# Analysis and Optimization of Tooth Surface Contact Stress of Gears with Tooth Profile Deviations, Meshing Errors and Lead Crowning Modifications Based on Finite Element Method and Taguchi Method

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## Abstract

**:**

## 1. Introduction

## 2. FEM Modeling

#### 2.1. Determination of Gear Engagement Position

#### 2.2. Multi-Point Constraint (MPC)

^{5}MPa, and the Poisson’s ratio μ = 0.3. Figure 4c shows the loading type and boundary conditions, and the two models have the same type of boundary conditions and linear pressure loads, in which the linear pressure is 10 N/m. Figure 4d,e show the analysis results of FEM, and the maximum stress on both models is 6.63 MPa. According to the stress nephogram and solution results, for the FEM model of MPC connection, when the size of the refined area is large enough, that is to say, when the distance between the MPC connection position and the stress analysis position is far enough, the solution results are basically consistent with the results of the entire refined mesh model. Therefore, in this paper, the mesh of the refined area and non-refined area is connected by the MPC, and in order to save the time of FEM analysis, and at the same time ensure the accuracy of the solution, the optimal size of the refined area will be studied and analyzed to determine the mesh transition boundary position.

#### 2.3. Determination of Mesh Transition Boundary Position

#### 2.3.1. Hertz Contact Theory

#### 2.3.2. Determination of Mesh Transition Boundary Position

## 3. TSCS Analysis of Gear with TPD, ME or LCM

#### 3.1. TSCS Analysis of Gear with TPD

_{n}is the upper limit value of the total TPD value of the grade n precision gear, and ${f}_{ni}$ is the actual TPD value of each node of the grade n precision gear ($i$ is the node number), so the TPDs of the refined areas of other precision grade gears can be obtained by Equation (7).

#### 3.2. TSCS Analysis of Gear with ME

_{2}, which is <$a$, and the second kind of ME is the rotation error around the X axis in the V plane, and the amount of ME is <BAB

_{1}, which is <$b$.

#### 3.3. TSCS Analysis of Gear with LCM

## 4. Analysis of the Influence of TPD, ME and LCM on TSCS by TM

#### 4.1. Preparations before the Tests

#### 4.2. Analysis of FEM Simulation Results

#### 4.3. The Interaction between TPD, ME and LCM

#### 4.4. Determining the Optimal Combination of Influence Factor Levels

#### 4.5. Comparative Analysis of Different Combinations of Factor Levels

## 5. Conclusions

- (1)
- In this paper, a 3D-FEM model of one pair of engaged teeth has been modeled and the mesh of the contact area has been refined by FEM software. In the model, the refined area mesh and the non-refined area mesh were connected by Multi-point constraint (MPC). At the same time, in order to save the time of the FEM solution on the premise of ensuring the accuracy of model analysis, the reasonable size of the refined area has been researched and confirmed;
- (2)
- In this paper, the FEM contact models of gears have been established and solved according to the three influence factors of TPD, ME and LCM, respectively. It is found that when there is only one single influence factor, the influence factor has a great influence on the TSCS. Compared with the ideal gear, the maximum TSCS of the gear with TPD can reach 2.45 times of that of the ideal gear, the maximum TSCS of the gear with ME can reach 1.88 times of that of the ideal gear, and the maximum TSCS of the gear with LCM can reach 1.65 times of that of the ideal gear;
- (3)
- In this paper, the Taguchi method has been used to research the influence degree of each factor on the TSCS when three factors exist at the same time. It is found that the TPD has the greatest influence on the TSCS, as the mean difference in TSCS between different grades of TPD can reach 1712 MPa. This is followed by the LCM, as the mean difference in TSCS between different LCM values can reach 421 MPa. The influence degree of ME is very limited, as the mean differences in TSCS value between different <$a$ and <$b$ are only 7 MPa and 10 MPa, respectively;
- (4)
- In this paper, the interactions between the influence factors have been researched, and it is found that the interactions between the TPD and other factors are not obvious, while the interactions between the LCM, <$a$ and <$b$ are very obvious, especially the interaction between the two kinds of ME. In addition, it has been verified that the LCM can effectively alleviate the phenomenon of the edge stress concentration of TSCS caused by ME. From Figure 28, it can be found that the maximum TSCS value of the gear with LCM is 11.73% less than that of the gear without LCM;
- (5)
- According to the type of gear researched in this paper, through the TM, the optimal combination of control factor levels has been determined, as follows: grade 2 for the TPD, −0.12° for the <$b$, 0.2° for the <$a$, and 3.5 μm for the LCM quantity. The gear contact fatigue life of the optimal combination of factor levels is much longer than that of the original combination;
- (6)
- For other types of gears and different influence factors, the research method and analysis process of this paper have certain reference value.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

