& l/2<x\le l\end{array}\right.$ (rounded flat)
Rail indentation	$z(x)=\frac{d}{2}(1-\cos \frac{2\pi x}{l})0\le x\le l$	2D	Consider	MBD	[7,9]
Radius deviation	$\Delta r(x)=d-\frac{1}{2R_w}{(x-l/2)}^{2}0\le x\le l$ (new flat)	2D	Consider	MBD	[1,2,3,13,14,18,19]
	$\left\{\begin{array}{l}\Delta r(x)=\frac{d}{2}(1-\cos \frac{2\pi x}{l})\\ d=l^2/16R_w\end{array}\right.0\le x\le l$ (rounded flat)
Discrete three-dimensional	$\left\{\begin{array}{l}{x}_i=\frac{l}{N}i(1\le i\le N)\\ y_{ij}=j{y}_i/M-y_i/2\\ z_{ij}=R_r-\frac{R_r\cos \frac{{\gamma }_i}{2}}{\cos (\frac{{\gamma }_i}{2}-{\gamma }_{ij})}\end{array}\right.$	3D	Not Consider	MBD	[17]
Center trajectory (time domain)	$z(t)=\left\{\begin{array}{l}-\frac{1}{2R}{V}^2{t}^{2}H(-Vt+\frac{l}{2})+(-\frac{1}{2R}{V}^2{t}^2+\frac{l}{R}Vt-\frac{l^2}{2R})H(Vt-\frac{l}{2})H(l-Vt)\left(\mathrm{stage}\mathrm{I}\right)\\ {(\frac{l}{l_{\mathrm{II}}})}^2\left(-\frac{1}{2R}{V}^2{t}^{2}H(-Vt+\frac{l_{\mathrm{II}}}{2})+(-\frac{1}{2R}{V}^2{t}^2+\frac{l_{\mathrm{II}}}{R}Vt-\frac{{l_{\mathrm{II}}}^2}{2R})H(Vt-\frac{l_{\mathrm{II}}}{2})H(l_{\mathrm{II}}-Vt)\right)\left(\mathrm{stage}\mathrm{II}\right)\\ \frac{1}{2}\left(\frac{l^2}{8R}\cos (\frac{2\pi Vt}{l_{\mathrm{III}}})+\frac{1}{2}c\left(1-\cos (\frac{4\pi Vt}{l_{\mathrm{III}}})\right)-\frac{l^2}{8R}\right)H(-Vt+l_{\mathrm{III}})\left(\mathrm{stage}\mathrm{III}\right)\end{array}\right.$	2D	Consider	MBD	[8]
Rectangular	None	3D	Not Consider	FEM	[10,11,20,21]
Roughness	None	3D	Consider	FEM	[16]

Table 1. The main methods for describing the geometry of a wheel flat.

Method	Formula	2D/3D	Consider Wear?	Simulation Type	Citation
Center trajectory	$z(x)=\left\{\begin{array}{ll}{x}^2/2R_w& 0\le x\le l/2\\ {(l-x)}^2/2R_w& l/2<x\le l\end{array}\right.$ (new flat)	2D	Consider	MBD	[5,6]
	$z(x)=\left\{\begin{array}{ll}4d/{(x/l)}^2& 0\le x\le l/2\\ 4d/{[(l-x)/l]}^2& l/2<x\le l\end{array}\right.$ (rounded flat)
Rail indentation	$z(x)=\frac{d}{2}(1-\cos \frac{2\pi x}{l})0\le x\le l$	2D	Consider	MBD	[7,9]
Radius deviation	$\Delta r(x)=d-\frac{1}{2R_w}{(x-l/2)}^{2}0\le x\le l$ (new flat)	2D	Consider	MBD	[1,2,3,13,14,18,19]
	$\left\{\begin{array}{l}\Delta r(x)=\frac{d}{2}(1-\cos \frac{2\pi x}{l})\\ d=l^2/16R_w\end{array}\right.0\le x\le l$ (rounded flat)
Discrete three-dimensional	$\left\{\begin{array}{l}{x}_i=\frac{l}{N}i(1\le i\le N)\\ y_{ij}=j{y}_i/M-y_i/2\\ z_{ij}=R_r-\frac{R_r\cos \frac{{\gamma }_i}{2}}{\cos (\frac{{\gamma }_i}{2}-{\gamma }_{ij})}\end{array}\right.$	3D	Not Consider	MBD	[17]
Center trajectory (time domain)	$z(t)=\left\{\begin{array}{l}-\frac{1}{2R}{V}^2{t}^{2}H(-Vt+\frac{l}{2})+(-\frac{1}{2R}{V}^2{t}^2+\frac{l}{R}Vt-\frac{l^2}{2R})H(Vt-\frac{l}{2})H(l-Vt)\left(\mathrm{stage}\mathrm{I}\right)\\ {(\frac{l}{l_{\mathrm{II}}})}^2\left(-\frac{1}{2R}{V}^2{t}^{2}H(-Vt+\frac{l_{\mathrm{II}}}{2})+(-\frac{1}{2R}{V}^2{t}^2+\frac{l_{\mathrm{II}}}{R}Vt-\frac{{l_{\mathrm{II}}}^2}{2R})H(Vt-\frac{l_{\mathrm{II}}}{2})H(l_{\mathrm{II}}-Vt)\right)\left(\mathrm{stage}\mathrm{II}\right)\\ \frac{1}{2}\left(\frac{l^2}{8R}\cos (\frac{2\pi Vt}{l_{\mathrm{III}}})+\frac{1}{2}c\left(1-\cos (\frac{4\pi Vt}{l_{\mathrm{III}}})\right)-\frac{l^2}{8R}\right)H(-Vt+l_{\mathrm{III}})\left(\mathrm{stage}\mathrm{III}\right)\end{array}\right.$	2D	Consider	MBD	[8]
Rectangular	None	3D	Not Consider	FEM	[10,11,20,21]
Roughness	None	3D	Consider	FEM	[16]

