# Numerical Analysis of Meshing of Loaded Misaligned Straight Bevel Gear Drives of Automobile Differential

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Computerized Design of Gear Model

## 3. Establishment of Finite Element Models

^{5}MPa and Poisson’s ratio μ = 0.3. The friction coefficient between two gears is 0.2 and the applied torque T = 50 Nm.

- (i)
- Nodes on the bottom rim of the driven gear are fixed.
- (ii)
- For the nodes on the bottom rim of the driving pinion, the radial and axial degrees of freedom are considered as fixed, and only the rational degree of freedom is set as free and coupled.
- (iii)
- The effect of force F at every node on the bottom rim of the driving pinion in the rotational direction on the pinion is equal to that of working torque T on the pinion. The force F is defined as the following:

_{d}= 7.0275 mm is the radius of the bottom rim of the driving pinion.

## 4. Determination of the Meshing Interval for Investigation

_{1}, tooth pair 0 enters mesh so that the total load is shared by both tooth pair −1 and tooth pair 0 in the double pair tooth contact zone II. When transferring point P

_{2}is reached, tooth pair −1 exits out of mesh. At the same time, the total load is suddenly transmitted to tooth pair 0 and the single pair tooth contact zone I for tooth pair 0 begins. As with the cycle, the load and motion will be transmitted from tooth pair 0 to tooth pair + 1.

_{1}and P

_{3}are selected as the critical points at the beginning and the end of a meshing cycle in the analysis, respectively.

## 5. Selection of Alignment Errors

- (i)
- Δγ is defined as the change of crossing angle.
- (ii)
- ΔP is defined as the axial displacement of the pinion.
- (iii)
- ΔG is defined as the axial displacement of the gear.
- (iv)
- ΔE is defined as the change center distance of the assembling straight bevel gears.

## 6. Results and Discussion

#### 6.1. Influence of Single Alignment Error on the Gear Mesh

#### 6.1.1. Influence of Single Alignment Error on the Contact Area

#### 6.1.2. Influence of the Single Alignment Error on the Transmission Error

_{d}is the radius of the bottom rim of the pinion, and ϕ

_{1}is the rotation angle of the driving pinion.

_{3}and ΔG has a more significant effect on the transmission error at the transferring point P

_{2.}

#### 6.1.3. Influence of Single Alignment Error on the Vibration and Noise of Gear Drives

^{2}

_{(e)rms}is the root mean square of the signals (in rms) caused by the transmission errors with some kind of alignment error, while ω

^{2}

_{(0)rms}is the root mean square of the signals (in rms) caused by the transmission errors under standard installation; ${(\Delta {\varphi}_{{}_{{}_{1(e)}}})}_{i}$ is the value of transmission error at the engagement position i with some kind of alignment error; ${(\Delta {\varphi}_{{}_{{}_{1(0)}}})}_{i}$ is the value of transmission error at the engagement position i under standard installation; engagement position i and engagement position i − 1 are two successive engagement positions.

#### 6.2. Influence of Combined Alignment Errors on the Gear Mesh

#### 6.2.1. Influence of Combined Alignment Errors on the Contact Zone

#### 6.2.2. Influence of Combined Alignment Errors on the Transmission Error

- (1)
- As shown in Figure 20a, it can be seen that, when ΔP and ΔG combine, the transmission error increases greatly in the whole meshing circle, compared with that in the single ΔP or ΔG.
- (2)
- As shown in Figure 20b, it can be seen that, when ΔP and ΔE (ΔE < 0) combine, the transmission error increases greatly in the whole meshing circle compared with that in the single ΔP. However, when ΔP and ΔE (ΔE > 0) combine, the transmission error decreases dramatically in the single pair tooth contact zone, compared with that in the single ΔP. However, the transmission error in the double pair tooth contact zone is basically in accordance with the single ΔP.
- (3)
- As shown in Figure 20c, it can be obtained that, when ΔG and ΔE < 0 combine transmission error increases greatly in the whole meshing circle compared with that in single ΔG. However, when ΔG and ΔE > 0 combine, the transmission error decreases dramatically in the whole meshing circle compared with that in single ΔG.
- (4)
- As shown in Figure 20d,e,f, it can be seen that, when Δγ and one of ΔP, ΔG, ΔE combines, the transmission errors are basically consistent compared with that in single ΔP, ΔG or ΔE. Therefore, it can be concluded that the transmission error is not sensitive to Δγ.
- (5)
- It can be seen from Figure 20 that it is the most dangerous type of alignment errors for increasing the transmission error dramatically, when ΔP, ΔG, ΔE (ΔE < 0) and Δγ.
- (6)
- The alignment errors have a cumulative effect on the length of contact line and the contact zone causing the change of gear stiffness. The reduction of the length of contact line will cause the decrease of gear stiffness. The transmission errors increase with the decreasing of gear stiffness.

