# Key Factors of the Initiation and Development of Polygonal Wear in the Wheels of a High-Speed Train

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## Abstract

**:**

## 1. Introduction

## 2. Characteristics of Polygonal Wear of High-Speed Wheels

- Wheels with different diameters have different proportions of polygonal wheels and different dominant orders (main wavelength) of polygonal wear (Figure 2). Furthermore, it seems that the smaller the diameter, the higher the proportion of the polygonal wheels. This depends on the depth of the decarburised layer on the surface of the wheel.
- The proportion of polygonal wheels exhibits a non-monotonic variation as a function of the wheel diameter, including three local maxima around 830–840 mm, 870–880 mm, and 910–920 mm, corresponding to three periods of high-rate development.

## 3. Simulation Model

#### 3.1. Rigid–Flexible Coupling Dynamical Model of Vehicle–Track

#### 3.2. Method for Online Searching Wheel–Rail Contact Points

_{r}denotes the angle of roll after the wheelset generates rigid motion and ψ

_{r}indicates the angle of yaw. The two parameters can be solved using a dynamic explicit integral algorithm. In addition, the transformation matrix between the natural system of coordinates and the wheelset system of coordinates is built using ϕ

_{r}and ψ

_{r}.

_{DR1}indicates the angle included between the axis of the wheelset and the horizontal line after the wheelset generates rigid motion and flexible deformation. This is written as

_{f}denotes the angle included between the axis of the wheelset and the horizontal line after the wheelset only produces flexible deformation. Figure 10 illustrates the solving method. First, two points (B

_{1}and B

_{2}) at both sides of the wheel are selected, and the connection line between the two points is drawn, and this line is parallel to the axis of the wheelset. Second, the wheelset generates flexible deformation, the two points move to the two new points B

_{01}and B

_{02}. Then, the positions of B

_{01}and B

_{02}are solved using the method described in Section 3.1. Finally, ϕ

_{f}can be solved.

#### 3.3. Archard Wear Model

## 4. Discussion on the Calculated Results of High-Order Polygonal Wear

## 5. Key Conditions of Initiation and Development of Polygonal Wear

#### 5.1. Integral Multiple Condition

#### 5.2. Effect of Different Diameters and Different Speeds

## 6. Conclusions

- The prominent feature of polygonal wear is that the dominant order and development rate of polygonal wear are very closely related to the wheel diameter, operation speed, and the resonance frequencies of the system, such as wheelsets.
- If the wheel circumference is nearly an integral multiple of the wavelength of the periodical wear along the rolling circle of the wheel, the polygonal wear initiates and develops very quickly. This is referred to as the integral multiple condition, which is a key or basic condition for polygonal wear. This “integral multiple” is on the order of the dominant polygonal wear.
- When the excited resonance frequency of the vehicle system is determined to be related to polygonal wear, the initiation and development of the polygonal wear depends on the operation speed and wheel diameter. Their relationship diagram is provided, and it can be used to predict polygonal wear development, modify the re-profiling schedule of polygonal wheels, and eliminate the polygonal wear on the wheels in time.
- The effect of flexibility of the wheelset is considered in the prediction model of the polygonal wear of high-speed wheels in this paper. Therefore, the high-frequency characteristics of the wheelset bending resonances can be more accurately expressed, and high-order polygonal wear related to the wheelset bending resonances at high frequencies can be numerically reproduced.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Test Items | Number Tested | Warning Count | Error Count | Warn+Err (%) |
---|---|---|---|---|

Aspect Ratio | 255,360 | 5232 | 0 | 2.05% |

Parallel Deviation | 255,360 | 0 | 0 | 0 |

Maximum Angle | 255,360 | 0 | 0 | 0 |

Jacobian Ratio | 255,360 | 0 | 0 | 0 |

Warping Factor | 255,360 | 0 | 0 | 0 |

**Figure A1.**The finite element model: (

**a**) Model 1 (the model in the manuscript, element count: 255,360, node count: 267,771); (

**b**) Model 2 (element count: 556,800, node count: 573,673); (

**c**) Model 3 (element count: 834,860, node count: 856,285).

Model 1 (the Model in the Manuscript) | Model 2 | Model 3 | |||
---|---|---|---|---|---|

Element count | 255,360 | 556,800 | The errors refer to model 1 (%) | 834,860 | The errors refer to model 1 (%) |

Node count | 267,771 | 573,673 | 856,285 | ||

Order | Frequency | ||||

1 | 72.13 | 72.12 | 0.01% | 72.12 | 0.01% |

2 | 87.08 | 87.02 | 0.07% | 86.95 | 0.15% |

3 | 143.6 | 143.3 | 0.21% | 142.9 | 0.49% |

4 | 234.1 | 233.3 | 0.34% | 232.1 | 0.85% |

5 | 284.7 | 284.0 | 0.25% | 283.2 | 0.53% |

6 | 347.3 | 346.1 | 0.35% | 344.1 | 0.92% |

7 | 386.0 | 385.4 | 0.16% | 384.9 | 0.28% |

8 | 576.4 | 575.7 | 0.12% | 574.7 | 0.29% |

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**Figure 3.**Sketch map: Amplitude spectrum of polygonal wear (site measurement) and threshold line of high-speed wheel roughness (referring to the ISO-3095).