2. Three-Dimensional Wheel Flat Model

According to the mechanism of wheel flat formation, a localized region of the wheel tread is worn away during sliding. Compared with a complete wheel, a wheel with a flat loses a three-dimensional patch. Empirically, the projection of a wheel flat on the tread surface is approximated as an ellipse, with the long axis of the ellipse parallel to the rolling direction and the short axis parallel to the lateral direction. Therefore, the three-dimensional patch corresponding to the newly formed wheel flat should resemble a semi-ellipsoidal crown, which changes shape with increasing abrasion.

Based on [8], this paper divides the patch patterns corresponding to wheel flats into three stages.

Stage I. New flat. Because of the initial formation of the wheel flat from abrasion, the patch morphology in this stage is a complete semi-ellipsoidal crown.

Stage II. Edge worn flat. The leading and trailing edges of the wheel flat begin to wear away, extending the length of the flat without changing its width or depth. In this stage, the patch edges start to show wear rounding, but the primary morphology remains a semi-ellipsoidal crown.

Stage III. Completely worn flat. The wheel flat is worn to the center, further extending in length compared with Stage II, with no change in width or depth. The patch morphology changes significantly to a near-saddle shape.

According to these three stages of wheel flat wear, its geometric modeling is divided into two categories as follows: new flat (Stage I) modeling and worn flat (Stages II and III) modeling.

2.1. New Flat Model

The 3D wheel flat parameters and coordinate system of three-dimensional views are defined in Figure 2, with the meanings of the parameters listed in

Table 2. Parameter table for a new flat.

Notation	Meaning	Control Parameter or Not
$L_0$	new flat length	×
$L_w$	new flat width	×
$d$	new flat depth	√
$d_0$	nominal depth	×
$R_w$	wheel nominal radius	√
$R_r$	curvature at rail head radius	√
$\alpha$	depth coefficient	√

.

For any point $A(x,y)$ on the wheel flat, which has depth $F_{\mathrm{I}}(x,y)$:

$$F_{\mathrm{I}}(x,y)=c\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}+d_0(x\in [-a,a],y\in [-b,b])$$

(1)

In Equation (1), $a=L_0/2$, $b=L_w/2$, and $c=d-d_0$.

The length of the new flat is determined by Equation (2):

$$L_0=2\sqrt{2R_w{d}_0-{d_0}^2}$$

(2)

where $d_0=d/\alpha$.

In the direction of the wheel flat width, considering the cutting effect of the rail head on the wheel tread, the width of the flat is determined by Equation (3).

$$L_w=2\sqrt{2R_{r}d-d^2}$$

(3)

According to Equations (2) and (3), the new flat aspect ratio can be written as Equation (4).

$$\frac{L_0}{L_w}=\sqrt{\frac{d_0(2R_w-d_0)}{d(2R_r-d)}}$$

(4)

Because $R_w>>d_0$, $R_r>>d$, Equation (4) is simplified to Equation (5).

$$\frac{L_0}{L_w}=\sqrt{\frac{R_w}{\alpha R_r}}$$

(5)

Based on Equation (5), the aspect ratio of the new wheel flat is related to the radius of curvature of the wheel–rail contact point at the abrasion.

2.2. Worn Flat Model

According to [8], during the edge-worn stage of a wheel flat, the material between the edges and the center of the flat cannot come into single-point contact with the rail. Thus, the initiation of wear and/or plastification from the edges is geometrically determined. At this stage, the wheel flat geometry is defined as shown in Figure 3.

The meanings of edge worn flat parameters are listed in

Table 3. Parameter table for an edge worn flat.

Notation	Meaning	Control Parameter or Not
$L_0$	new flat length	×
$L_w$	new flat width	×
$l$	worn region length	×
$L$	edge worn flat length	×
$d$	edge worn flat depth	√
$d_0$	nominal depth	×
$R_w$	wheel nominal rolling circle radius	√
$R_r$	curvature at rail head radius	√
$\alpha$	depth coefficient	√
$\beta$	length coefficient	√
$\gamma$	worn coefficient	√

.

