# Influence of Wheel-Rail Contact Algorithms on Running Safety Assessment of Trains under Earthquakes

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## Abstract

**:**

## 1. Introduction

## 2. Wheel-Rail Contact Model

#### 2.1. Wheel-Rail Contact Geometry Calculation

_{w}-X

_{w}Y

_{w}Z

_{w}). In the absolute coordinate system, O is positioned at the center of the track, with the OX axis pointing in the direction of rolling, the OZ axis pointing vertically, and the OY axis aligning with the track transverse direction as determined by the right-hand rule. In the wheelset coordinate system, O

_{w}is situated at the center of mass of the wheelset, and the O

_{w}Y

_{w}-axis aligns with the wheelset’s axis of revolution. The basic idea of this method is as follows. First, the wheel profile is discretized along the transverse direction to obtain several rolling circles. A coordinate expression of the potential contact points on the wheel in the absolute coordinate system O-XYZ is obtained:

#### 2.2. Wheel-Rail Contact Force Calculation

**j**,

**k**) is the unit vector in the absolute coordinate system; ${\mathit{F}}_{Ni}$, ${\mathit{F}}_{xi}$, ${\mathit{F}}_{yi}$, and ${\mathit{M}}_{i}$ are the contact force and spin moment vectors at the ith wheel-rail contact point.

## 3. Dynamic Analysis Model of Train-Track Coupling System under Earthquakes

#### 3.1. Train Subsystem

#### 3.2. Track Subsystem

#### 3.3. Train-Track Coupling

## 4. Comparison of Different Wheel-Rail Contact Models

_{A}and CHN60 were adopted as the wheel and rail profiles, respectively, as presented in Figure 10.

#### 4.1. Comparison under Ordinary Conditions

^{10}N/m

^{3/2}. In contrast, when calculating based on Model 4, this value is maintained at 9.20 × 10

^{10}N/m

^{3/2}due to the straight segment of the wheel tread keeping in contact with the R300 arc segment of the rail head. Figure 13d shows a slight difference in ${\delta}_{c}$ calculated based on Models 1 and 4. Moreover, although the empirical formula causes a calculation error for ${K}_{nr}$ and ${\delta}_{c}$, it has largely no impact on the contact force. Thus, it can be concluded that under this condition, it is applicable to adopt the normal compression algorithm based on vertical penetration, the empirical formula for calculating the normal contact stiffness, and only consider single-point contact.

#### 4.2. Comparison under Seismic Conditions

#### 4.2.1. Wheel-Rail Contact Dynamics

^{10}N/m

^{3/2}. When contact occurs between the R450 arc segment of the wheel tread and the R300 arc segment of the rail head, ${K}_{nr}$ decreases to 8.21 × 10

^{10}N/m

^{3/2}. Figure 17c,e show that the theoretical formula also causes a difference in ${\delta}_{c}$, but has no significant effect on ${F}_{N}$, which is consistent with the phenomenon observed under ordinary conditions. For the contact in region 2, the solutions to ${K}_{nr}$ based on Models 3 and 4 are significantly different, as presented in Figure 17b. In this case, the contact mainly occurs between the R14 arc segment of the wheel flange root and the R13 arc segment of the rail gauge corner. The calculation result of ${K}_{nr}$ based on the theoretical formula is 5.13 × 10

^{10}N/m

^{3/2}, which is approximately half of that based on the empirical formula. A different calculation formulae for ${K}_{nr}$ will cause significant differences in ${\delta}_{c}$, as shown in Figure 17d. Consequently, there is a noticeable difference after 8 s, as shown in Figure 17f. Therefore, to ensure the reliability of the normal contact force calculation, the theoretical calculation formula for the normal contact stiffness should be introduced.

#### 4.2.2. Running Safety Assessment

## 5. Model Validation

## 6. Conclusions

- Using different wheel-rail contact algorithms will significantly affect the calculation accuracy of wheel-rail force in the case of flange-root contact under earthquakes. The most significant influence is due to the normal compression algorithm. Using an algorithm based on vertical penetration can lead to a maximum relative error of 339.50% for the case considered in this study. The consideration of the number of wheel-rail contact points also has a notable impact, with a maximum relative error of 35.00% caused by only considering single point contact. The influence of the normal contact stiffness algorithm is the least significant, with a maximum relative calculation error of 23.55% caused by using the empirical formula.
- Using different wheel-rail contact algorithms will have a significant impact on the indices of running safety assessment under earthquakes. Using wheel-rail normal compression algorithm based on vertical penetration will significantly underestimating the train running safety margin, while only considering the wheel-rail single point contact will overestimate the train running safety margin, and using the wheel-rail normal contact empirical formula has little impact.
- To ensure the accuracy of running safety assessment of trains under earthquakes, it is recommended to use the normal compression algorithm based on normal penetration and consider multi-point contact in wheel-rail contact modelling.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Determination of wheel-rail contact points and normal compression amounts based on different searching methods: (