**Figure 8.**Bending modes of wheelset (0–800 Hz): (

**a**) First bending mode at 87 Hz; (

**b**) Second bending mode at 143 Hz; (

**c**) Third bending mode at 284 Hz; (

**d**) Fourth bending mode at 576 Hz.

**Figure 9.**Modelling flexible wheelset: (

**a**) Static wheelset; (

**b**) The wheelset after rigid motion; (

**c**) The wheelset after flexible deformation; (

**d**) dummy wheelsets; (

**e**) the contact geometry relation between the dummy wheelset and the two rails.

**Figure 12.**(

**a**) Vertical irregularity; (

**b**) Lateral irregularity; (

**c**) Normal contact forces; (

**d**) Lateral creepages.

**Figure 13.**Results of wear loss along the circumferences of the two wheels with rolling of 50 circles. (

**a**) Flexible wheelset and (

**b**) Rigid wheelset.

**Figure 16.**Description of basic condition for wheel OOR generation [12].

**Figure 18.**Polygonal wear results obtained by superposing the uneven wear of the wheels rolling with cycles, (

**a**) D = 920 mm, (

**b**) D = 900 mm.

**Figure 19.**Polygonal wear development with different operation speeds and different wheel diameters.

Vehicle System | Track System | ||
---|---|---|---|

Parameter | Value | Parameter | Value |

Car body mass (kg) | 44,039 | Young’s modulus of rail (N/m^{2}) | 2.059 × 10^{11} |

Bogie mass (kg) | 2439 | Shear modulus of rail (N/m^{2}) | 7.9 × 10^{10} |

Wheelset mass (kg) | 1881 | Density of rail (kg/m^{3}) | 7860 |

Density of wheelset (kg/m^{3}) | 7860 | Cross-sectional area (m^{2}) | 7.745 × 10^{−3} |

Primary suspension vertical stiffness (MN/m) | 9 | Bending moment about y-axis (m^{4}) | 3.217 × 10^{−5} |

Primary suspension vertical damping (kN·s/m) | 10.5 | Bending moment about z-axis (m^{4}) | 5.26 × 10^{−6} |

Primary suspension lateral stiffness (MN/m) | 5.45 | Polar moment of inertia (m^{4}) | 3.741 × 10^{−5} |

Primary suspension lateral damping (kN·s/m) | 5.0 | Lateral shear coefficient | 0.4570 |

Secondary suspension vertical stiffness (MN/m) | 0.24 | Vertical shear coefficient | 0.5329 |

Secondary suspension vertical damping (kN·s/m) | 25 | Density of slab (kg/m^{3}) | 2800 |

Secondary suspension lateral stiffness (MN/m) | 0.13 | Young’s modulus of slab (N/m^{2}) | 3.6 × 10^{10} |

Secondary suspension lateral damping (kN·s/m) | 15 | Rail pad vertical stiffness (MN/m) | 40 |

Distance between two bogie frames (m) | 8.9 | Rail pad vertical damping (kN·s/m) | 30 |

Wheelbase (m) | 2.5 | Sleeper spacing (m) | 0.63 |

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**MDPI and ACS Style**

Wu, Y.; Jin, X.; Cai, W.; Han, J.; Xiao, X.
Key Factors of the Initiation and Development of Polygonal Wear in the Wheels of a High-Speed Train. *Appl. Sci.* **2020**, *10*, 5880.
https://doi.org/10.3390/app10175880

**AMA Style**

Wu Y, Jin X, Cai W, Han J, Xiao X.
Key Factors of the Initiation and Development of Polygonal Wear in the Wheels of a High-Speed Train. *Applied Sciences*. 2020; 10(17):5880.
https://doi.org/10.3390/app10175880

**Chicago/Turabian Style**

Wu, Yue, Xuesong Jin, Wubin Cai, Jian Han, and Xinbiao Xiao.
2020. "Key Factors of the Initiation and Development of Polygonal Wear in the Wheels of a High-Speed Train" *Applied Sciences* 10, no. 17: 5880.
https://doi.org/10.3390/app10175880