Research and Analysis on Contact Resistance of Wheel and Insulated Rail Joint in High-Speed Railway Stations

Abstract

Insulated rail joint (IRJ) as one of the components of the track circuit will be burned once the electric arc is generated when the train wheel passes through the insulated rail joint and the track circuit occurs the red-light band fault, which has a great impact on the safety of the train. Reducing the voltage at both ends of the insulated rail joint can effectively prevent the occurrence of electric arc. Clarifying the wheel-IRJ contact resistance is the basis for analyzing the voltage of wheel-IRJ. The effective apparent contact area of the wheel-IRJ contact was derived through the mathematical relation. Considering the surface roughness parameters, based on the electrical contact theory and the Greenwood-Williamson (GW) model, the mathematical models of the wheel-IRJ contact resistance and current density were established, and their influencing factors were analyzed. The results show that the contact resistance increased first and then decreased. The influences of load, roughness of the contact interface, and electric arc heat on the contact resistance were greater. Considering the influencing factors of contact resistance and current density, corresponding protective measures were proposed to reduce the potential difference at both ends of the insulated rail joint.

1. Introduction

In recent years, the problem of pulling arc and burning of mechanical insulated rail joint (IRJ) in high-speed railway stations has become more prominent [1,2,3,4,5]. As an integral part of the track circuit in the station, the insulated rail joint and rail are burnt out once the electric arc is drawn [1,2,3], causing faults such as red-light band. In addition, glued insulation is widely used in high-speed railways, which makes its replacement difficult and costly. Therefore, reducing the impact of the electric arc on the insulated rail joints and rails is an important part of ensuring the normal operation of high-speed railways.

In high-speed railways, the traction power of the electrical multiple unit (EMU) is large, the transient current at start-up is large, and the rail potential is also high, which will generate an electric arc when the train passes through the insulated rail joint at the return current cut-off point (the neutral point of the choke transformer on both sides of the insulation section is disconnected). Related studies show that the high potential difference between the two ends of the insulated rail joint is the main reason for the arcing of the insulated rail joint [2,6]. Many researchers put forward some protective measures from the perspective of reducing potential difference [4,5,7]. The voltage across the insulated rail joint depends on the current flowing through the insulated rail joint and the track impedance. Wheel–rail contact is a dynamic electrical contact. One of the most important characteristics of electrical contact is contact resistance, which is a part of the rail impedance. The contact resistance formed when the wheel passes through the insulated rail joint not only has a great impact on the whole traction circuit [8], but also is the main source of joule heat of the wheel-IRJ contact. Therefore, it is very important to clarify the change rule and influence factors of wheel-IRJ contact resistance.

The most direct way to obtain the wheel-IRJ contact resistance is to collect the voltage and current data on site. In view of the particularity of high-speed railway operation, the wheel-IRJ contact resistance can be studied by establishing a model of the wheel-IRJ contact resistance. Since contact resistance is affected by microscopic parameters (size, dimensions, and density of conductive spot, etc.), surface film, surface roughness, and environmental temperature [9], Kui Li et al. [10] used finite elements to calculate the current field to analyze the contact resistance, and Wanbin Ren et al. [11] calculated the contact resistance by building a simulation model for electric field analysis. These methods can fully reflect the influence of micro-parameters on contact resistance due to considering the distribution of current lines in the contact area. Xiangzhao Jin et al. [12] analyzed the contact pressure by finite element structural field and calculated the contact resistance of gas insulated switchgear joints by combining Bahrami’s rough surface elastic-plastic mechanical contact model. The wheel-IRJ contact is a multi-physics field coupling problem. The finite element method can be used to simulate the coupling of mechanical, thermal, and electrical behaviors of electrical contact [13]. However, the process of wheels passing through the insulated rail joint is a discontinuous contact problem. On the one hand, Hertz theory is no longer applicable [14]. On the other hand, although finite elements can deal with nonlinear problems well, there are problems related to contact analysis not being easy to converge and simulation speed no being easy to be too high [15].

