# Motion Periods of Planet Gear Fault Meshing Behavior

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## Abstract

**:**

## 1. Introduction

- The faulty tooth initially meshing with the ring gear;
- The faulty tooth initially meshing with the sun gear.

## 2. Motion Periods of the Planet Gear Fault-Meshing Position

#### 2.1. Influences of the Fault-Meshing Behavior

#### 2.2. Motion Period of Planet Gear Fault-Meshing Behavior

#### 2.2.1. Initial Fault-Meshing Position at Ring Gear

#### 2.2.2. Initial Fault-Meshing with Sun Gear

## 3. Experimental Study

## 4. Conclusions

- Condition 1: Fault meshing initially occurred on the ring gear$${n}_{planet-ring}=\frac{LCM\{{Z}_{ring}-{Z}_{planet},{Z}_{planet}\}}{{Z}_{planet}}$$$${n}_{carrier-ring}=\frac{LCM\{{Z}_{ring}-{Z}_{planet},{Z}_{planet}\}}{{Z}_{ring}-{Z}_{planet}}$$$${t}_{tidal-ring}=\frac{{n}_{planet-ring}}{{f}_{planet}}=\frac{{n}_{carrier}}{{f}_{carrier}}$$
- Condition 2: Fault meshing initially occurred at the sun gear$${n}_{sun}=\frac{LCM\{{Z}_{ring}+{Z}_{sun},{Z}_{sun}\}}{{Z}_{sun}}{n}_{carrier-ring}$$$${n}_{planet-sun}=\frac{LCM\{{Z}_{ring}+{Z}_{sun},{Z}_{sun}\}}{{Z}_{ring}+{Z}_{sun}}{n}_{planet-ring}$$$${n}_{carrier-sun}=\frac{LCM\{{Z}_{ring}+{Z}_{sun},{Z}_{sun}\}}{{Z}_{sun}}{n}_{carrier-ring}$$$${t}_{tidal-sun}=\frac{{n}_{sun}}{{f}_{sunrot}}=\frac{{n}_{planet-sun}}{{f}_{planet}}=\frac{{n}_{carrier-sun}}{{f}_{carrier}}$$

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

GCD | Greatest Common Divisor |

LCM | Least Common Multiple |

## Appendix A. Geometrical validation of the motion periods

Known Parameters | Values/Unit | Description |
---|---|---|

${\theta}_{carrier}$ | $\frac{9}{16}/cycle$ | Rotation angle of carrier between two times of fault-meshing |

${\theta}_{planet}$ | 1/cycle | Rotation angle of the faulty planet gear between two times of fault-meshing |

${\theta}_{sun}$ | $\frac{8}{7}$/cycle | Rotation angle of sun gear between two times of fault-meshing |

$LCM\{{Z}_{ring}-{Z}_{planet},{Z}_{planet}\}$ | 576 | LCM between ${Z}_{planet}$ and $({Z}_{ring}-{Z}_{planet})$ |

$LCM\{{Z}_{ring}+{Z}_{sun},{Z}_{sun}\}$ | 896 | LCM between $2({Z}_{planet}+{Z}_{sun})$ and ${Z}_{sun}$ |

Undetermined Parameters | Values/Unit | Description |
---|---|---|

${m}_{ring}$ | 16/times | Total fault-meshing times of a motion period |

${n}_{planet-ring}$ | 16/cycles | Number of rotations of the faulty planet gear in a motion period |

${n}_{carrier-ring}$ | 9/cycles | Number of rotations of carrier in a motion period |

Undetermined Parameters | Values/Unit | Description |
---|---|---|

${m}_{sun}$ | 112/times | Total fault-meshing times of a motion period |

${n}_{sun}$ | 288/cycles | Number of rotations of the sun gear in a motion period |

${n}_{carrier-sun}$ | 63/cycles | Number of rotations of carrier in a motion period |

${n}_{planet-sun}$ | 112/cycles | Number of rotations of faulty planet gear in a motion period |

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**Figure 3.**Possible fault-meshing position and the corresponding propagating distance of transfer path1.

**Figure 4.**Possible distances between the fault meshing positions and the sensor in Figure 3.

${\mathit{Z}}_{\mathit{sun}}$ | ${\mathit{Z}}_{\mathit{planet}}$ | ${\mathit{Z}}_{\mathit{ring}}$ | N |
---|---|---|---|

28 | 36 | 100 | 4 |

Rotational Speed | Time Duration of ${\mathit{t}}_{\mathit{tidal}-\mathit{ring}}$ |
---|---|

1800 rpm | $\frac{48}{35}$ s |

3000 rpm | $\frac{144}{175}$ s |

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**MDPI and ACS Style**

Zhang, M.; Wang, K.; Li, Y. Motion Periods of Planet Gear Fault Meshing Behavior. *Sensors* **2018**, *18*, 3802.
https://doi.org/10.3390/s18113802

**AMA Style**

Zhang M, Wang K, Li Y. Motion Periods of Planet Gear Fault Meshing Behavior. *Sensors*. 2018; 18(11):3802.
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**Chicago/Turabian Style**

Zhang, Mian, Kesheng Wang, and Yaxin Li. 2018. "Motion Periods of Planet Gear Fault Meshing Behavior" *Sensors* 18, no. 11: 3802.
https://doi.org/10.3390/s18113802