# A Novel Tooth Modification Methodology for Improving the Load-Bearing Capacity of Non-Orthogonal Helical Face Gears

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generation of Non-Orthogonal Helical Face Gear Pair

#### 2.1. Generation of Tooth Surface of Face Gear

_{2}, S

_{g}, and S

_{S}are rigidly connected to the face gear, the disk grinding wheel and the virtual cutter, and Z

_{2}, Z

_{g}, and Zs are the corresponding rotation axes, respectively.

_{2}and Os of the two coordinate systems S

_{2}and S

_{S}is L

_{0}, and the angle γ

_{m}is the shaft angle. The center distance between the disk grinding wheel and the virtual cutter is E

_{g}, and E

_{g}is defined as ${E}_{g}={r}_{g}-{r}_{ps}$, where r

_{ps}is the radius of the pitch circle of the virtual cutter, and r

_{g}is the radius of the pitch circle of the disk grinding wheel; l

_{g}is the moving distance of the center of the disk grinding wheel. There are three kinds of motion during the grinding process: the disc-shaped grinding wheel swings around the rotation axis Z

_{S}of the virtual gear shaping cutter at an angular velocity ω

_{s}, the face gear rotates around its own rotation axis Z

_{2}at an angular velocity ω

_{2}, and the grinding wheel rotates around its own rotation axis X

_{g}at a high speed ω

_{g}to form a cutting motion.

_{S}to the coordinate system S

_{2}[35]. Then, the tooth surface of the face gear is generated using the meshing equation.

_{c}

_{2}represents the modification parameter. The parameters u

_{2}and l

_{g}are associated with the flank characteristics of the cutting tool.

_{a}to S

_{si}, and ${L}_{sa}$ is a 3 × 3 submatrix of ${M}_{sa}$. ${M}_{sa}$ is determined via Equation (5).

_{g}at a high speed ω

_{g}to form a cutting motion.

_{g}is the rotation angle of the face gear; M

_{gs}is a 4 × 4 coordinate transformation matrix from S

_{s}to S

_{g}; and L

_{gs}is a 3 × 3 submatrix of M

_{gs}. M

_{gs}is determined via Equation (12).

_{g}to S

_{2}, and ${L}_{2g}$ is a 3 × 3 submatrix of ${M}_{2g}$. ${f}_{1}$, ${f}_{1}$ are the meshing equations of face gears in the coordinate system S

_{s}. ${\stackrel{\rightharpoonup}{v}}_{g1}$ is the grinding wheel’s center speed, while ${\stackrel{\rightharpoonup}{v}}_{g2}$ is the non-orthogonal helical face gears’ relative speed to the grinding wheel. ${M}_{2g}$ is determined via Equation (17).

#### 2.2. Generation of Novel Double-Crowned Tooth Surface of Pinion

_{a}

_{1}, S

_{b}

_{1}, and S

_{t}

_{1}is connected to the rack-cutter. The tooth modification along the tooth profile direction can be realized based on the tooth modification of the rack-cutter.

_{1}. ΔL

_{1}is the rack-cutter’s extra translation motion parameter, which is related to the intentional designed high-order transmission error. S

_{n}

_{1}is the fixed coordinate system. r

_{p}

_{1}is the pitch radius of the pinion. S

_{t}

_{1}and S

_{1}are the coordinate systems of the rack-cutter and generated pinion, respectively.

_{t}

_{1}to S

_{1}and ${\left[L\right]}_{1,t1}$ is a 3 × 3 submatrix of ${\left[M\right]}_{1,t1}$; ${\stackrel{\rightharpoonup}{r}}_{t1}$ and ${\stackrel{\rightharpoonup}{n}}_{t1}$ are vectors in the coordinate system S

_{t}

_{1}, both obtained via Equation (23).

_{1}is the modification parameter for the tooth profile modification of the cutter; u

_{1}and l

_{1}are the related tooth flank parameters of the cutter.

## 3. Analysis of Meshing Performance of Face Gears

#### 3.1. Tooth Contact Analysis

_{1}being the fixed coordinate system of the cylindrical gear and S

_{2}being the fixed coordinate system of the face gear. In the figure, B = r

_{ps}− r

_{p}

_{1}and γ

_{f}= γ

_{m}+ Δγ, where γ

_{m}is the shaft angle, and Δγ is the shaft angle error.

_{1}and face gear ∑

_{2}; ${\stackrel{\rightharpoonup}{n}}_{1}^{(f)}$ and ${\stackrel{\rightharpoonup}{n}}_{2}^{(f)}$ are unit normal vectors for the pinion’s tooth surface ${\stackrel{\rightharpoonup}{r}}_{1}^{(f)}$ and the face gear ${\stackrel{\rightharpoonup}{n}}_{2}^{(f)}$, respectively; ${M}_{fi}$ (i = 1, 2) is a 4 × 4 matrix; ${L}_{fi}$ (i = 1, 2) is a 3 × 3 submatrix of ${M}_{fi}$ (i = 1, 2).

