# Numerical Simulations of the Driving Process of a Wheeled Machine Tire on a Snow-Covered Road

^{*}

## Abstract

**:**

## 1. Introduction

## 2. FE Model of Tire

#### 2.1. Process of Establishing Tire FE Model

- Draw the tire’s cross-section, leaving out the tread, and then perform geometric cleaning. The tire’s cleaned cross-section is depicted in Figure 1a.
- Divide the FE mesh on the cross-sectional view of the tire in Figure 1a, as shown in Figure 1b in 2D. It is noted that the mesh size has a significant impact on the model’s computing time. A very small grid will increase the computation time and reduce the effectiveness of the computer. Therefore, we must choose a suitable mesh to shorten the computation times and improve the computing efficiency in order to ensure the model’s accuracy.
- On the two-dimensional mesh, we obtain the complete three-dimensional FE mesh model of the smooth tire by rotating it 360° and stitching the corresponding nodes, as shown in Figure 1c.
- Due to the periodicity of the pattern, we first establish a geometric model of the tread in a cycle, and then rotate the tread mesh for one revolution to obtain the final finite element mesh model of the tire tread using the rotation function. Figure 2 demonstrates the final FE model of the tire tread.
- Once the FE model of the tire body and tire tread are established, both the two components need to be adhered by means of a tied algorithm. Note that the nodes on the external surface of the tire body are set as the master surface, while the faces that match it on the tire tread are the slave surface, and the nodes on the slave surface are slave nodes, naturally. The resulting FE model of the tire is shown in Figure 3.

#### 2.2. Model Parameters

_{ij}is a material shear property, which indicates the shear properties of the material. It is necessary to obtain these parameters by means of appropriate mechanical experiments. The complex structure composed of the cord layer, belt layer, and bead bundle can be regarded as an anisotropic material based on the matrix of the rubber. Therefore, this part is described using a typical anisotropic material model, also referred to in [37,38], and the specific parameter values of the anisotropic material are shown in Table 3.

#### 2.3. Verification of Tire FE Model using Stiffness Experiments

## 3. Model of Snow-Covered Road

#### 3.1. SPH Method

^{αβ}is the total stress tensor of the SPH model, and f

^{α}is the acceleration component caused by external forces. During calculation, the total stress is decomposed into hydrostatic stress and deviatoric stress. In order to enhance the stability of the numerical simulation, it is necessary to add a kind of artificial viscous term into the control equation in order to decrease the nonphysical vibration. The artificial viscosity in the model is between 0 and 0.01 [41]. Furthermore, the interaction of the SPH particles in the neighboring regions is decided by the smoothing length. Using Equation (3), the smallest smoothing length H

_{min}and the maximal smoothing length H

_{max}are obtained. Here, h

_{0}is the initial smoothing length, and h(t) is the instantaneous smoothing length:

_{d}is the interparticle force.

#### 3.2. Parameter Calibration of SPH for Snow

## 4. Numerical Simulation of Tire Driving on Snow-Covered Road

#### 4.1. SPH-FEM Coupled Model

_{R}. The angular speed is raised to its highest level, and it remains stable throughout the whole process. The gravitational field is applied to all the nodes in the model. Under S

_{R}= 0, the driving process of the tire on the snow-covered road is shown in Figure 11, and the simulation and experimental comparison of the tire’s final trace on the surface of the road are shown in Figure 12.

_{L}is acquired at various slip rates. The simulation result of the μ

_{L}–S

_{R}curve is compared with the theoretical result, which is derived from [44], as shown in Figure 13. As the slip rate reaches 0.2, when the longitudinal adhesion coefficient is at its maximum, the adhesive strength is the greatest, and the braking property is the best. But as the sliding rate goes up, the tire’s shear action on the snow-covered ground becomes weaker, causing the tire’s adhesive coefficient to drop. The simulation adhesion coefficient–slip ratio curve was highly correlated with the theoretical curve, further verifying the correctness of the FEM-SPH model developed in the paper for the tire–snow-covered road. Nevertheless, the longitudinal adhesion coefficient derived from the simulated results is always lower than the theoretical one, because the real snow particles do not have the same dimensions and shapes, and the accumulation of snow particles results in a mutually interlocking effect, which greatly improves the shear resistance of the real snow. On the contrary, the SPH model is made up of uniform ball grains, which cannot completely describe the mechanics characteristics of actual snow particles.