**a**) wheel-tread contact; (

**b**) wheel-flange contact.

**Figure 4.**Distribution range of wheel-rail contact point positions in wheel-tread contact and flange contact.

**Figure 7.**Vehicle model with corresponding DOFs: (

**a**) Side view; (

**b**) Front view; (

**c**) Top view; (

**d**) Sign convention of vehicle.

**Figure 11.**Samples of rail irregularities: (

**a**) Lateral irregularities; (

**b**) Vertical irregularities; (

**c**) Torsional irregularities.

**Figure 13.**Time histories of wheel-rail dynamic responses based on different models: (

**a**) Vertical wheel-rail force; (

**b**) Lateral wheel-rail force; (

**c**) Normal contact stiffness; (

**d**) Normal compression amount.

**Figure 14.**Time histories of wheel-rail contact forces based on Models 1 and 2: (

**a**) Vertical wheel-rail force; (

**b**) Lateral wheel-rail force.

**Figure 15.**Wheel-rail contact state at time t = 3.80 s simulated based on Models 1 and 2: (

**a**) Simulation result based on Model 1; (

**b**) Simulation result based on Model 2.

**Figure 16.**Time histories of wheel-rail dynamic responses based on Models 2 and 3: (

**a**) Normal compression amount in region 1; (

**b**) Normal compression amount in region 2; (

**c**) Normal contact force in region 1; (

**d**) Normal contact force in region 2; (

**e**) Wheel jump amount.

**Figure 17.**Time histories of wheel-rail dynamic responses based on Models 3 and 4: (

**a**) Normal contact stiffness in region 1; (

**b**) Normal contact stiffness in region 2; (

**c**) Normal compression amount in region 1; (

**d**) Normal compression amount in region 2; (

**e**) Normal contact force in region 1; (

**f**) Normal contact force in region 2.

**Figure 18.**Maximum wheel-rail forces calculated based on different models: (

**a**) Left vertical wheel-rail force; (

**b**) Left lateral wheel-rail force; (

**c**) Right vertical wheel-rail force; (

**d**) Right lateral wheel-rail force.

**Figure 19.**The vehicle dynamic analysis model established by Nishimura et al. [33]: (

**a**) vehicle model; (

**b**) track model.

**Figure 20.**Time histories of wheel-rail contact force under excitation with frequency of 0.5 Hz and amplitude of 320 mm: (

**a**) Vertical wheel-rail force; (

**b**) Lateral wheel-rail force.

**Figure 21.**Time histories of wheel-rail contact force under excitation with frequency of 0.8 Hz and amplitude of 105 mm: (

**a**) Vertical wheel-rail force; (

**b**) Lateral wheel-rail force.

**Figure 22.**Time histories of wheel-rail contact force under excitation with frequency of 1.5 Hz and amplitude of 100 mm: (

**a**) Vertical wheel-rail force; (

**b**) Lateral wheel-rail force.

Wheel-Rail Contact Model | Consideration of Contact Point | Calculation Basis for Normal Compression Amount | Algorithm for Normal Contact Stiffness |
---|---|---|---|

Model 1 | Single-point contact | Vertical penetration | Empirical formula |

Model 2 | Multipoint contact | Vertical penetration | Empirical formula |

Model 3 | Multipoint contact | Normal penetration | Empirical formula |

Model 4 | Multipoint contact | Normal penetration | Theoretical formula |

Running Safety Indices | Q/P | ∆P/P | ∑Q (kN) |
---|---|---|---|

Limit | 0.8 | 0.8 | 55 |

Running Safety Indices | Model 1 | Model 2 | Model 3 | Model 4 |
---|---|---|---|---|

Q/P | 0.83 | 0.87 | 0.78 | 0.78 |

∆P/P | 0.95 | 1.0 | 0.80 | 0.79 |

∑Q (kN) | 228.79 | 250.99 | 49.33 | 46.65 |

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**MDPI and ACS Style**

Cai, G.; Zhu, Z.; Gong, W.; Zhou, G.; Jiang, L.; Ye, B.
Influence of Wheel-Rail Contact Algorithms on Running Safety Assessment of Trains under Earthquakes. *Appl. Sci.* **2023**, *13*, 5230.
https://doi.org/10.3390/app13095230

**AMA Style**

Cai G, Zhu Z, Gong W, Zhou G, Jiang L, Ye B.
Influence of Wheel-Rail Contact Algorithms on Running Safety Assessment of Trains under Earthquakes. *Applied Sciences*. 2023; 13(9):5230.
https://doi.org/10.3390/app13095230

**Chicago/Turabian Style**

Cai, Guanmian, Zhihui Zhu, Wei Gong, Gaoyang Zhou, Lizhong Jiang, and Bailong Ye.
2023. "Influence of Wheel-Rail Contact Algorithms on Running Safety Assessment of Trains under Earthquakes" *Applied Sciences* 13, no. 9: 5230.
https://doi.org/10.3390/app13095230