Holm, the founder of electrical contact, pointed out that the contact resistance is determined by the contraction resistance and the film resistance, which are in series on the circuit. Shangpeng Sun et al. [16] proposed a calculation method of shunt resistance of track circuit based on the electrical contact theory and found that the pollution film has a great impact on the wheel–rail contact resistance. Pengfei Su [17] used the finite element method to calculate the macro contact area of wheel–rail and studied the contact resistance of wheel–rail static contact according to Hertz theory and electrical contact theory. None of the above studies considered the effect of surface roughness parameters on wheel–rail contact resistance. Hui Li et al. [18] established the contact resistance model of crimped IGBT devices based on the electrical contact theory, taking into account factors such as surface morphology and contact pressure. However, the crimp-type IGBT device forms an electrical connection between the internal chip and the external electrode by applying pressure, which is different from wheel-IRJ contact. Based on the Greenwood-Williamson (GW) contact model, Dayu Wang et al. [9,19] established a mathematical model of contact resistance affected by macro parameters, such as contact pressure with roughness parameters as the intermediate quantity. Currently, there are the statistical model [20], fractal model [21], and finite element model [22] for the rough surface contact model. The GW model regards the contact surface of two contact objects as an equivalent rough surface and an ideal smooth plane, which is suitable for analyzing the wheel–rail contact situation. Under the condition that the contact radius is greater than the thickness of insulated endplate, Xiaoming Wang et al. [23] simulated and analyzed the change in contact resistance between wheel and insulated rail joint based on the electrical contact theory. However, they did not consider the influence of surface roughness on contact resistance. The relevant standards point out that the insulated endplate thickness is generally 6–8 mm [24], so there is more than a relationship between the wheel–rail contact spot radius and the thickness of the insulated endplate. To study the contact resistance with the plating layer, Sawada et al. [25,26] separated the relationship between contact force and contact resistance into contact force–contact area and contact area–contact resistance. For the wheel–rail contact, the static macro contact area can be obtained by finite element simulation, but the contact area of the wheel rolling through the insulated rail joint will change due to the existence of the insulated rail joint between the two track sections. The effective apparent contact area between the wheel and the insulated rail joint can be accurately obtained through mathematical derivation.

In this paper, according to the three relations between the contact radius of the wheel–rail contact spot and the thickness of the insulated endplate of the insulated rail joint, the effective apparent contact area of the wheel-IRJ contact is deduced by mathematical method. Considering the macroscopic parameters, such as surface roughness, a mathematical model of contact resistance and current density of wheel-IRJ is established based on the electrical contact theory, and its variation law and its influencing factors are analyzed. From the point of view of contact resistance and current density, the protective measures to reduce the potential difference at both ends of the insulated rail joint are proposed.

2. Theoretical Basis

2.1. Electrical Contact Theory

The surface of any fine machined object is not smooth. When the current passes through two contact objects, the current line will constrict. Holm’s theory shows that the constriction resistance R of two contact objects can be defined as

$$R=\frac{\rho }{2\alpha }$$

(1)

where $\rho$is the resistivity, and $\alpha$is the radius of circular conductive spot.

In addition, the conductive spot surface has a contaminated film layer caused by various factors, such as the environment, temperature and humidity, which create an additional film resistance between the two contact objects. The contact resistance between two contact objects is the sum of constriction resistance and film resistance [27].

2.2. GW Rough Surface Contact Model

Greenwood et al. proposed a GW rough surface contact model [20]. To facilitate the problem analysis, we assumed that the two contact surfaces were a plane and a nominally flat surface, as shown in Figure 1. The contact was elastic, and the influence of surface film was not considered. All asperities had the same curvature radius on the apparent contact surface, and the asperity height obeyed a Gaussian distribution. The solid straight line in Figure 1 is the smooth surface, and the dashed straight line is the reference plane of the rough surface.

Then, the number n of rough asperities involved in contact can be defined as

$$n=\eta A_n{{\displaystyle \int }}_h^{\infty }\frac{1}{\sigma \sqrt{2\pi }}{\mathrm{e}}^{-\frac{z^2}{2{\sigma }^2}}dz$$

(2)

The actual total contact area $A_r$can be described by

$$A_r=\pi \eta A_n\beta {{\displaystyle \int }}_h^{\infty }\left(z-h\right)\frac{1}{\sigma \sqrt{2\pi }}{\mathrm{e}}^{-\frac{z^2}{2{\sigma }^2}}dz$$

(3)

The load W can be expressed by

$$W=\frac{4}{3}\pi \eta A_n{E}^{\prime }{\beta }^{\frac{1}{2}}{{\displaystyle \int }}_h^{\infty }{\left(z-h\right)}^{\frac{3}{2}}\frac{1}{\sigma \sqrt{2\pi }}{\mathrm{e}}^{-\frac{z^2}{2{\sigma }^2}}dz$$

(4)

where $\eta$ is the surface density of rough surface, $\sigma$ is the standard deviation of the height of rough surface, $A_n$ is the nominal contact area, $h$ is the distance between the smooth surface and the rough reference surface, $z$ is the asperities height, $\beta$ is the asperities radius, $1/E^{\prime }=1/2·((1-{\nu }_1{}^2)/E_1+\left(1-{\nu }_2{}^2)/E_2\right)$, $E_1$, and $E_2$ are, respectively, Young’s modulus of wheel and rail, ${\nu }_1$ and ${\nu }_2$ are, respectively, Poisson’s ratio of wheel and rail [20].