3D | Three-dimensional |

FEM | Finite element method |

LCM | Lead crowning modifications |

ME | Meshing errors |

MPC | Multi-point constraint |

TM | Taguchi method |

TPD | Tooth profile deviations |

TSCS | Tooth surface contact stress |

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Normal Modulus | Tooth Number | Pressure Angle | Addendum Coefficient | Clearance Coefficient | Helix Angle | Tooth Width | Modification Coefficient | Poisson Ratio | Elastic Modulus |
---|---|---|---|---|---|---|---|---|---|

2 mm | 20 | 20° | 1 | 0.25 | 0° | 5 mm | 0 | 0.25 | 2.07 × 10^{5} MPa |

Type of Main Elements | Contact Surface Element | Target Surface Element | Friction Coefficient | Material Density |
---|---|---|---|---|

Solid185 | Conta173 | Targe170 | 0.1 | 7.8 × 10^{−9} t/mm^{3} |

$\mathit{w}\times \mathit{h}$ | The Maximum TSCS (MPa) | The Errors of TSCS | Time (min) |
---|---|---|---|

Hertz contact theory | 1653.75 | ||

$8b\times 8b$ | 1676.23 | 1.36% | 75 |

$4b\times 4b$ | 1682.93 | 1.76% | 9 |

$4b\times 3b$ | 1693.59 | 2.41% | 6 |

$4b\times 2b$ | 1754.98 | 6.12% | 4 |

$3b\times 4b$ | 1712.73 | 3.57% | 6 |

Symbol | Description | Value |
---|---|---|

${\sigma}_{H}$ | Contact stress (MPa) | 1657.28 |

${Z}_{D}$ | Single pair tooth contact factors for the wheel | 1.0 |

${Z}_{H}$ | Zone factor | 2.5 |

${Z}_{E}$ | Elasticity factor | 187.5 |

${Z}_{\epsilon}$ | Contact ratio factor | 1.0 |

${Z}_{\beta}$ | Helix angle factor | 1.0 |

${F}_{t}$ | Tangential force at the working pitch circle (N) | 1250 |

${d}_{1}$ | Diameter (mm) | 40 |

b | Face width (mm) | 5 |

u | Gear ratio | 1.0 |

${K}_{A}$ | Application factor | 1.0 |

${K}_{\gamma}$ | Mesh load factor | 1.0 |

${K}_{V}$ | Dynamic factor | 1.0 |

${K}_{H\beta}$ | Face load factor | 1.0 |

${K}_{H\alpha}$ | Transverse load factor | 1.0 |

Pitch Circle Diameter (mm) | Module (mm) | Precision Grade | |||||||
---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||

20 $<$ d $\le $ 50 | 0.5 $\le $ m $\le $ 2 | 0.9 μm | 1.2 μm | 1.8 μm | 2.5 μm | 3.5 μm | 5.0 μm | 7.0 μm | 10.0 μm |

Precision Grades | Torques (N·m) | |||||||
---|---|---|---|---|---|---|---|---|

8 | Comparison with Ideal Gears | 10 | Comparison with Ideal Gears | 12 | Comparison with Ideal Gears | 14 | Comparison with Ideal Gears | |

Ideal | 937.19 | - | 1063.94 | - | 1166.77 | - | 1260.01 | - |

2 | 1270.99 | 35.62% | 1409.09 | 32.44% | 1527.70 | 30.93% | 1627.02 | 29.13% |

4 | 1635.07 | 74.47% | 1775.86 | 66.91% | 1902.75 | 63.08% | 2015.56 | 59.96% |

6 | 2294.29 | 144.81% | 2485.38 | 133.60% | 2627.90 | 125.23% | 2770.99 | 119.91% |

