# Research Progress of High-Speed Wheel–Rail Relationship

## Abstract

**:**

## 1. Introduction

## 2. Basic Theory of Wheel/Rail Rolling Contact

_{1}and A

_{2}in Figure 4. They are next to the leading edge of the elliptical contact patch. Especially for trains running on a tangent track, the model can effectively consider the lateral and longitudinal coupling of the wheel–rail system and carry out the rapid dynamic behavior simulation of large vehicle-track coupling system. However, when a train runs on a curved track, the outer wheel rim will contact the inner corner of the outer rail and in this case, the wheel–rail contact angle is large. The larger the wheel–rail contact angle, the greater the spin creepage between the wheel and rail, the greater the influence on the longitudinal and lateral creep forces of the wheel and rail and the greater the error of the dynamics simulation results [3]. This is because the model does not consider the influence of wheel/rail spin creepage on the wheel/rail creep forces.

_{m1}, f

_{m2}and f

_{m3}), as shown in Figure 11b. The peak denoted by f

_{ir3}represents the response to the harmonic excitation. It should be noted that the bending vibration of the wheelset is easy to cause the polygonal wear of the wheel under certain conditions [9,10].

_{s}is the response to the discrete sleeper support excitation.

_{i}and u

_{j}(i, j = 1, 2, 3) are the components of $u$, $\Omega $ is the rolling angular speed of the wheelset along the track, $K$ is the stiffness matrix, $Q$ is the generalized force matrix acting on the wheelset, 𝜌 is the wheelset material density, ${\tilde{u}}_{i}$ (i = 1, 2, 3) is the component of transfer matrix of $u$ and $E$ is a constant matrix of three times three, and in it E

_{11}= E

_{33}= 1 and the other elements are zero. An analytical transfer function of wheelset was also given in [55] and omitted here.

_{band-i}(i = 1, 2, …, 7) indicates the band frequency of the wheelset. Obviously, the rotational angular velocity makes the axle-bending frequencies bifurcate, and with the increase in the bending frequency, the frequency bifurcation of the wheelset-bending vibration tends to decrease. f

_{(n,m)}indicates the frequency of nodal-diameter and pitch-circle modes. The foot symbols, n and m, represent the number of pitch-diameters and pitch-circles, respectively. Obviously, only f

_{(4,0)}is forked. f

_{um-i}(i = 1, 2, 3) denotes the frequency of the umbrella mode. Wheelset rotation has little effect on the wheel’s umbrella mode frequency.

## 3. Optimal Matching Design of High-Speed Wheel–Rail

_{R}represents the number of wheel/rail rolling contact, the red curve Y

_{m}(N

_{R}) represents the nominal hardness function of the contact surface materials, the blue curve Y

_{s}(N

_{R}) represents the actual hardness function of the materials, Y

_{s}(N

_{R}) is lower than Y

_{m}(N

_{R}) because the wheel/rail contact surface materials are constantly worn away, Y

_{0}(HB) is the initial hardness of wheel/rail surface materials, h

_{W}(N

_{R}) is the wear depth of the contact surface and K in the Figure represents the intersection point of the wear depth and the actual hardened layer depth. When the wear depth is equal to the hardened layer depth, if the wheel/rail continue to roll slip wear, the wear will increase rapidly.

## 4. Adhesion Theory and Mechanism of High-Speed Wheel–Rail in Rolling Contact

_{1}, δ

_{2}) of the contact surfaces and the fluid between the wheel–rail interface, respectively. Figure 24d shows three texture states of rough surface (γ < 1, =1 and >1). γ < 1, =1 and >1 represents the distribution state of roughness texture along longitudinal, longitudinal and transverse and lateral directions, respectively. This wheel–rail contact state seriously affects the wheel–rail adhesion coefficient or wheel–rail adhesion force F.

## 5. Wear and Rolling Contact Fatigue of High-Speed Wheel–Rail

#### 5.1. High-Speed Wheel/Rail Wear

#### 5.1.1. Transverse Wear of Wheel/Rail

_{l1}, r

_{l2}, r

_{r1}and r

_{r2}are the rolling radii at four different points on the left and right wheel treads, located on both sides of the hollow wear;

**y**is the lateral displacement of the wheelset center with respect to the track central line. Statistical analysis of field test data shows that 50% of hollow wear of high-speed wheels have the depth of about 0.15–0.25 mm, and the width is about 30 mm–60 mm, depending on the operating mileage. The hollow wear center is close to the wheel nominal rolling circle. When a train was running at high speeds, the transversal oscillation of bogie and wheelset at 7–10 Hz was very sensitive to the hollow wear of the wheelset, as shown in Figure 28. Faced with the bogie transversal oscillation increasing the horizontal acceleration level of the axle box, the train operation at high speeds had to slow down. The solution of this problem puzzled railway engineering experts for some time. However, it is not difficult to understand the mechanism of the hollow wear formation on high-speed wheelsets, as shown in Figure 28, which leads to the wheelset hunting motion of 7–10 Hz. When the hollow worn wheelset rolls over a pair of rails, if the heads of the two rails are lightly flattened due to wear or the radii of the arcs of the rail heads become larger, the two-point contact forms between the wheel/rail on each side. In this case, a small lateral wobble as the wheelset rolls along the track at high speeds will result in discontinuous or jumping changes in the instantaneous rolling radii of the left and right wheels. Then, the change in diameter difference between in the left and right wheels, or equivalent conicity of the wheelset, is discontinuous with