#### 4.2. Numerical Simulation Examples

#### 4.2.1. Tire Tread

#### 4.2.2. Wheel Load

#### 4.2.3. Tire Inflated Pressure

#### 4.2.4. Snowfall

## 5. Conclusions

- The SPH model provides an accurate description of the fluidity, failure, and discontinuity of snow particles. The established FE model may also reflect the mechanical property of a tire.
- The tire treads model can obviously improve the operating capacity of the tire on the snow-covered road. The deeper the tread, the higher the performance of the tire.
- A positive correlation exists between the traction property and the wheel load. Therefore, a relatively high tire load can improve the tire’s coefficient of friction on the snowy ground, thus improving the tire’s performance on the snowy road.
- A lower inflated pressure will make it easier for the tire to attach to the snow, which will improve the traction of the tire while traveling. Therefore, it is suggested that the inflated pressure should be reduced correctly in order to increase the grip of the tires on the snow-covered road.
- Finally, the depth of snow on the road is a very important driver of road safety. Consequently, when applying anti-lock devices while driving on a snow-covered road, it is possible to effectively increase the coefficient of friction between the tire and the road surface, thereby reducing the risk of operation.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Method | Merits | Demerits | |
---|---|---|---|

Mesh-based | Lagrange | This method is mature and computationally efficient [34]. | The shortcoming of this method is that it cannot be used to describe the large deformation problem, and breaking frequently happens during the computation [10]. |

Euler | This approach can be used to model large deformation problems [35]. | This method is computationally inefficient [10]. | |

Particle-based | DEM | This method can be used to simulate the mechanical properties of discrete snow and to describe the spray phenomenon [34]. | This method is computationally inefficient [34]. |

SPH | The SPH method has the advantage of computational accuracy in describing the liquidity and large deformation of snow [10]. |

**Table 2.**Material parameters of rubber from [37].

Rubber Material Component | Density, kg∙m^{3} | Yeoh Strain Energy Potential Constants | |||
---|---|---|---|---|---|

C_{10} | C_{20} | C_{30} | D_{i} (i = 1,2,3) | ||

Tread rubber | 1.13 × 10^{3} | 2.37 | −9.87 | 43.66 | 0 |

Belt rubber | 7.53 × 10^{3} | 5.65 | −35.12 | 145.76 | 0 |

Carcass rubber | 3.85 × 10^{3} | 3.67 | −14.73 | 77.87 | 0 |

Inner liner rubber | 1.14 × 10^{3} | 2.13 | −14.91 | 67.67 | 0 |

Sidewall rubber | 1.08 × 10^{3} | 2.26 | −10.00 | 45.30 | 0 |

Apex rubber | 1.15 × 10^{3} | 2.15 | −7.84 | 33.91 | 0 |

Bead filler rubber | 1.17 × 10^{3} | 4.76 | −12.97 | 65.78 | 0 |

Bead rubber | 1.23 × 10^{3} | 4.55 | −39.05 | 179.76 | 0 |

Parameter, Unit | Belt Layer | Cord Ply Layer | Bead |
---|---|---|---|

Density, kg∙m^{−3} | 4.99 × 10^{3} | 1.14 × 10^{3} | 7.80 × 10^{3} |

Young’s modulus, MPa | |||

E_{a} | 80,641.06 | 22,002.34 | 149,403.06 |

E_{b} | 57.49 | 24.27 | 281.65 |

E_{c} | 57.49 | 24.27 | 281.65 |

Poisson’s ratio | |||

V_{ba} | 2.67 × 10^{−4} | 4.49 × 10^{−4} | 6.26 × 10^{−4} |

V_{ca} | 2.67 × 10^{−4} | 4.49 × 10^{−4} | 6.26 × 10^{−4} |

V_{cb} | 0.49 | 0.49 | 0.49 |

Shear modulus, MPa | |||

G_{ab} | 13.99 | 6.01 | 64.96 |

G_{bc} | 19.29 | 8.14 | 94.51 |

G_{ca} | 13.99 | 6.01 | 64.96 |

Part | Rigid Plate | PVC |
---|---|---|

Density, g∙mm^{−3} | 7.85 × 10^{−3} | 1.17 × 10^{−3} |

Young’s modulus, Pa | 2.1 × 10^{11} | |

Poisson’s ratio | 0.30 | 0.49 |

Parameter | Value |
---|---|

Static friction coefficient (SPH–Road) | 0.60 |

Kinetic friction coefficient (SPH–Road) | 0.40 |

Static friction coefficient (SPH–Tire) | 0.30 |

Kinetic friction coefficient (SPH–Tire) | 0.15 |

Smoothing length | 1.20 |

Scaling factor for smoothing length | 0.20 |

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**MDPI and ACS Style**

Wang, D.; Wang, H.; Xu, Y.; Zhou, J.; Sui, X.
Numerical Simulations of the Driving Process of a Wheeled Machine Tire on a Snow-Covered Road. *Machines* **2023**, *11*, 657.
https://doi.org/10.3390/machines11060657

**AMA Style**

Wang D, Wang H, Xu Y, Zhou J, Sui X.
Numerical Simulations of the Driving Process of a Wheeled Machine Tire on a Snow-Covered Road. *Machines*. 2023; 11(6):657.
https://doi.org/10.3390/machines11060657

**Chicago/Turabian Style**

Wang, Di, Hui Wang, Yan Xu, Jianpin Zhou, and Xinyu Sui.
2023. "Numerical Simulations of the Driving Process of a Wheeled Machine Tire on a Snow-Covered Road" *Machines* 11, no. 6: 657.
https://doi.org/10.3390/machines11060657