# Design and Load Distribution Analysis of the Mismatched Cycloid-Pin Gear Pair in RV Speed Reducers

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## Abstract

**:**

## 1. Introduction

## 2. Tooth Profile Geometry of Cycloid Gear

#### 2.1. Tooth Profile Generation and Modification

#### 2.2. Curvature Radius of the Cycloid Gear Tooth Profile

## 3. Improved Load Distribution Modelling

- The model is discussed in the two-dimensional plane because the teeth are straight and parallel to the shaft axis.
- The model is established under the quasi-static condition, neglecting the inertia forces.
- Friction can be neglected to avoid the uncertainness about the Coulomb frictional coefficient.
- The influence of damaging will be negligible to avoid the uncertainness of the damaging when the stresses are beyond the allowable value.

- Step 1
- Defining the modified tooth profile of the cycloid gear with the design parameters and the modifiaction ammounts.
- Step 2
- Applying the unloaded tooth contact analysis (TCA) to determine the meshing information when two contact conditions are satisfied by transferring the tooth and pin profiles into the fixed coordinate system.
- Step 3
- Determining the the initial guesses of angular parameters based on the geometrical and kinetic relationships; the values will be updated after every iteration. The system of nonlinear equations is then solved by the fsolve fuction provided in Matlab.
- Step 4
- Considering the meshing point as a linear spring with a contact stiffness along the line of action where the contact deformation of the pin-groove pair is introduced.
- Step 5
- The force and torque system equations are solved by the fsolve fuction provided in Matlab simultaneously by using an iterative technique after the initial load and angular displacement are given.
- Step 6
- After several iterations, load and angular displacement changes will converge within 1%, and then the iterative procedure will be terminated.
- Step 7
- Outputting the predicted rasults, including the distributed loads among pins, the contact stress, the transmission error, and the variation of the instantaneous gear ratio through a mesh cycle.

#### 3.1. Unloaded Tooth Contact Analysis

#### 3.2. Initial Guesses Determination

#### 3.3. Compatibility and Equilibrium Conditions

#### 3.4. Hertzian Contact Stiffness

#### 3.5. Contact Stress, Loaded TE, and Gear Ratio

## 4. Analysis Results and Discussion

#### 4.1. Comparison and Verification

**√**) or not (

**×**), as shown in Table 2.

#### 4.2. Effect of the Position Deviations of Ring Pins

## 5. Conclusions

- 1.
- In order to improve the precision of the prediction model, the contact deformation of the pin-groove pairs under the cycloid drive design stage should be considered indispensably, which is verified by the comparison with those obtained by the current model. It has some effects on the maximum contact stress and the mean and peak-to-peak values of the loaded TE.
- 2.
- The obtained results show the necessity of the consideration of the tooth profile modification in the design and analysis of the cycloid-pin gear pair. It was observed to significantly influence the distributed load, contact stress, loaded TE, and real gear ratio of the cycloid drive. The number of contacting pins decreases from 20 to 9. The stress increases to 149 MPa. The absolute mean values of the loaded TE increase by about 30 arc seconds when the tooth profile modification is present.
- 3.
- Even minute random angular and radial position errors of the ring pins with the ranges of ±0.05 mrad and ±4 μm of the pinion have considerable effects on the tooth contact parameters. Large discrepancies are observed, the maximum contact stress reaches 1887 MPa beyond the strength limit, the peak-to-peak values of the loaded TE increase from 0.21 arc seconds to 5.38 arc seconds, and the peak-to-peak value of the gear ratio increases to 44 % of the ideal gear ratio. The irregular change of meshing clearances caused by the combined position error has a very strong influence on the tooth contact, which will worsen the contact strength, transmission accuracy, and torque ripple of the RV speed reducer.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) The schematic view of an RV speed reducer; (

**b**) the 3D model of a cycloid-pin gear pair.

**Figure 2.**The manufacturing errors of the ring gear: (

**a**) the pin-groove radial error; (

**b**) the pin-groove position error.

**Figure 3.**Coordinate systems for the tooth profile modification of cycloid gears: (

**a**) pin radius modification; (

**b**) pin position modification.

**Figure 8.**Comparison of the loads distributed among pins and pin-grooves at (

**a**) 0°, (

**b**) 60°, (

**c**) 120°, and (

**d**) 180° crankshaft angles for four cases.

**Figure 12.**Comparison of the load distribution on ring pins at (

**a**) 0°, (

**b**) 60°, (

**c**) 120°, and (

**d**) 180° crankshaft angles for three cases.

**Figure 13.**Comparison of the time-varying Hertzian contact stress on all (

**a**) the pins and (

**b**) the corresponding pin-grooves.

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

${n}_{2}$ | Tooth number | 39 |

${n}_{1}$ | Pin number | 40 |

$a$ | Pin position radius | 82 mm |

${\rho}_{p}$ | Pin radius | 3.5 mm |

${\rho}_{g}$ | Pin-groove radius | 3.505 mm |

$e$ | Eccentricity | 1.5 mm |

$B$ | Cycloid gear width | 12 mm |

$\Delta a$ | $\mathrm{TPM}\mathrm{amount}\mathrm{of}a$ | 0 μm |

$\Delta \rho $ | $\mathrm{TPM}\mathrm{amount}\mathrm{of}\rho $ | 8 μm |

${\epsilon}_{e}$ | Eccentricity error | 0 μm |

${\epsilon}_{\rho}$ | Radial error | 0 μm |

Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|

TPM | √ | √ | × | × |

${\delta}_{p}$ | × | √ | × | √ |

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**MDPI and ACS Style**

Li, X.; Tang, L.; He, H.; Sun, L.
Design and Load Distribution Analysis of the Mismatched Cycloid-Pin Gear Pair in RV Speed Reducers. *Machines* **2022**, *10*, 672.
https://doi.org/10.3390/machines10080672

**AMA Style**

Li X, Tang L, He H, Sun L.
Design and Load Distribution Analysis of the Mismatched Cycloid-Pin Gear Pair in RV Speed Reducers. *Machines*. 2022; 10(8):672.
https://doi.org/10.3390/machines10080672

**Chicago/Turabian Style**

Li, Xuan, Linggao Tang, Haidong He, and Lining Sun.
2022. "Design and Load Distribution Analysis of the Mismatched Cycloid-Pin Gear Pair in RV Speed Reducers" *Machines* 10, no. 8: 672.
https://doi.org/10.3390/machines10080672