_{2}is the rotation angle of the face gear; φ

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

_{f}, the distance from the tooth contact point to the coordinate origin is the same, and the top edge tangent vector of the face gear is perpendicular to the tooth normal vector of the pinion.

#### 3.2. Calculation of Hertzian Contact Stress

_{1}and δ

_{2}are the shape variables of surface 1 and surface 2, respectively.

_{n}is the contact point’s normal load, a is the ellipse’s semi-professional axis; b is its semi-minor axis; e is the eccentricity of the ellipse, and K(e) and E(e) are the elliptic integrals of the first and second kinds, respectively; p

_{0}is the maximum contact stress.

## 4. Designation of Intentional High-Order Transmission Error

_{A}and P

_{E}denote the positions of the gears entering and exiting meshing, respectively. The points P

_{B}and P

_{D}are the two peaks of the curve. The point P

_{C}is the transition point in the middle, and the positional relationship between P

_{B}, P

_{C}

_{,}and P

_{D}is determined by the two parameters of λ

_{B}and λ

_{D}. T

_{m}denotes the meshing period of the gears.

_{1}is the rotation angle of the pinion; δφ

_{2}is the transmission error; a

_{0}~a

_{6}are the coefficients of the HTE function.

## 5. Numerical Examples and Discussions

- (1)
- Parameter calculation. According to the tooth height of the gear shaping cutter, the tooth height parameter z
_{2}of the face gear in the coordinate system S_{2}is derived; - (2)
- Calculate the tooth width. Find the minimum inner diameter R
_{1}of the gear undercut and the maximum outer diameter R_{2}without tooth tip sharpening, and select an appropriate tooth width within the range of R_{1}and R_{2}as the known quantity y_{2}; - (3)
- Discrete y
_{2}and z_{2}. Through discretization, i discrete values of y_{2i}(y_{21}, y_{22}, y_{23}, …, y_{2i}) and j discrete values of z_{2j}(z_{21}, z_{22}, z_{23}, …, z_{2}_{j}) are obtained; based on the y_{2i}and z_{2j}, which are used as the input values and substituted into the tooth surface equation, we can obtain i × j values of θ_{Sij}(θ_{Si}_{1}, θ_{Si}_{2}, θ_{Si}_{3}, …, θ_{Sij}) and i × j values of φ_{Sij}(φ_{Si}_{1}, φ_{Si}_{2}, φ_{Si}_{3}, …, φ_{Sij}); - (4)
- Visualization of the working tooth surface. Back-substitute the i × j group (θ
_{Sij}, φ_{Sij}) into the non-orthogonal asymmetric surface gear tooth surface equation to obtain i × j discrete coordinate points (x_{ij}, y_{ij}, z_{ij}) on the corresponding tooth surface, and apply MatLab instructions to generate work surfaces for gears with non-orthogonal faces.

#### 5.1. Tooth Modification

#### 5.2. Tooth Contact Analysis

#### 5.3. Loaded Tooth Contact Analysis

## 6. Conclusions

- (1)
- This paper proposes a new bidirectional gear modification method. The tooth modification is determined by the modified rack-cutter, and its feed motion is related to an intentionally designed transmission error. The novelty of the tooth modification design is that the transmission error can be predesigned.
- (2)
- The performance of the introduced novel tooth modification is studied through TCA and LTCA. Under the non-misalignment error working conditions, the contact stress and bending stress of the novel tooth modification decrease by as much as 34.83% and 12.72%, which shows better meshing performance compared to the traditional tooth modification.
- (3)
- Under the misalignment error working conditions, the contact stress and bending stress of the novel tooth modification decrease by as much as 26.24% and 7.98%. The introduced new gear modification method has lower tooth profile contact stress and tooth root bending stress both with and without misalignment errors.
- (4)
- The introduced novel tooth modification in this paper is universal, and not limited to face gears but can be extended to other types of gears.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a_{1} | modification parameter for tooth profile modification |

S_{i} | coordinate system i |

u_{i}, l_{i} | surface parameter of ∑i |

[M]_{i,j} | coordinate transmission matrix (from S_{j} to S_{i}) |

${\stackrel{\rightharpoonup}{r}}_{i}$$,{\stackrel{\rightharpoonup}{n}}_{i}$ | position vector and unit normal vector of surface ∑i |