From Figure 3, the edge worn flat contains the following two regions: the worn region and the non-worn region. In the nominal rolling section, the endpoint $A(x_1,y_1)$, endpoint $B(x_2,y_2)$ of the worn region are defined. In this case, the depth function of the non-worn region degrades to $F_{\mathrm{I}}(x)$, the depth function of the worn region is defined as $g(x)$, and the analytic equation of the rolling circle outside wheel flat region is defined as $r(x)$. Considering the cubic polynomial approximation for the worn region, the boundary conditions are shown in Equation (6).

$$\left\{\begin{array}{l}r(x_1)=g(x_1)\\ r^{\prime }(x_1)=g^{\prime }(x_1)\\ F_{\mathrm{I}}(x_2)=g(x_2)\\ {F^{\prime }}_{\mathrm{I}}(x_2)=g(x_2)\end{array}\right.$$

(6)

In Equation (6), $r(x)$, $g(x)$, and $F(x)$, respectively, are defined as follows:

$$r(x)=R_w-\sqrt{{R_w}^2-x^2}$$

(7)

$$g(x)=a_1{x}^3+a_2{x}^2+a_{3}x+a_4$$

(8)

$$F_{\mathrm{I}}(x)=(1-\frac{1}{\alpha })(\sqrt{1-\frac{{\alpha }^2{x}^2}{2\alpha R_{w}d-d^2}}+\frac{1}{\alpha })d$$

(9)

For $x_1$, $x_2$, $x_1=-L/2$, $x_2=-L/2+l$, $\beta =L/L_0$, and $\gamma =l/L_0$.

The polynomial coefficients of the worn region can be fully determined according to Equation (6).

For any section parallel to the rolling circle, the length coefficient of the edge worn flat is replaced by ${\beta }^{\prime }$, ignoring the change in the radius of the rolling circle due to the taper of the wheel tread in the wheel flat region.

$${\beta }^{\prime }=\sqrt{(1-\frac{\Delta y^2}{2R_{r}d-d^2})}\beta (\Delta y\in [0,\frac{L_w}{2}])$$

(10)

The worn coefficient is linearly interpolated based on the section location, as shown in Equation (11).

$${\gamma }^{\prime }=\frac{L_w-2\Delta y}{L_w}\gamma$$

(11)

According to Equations (6)–(11), the depth at any position within the worn region of the wheel flat $g(x,y)$ can be obtained.

In summary, the depth function of the edge worn wheel flat $F_{\mathrm{II}}(x,y)$ is shown in Equation (12).

$$F_{\mathrm{II}}(x,y)=\left\{\begin{array}{ll}g(x,y)& x,y\in S_w\\ F_{\mathrm{I}}(x,y)& x,y\in S_n\end{array}\right.$$

(12)

where $S_w$ represents the feasible domain of the worn region and $S_n$ represents the feasible domain of the non-worn region.

The completely worn stage is a special case of the edge worn stage, in which the worn region length $l$ is equal to the edge worn flat length $L$, resulting in an analytic expression that matches exactly with that of the edge worn stage.

2.3. Geometric Constraints on Wheel Flats

During the edge worn or completely worn stage, the profile of the flat is geometrically determined, resulting in the following constraints for any section in the wheel flat region during the wear process.

For any $y\in [-L_w/2,L_w/2]$, $x\in \left[-{\beta }^{\prime }{L}_0/2,-{\beta }^{\prime }{L}_0/2+{\gamma }^{\prime }{L}_0\right]\cup \left[{\beta }^{\prime }{L}_0/2-{\gamma }^{\prime }{L}_0,{\beta }^{\prime }{L}_0/2\right]$, the existence of constant constraints is shown in Equation (13), where $r(x,y)$ indicates the profile function of any rolling circle in the lateral direction of the wheel tread.

$$\left\{\begin{array}{l}g(x,y)\ge r(x,y)\\ g(x,y)\ge F_{\mathrm{I}}(x,y)\end{array}\right.$$

(13)

According to Equation (13), the constraint relationship between the worn coefficient and the length coefficient can be established. In this study, the feasible domains of worn coefficients for different length coefficients are determined using grid search, as illustrated in Figure 4.

The wheel flat region is discretized into 50 parallel sections along the lateral direction of the wheel tread. Within each section, the wheel flat profile is further discretized into 50 points. $\beta \in (1,5]$ is defined for each specific value of $\beta$, where $\gamma \in (\frac{\beta -1}{2},\frac{\beta }{2}]$. Therefore, the feasible domain search space corresponding to each section is subdivided into a 20 × 20 grid based on the control parameters. Each grid cell corresponds to a specific combination of the control parameters, denoted as ${\beta }_0$ and ${\gamma }_0$. Subsequently, each grid cell is systematically evaluated to determine if all discretized points within the current section satisfy Equation (13) within the numerical computation tolerance. This evaluation process is translated into solving a convex optimization problem. If all points in a grid cell meet the criteria of Equation (13), the cell is marked as valid; otherwise, it is marked as invalid. Finally, the intersection of all valid grid cells across sections defines the distribution of labeled cells, and the boundary of the feasible domain is refined using nearest neighbor interpolation.