$\mathbf{ME}\text{}(\xb0)$ | Torques (N·m) | |||||||
---|---|---|---|---|---|---|---|---|

8 | Comparison with Ideal Gears | 10 | Comparison with Ideal Gears | 12 | Comparison with Ideal Gears | 14 | Comparison with Ideal Gears | |

Ideal | 937.19 | - | 1063.94 | - | 1166.77 | - | 1260.01 | - |

a = 0.2 | 1266.82 | 35.17% | 1343.96 | 26.32% | 1412.22 | 21.04% | 1499.08 | 18.97% |

a = 0.4 | 1530.68 | 63.33% | 1615.63 | 51.85% | 1692.28 | 45.04% | 1755.10 | 39.29% |

a = 0.6 | 1769.67 | 88.83% | 1853.45 | 74.21% | 1949.32 | 67.07% | 2049.15 | 62.63% |

b = −0.04 | 1081.01 | 15.35% | 1182.92 | 11.18% | 1233.93 | 5.76% | 1314.95 | 4.36% |

b = −0.08 | 1283.69 | 36.97% | 1399.51 | 31.54% | 1489.06 | 27.62% | 1518.91 | 24.54% |

b = −0.12 | 1431.54 | 52.75% | 1520.48 | 42.91% | 1588.72 | 36.16% | 1664.76 | 32.12% |

LCM Quantity (μm) | Torques (N·m) | |||||||
---|---|---|---|---|---|---|---|---|

8 | Comparison with Ideal Gears | 10 | Comparison with Ideal Gears | 12 | Comparison with Ideal Gears | 14 | Comparison with Ideal Gears | |

Ideal | 937.19 | - | 1063.94 | - | 1166.77 | - | 1260.01 | - |

2 | 1285.64 | 37.18% | 1394.60 | 31.08% | 1495.83 | 28.20% | 1587.48 | 25.99% |

4 | 1444.08 | 54.09% | 1559.23 | 46.55% | 1668.85 | 43.03% | 1754.07 | 39.21% |

6 | 1541.64 | 64.50% | 1660.51 | 56.07% | 1768.54 | 51.58% | 1851.27 | 46.93% |

Control Factors | TPD | $<\mathit{a}$ | $<\mathit{b}$ | LCM |
---|---|---|---|---|

Level 1 | Grade 2 | 0.2° | −0.04° | 3.5 μm |

Level 2 | Grade 4 | 0.4° | −0.08° | 7 μm |

Level 3 | Grade 6 | 0.6° | −0.12° | 10.5 μm |

Serial Number | Tooth Profile Precision Grade | $<\mathit{b}$ (°) | $<\mathit{a}$ (°) | Lead Crowning (μm) |
---|---|---|---|---|