Using Equations (2)–(4), we deduced the relationship between the actual contact area and the load under the GW rough surface contact as

$$\frac{A_r}{W}=\frac{3\pi \sigma {\beta }^{\frac{1}{2}}\sqrt{2\pi }}{4E^{\prime }}\frac{1}{{{\displaystyle \int }}_h^{\infty }{\left(z-h\right)}^{\frac{1}{2}}{\mathrm{e}}^{-\frac{z^2}{2{\sigma }^2}}dz}$$

(5)

According to Figure 1, at any roughness with a given height of $z$, when $z$ > $h$, two objects contact, and the contact radius was $\alpha =\sqrt{\left(z-h\right)\beta }$. According to Equations (1) and (2), the contact resistance $R_c$ under GW rough contact can be obtained as follows:

$$R_c=\frac{\rho }{2\sum \sqrt{\left(z-h\right)\beta }}$$

(6)

3. Mathematical Model

3.1. Mathematical Model of Wheel-IRJ Contact Resistance

The train exerts a certain force on the track. In this study, the relationship between force and contact resistance was studied in two parts: the force–contact area and contact area–contact resistance for the wheel-IRJ contact resistance.

When the wheel had a small lateral displacement, and operated on a straight track, it was assumed that the wheel–rail apparent contact spot was an equivalent circle of radius $r_a$. The wheel–rail conductive spots were uniformly distributed and circular in the apparent contact spot. Wheels and rails had equal resistivity, $\rho$.

The apparent contact spot areas with different loads are listed in

Table 1. Apparent contact spot area of wheel–rail under different loads.

Load/t	Apparent Contact Spot Area/mm2	${\mathit{r}}_{\mathit{a}} (\mathbf{Approximate} \mathbf{Value})/\mathbf{mm}$
9	98	5.6
12	118	6.1
15	144	6.8
21	182	7.6
24	204	8.0
27	226	8.5

[17]. In this paper, the insulated endplate thickness $d$ is 8 mm. According to Table 1, there are three different relationships between $r_a$ and $d$. These different relationships affect the wheel-IRJ contact area.

Figure 2 shows the relationship between the wheel–rail circular contact spot and the insulated endplate thickness. Under different relations between $r_a$ and d, the effective apparent contact area $S_{gx}$ (x = 1, 2, …) of the wheel rolling from the initial position (the circular contact spot of the wheel–rail is just about to contact with the insulated endplate but not yet) to completely cross the insulated rail joint is as follows.

• When $r_a>d$, the dynamic contact process between wheel and insulated rai joint is shown in Figure 3. The wheel rolling from the initial position to $r_a$ for $S_{gx} \left(x=1, 2, \dots \right)$ is consistent with reference [23].

$$S_1=\frac{\pi r_a{}^2}{2}-2{{\displaystyle \int }}_0^{T_1}{\left(r_a{}^2-{\left(r_a-vt\right)}^2\right)}^{\frac{1}{2}}vdt, \left(T_1=\frac{d}{v}\right)$$

(7)

$$S_2=2{{\displaystyle \int }}_{T_1}^{T_2}{\left(r_a{}^2-{\left(r_a-vt+d\right)}^2\right)}^{\frac{1}{2}}vdt, \left(T_2=\frac{r_a}{v}\right)$$

(8)

Rolling $2d$ is shown in Figure 3, $T_3=2d/v$, $T_2<t\le T_3$

$$S_{g3}=\frac{\pi r_a{}^2}{2}-2{{\displaystyle \int }}_{T_2}^t{\left(r_a{}^2-{\left(vt-r_a\right)}^2\right)}^{\frac{1}{2}}vdt+S_2+2{{\displaystyle \int }}_{T_2}^t{\left(r_a{}^2-\left(r_a-(vt-d\right){)}^2\right)}^{\frac{1}{2}}vdt$$

(9)

$$S_3=\frac{\pi r_a{}^2}{2}-2{{\displaystyle \int }}_{T_2}^{T_3}{\left(r_a{}^2-{\left(vt-r_a\right)}^2\right)}^{\frac{1}{2}}vdt$$

(10)

$$S_4=S_2+2{{\displaystyle \int }}_{T_2}^{T_3}{\left(r_a{}^2-\left(r_a-(vt-d\right){)}^2\right)}^{\frac{1}{2}}vdt$$

(11)

Rolling $(r_a+d)$ is shown in Figure 3, $T_4=(r_a+d)/v$, $T_3<t\le T_4$

$$S_{g4}=S_3-2{{\displaystyle \int }}_{T_3}^t{\left(r_a{}^2-{\left(vt-r_a\right)}^2\right)}^{\frac{1}{2}}vdt+S_4+2{{\displaystyle \int }}_{T_3}^t{\left(r_a{}^2-\left(r_a-(vt-d\right){)}^2\right)}^{\frac{1}{2}}vdt$$