The feasible domain of control parameters is shown in Figure 5.

In Figure 5, the entire search space is partitioned into three regions by the upper and lower boundaries. The yellow-marked region (Stage II) corresponds to the feasible domain of the control parameters, depicted by Equation (14), where ${\gamma }_L$ and ${\gamma }_U$ denote the values of the upper and lower boundaries, respectively.

$$\left\{\begin{array}{l}\gamma \in [{\gamma }_L,{\gamma }_U](\beta <{\beta }_C)\\ \gamma \in [{\gamma }_L,{\gamma }_U)(\beta \ge {\beta }_C)\end{array}\right.$$

(14)

In the feasible domain, point A corresponds to the control parameter $\beta =1$,$\gamma =0$, indicating that the length of the worn region of the wheel flat is 0, representing the new flat stage. As the wheel flat undergoes edge wear, the length of the flat increases, at which time the value of $\beta$ increases, and the range of $\gamma$ expands accordingly. When $\beta ={\beta }_C$(${\beta }_C=1.8$), it reaches ${\gamma }_U=\beta /2$ for the first time and maintains this relationship thereafter. Therefore, point C marks the critical transition between the edge-worn stage and the completely worn stage. Subsequently, at Point D, $\beta >{\beta }_C$ and $\gamma \in [{\gamma }_L,{\gamma }_U)$ correspond to Stage II, and at Point F, $\beta >{\beta }_C$ and $\gamma =\beta /2$ correspond to Stage III.

Moreover, the critical value of the worn and fresh wheel flat of the haversine method is $\pi /2$, which closely matches the critical value of the model presented in this paper, indicating consistency in depicting the completely worn stage.

The search space contains two types of error regions in addition to the feasible domain. Point B represents the first type of error, violating the $g(x,y)\ge F_{\mathrm{I}}(x,y)$ constraint in Equation (13), while Point E corresponds to the second type of error, violating $g(x,y)\ge r(x,y)$ in Equation (13).

3. Complex 3D Wheel Flat Surface Reconstruction

The simulation of vehicle–track dynamics based on the measured wheel roughness is more accurate when compared with the analytical form of the geometric modeling. When the wheel contains a wheel flat, the reconstruction of the complex surface of the flat through the measured roughness data of the wheel lays the foundation for accurately reproducing the vehicle–track dynamics response law under the flat condition.

In recent years, the non-uniform rational B-splines (NURBS) method has gained popularity as a mathematical description of shapes in computer-aided geometric design (CAGD). This method combines the advantages of Bézier’s method and offers great flexibility in designing complex surfaces such as wheel flats. In the design of profiles of wheel and rail, and wear prediction, the cubic B-spline is primarily used [22,23,24,25,26,27]. In addition, the B-spline technique plays an important role in the profile design of high-speed turnout switch panels [28]. This paper focuses on establishing the complex surface reconstruction algorithm of wheel flats based on the cubic B-spline.

The reconstruction of the wheel flat surface consists of two stages including the computation of the flat control vertex matrix and the computation of the flat interpolation point matrix.

3.1. Theoretical Derivation

3.1.1. Control Vertex Matrix Computation

The computation process for the control vertex matrix involves the following steps.

Step 1. Construct the wheel roughness sample point matrix $Q$, where each element $q_{ij}(i=0,1,2,\dots r;j=0,1,2,\dots s)$ represents a sample point. The wheel flat region is divided into $s+1$ parallel sections along the width direction, with each section sampled at equal intervals using a wheel out-of-roundness testing instrument. Given that the number of sampling points in each section may vary, it is necessary to resample the data points to align them uniformly, where each section contains $r+1$ data points. The ordering direction of the data points in each section is defined as $u$, and the ordering direction of each section is defined as $v$.

Step 2. Construct node vectors.

The data points in each section are parameterized using normalized cumulative chord length, defined as follows:

$$\left\{\begin{array}{l}{u}_0=0\\ u_i=u_{i-1}+\left|\Delta {\stackrel{\rightharpoonup }{q}}_{i-1}\right|,i=1,2,\dots r\\ \Delta {\stackrel{\rightharpoonup }{q}}_{i-1}={\stackrel{\rightharpoonup }{q}}_i-{\stackrel{\rightharpoonup }{q}}_{i-1}\end{array}\right.$$

(15)

The parameterized sequences of all sections are averaged to obtain the $u$-parameter segmentation sequence ${\tilde{u}}_i(i=0,1,2,\dots ,r)$. The centroid of each section data point is computed, and the sequence of centroids is parameterized by the normalized cumulative chord length to obtain the $v$-parameter segmentation sequence ${\tilde{v}}_j(j=0,1,\dots ,s)$.

