# Optimization Design for the Planetary Gear Train of an Electric Vehicle under Uncertainties

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Multi-Objective Uncertainty Optimization Problem

^{th}subinterval and the subinterval number, respectively. The subinterval number can be determined by referring to the number of uncertain parameters. The interval range of constraint function ${G}_{i}\left(U\right)$ is expressed as follows:

#### 2.2. Improved Evolutionary Algorithm

_{v}.

^{th}optimized variable respectively.

^{th}iteration; $pc\left(0\right)$ and $pm\left(0\right)$ are the initial crossover probability and mutation probability respectively; ${n}_{t}$ is the total evolutionary generation.

#### 2.3. Multi-Criteria Decision Making (MCDM) Method

#### 2.4. Main Processes of MUOD

## 3. Design Requirements of the Planetary Gear Train

#### 3.1. Main Design Variables and Optimization Objectives

#### 3.2. Main Design Constraints

#### 3.2.1. Equally Spaced Planets

#### 3.2.2. Equally Spaced Planets

#### 3.2.3. Tooth Width Coefficient

#### 3.2.4. Minimum Teeth of No-Undercut

#### 3.2.5. Concentric Constraint

#### 3.2.6. Adjacency Constraint

#### 3.2.7. Contact Stress Requirement

#### 3.2.8. Bending Stress Requirement

## 4. Optimization Results and Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Optimization Results

Ranking | $\mathit{b}$ (mm) | $\mathit{\beta}$ (°) | ${\mathit{m}}_{\mathit{n}}$ (mm) | ${\mathit{Z}}_{\mathit{p}}$ | ${\mathit{Z}}_{\mathit{r}}$ | ${\mathit{Z}}_{\mathit{s}}$ | ${\mathit{c}}_{\mathit{i}}$ |
---|---|---|---|---|---|---|---|

1 | 31.60 | 24.91 | 2 | 22 | 64 | 20 | 0.9333 |

2 | 31.63 | 25.32 | 2 | 22 | 64 | 20 | 0.9195 |

3 | 31.63 | 25.46 | 2 | 22 | 64 | 20 | 0.9153 |

4 | 32.00 | 22.52 | 2 | 25 | 71 | 21 | 0.6758 |

5 | 32.00 | 22.55 | 2 | 25 | 71 | 21 | 0.6753 |

6 | 31.94 | 22.69 | 2 | 25 | 71 | 21 | 0.6752 |

7 | 32.00 | 22.56 | 2 | 25 | 71 | 21 | 0.6751 |

8 | 31.96 | 22.66 | 2 | 25 | 71 | 21 | 0.6751 |

9 | 31.95 | 22.69 | 2 | 25 | 71 | 21 | 0.6748 |

10 | 32.01 | 22.83 | 2 | 25 | 71 | 21 | 0.6709 |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

Ranking | $\mathit{b}$ (mm) | $\mathit{\beta}$ (°) | ${\mathit{m}}_{\mathit{n}}$ (mm) | ${\mathit{Z}}_{\mathit{p}}$ | ${\mathit{Z}}_{\mathit{r}}$ | ${\mathit{Z}}_{\mathit{s}}$ | ${\mathit{c}}_{\mathit{i}}$ |
---|---|---|---|---|---|---|---|

1 | 34.81 | 22.38 | 2 | 24 | 70 | 22 | 0.9333 |

2 | 34.81 | 22.39 | 2 | 24 | 70 | 22 | 0.9331 |

3 | 34.87 | 22.38 | 2 | 24 | 70 | 22 | 0.9247 |

4 | 34.86 | 22.51 | 2 | 24 | 70 | 22 | 0.9241 |

5 | 34.87 | 22.59 | 2 | 24 | 70 | 22 | 0.9201 |

6 | 34.87 | 22.60 | 2 | 24 | 70 | 22 | 0.9200 |

7 | 34.87 | 22.61 | 2 | 24 | 70 | 22 | 0.9193 |

8 | 34.95 | 22.67 | 2 | 24 | 70 | 22 | 0.9106 |

9 | 35.11 | 22.70 | 2 | 24 | 70 | 22 | 0.8968 |

10 | 35.11 | 22.74 | 2 | 24 | 70 | 22 | 0.8951 |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

Ranking | $\mathit{b}$ (mm) | $\mathit{\beta}$ (°) | ${\mathit{m}}_{\mathit{n}}$ (mm) | ${\mathit{Z}}_{\mathit{p}}$ | ${\mathit{Z}}_{\mathit{r}}$ | ${\mathit{Z}}_{\mathit{s}}$ | ${\mathit{c}}_{\mathit{i}}$ |
---|---|---|---|---|---|---|---|

1 | 32.56 | 22.31 | 2 | 22 | 64 | 20 | 0.9333 |

2 | 32.56 | 22.50 | 2 | 22 | 64 | 20 | 0.9292 |

3 | 32.69 | 22.57 | 2 | 22 | 64 | 20 | 0.9214 |

4 | 32.73 | 22.67 | 2 | 22 | 64 | 20 | 0.9175 |

5 | 32.69 | 23.01 | 2 | 22 | 64 | 20 | 0.9117 |

6 | 32.69 | 23.02 | 2 | 22 | 64 | 20 | 0.9114 |

7 | 32.69 | 23.03 | 2 | 22 | 64 | 20 | 0.9114 |

8 | 32.73 | 23.16 | 2 | 22 | 64 | 20 | 0.9064 |

9 | 32.73 | 23.21 | 2 | 22 | 64 | 20 | 0.9055 |

10 | 32.73 | 23.28 | 2 | 22 | 64 | 20 | 0.9039 |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