#### 6.2.3. Influence of Combined Alignment Errors on the Vibration and Noise of Gear Drives

_{2,}and the reduction of the fluctuation of transmission error at the transferring point P

_{2}leads to the decrease of the vibration and noise of gear drives.

## 7. Conclusions

- (1)
- The alignment errors will cause the straight bevel gear pair contacting at one end and gaping at the other end of the teeth and make the contact zone and the contact line of the straight bevel gear pair change greatly. Furthermore, this change has a direct effect on the transmission error, vibration, and noise of the gear drives.
- (2)
- The contact zone and the contact line of the straight bevel gear pair are sensitive to ΔP, ΔG, and ΔE, resulting in the uneven load distribution and stress concentration on the tooth pair. However, the contact zone and the contact line of the straight bevel gear pair are not sensitive to the alignment error Δγ.
- (3)
- The single alignment error ΔP, ΔG or ΔE has great influence on the transmission error, vibration, and noise of the straight bevel gear drives, and the single alignment error Δγ has great influence on the vibration and noise of the straight bevel gear drives, but it does not have a great effect on the transmission error of the straight bevel gear drives.
- (4)
- Since the combined alignment errors have great influence on the meshing performance of the straight bevel gear, it is important to control the combined alignment errors to ensure the straight bevel gear drives with low vibration and noise. In the practical assembly of the straight bevel gear pair, it is feasible to improve the meshing performance of the straight bevel gear through a better combination of the alignment errors such as the combination of ΔP and ΔE (ΔE > 0) or the combination of ΔG and ΔE (ΔE > 0).
- (5)
- The most dangerous type of alignment error is ΔP, ΔG, ΔE (ΔE < 0) and Δγ. Under this type of alignment error, the length of contact line decreases along longitudinal direction dramatically, leading to a great increase in transmission error, vibration, and noise of the straight bevel gear drives.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 10.**Distribution of contact stress on tooth surface for the pinion (