Finally, the segmentation sequences are converted into node vectors $U=\left[u_0,u_1,\dots u_{m+4}\right]$ and $V=\left[v_0,v_1,\dots v_{n+4}\right]$, where $m=r+2$, $n=s+2$.

Step 3. Compute the control vertex matrix. The computation process is illustrated in Figure 6.

The computation of the control vertex matrix involves two inversion processes. Firstly, a cubic spline curve control vertex inversion process is applied to each column of data points in the sample point matrix to derive the control vertex matrix of intermediate vertices. Subsequently, another inversion process is conducted for each row of the intermediate control vertex matrix to obtain the final control vertex matrix.

The inversion process follows a simplified form of the matrix equation as described in [30].

$$\left[\begin{array}{ccccc}{b}_1& c_1& a_1& & \\ a_2& b_2& c_2& & \\ & \ddots & \ddots & \ddots & \\ & & a_{n-2}& b_{n-2}& c_{n-2}\\ & & c_{n-1}& a_{n-1}& b_{n-1}\end{array}\right]\left[\begin{array}{c}{d}_1\\ d_2\\ ⋮\\ d_{n-2}\\ d_{n-1}\end{array}\right]=\left[\begin{array}{c}{e}_1\\ e_2\\ ⋮\\ e_{n-2}\\ e_{n-1}\end{array}\right]$$

(16)

Equation (16) details the elements of each row of the coefficient matrix, excluding the first and last rows, as presented in Equation (17).

$$\left\{\begin{array}{l}{a}_i=\frac{{({\Delta }_{i+2})}^2}{{\Delta }_i+{\Delta }_{i+1}+{\Delta }_{i+2}}\\ b_i=\frac{{\Delta }_{i+2}({\Delta }_i+{\Delta }_{i+1})}{{\Delta }_i+{\Delta }_{i+1}+{\Delta }_{i+2}}+\frac{{\Delta }_{i+1}({\Delta }_{i+2}+{\Delta }_{i+3})}{{\Delta }_{i+1}+{\Delta }_{i+2}+{\Delta }_{i+3}}\\ c_i=\frac{{({\Delta }_{i+2})}^2}{{\Delta }_{i+1}+{\Delta }_{i+2}+{\Delta }_{i+3}}\\ e_i=({\Delta }_{i+1}+{\Delta }_{i+2})q_{i-1}\end{array}\right.,(i=2,3,\dots ,n-2)$$

(17)

In this study, free endpoint boundary conditions are employed, where:

$$\left\{\begin{array}{l}{b}_1={\Delta }_1+2{\Delta }_2+2{\Delta }_3+{\Delta }_4\\ c_1=-({\Delta }_1+{\Delta }_2+{\Delta }_3)\\ a_1=0\\ e_1=({\Delta }_2+{\Delta }_3+{\Delta }_4)q_0\end{array}\right.\left\{\begin{array}{l}{c}_{n-1}=0\\ a_{n-1}=-({\Delta }_n+{\Delta }_{n+1}+{\Delta }_{n+2})\\ b_{n-1}={\Delta }_{n-1}+2{\Delta }_n+2{\Delta }_{n+1}+{\Delta }_{n+2}\\ e_{n-1}=({\Delta }_{n-1}+{\Delta }_n+{\Delta }_{n+1})q_{n-2}\end{array}\right.$$

(18)

The vertex vector of the inverse process can be solved using Equations (16)–(18).

3.1.2. Interpolation Point Matrix Computation

The interpolation point matrix computation process consists of the following steps.

Step 1. Determine the node vectors of a non-uniform B-spline surface.

In the $u$-direction, the corresponding node vectors are determined using the Hadley–Judd method based on the control polygons of each section, where:

$$\left\{\begin{array}{l}{u}_k=0\\ u_i={\displaystyle \sum _{j=k+1}^i\left(u_j-u_{j-1}\right)}\\ u_{n+1}=1\end{array}\right.,i=k+1,k+2,\dots ,n$$

(19)

In Equation (19), $u_i$ represents the value of each node in the node vector in the $u$-direction, and $u_i-u_{i-1}$ is shown as Equation (20).

$$u_i-u_{i-1}=\frac{{\displaystyle \sum _{j=i-k}^{i-1}\left|{\stackrel{\rightharpoonup }{d}}_j-{\stackrel{\rightharpoonup }{d}}_{j-1}\right|}}{{\displaystyle \sum _{i=k+1}^{n+1}{\displaystyle \sum _{j=i-k}^{i-1}\left|{\stackrel{\rightharpoonup }{d}}_j-{\stackrel{\rightharpoonup }{d}}_{j-1}\right|}}},i=k+1,k+2,\dots ,n+1$$

(20)

In Equation (20), ${\stackrel{\rightharpoonup }{d}}_j$ represents the vector pointing from the origin to the control vertex $d_j$, and $\left|{\stackrel{\rightharpoonup }{d}}_j-{\stackrel{\rightharpoonup }{d}}_{j-1}\right|$ denotes the side length of the control polygon.