1 | Grade 2 | −0.04 | 0.2 | 3.5 |

2 | Grade 2 | −0.04 | 0.4 | 7.0 |

3 | Grade 2 | −0.04 | 0.6 | 10.5 |

4 | Grade 4 | −0.08 | 0.2 | 3.5 |

5 | Grade 4 | −0.08 | 0.4 | 7.0 |

6 | Grade 4 | −0.08 | 0.6 | 10.5 |

7 | Grade 6 | −0.12 | 0.2 | 3.5 |

8 | Grade 6 | −0.12 | 0.4 | 7.0 |

9 | Grade 6 | −0.12 | 0.6 | 10.5 |

10 | Grade 4 | −0.12 | 0.2 | 7.0 |

11 | Grade 4 | −0.12 | 0.4 | 10.5 |

12 | Grade 4 | −0.12 | 0.6 | 3.5 |

13 | Grade 6 | −0.04 | 0.2 | 7.0 |

14 | Grade 6 | −0.04 | 0.4 | 10.5 |

15 | Grade 6 | −0.04 | 0.6 | 3.5 |

16 | Grade 2 | −0.08 | 0.2 | 7.0 |

17 | Grade 2 | −0.08 | 0.4 | 10.5 |

18 | Grade 2 | −0.08 | 0.6 | 3.5 |

19 | Grade 6 | −0.08 | 0.2 | 10.5 |

20 | Grade 6 | −0.08 | 0.4 | 3.5 |

21 | Grade 6 | −0.08 | 0.6 | 7.0 |

22 | Grade 2 | −0.12 | 0.2 | 10.5 |

23 | Grade 2 | −0.12 | 0.4 | 3.5 |

24 | Grade 2 | −0.12 | 0.6 | 7.0 |

25 | Grade 4 | −0.04 | 0.2 | 10.5 |

26 | Grade 4 | −0.04 | 0.4 | 3.5 |

27 | Grade 4 | −0.04 | 0.6 | 7.0 |

Serial Number | Maximum TSCS (MPa) | Serial Number | Maximum TSCS (MPa) | Serial Number | Maximum TSCS (MPa) |
---|---|---|---|---|---|

1 | 1680.91 | 10 | 2272.12 | 19 | 3888.53 |

2 | 1854.25 | 11 | 2459.15 | 20 | 3070.89 |

3 | 1931.67 | 12 | 2135.32 | 21 | 3636.82 |

4 | 2118.22 | 13 | 3719.14 | 22 | 1962.06 |

5 | 2322.13 | 14 | 3819.85 | 23 | 1683.50 |

6 | 2420.97 | 15 | 3229.20 | 24 | 1850.24 |

7 | 3102.55 | 16 | 1907.33 | 25 | 2454.60 |

8 | 3753.72 | 17 | 1999.51 | 26 | 2177.90 |

9 | 3861.61 | 18 | 1806.02 | 27 | 2297.35 |

Level | Tooth Profile Accuracy | $<\mathit{b}$$\text{}(\xb0)$ | $<\mathit{a}$$\text{}(\xb0)$ | Lead Crowning (μm) | ||||
---|---|---|---|---|---|---|---|---|

SN Ratios | Means | SN Ratios | Means | SN Ratios | Means | SN Ratios | Means | |

1 | −65.34 | 1853 | −67.85 | 2574 | −67.83 | 2567 | −67.09 | 2334 |

2 | −67.20 | 2295 | −67.89 | 2574 | −67.85 | 2571 | −68.01 | 2624 |

3 | −71.01 | 3565 | −67.81 | 2564 | −67.87 | 2574 | −68.45 | 2755 |

Delta | 5.66 | 1712 | 0.08 | 10 | 0.04 | 7 | 1.36 | 421 |

Rank | 1 | 3 | 4 | 2 |

Table | $<\mathit{b}$$\text{}(\xb0)$ | $<\mathit{a}$$\text{}(\xb0)$ | LCM (μm) | TSCS (MPa) |
---|---|---|---|---|

2 | −0.04 | 0.2 | 3.5 | 1614.97 |

0.4 | 1618.91 | |||

0.6 | 1622.06 | |||

−0.08 | 0.2 | 1615.59 | ||

0.4 | 1619.53 | |||

0.6 | 1622.67 | |||

−0.12 | 0.2 | 1605.57 | ||

0.4 | 1609.51 | |||

0.6 | 1612.66 |

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**MDPI and ACS Style**

Li, Q.; Xie, L.
Analysis and Optimization of Tooth Surface Contact Stress of Gears with Tooth Profile Deviations, Meshing Errors and Lead Crowning Modifications Based on Finite Element Method and Taguchi Method. *Metals* **2020**, *10*, 1370.
https://doi.org/10.3390/met10101370

**AMA Style**

Li Q, Xie L.
Analysis and Optimization of Tooth Surface Contact Stress of Gears with Tooth Profile Deviations, Meshing Errors and Lead Crowning Modifications Based on Finite Element Method and Taguchi Method. *Metals*. 2020; 10(10):1370.
https://doi.org/10.3390/met10101370

**Chicago/Turabian Style**

Li, Qiang, and Liyang Xie.
2020. "Analysis and Optimization of Tooth Surface Contact Stress of Gears with Tooth Profile Deviations, Meshing Errors and Lead Crowning Modifications Based on Finite Element Method and Taguchi Method" *Metals* 10, no. 10: 1370.
https://doi.org/10.3390/met10101370