(12)

Rolling 2$r_a$ is shown in Figure 3, $T_5=2r_a/v$, $T_4<t\le T_5$

$$S_{g5}=S_2-2{{\displaystyle \int }}_{T_4}^t{\left(r_a{}^2-{\left(vt-r_a\right)}^2\right)}^{\frac{1}{2}}vdt+\frac{\pi r_a{}^2}{2}+2{{\displaystyle \int }}_{T_4}^t{\left(r_a{}^2-{\left(vt-r_a-d\right)}^2\right)}^{\frac{1}{2}}vdt$$

(13)

Rolling $(2r_a+d)$ is shown in Figure 3, $T_6=(2r_a+d)/v$, $T_5<t\le T_6$

$$S_{g6}=2{{\displaystyle \int }}_{T_5}^t{\left(r_a{}^2-{\left(vt-r_a-d\right)}^2\right)}^{\frac{1}{2}}vdt+S_1+\frac{\pi r_a{}^2}{2}$$

(14)

where $v$ is the train running speed. $Tx \left(x=1, 2, \cdots \right)$ represents the time when the wheel rolls the corresponding distance from the initial position. $Sgx \left(x=1, 2, \cdots \right)$ is the apparent contact area of wheel-IRJ for each rolling distance.

2.

When $r_a$ < $d$, the dynamic contact process for wheel-IRJ is shown in Figure 4.

Rolling $r_a$ is shown in Figure 4, $T_1=r_a/v$, $0<t\le T_1$

$$S_{g1}=\pi r_a{}^2-2{{\displaystyle \int }}_0^t{\left(r_a{}^2-{\left(r_a-vt\right)}^2\right)}^{\frac{1}{2}}vdt$$

(15)

Rolling $d$ is shown in Figure 4, $T_2=d/v$, $T_1<t\le T_2$

$$S_{g2}=\frac{\pi r_a{}^2}{2}-2{{\displaystyle \int }}_{T_1}^t{\left(r_a{}^2-{\left(vt-r_a\right)}^2\right)}^{\frac{1}{2}}vdt$$

(16)

$$S_5=\frac{\pi r_a{}^2}{2}-2{{\displaystyle \int }}_{T_1}^{T_2}{\left(r_a{}^2-{\left(vt-r_a\right)}^2\right)}^{\frac{1}{2}}vdt$$

(17)

Rolling 2$r_a$ is shown in Figure 4, $T_3=2r_a/v$, $T_2<t\le T_3$

$$S_{g3}=S_5-2{{\displaystyle \int }}_{T_2}^t{\left(r_a{}^2-{\left(vt-r_a\right)}^2\right)}^{\frac{1}{2}}vdt+2{{\displaystyle \int }}_{T_2}^t{\left(r_a{}^2-{\left(r_a-\left(vt-d\right)\right)}^2\right)}^{\frac{1}{2}}vdt$$

(18)

$$S_6=2{{\displaystyle \int }}_{T_2}^{T_3}{\left(r_a{}^2-{\left(r_a-\left(vt-d\right)\right)}^2\right)}^{\frac{1}{2}}vdt$$

(19)

Rolling $(r_a+d)$ is shown in Figure 4, $T_4=(r_a+d)/v$, $T_3<t\le T_4$

$$S_{g4}=2{{\displaystyle \int }}_{T_3}^t{\left(r_a{}^2-{\left(r_a-\left(vt-d\right)\right)}^2\right)}^{\frac{1}{2}}vdt+S_6$$

(20)

Rolling $(2r_a+d)$ is shown in Figure 4, $T_5=(2r_a+d)/v$, $T_4<t\le T_5$

$$S_{g5}=2{{\displaystyle \int }}_{T_4}^t{\left(r_a{}^2-{\left(vt-r_a-d\right)}^2\right)}^{\frac{1}{2}}vdt+\frac{\pi r_a{}^2}{2}$$

(21)

3.

When$r_a=d$, the contact process of the wheel rolling from the initial position to completely cross the insulated rail joint is shown in Figure 5.