The node vector $U=\left[u_0,u_1,\dots u_{m+4}\right]$ in the $u$-direction is computed as the average of all node vectors. Similarly, the node vector in the $v$-direction is $V=\left[v_0,v_1,\dots v_{n+4}\right]$.

Step 2. Compute the interpolation points of the wheel flat surface.

For any point $\left(u,v\right)$ in the domain of definition, the De Boor algorithm is first applied to $m+1$ control polygons along the $v$-direction. Subsequently, the resulting $m+1$ points are arranged into an intermediate polygon, followed by executing the De Boor algorithm again for the parameter $u$ to determine the depth $p\left(u,v\right)$ of any interpolated point on the wheel flat surface.

In this study, the De Boor algorithm is employed in its recursive form, presented as follows:

$$\left\{\begin{array}{l}p(u)={\displaystyle \sum _{j=0}^n{d}_j{N}_{j,k}(u)={\displaystyle \sum _{j=i-k}^{i-l}{d_j}^l{N}_{j,k-l}(u)=\cdots d_{i-k}^k,u\in [u_i,u_{i+1}]\subset [u_k,u_{n+1}]}}\\ d_j^l=\left\{\begin{array}{ll}{d}_j& l=0\\ (1-{\alpha }_j^l)d_j^{l-1}+{\alpha }_j^l{d}_{j+1}^{l-1}& j=i-k,i-k+1,\cdots ,i-l;l=1,2,\cdots k\end{array}\right.\\ {\alpha }_j^l=\frac{u-u_{j+l}}{u_{j+k+l}-u_{j+l}}\\ \mathrm{define}\frac{0}{0}=0\end{array}\right.$$

(21)

3.2. Example of Wheel Flat Surface Reconstruction

The reconstruction process of the wheel flat surface is shown in Figure 7.

Based on this process, the algorithm for reconstructing the wheel flat surface was tested in this study. According to the speed limits specified in

Table 4. Speed limit table for CRH3C EMU due to wheel damage [31].

Type of Wheel Damage	Defect Level	Defect Parameter (mm)	Speed Limit (km/h)
Wheel tread abrasion	I	$0.25<d<0.5$	$v\le 200$
	II	$0.5\le d<1$	$v\le 120$
	III	$d>1$	$v\le 80$

, the depth of the wheel flat was categorized into three levels as follows: 0.2 mm, 0.4 mm, and 0.6 mm, each machined onto the wheel tread. The shape of the machined flat was controlled using Equations (2) and (3).

After machining, each depth of the wheel flat was divided into 10 parallel sections, and the wheel roughness of each section was tested followed by reconstruction. The results of the wheel flat surface reconstruction are presented in

Table 5. The results of the wheel flat surface reconstruction.

	d = 0.2 mm	d = 0.4 mm	d = 0.6 mm
Wheel flat
Size	25 × 20	40 × 30	50 × 40
Roughness
Reconstruction

.

The findings in Table 5 indicate a close match between the size and depth distribution of the reconstructed wheel flat surface and the actual machining results. Specifically, the maximum depths of the reconstruction model were 0.18 mm, 0.038 mm, and 0.014 mm for depths originally machined at 0.2 mm, 0.4 mm, and 0.6 mm, respectively, differing from the machining depths by only 2.3% in the case of the 0.6 mm depth. However, because of the limited number of sections and measurement accuracy, especially near the edge sections where the measurement instrument’s accuracy is challenged, the reconstruction results exhibit some roughness at the endpoints. Furthermore, the interpolation results for depths at individual points within the wheel flat region tend to skew higher.

4. Wheel–Track Coupled Dynamics Model Considering Wheel Flat Geometry

4.1. Wheel–Track Coupled Dynamics Model

To investigate the dynamic response of the wheel–rail system under varying wheel flat geometries, a coupled dynamic model of the wheel–rail system was developed, illustrated in Figure 8.

The vehicle model comprises a car body and two bogies, each equipped with two wheel pairs. A wheel flat is simulated on the right wheel of the first wheel pair. Vertical dynamics are modeled using a Ruzicka approach, incorporating primary and secondary suspension vertical hydraulic dampers and rubber node stiffness at both ends. Vertical dampers are simplified as Maxwell models in a spring–damper series configuration. The dynamic equations of the entire vehicle system are expressed as Equation (22) [17].

$$\left[M\right]\left\{\ddot{X}\right\}+\left[C\right]\left\{\dot{X}\right\}+\left[K\right]\left\{X\right\}=\left\{P\right\}$$

(22)

In Equation (22), $\left[M\right]$, $\left[C\right]$, and $\left[K\right]$ denote matrices of mass, damping, and stiffness. $\left\{\ddot{X}\right\}$, $\left\{\dot{X}\right\}$, and $\left\{X\right\}$ represent vectors of acceleration, velocity, and displacement. $\left\{P\right\}$ is the force vector of the vehicle system.

In the track system, the rails are modeled as Timoshenko beams, discretely supported on sleepers by rail pads and fasteners, which are represented as a combination of springs and dampers. Considering the elastic properties of the ballast, springs and dampers are inserted between the sleepers and the subgrade. Additionally, no stochastic excitation is introduced into the track system, except for the wheel flat.