β | base helix angle |

ΔL_{1} | parameter of additional translation motion of rack-cutter |

Δγ | misalignment angle error |

δφ_{2} | transmission error |

θ_{1} | rotation angle of generated pinion |

p_{0} | maximum contact stress |

φ_{1}, φ_{2} | rotation angle of pinion and face gear |

Abbreviations | |

Nov-mod | novel modification |

Non-mod | non-modification |

Tra-mod | traditional modification |

TE | transmission error |

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**Figure 2.**Tooth profile of the rack-cutter. (

**a**) Tooth profile schematic; (

**b**) schematic diagram of helix angle; (

**c**) machining schematic; (where s

_{0}is the pitch of the round segment; $\alpha $ is the pressure angle of the rack-cutter; ${\beta}_{2}$ is the helix angle).

**Figure 3.**Generation of coordinate system for the disk grinding wheel. (

**a**) Grinding wheel processing; (

**b**) coordinate system machining.

**Figure 5.**Tooth modification of the rack-cutter of pinion. (

**a**) Second-order parabolic tooth profile; (

**b**) schematic diagram of helix angle.

**Figure 14.**Contact patterns and TEs of non-mod, tra-mod, and nov-mod without misalignment error. (

**a**) Contact patterns; (

**b**) TEs.

**Figure 16.**Variations in contact pattern and TE of non-mod with misalignment error. (

**a**) Contact pattern; (

**b**) TEs.

**Figure 17.**Variations in contact pattern and TE of tra-mod with misalignment error. (

**a**) Contact pattern; (

**b**) TEs.

**Figure 18.**Variations in contact pattern and TE of nov-mod with misalignment error. (

**a**) Contact pattern; (

**b**) TEs.

**Figure 19.**Comparison of the contact stresses without misalignment error. (

**a**) Non-mod; (

**b**) tra-mod; (

**c**) nov-mod; (

**d**) comparison of contact stresses.

**Figure 20.**Comparison of the bending stresses without misalignment error. (

**a**) Non-mod; (

**b**) tra-mod; (

**c**) nov-mod; (

**d**) comparison of the bending stresses.

**Figure 21.**Comparison of the contact stresses with misalignment error, Δγ = 1.5′. (

**a**) Non-mod; (

**b**) tra-mod; (

**c**) nov-mod; (

**d**) comparison of the contact stresses.

**Figure 22.**Comparison of the bending stresses with misalignment error, Δγ = 1.5′. (

**a**) Non-mod; (

**b**) tra-mod; (

**c**) nov-mod; (

**d**) comparison of the contact stresses.

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

Pinion tooth number | 25 |

Cutter tooth number | 28 |

Face gear tooth number | 160 |

Normal module (mm) | 6.35 |

Pressure angle (degree) | 25 |

Helix angle (degree) | 15 |

Shaft angle (degree) | 100 |

Inner radius (mm) | 510 |

External radius (mm) | 600 |

**Table 2.**Comparison of the maximum contact stress and maximum bending stresses without misalignment error.

Items | Contact Stress | Bending Stress | ||
---|---|---|---|---|

Results (MPa) | Variation | Results (MPa) | Variation | |

Non-mod | 890.7 | — | 51.1 | — |

Tra-mod | 739.7 | −16.95% | 58.5 | +14.48% |

Nov-mod | 580.4 | −34.83% | 44.6 | −12.72% |

**Table 3.**Comparison of the maximum contact stress and maximum bending stresses with misalignment error, Δγ = 1.5′.

Items | Contact Stress | Bending Stress | ||
---|---|---|---|---|

Results (MPa) | Variation | Results (MPa) | Variation | |

Non-mod | 940.9 | — | 63.9 | — |

Tra-mod | 774.2 | −17.73% | 80.6 | +26.13% |

Nov-mod | 694.5 | −26.24% | 58.8 | −7.98% |

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## Share and Cite

**MDPI and ACS Style**

Jia, C.; Li, B.; Xu, J.
A Novel Tooth Modification Methodology for Improving the Load-Bearing Capacity of Non-Orthogonal Helical Face Gears. *Machines* **2023**, *11*, 1077.
https://doi.org/10.3390/machines11121077

**AMA Style**

Jia C, Li B, Xu J.
A Novel Tooth Modification Methodology for Improving the Load-Bearing Capacity of Non-Orthogonal Helical Face Gears. *Machines*. 2023; 11(12):1077.
https://doi.org/10.3390/machines11121077

**Chicago/Turabian Style**

Jia, Chao, Bingquan Li, and Junhong Xu.
2023. "A Novel Tooth Modification Methodology for Improving the Load-Bearing Capacity of Non-Orthogonal Helical Face Gears" *Machines* 11, no. 12: 1077.
https://doi.org/10.3390/machines11121077