Rolling $r_a$ is shown in Figure 5, $T_1=r_a/v$, $0<t\le T_1$

$$S_{g1}=\pi r_a{}^2-2{{\displaystyle \int }}_0^t{\left(r_a{}^2-{\left(r_a-vt\right)}^2\right)}^{\frac{1}{2}}vdt$$

(22)

Rolling 2$r_a$ is shown in Figure 5, $T_2=2r_a/v$, $T_1<t\le T_2$

$$S_{g2}=\frac{\pi r_a{}^2}{2}-2{{\displaystyle \int }}_{T_1}^t{\left(r_a{}^2-{\left(vt-r_a\right)}^2\right)}^{\frac{1}{2}}vdt+2{{\displaystyle \int }}_{T_1}^t{\left(r_a{}^2-{\left(d+r_a-vt\right)}^2\right)}^{\frac{1}{2}}vdt$$

(23)

Rolling 3$r_a$ is shown in Figure 5, $T_3=3r_a/v$, $T_2<t\le T_3$

$$S_{g3}=2{{\displaystyle \int }}_{T_2}^t{\left(r_a{}^2-{\left(vt-r_a-d\right)}^2\right)}^{\frac{1}{2}}vdt+\frac{\pi r_a{}^2}{2}$$

(24)

The existence of rain, grease, and other pollutants and their oxides on the rail surface makes the wheel–rail contact with a third medium. The influence of the polluted film layer was not considered temporarily in this study. That is, the wheel–rail contact was assumed to be a complete metal–metal contact. At this point, the wheel–rail contact resistance can be defined by [16]

$$R_{wr}=f\left(\xi \right)\frac{\rho }{2r_a}$$

(25)

where $f\left(\xi \right)=0.939{\xi }^{-0.174} (\xi >0)$, $\xi =S_r/S_a$, $S_a$is the apparent contact spot area, and $S_r=F/H$is the actual contact spot area of wheel–rail. When calculating the wheel-IRJ contact resistance using Equation (25), the actual contact area of the wheel-IRJ can no longer be calculated according to $S_r=F/H$because the presence of the insulated rail joint makes the actual contact area change. Assuming that the running condition of the wheel is unchanged when it passes through the insulated rail joint, the actual contact spot area of the wheel-IRJ is expressed by $S_r=S_{gx}/S_a·F/H$. $F$ is the vertical force on the wheel, and $F=1.25Qg$. $Q$ is the mass acting on the rail for each wheel. $H$ is the rail hardness.

3.2. Mathematical Model of Wheel–IRJ Contact Resistance Based on GW Model

The wheel–rail contact surface is an uneven and rough surface. Under the previous assumptions, the wheel–rail contact used the GW rough surface contact model. According to Equation (6), combined with Equation (25), we obtained the wheel-IRJ contact resistance $R_{wrc}$ under GW rough surface contact as:

$$R_{wrc}=f\left({\xi }_c\right)\frac{\rho }{2\sum \sqrt{\left(z-h\right)\beta }}$$

(26)

$f\left(\xi \right)=0.939{\xi }^{-0.174}$of Equation (25) was fitted according to the ratio of Holm’s actual contact area to the apparent contact area and the ratio of Holm’s contraction resistance to contraction resistance at the apparent contact area. When Equation (26) is used to calculate the contact resistance under rough contact between wheel and insulated rail joint, the calculation results are not accurate if $f\left(\xi \right)$of Equation (25) is still used. $f\left({\xi }_c\right)$ of the wheel-IRJ rough contact can be fitted using Equations (3) and (26), as follows:

$$f\left({\xi }_c\right)=0.1666{\xi }_c{}^{-0.5}$$

(27)

where, at rough contact, ${\xi }_c=S_{rc}/S_a$, $S_{rc}=S_{gx}/S_a·A_r$.

3.3. Wheel–IRJ Current Density under Rough Surface Contact

The current density $J$ under a rough surface contact can be defined by [28]:

$$J=\frac{I}{A_r}$$

(28)

where $I$ is the current flowing through the wheel–rail contact surface.

4. Results

Using the data in Table 1, according to Equation (5) in Section 2.2, Equations (7)–(25) in Section 3.1, and Equations (26) and (27) in Section 3.2, we analyzed the wheel-IRJ contact resistance and the influences of the load, train running speed, insulated endplate thickness, and electric arc heat on the wheel-IRJ contact resistance. The current density of the wheel-IRJ contact were calculated according to Equation (28). $\rho$ takes 2.1 × 10⁻⁷ Ω·m, and $H$ takes 280 × 10⁻² kN/mm [16].

4.1. Influence of Load on Contact Resistance

Under the condition of 9 t, the change in wheel–rail contact resistance when the train wheel runs from the initial position shown in Figure 4 to the complete crossing of the insulated rail joint is shown in Figure 6. $v$ was taken as 49.7 km/h [23]. It can be seen from Figure 6 that the contact resistance of wheel-IRJ contact at the initial position is 2.334 × 10⁻⁵ Ω.