**y**, and they change periodically and reversely. From Figure 28, the radius difference is written as ∆r

_{i}=r

_{ri}− r

_{li}(i = 1, 2) and the equivalent conicity is written as λ = ∣Δr

_{i}∣/2

**y**. If the lateral displacement

**y**<0, the subscript i = 1, and if

**y**> 0, i = 2. Usually, r

_{l1}< r

_{l2}and r

_{r1}> r

_{r2}or they are not equal. The longitudinal creepages and the longitudinal creep forces between the left and right wheels/rails depend on the radius difference ∆r

_{i}. Their variation characteristics with

**y**are similar to that of wheel diameter difference. The longitudinal creep forces on the left and right wheels form a moment of couple that will alternately excite the periodical yaw motion of the wheelset at 7–10 Hz [63].

#### 5.1.2. Uneven Wear in the Rolling Direction of High-Speed Wheel/Rail

#### 5.2. Rolling Contact Fatigue of High-Speed Wheel–Rail

## 6. High-Speed Wheel–Rail Noise

## 7. Conclusions

## 8. Further Research on Relationship of Wheel and Rail

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) New wheel and the new rail contact forming an elliptical contact spot, and (

**b**) seriously worn wheel/rail contact forming a lateral slender contact area.

**Figure 2.**Division of the slip–slip area of the wheel–rail rectangular contact patch, first established by Carter.

**Figure 5.**Three-dimensional elasto-plastic finite element calculation model of wheel–rail in rolling contact.

**Figure 9.**(

**a**) Tangential force density distribution in wheel–rail patch during wheel rolling over a vertical crack; (

**b**) friction distribution on crack surface of rail.

**Figure 11.**Comparison of wheel/rail vertical forces calculated by two vehicle-track coupling dynamics models (

**a**) without wheelset deformation; (

**b**) with wheelset deformation.

**Figure 18.**Equivalent conicities and critical hunting speeds of old and new wheelsets with operation mileage.

**Figure 19.**(

**a**) Serious pit wear on the lateral tread of wheel; (

**b**) wheel tread spalling and flange root checking [16].

**Figure 20.**Total wear volume of wheel/rail specimens vs. hardness ratio of wheel material to rail material.

**Figure 21.**Hardness and wear of wheel/rail contact surface material vs. number of wheel/rail rolling extrusion.

**Figure 23.**Adhesion coefficient with velocity at different roughness levels and water mediums [81].

**Figure 24.**Adhesion physical model considering the effect of elastic–fluid and elastoplastic micro-roughness between wheel and rail and wheel rolling speed.

**Figure 25.**Comparison between experimental results and numerical results with considering thermal effect.

**Figure 27.**V-track test rig [94].

**Figure 29.**(

**a**) Photo of polygonal wear of a high-speed wheel; (

**b**) comparison of roundness of polygonal worn wheel before and after repair.

**Figure 30.**Irregularity spectra of the polygon wear of the wheels of a whole high-speed train. (

**A**). the eccentric wear of the wheels. (

**B**). 14-order 400 Hz (

**C**). 2-order 590 Hz.

**Figure 31.**(

**a**) Two vibration modes of bogie frame at 585 Hz and 595 Hz, (

**b**) the fourth-order bending mode of wheelset at 610 Hz.

**Figure 32.**(

**a**) Electric locomotive wheel with disc braking system; (

**b**) locomotive wheel with brake shoe system.

**Figure 35.**(

**a**) Photo of rail corrugation at a tangent track of a high-speed line; (

**b**) photo of the high-speed rail after grinding.

**Figure 36.**Photos of crescent-shaped crack on wheel tread and state after cutting at different depths in the radius direction.

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**MDPI and ACS Style**

Jin, X.
Research Progress of High-Speed Wheel–Rail Relationship. *Lubricants* **2022**, *10*, 248.
https://doi.org/10.3390/lubricants10100248

**AMA Style**

Jin X.
Research Progress of High-Speed Wheel–Rail Relationship. *Lubricants*. 2022; 10(10):248.
https://doi.org/10.3390/lubricants10100248

**Chicago/Turabian Style**

Jin, Xuesong.
2022. "Research Progress of High-Speed Wheel–Rail Relationship" *Lubricants* 10, no. 10: 248.
https://doi.org/10.3390/lubricants10100248