The main parameters of the vehicle and track model are shown in Appendix A.

4.2. Wheel and Rail Profiles

In this paper, the rail profile is CHN60 and the wheel profile is LMB₁₀.

To obtain the time-domain solution of the wheel–rail system response under different forms of wheel flats, the vertical runout of the wheel center of mass is typically simulated by the radius deviation function of the nominal rolling circle. In the circumferential direction of the wheel, the longitudinal symmetric section ($y=0$) of the wheel flat geometry is extracted and discretized into a radius deviation sequence by merging it with the nominal rolling circle, as shown in Figure 9a. As the position of the wheel–rail contact point varies, the corresponding radius deviation sequence within the rolling circle section is updated synchronously.

To investigate the effect of different wheel flat geometries on the wheel–rail contact geometry during the rolling process, the flat region is discretized along its length direction. It is then combined with the complete wheel tread to form a sequence of discrete profiles. Each discrete profile serves as an iterative step in the contact geometry computation process, as shown in Figure 9b.

5. Simulation and Discussion

5.1. Three-Dimensional Wheel Flat Dynamics Simulation

In this section, an explicit method is used to integrate the vehicle–track coupled dynamics model constructed in Figure 8 to obtain the time domain values of wheel–rail impact force and acceleration in the model. During the integration process, a fixed time step is employed to ensure that the total number of integration steps is consistent with the total number of iterative steps in Figure 9b.

Within each time step, the wheel tread profile corresponding to the current time step is first obtained according to Figure 9b to solve for the wheel–rail contact position. Using the form shown in Figure 9a, the wheel circumferential input function is constructed and merged with the rolling circle profile corresponding to the contact position. This is then integrated and solved to obtain the wheel–rail impact force and acceleration of the current time step, while retaining the state information of the displacement, velocity, acceleration, and other relevant data from all the components in Figure 8, as well as the wheel–rail contact geometry information at the end of the current time step. When the next time step begins, all state information from the previous step is used as input, the wheel tread profile is updated again, and the cycle repeats until all time steps have been computed.

To accelerate the computation process, the simulation was conducted on a hardware environment with a 2.66 GHz processor, 64 GB of RAM, and enabled GPU parallel computing.

5.2. Effect of Flat Geometry on Wheel–Rail Impact Force

According to the simulation methodology proposed in Section 5.1, the time-domain solution for the wheel–rail vertical force when the wheel has a flat is presented in Figure 10. The wheel–rail vertical force exhibits periodic impacts in the time domain, characterized by high-frequency vibrations with very short impact cycles. As the wheel enters the flat region, a brief wheel–rail separation occurs, followed by the force reaching its maximum peak value, referred to as the P1 force, and then gradually decaying to near the steady state value.

The time-domain solution in Figure 10 shows good consistency with the wheel–rail impact force patterns computed in [1,2,17], validating the accuracy of the dynamic model. The discrepancy in the amplitude of the time-domain results compared with the literature is due to the larger axle weight of the vehicle used in this paper, which results in a higher impact force.

Based on the time-domain characteristics of the wheel–rail force, Figure 11 compares the P1 force variation with velocity for different wheel flat geometries at various depths. The velocity dependence of the maximum wheel–rail impact force is consistent across different wheel flat geometries and depths. The impact force at the new flat stage is larger than at the worn flat stage, with the maximum peak value corresponding to a lower velocity in the new flat stage. This is because the length of the new flat is smaller than that of the worn flat, and the horizontal velocity corresponding to the same drop in the wheel center of mass height is lower in the new flat stage. The maximum impact force is negatively correlated with the minimum circumferential curvature of the flat, and the curvature discontinuity in the new flat leads to a singularity in contact force, as detailed in [32].

The geometric description method for the wheel flat proposed in this paper shows good agreement with the surface reconstruction method based on measured roughness, indicating that the basic assumption of the worn stage in the analytical description method is reasonable and the computational accuracy meets the requirements.

In addition, since the analytical form is parametrically controllable, when $\beta$ is close to $\pi /2$ ( $\beta =1.8$) and $\gamma =\beta /2$ , the maximum impact force in stage III differs from the haversine model by only 0.51%, 0.97%, and 1.3% for the three depth conditions, respectively. This indicates that the analytical model proposed in this paper exhibits a high degree of flexibility.

5.3. Effect of the Wear Degree of Wheel Flats on Wheel–Rail Impact Force

For a given length of the wheel flat, the range of values for the worn coefficient is determined based on the feasible domain of the control parameters. This section further investigates the variation in the wheel–rail vertical impact force under different values of the worn coefficient, as shown in Figure 12.

As the depth of the flat increases, the bandwidth of the wheel–rail impact force map in the velocity-worn coefficient domain gradually widens, resulting in a corresponding increase in the impact force amplitude. According to Figure 12d–f, the velocity interval corresponding to the critical regime shifts to higher values with increasing depth, a phenomenon discussed in the existing literature.