Hang Yang et al. [29] found that, when the voltage at both ends of the insulated rail joint is ≤20 V, no arc will occur by onsite collecting the current and voltage at the return cut-off point. Pengfei Su [17] found that, when the current flowing through the wheel–rail was 61 A, a voltage of 20 V was generated at the wheel–rail contact. Shiwu Yang et al. [2] simulated and tested the voltage at both ends of the insulated rail joint and the current in the circuit when different degrees of electric arc occurred in the laboratory. It can be concluded that, when the current is 61 A and the voltage is 20 V, no electric arc occurs when the wheel leaves the insulated rail joint. Pengfei Su [17] used Hertz theory and electrical contact theory to solve for a static wheel–rail contact resistance of 2.09 × 10⁻⁵ Ω under the condition that the voltage is 20 V and the current is 61 A. The wheel-IRJ contact resistance at the initial position calculated according to the model built in this paper is in the same order of magnitude as that in the literature [17], and the difference is 2.4 × 10⁻⁶, which is very small. The contact resistance is inversely proportional to the contact area. It can be seen from the contact area change relationship between the wheel and the insulated rail joint in Figure 3, Figure 4 and Figure 5 that the change trend of the wheel-IRJ contact resistance conforms to the actual situation.

The wheel-IRJ contact area and contact resistance under different loads are shown in

Table 2. Contact area and contact resistance of wheel-insulated rail joint (IRJ) under different loads.

Load/t	$\mathbf{Min} {\mathit{S}}_{\mathit{g}\mathit{x}}/{\mathbf{mm}}^2$	$\mathbf{Min} {\mathit{S}}_{\mathit{r}}/{\mathbf{mm}}^2$	$\mathbf{Max} {\mathit{R}}_{\mathit{w}\mathit{r}}/\mathsf{\Omega }$
9	17.02	3.42	3.165 × 10−5
12	27.45	6.106	2.693 × 10−5
15	42.36	9.652	2.331 × 10−5
21	66.08	16.68	1.963 × 10−5
24	78.62	20.23	1.842 × 10−5
27	95.51	24.96	1.706 × 10−5

.

It can be seen from Table 2 that the wheel-IRJ effective apparent contact area and actual contact area increased with an increase in load. This is because as the load increased, the deformation variable of the initial contact asperities of the wheel–rail increased, and the contact range of the wheel–rail became larger, so that the original non-contacted part outside wheel–rail contact spot also came into contact. The contact resistance decreased with increasing load, and when the load increased to 21 t, the contact resistance decreased a smaller extent, which was related to the fact that the wheel–rail contact spot area increased with increasing load, but the trend was delayed, as shown in Table 1. However, the greater the train load is, the higher the traction level is required, which inevitably results in greater traction current and increases the potential difference between the ends of the insulated rail joints at the return cut-off point. Relevant research had shown that the insulated rail joint of the warning signal was also burnt in heavy-haul railway [30]. Therefore, from the perspective of the influence of the load on the contact resistance, the return cut-off point should be reduced when the track circuit is laid out in the high-speed railway station. On the premise of ensuring the normal operation of the track circuit, the potential difference at both ends of the insulated rail joint can be reduced by two methods. One is to connect the neutral points of the two choke transformers at the return cut-off point directly using the central connecting plate. The other is to use the impedance matching principle to connect the neutral point. The impedance matching principle is used to ensure that the traction return current can flow smoothly across the insulated rail joint while making the signal current of the track circuit flow only in this track section.

4.2. Influence of Train Running Speed on Contact Resistance

The wheel-IRJ contact area shown in

Table 3. Minimum contact area under different speeds (unit: mm²).

Load/t	$\mathit{v}=80/(\mathbf{km}/\mathbf{h})$		$\mathit{v}=120/(\mathbf{km}/\mathbf{h})$		$\mathit{v}=200/(\mathbf{km}/\mathbf{h})$
	${\mathit{S}}_{\mathit{g}\mathit{x}}$	${\mathit{S}}_{\mathit{r}}$	${\mathit{S}}_{\mathit{g}\mathit{x}}$	${\mathit{S}}_{\mathit{r}}$	${\mathit{S}}_{\mathit{g}\mathit{x}}$	${\mathit{S}}_{\mathit{r}}$
9	17.01	3.418	17.02	3.42	17.05	3.425
15	42.37	9.655	42.37	9.655	42.43	9.667
24	78.62	20.23	78.62	20.23	78.67	20.25

was obtained by changing the train running speed.

As can be seen from Table 3, the train running speed had basically no effect on the wheel-IRJ contact area.

4.3. Influence of Insulated Endplate Thickness on Contact Resistance

The wheel-IRJ contact resistances at different insulated endplate thicknesses are listed in

Table 4. Maximum contact resistance under different insulated endplate thickness (unit: Ω).