**top**) and the gear (

**bottom**) with the different single alignment error. (

**a**) ΔP = 0, (

**b**) ΔP = 0.1, (

**c**) ΔG = 0.1, (

**d**) ΔE = −0.02, (

**e**) ΔE = 0.02, (

**f**) Δγ = 0.5.

**Figure 18.**Distribution of contact stress on tooth surface for the pinion (

**top**) and the gear (

**bottom**) with combined alignment errors. (

**a**) ΔP = 0.1 and ΔG = 0.1, (

**b**) Δγ = 0.5 and ΔG = 0.1, (

**c**) Δγ = 0.5 and ΔP = 0.1, (

**d**) ΔG = 0.1 and ΔE = 0.02, (

**e**) ΔG = 0.1 and ΔE = −0.02, (

**f**) ΔP = 0.1 and ΔE = 0.02, (

**g**) ΔP = 0.1 and ΔE = −0.02, (

**h**) Δγ = 0.5 and ΔE = 0.02, (

**i**) Δγ = 0.5 and ΔE = −0.02.

**Figure 19.**Distribution of contact stress on tooth surface for the pinion (

**top**) and the gear (

**bottom**) with combined alignment errors. (

**a**) ΔP = 0.1, ΔG = 0.1 and ΔE = 0.02, (

**b**) ΔP = 0.1, ΔG = 0.1 and ΔE = −0.02, (

**c**) ΔP = 0.1, ΔG = 0.1 and Δγ = 0.5, (

**d**) ΔG = 0.1, ΔE = 0.02, Δγ = 0.5, (

**e**) ΔG = 0.1, ΔE = −0.02 and Δγ = 0.5, (

**f**) ΔP = 0.1, ΔE = 0.02 and Δγ = 0.5, (

**g**) ΔP = 0.1, ΔE = −0.02 and Δγ = 0.5, (

**h**) ΔP = 0.1, ΔG = 0.1, ΔE = 0.02 and Δγ = 0.5, (

**i**) ΔP = 0.1, ΔG = 0.1, ΔE = −0.02 and Δγ = 0.5.

**Figure 20.**Variation of transmission errors in the meshing cycle with combined alignment errors. (

**a**) based on ΔP and ΔG, (

**b**) based on ΔP and ΔE, (

**c**) based on ΔG and ΔE, (

**d**) based on ΔP and Δγ, (

**e**) based on ΔG and Δγ, (

**f**) based on ΔE and Δγ.

**Figure 21.**(

**a**,

**b**) Variation of MFTE and Variation of noise of gear drives under combined alignment errors based on ΔP and ΔG, respectively.

**Figure 22.**(

**a**,

**b**) Variation of MFTE and Variation of noise of gear drives under combined alignment errors based on ΔP, ΔE and Δγ, respectively.

**Figure 23.**(

**a**,

**b**) Variation of MFTE and Variation of noise of gear drives under combined alignment errors based on ΔG, ΔE and Δγ, respectively.

**Figure 24.**(

**a**,

**b**) Variation of MFTE and Variation of noise of gear drives under combined alignment errors based on ΔE and Δγ, respectively.

Parameter | Driving Pinion | Driven Gear |
---|---|---|

Module [mm] | 3.7792 | 3.7792 |

Shaft angle [deg] | 90 | 90 |

Number of teeth | 10 | 14 |

Pressure angle [deg] | 22.5 | 22.5 |

Modification coefficient of height | −0.1812 | −0.1812 |

Modification coefficient of thickness | 0.05 | −0.05 |

Addendum coefficient | 0.8 | 0.8 |

Headspace coefficient | 0.188 | 0.188 |

Face width [mm] | 9.75 | 9.75 |

Gear center hole [mm] | 7.0275 | 10 |

Basic Type of Alignment Error | Δγ (deg) | ΔP (mm) | ΔG (mm) | ΔE (mm) |
---|---|---|---|---|

Value | −0.01 | −0.01 | −0.01 | −0.02 |

0.1 | 0.01 | 0.01 | −0.01 | |

0.2 | 0.05 | 0.05 | 0.01 | |

0.5 | 0.1 | 0.1 | 0.02 | |

1 |

Combined Alignment Errors | ΔPΔGΔγ | ΔPΔEΔγ | ΔGΔEΔγ | ΔPΔGΔE | ΔPΔGΔEΔγ | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Value | 0.1 | 0.1 | 0.1 | −0.02 | 0.1 | −0.02 | 0.1 | 0.1 | 0.1 | 0.1 | −0.02 | 0.5 | ||||

0.1 | 0.1 | 0.5 | 0.1 | 0.02 | 0.1 | 0.02 | 0.1 | 0.1 | −0.02 | 0.1 | 0.1 | 0.02 | 0.5 | |||

0.1 | −0.02 | 0.5 | 0.1 | −0.02 | 0.5 | 0.1 | 0.1 | 0.02 | ||||||||

0.1 | 0.02 | 0.5 | 0.1 | 0.02 | 0.5 |

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**MDPI and ACS Style**

Chen, Q.; Wang, S.; Li, P.; Li, X.; Liu, J.; Hu, D.; Zhao, Z.; Xiong, X.
Numerical Analysis of Meshing of Loaded Misaligned Straight Bevel Gear Drives of Automobile Differential. *World Electr. Veh. J.* **2022**, *13*, 41.
https://doi.org/10.3390/wevj13020041

**AMA Style**

Chen Q, Wang S, Li P, Li X, Liu J, Hu D, Zhao Z, Xiong X.
Numerical Analysis of Meshing of Loaded Misaligned Straight Bevel Gear Drives of Automobile Differential. *World Electric Vehicle Journal*. 2022; 13(2):41.
https://doi.org/10.3390/wevj13020041

**Chicago/Turabian Style**

Chen, Qianjin, Shuiming Wang, Pengfei Li, Xinguang Li, Jianhua Liu, Dewu Hu, Zhigang Zhao, and Xiaoshuang Xiong.
2022. "Numerical Analysis of Meshing of Loaded Misaligned Straight Bevel Gear Drives of Automobile Differential" *World Electric Vehicle Journal* 13, no. 2: 41.
https://doi.org/10.3390/wevj13020041