Based on this pattern, the variation in the wheel–rail vertical force with the worn coefficient in the subcritical and transcritical regimes is further examined, as shown in Figure 12g–i. In the subcritical regime, the wheel–rail vertical force gradually decreases as the worn coefficient increases, while in the transcritical regime, the wheel–rail vertical force initially increases and then decreases with increasing worn coefficient.

Furthermore, this paper qualitatively analyzes the possible reasons for the differences in monotonicity based on the analytical form of the wheel–rail impact force proposed in [32]. In the subcritical regime, the impact force is primarily influenced by the stiffness of the track. The magnitude of the impact force is inversely proportional to the minimum circumferential curvature of the wheel flat. As the worn coefficient increases, the distribution of the circumferential curvature changes, resulting in an increase in its minimum value. Consequently, the impact force exhibits a nonlinear decrease. In the transcritical regime, the wheel–rail impact force is mainly influenced by wheel–rail inertia. Here, the minimum circumferential curvature is replaced by an effective value, which is jointly determined by the velocity and the minimum circumferential curvature. Thus, the monotonicity of the impact force changes as the worn coefficient increases.

5.4. Effect of the Wear Degree of Wheel Flats on Wheel–Rail Contact Geometry

According to the wheel tread profile discretization method shown in Figure 9b, this section investigates the effects of different worn coefficients on the wheel–rail contact geometry. Taking $d=0.6\mathrm{mm}$ as an example, and without considering other geometric irregularities, the change in coordinates of the wheel–rail contact points in each iterative step during the forward rolling process of the complete wheel on the left side and the wheel containing a flat on the right side, when the wheel pair lateral displacement is 0, is shown in Figure 13.

In Figure 13, the geometric model of the wheel flat is spatially symmetric, resulting in a symmetric process of contact point coordinate changes. The presence of a wheel flat on the right wheel affects the wheel–rail contact position of the complete wheel on the left side. As the iterative steps increase, the left wheel–rail contact point moves away from the flange and then gradually returns to the initial contact position, with continuous coordinate changes throughout the process. The right wheel–rail contact point also changes, shifting laterally towards the flange in step 5, after which the contact position remains almost constant until it returns to the initial position in step 56. Comparing the jump durations for different wear degrees, it is found that the maximum wear degree increased by 9.8% compared with the minimum wear degree.

Different worn coefficients affect the start and stop moments of the wheel’s lateral jump. During the contact point computation, the defect influences the contact point position when the defect depth on the tread exceeds the accuracy of the contact point search algorithm, which is a fixed value. In the same iterative step, the greater the maximum depth of the wheel flat as the worn coefficient increases, the fewer iterative steps are required to exceed the fixed value, leading to an earlier start moment of the lateral jump.

6. Conclusions

This paper focuses on the modeling and simulation methods for the spatially complex morphology of wheel flats and systematically proposes a three-dimensional spatial model of flats using both analytical and data-driven approaches. A wheel–track coupled dynamics model that considers wheel flat geometry is established, revealing the influence of the control parameters of wheel flat geometry on the wheel–rail impact response and the wheel–rail contact geometry.

The control parameters of the analytical form of the wheel flat spatial model proposed in this paper are adjustable, making it highly suitable for predicting the impact response at different stages. By combining dynamic indices such as the P1 force, a more accurate safety threshold for the three-dimensional scale of the wheel flat can be established. Furthermore, when spatial sampling points in the wheel flat region are available, the spatial geometric model established through the reconstruction method can be utilized in dynamic simulations to produce high-precision time–frequency response data. This approach reproduces bench test results through numerical computation and offers a new method for evaluating the reliability of key vehicle components under extreme working conditions, such as wheel flats.

Overall, the main conclusions are as follows.

The analytical modeling method for three-dimensional wheel flats considering multiple worn stages proposed in this paper can effectively characterize each worn morphology during the life cycle of a flat in a parameter-driven approach. By using measured wheel roughness data, an effective reconstruction method for the real morphology of wheel flats is proposed based on the NURBS technique, achieving a minimum reconstruction error of only 2.3%.

According to the simulation results, by adjusting the control parameters, the analytical model proposed in this paper can effectively approximate the existing haversine wheel flat modeling method, with an average error of only 0.927% under different depth conditions. This modeling method, which considers multiple worn stages, allows for a detailed investigation of the effects of different worn states on the wheel–rail impact response and contact geometry through numerical simulation. Based on the method, the following patterns were found: In the subcritical regime, the wear degree of the wheel flat predominantly affects the wheel–rail impact force, while in the transcritical regime, both wear degree and velocity determine the magnitude of the wheel–rail impact force. As the degree of wear increases, the wheel lateral jump occurs earlier, with the maximum wear degree increasing the jump duration by 9.7% compared with the minimum wear degree.

The spatial modeling method of flats proposed in this paper is highly flexible and provides a new technical approach to simulating the dynamic behavior of wheel–rail contact accurately when a flat is present.