Load/t	$\mathit{d}=6 \mathbf{mm}$	$\mathit{d}=8 \mathbf{mm}$	$\mathit{d}=10 \mathbf{mm}$
9	2.801 × 10−5	3.165 × 10−5	4.084 × 10−5
15	1.95 × 10−5	2.331 × 10−5	2.851 × 10−5
24	1.691 × 10−5	1.842 × 10−5	2.317 × 10−5
27	1.624 × 10−5	1.706 × 10−5	2.197 × 10−5

.

It can be seen from Table 4 that the insulated endplate thickness had a significant influence on the wheel-IRJ contact resistance. When $d>8$ mm, the contact resistance increased significantly, and the influence was greater when the load was small. According to the relationship between the wheel–rail contact spot radius and the insulated endplate thickness, when the thickness of insulated end plate was larger, the effective apparent contact area of the wheel-IRJ contact was smaller, resulting in an increase in contact resistance. However, in the range of 6 to 8 mm, required by the standard, there was little difference in the wheel-IRJ contact resistance.

The trend of the wheel-IRJ contact resistance at different insulated endplate thicknesses for a load of 9 t was shown in Figure 7. From Figure 7, it can be seen that the change in the contact resistance was steeper when the insulated endplate thickness was 10 mm than when it was 6 mm. The rapid change leads to an increase in the resistance change in the wheel-IRJ, which directly affects the whole traction circuit, and it also aggravates the impact current at the moment when the wheel leaves the insulated rail joint [6,31] and then aggravates the generation of electric arc.

Considering the contact resistance and its change speed, it is recommended to use 8 mm for insulated end plates thickness on high-speed lines.

4.4. Influence of Electric Arc Heat on Contact Resistance

The maximum electric arc temperature generated when the wheel leaves the insulated rail joint was positively correlated with the current flowing through the wheel–rail contact surface, as shown in Figure 8.

The electric arc heat changed the material parameters of the wheel and rail. According to Equation (25), combined with the rail resistivity at different temperatures [32], wheel-IRJ contact resistance at different temperatures is shown in

Table 5. Wheel-IRJ contact resistance at different temperatures.

Temperature/°C	Maximum Contact Resistance/Ω
20	3.165 × 10−5
200	4.522 × 10−5
400	7.536 × 10−5
1200	1.809 × 10−4

.

As shown in Figure 8, the electric arc heat increased with an increase in the current flowing through the wheel–rail contact surface. When the electric arc heat was less than the melting point of the rail (1500 °C), the contact resistance increased with an increase in temperature, as shown in Table 5. However, relevant scholars have found that the electric arc generated at insulated rail joints can instantaneously form a high temperature of 4000 °C–6500 °C [2]. Such a high temperature directly damages the rail and insulated rail joints by thermal erosion, which worsens the wheel–rail contact relationship, and even it affects traffic safety. Thermal erosion of insulated rail joints and rail by electric arc is one of the reasons for IRJ failure in high-speed rail stations.

4.5. Influence of Surface Roughness on Contact Resistance

Rough surface parameters are listed in

Table 6. Rough surface contact parameters.

Surface Roughness Condition	$\mathit{\eta }$/mm−2	$\mathit{\beta }$/mm	$\mathit{\sigma }$/mm
intermediate roughness	1180	0.170	0.3 × 10−3
roughness	1150	0.055	1 × 10−3
very rough	660	0.030	2.95 × 10−3

[20,33]. The wheel-IRJ contact resistance under GW rough surface contact obtained according to Equations (2)–(5), and Equations (26) and (27) are shown in

Table 7. Maximum contact resistance in GW rough surface contact (unit: Ω).

Surface Roughness Condition	15 t		21 t
	${\mathit{S}}_{\mathit{r}}$	${\mathit{S}}_{\mathit{r}\mathit{c}}$	${\mathit{S}}_{\mathit{r}}$	${\mathit{S}}_{\mathit{r}\mathit{c}}$
intermediate roughness	4.05 × 10−5	5.767 × 10−5	2.74 × 10−5	3.902 × 10−5
roughness	4.001 × 10−5	1.02 × 10−4	2.707 × 10−5	6.906 × 10−5
very rough	5.469 × 10−5	2.138 × 10−4	3.719 × 10−5	1.447 × 10−4

, where the wheel-IRJ actual contact area was calculated using $S_r$ and $S_{rc}$ respectively.