Ranking | $\mathit{b}$ (mm) | $\mathit{\beta}$ (°) | ${\mathit{m}}_{\mathit{n}}$ (mm) | ${\mathit{Z}}_{\mathit{p}}$ | ${\mathit{Z}}_{\mathit{r}}$ | ${\mathit{Z}}_{\mathit{s}}$ | ${\mathit{c}}_{\mathit{i}}$ |
---|---|---|---|---|---|---|---|

1 | 34.84 | 22.33 | 2 | 23 | 67 | 21 | 0.9333 |

2 | 34.86 | 22.33 | 2 | 23 | 67 | 21 | 0.9319 |

3 | 35.15 | 23.05 | 2 | 23 | 67 | 21 | 0.8859 |

4 | 35.15 | 23.14 | 2 | 23 | 67 | 21 | 0.8823 |

5 | 35.17 | 23.17 | 2 | 23 | 67 | 21 | 0.8798 |

6 | 35.17 | 23.18 | 2 | 23 | 67 | 21 | 0.8795 |

7 | 35.27 | 23.28 | 2 | 23 | 67 | 21 | 0.8699 |

8 | 35.27 | 23.42 | 2 | 23 | 67 | 21 | 0.8652 |

9 | 35.33 | 23.46 | 2 | 23 | 67 | 21 | 0.8592 |

10 | 35.38 | 23.49 | 2 | 23 | 67 | 21 | 0.8558 |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

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**Figure 3.**Schematic diagram of two kinds of gear meshing: (

**a**) external meshing; (

**b**) internal meshing; and (

**c**) load distribution coefficient.

Parameters | Values |
---|---|

Body size (length, width, height) (mm) | 7232, 2240, 2820 |

Wheelbase (mm) | 3935 |

Curb mass (kg) | 5000 |

Full load mass (kg) | 8500 |

Front/rear wheel track (mm) | 1901/1630 |

Rolling radius (mm) | 373 |

Maximum speed (km/h) | 100 |

Maximum climbing degree | 30% |

Maximum speed in 30 min (km/h) | 90 |

Design Variables | Lower Bound | Upper Bound |
---|---|---|

${Z}_{s}$ | 20 | 30 |

${Z}_{p}$ | 20 | 30 |

${Z}_{r}$ | 60 | 80 |

$\beta $ | 20 | 30 |

$b$ | 30 | 50 |

${m}_{n}$ | / | / |

Uncertainties | Case 1 | Case 2 | Case 3 |
---|---|---|---|

${\beta}^{IR}$ (°) | 2 | 3 | 4 |

${b}^{IR}$ (mm) | 2 | 3 | 4 |

${E}^{IR}$ (GPa) | 10.5 | 21 | 31.5 |

${T}_{t}^{IR}$ (N·m) | 393.6 | 590.4 | 787.2 |

**Table 4.**Reliability-based possibility degree $\lambda $ of four solution sets under three uncertainty cases.

Constraint | MDOD (Case 1) | MUOD (Case 1) | MDOD (Case 2) | MUOD (Case 2) | MDOD (Case 3) | MUOD (Case 3) |
---|---|---|---|---|---|---|

Lower bound of tooth width coefficient | 1 | 1 | 1 | 1 | 1 | 1 |

Upper bound of tooth width coefficient | 0.64 | 0.80 | 0.60 | 0.81 | 0.57 | 0.81 |

Minimum teeth of no-undercut for ${Z}_{s}$ | 1 | 1 | 1 | 1 | 1 | 1 |

Minimum teeth of no-undercut for ${Z}_{p}$ | 1 | 1 | 1 | 1 | 1 | 1 |

Minimum teeth of no-undercut for ${Z}_{r}$ | 1 | 1 | 1 | 1 | 1 | 1 |

Adjacency constraint | 1 | 1 | 1 | 1 | 1 | 1 |

Contact stress | 1 | 1 | 1 | 1 | 1 | 1 |

Bending stress | 1 | 1 | 0.84 | 0.85 | 0.76 | 0.98 |

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**MDPI and ACS Style**

Xu, X.; Chen, J.; Lin, Z.; Qiao, Y.; Chen, X.; Zhang, Y.; Xu, Y.; Li, Y.
Optimization Design for the Planetary Gear Train of an Electric Vehicle under Uncertainties. *Actuators* **2022**, *11*, 49.
https://doi.org/10.3390/act11020049

**AMA Style**

Xu X, Chen J, Lin Z, Qiao Y, Chen X, Zhang Y, Xu Y, Li Y.
Optimization Design for the Planetary Gear Train of an Electric Vehicle under Uncertainties. *Actuators*. 2022; 11(2):49.
https://doi.org/10.3390/act11020049

**Chicago/Turabian Style**

Xu, Xiang, Jiawei Chen, Zhongyan Lin, Yiran Qiao, Xinbo Chen, Yong Zhang, Yanan Xu, and Yan Li.
2022. "Optimization Design for the Planetary Gear Train of an Electric Vehicle under Uncertainties" *Actuators* 11, no. 2: 49.
https://doi.org/10.3390/act11020049