Comparing Table 2 and Table 7, it can be seen that the contact resistance under GW rough surface contact was of the same order of magnitude as the contact resistance calculated based on the electrical contact theory. The contact resistance under different roughness contacts was greater than that based on electrical contact theory. Thus, the rough surface contact increases the wheel-IRJ contact resistance, and the rougher the contact surface, the greater the contact resistance. With the same rough surface contact, increasing the load caused the contact area gradually to increase and contact resistance to decrease. However, the contact stiffness of a rough surface varied with the normal load [33]. Using $S_r=S_{gx}/S_a·F/H$ to calculate the wheel-IRJ contact resistance under rough contact will be inaccurate.

The size of the contact resistance affects the voltage at the ends of the insulated rail joint at the return cut-off point. At the same time, the greater the contact resistance, the less likely the traction current is to be conducted back through the wheel–rail contact spot. According to the influence of load and surface roughness on the contact resistance, the voltage at both ends of insulated rail joints can be reduced by properly increasing the wheel–rail contact load and reasonably designing the wheel–rail surface roughness to ensure safe train operation.

4.6. Current Density Change in Wheel-IRJ under Rough Surface Contact

Figure 9 shows the current density curve of the wheel-IRJ rough contact, wherein the surface roughness condition is roughness, as shown in Table 6 at 9 t and 15 t. Figure 10 shows the current density distribution for the wheel-IRJ in very rough conditions, as shown in Table 6, contact at 15 t.

As shown in Figure 9, under the same load, the current density under rough contact increased with an increase in current flowing through the wheel–rail contact surface. At the same current value, as the load increased, the actual contact area increased, causing the current density to decrease significantly. Comparing Figure 9 and Figure 10, it can be seen that the rougher the contact surface, the smaller the area where conduction actually took place under the same load, which caused the current density to increase significantly with the increase in contact surface roughness.

The size of the contact area ultimately affects the voltage at the ends of insulated rail joints. In high-speed railway station, the traction return conducted by the wheel–rail contact is larger. Additionally, a rougher surface reduces the actual contact area. These conditions led to an increase in the current intensity per unit area of the wheel-IRJ contact. This corresponded to an increase in the current density. Whether the current density increases due to the increase in current or the decrease in contact area, the wheel–rail contact voltage will increase. According to the electric arc generation mechanism [34], the higher the voltage, the easier it is to generate an electric arc. Therefore, the higher the current density, the easier it is for electric arcs to occur. The influence of the electric arc on the insulated rail joints at the return cut-off point is inevitable once an electric arc occurs.

From the current density point of view, on the one hand, from breaking the electric arc conditions, under the premise of ensuring the normal operation of the track circuit, the traction return current can flow across the insulated rail joints by connecting the neutral point of the choke transformer in the adjacent track section or using impedance matching technology. On the other hand, the potential difference between the two ends of the insulated rail joints can be reduced by appropriately increasing the contact load and properly reducing the contact interface roughness by considering the influence of surface roughness on the adhesion characteristics of the wheel–rail contact interface.

5. Conclusions

The wheel-IRJ contact resistance is the basic parameter for analyzing the potential difference at both ends of the insulated rail joint. The increase in voltage at both ends of the insulated rail joint will cause the occurrence of electric arc, which will burn the insulated rail joint. The wheel-IRJ contact is a dynamic electrical contact. In this paper, the effective apparent contact area of the wheel rolling through the insulated rail joint is derived by mathematical method. Considering the surface roughness parameters, the mathematical models of the wheel-IRJ contact resistance and current density are established based on the electrical contact theory and GW model. The results show that the wheel-IRJ contact resistance increases first and then decreases. The train running speed has little influence on the wheel-IRJ contact resistance. The load, roughness of the contact interface, and electric arc heat significantly influence the contact resistance. The contact resistance increased as the load increased, but the increasing trend of the contact resistance slows down when it increases to 21 t. The rougher the contact interface, the greater the contact resistance. The higher the current density is, the easier it is to generate electric arc when the wheel contact with the insulated rail joint. Electric arc heat can deteriorate the wheel–rail contact relationship, and electric arc heat erosion is one of the main reasons for the failure of insulated rail joint in high-speed railway stations. To reduce the impact of contact resistance on the potential difference of insulated rail joints, it is recommended to use 8 mm thickness of insulated endplates in high-speed railways and heavy load lines. From the perspective of contact resistance and current density, the potential difference between the two ends of the insulation section is reduced by taking methods, such as appropriately increasing the wheel–rail contact load, appropriately reducing the contact surface roughness under the premise of ensuring train safety, and reducing the setting of the insulated rail joint at the return cut-off point or connecting the neutral point of the choke transformer of the adjacent rail section with the help of impedance matching technology under the premise of ensuring the normal operation of the track circuit. The research results in this paper provide the basis for the analysis of voltage of insulated rail joint and traction circuit. However, due to time constraint, the effect of high temperature generated by electric arc on mechanical performance of the rail was not considered during the analysis, which we will further explored in the future.