# Optimal Vibration Suppression Modification Method for High-Speed Helical Gear Transmission of Battery Electric Vehicles under Full Working Conditions

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## Abstract

**:**

## 1. Introduction

- (1)
- obtain the optimal tooth profile under full working conditions by combining TCA, LTCA, and genetic optimization algorithms;
- (2)
- study the effect of different modification methods under full working conditions;
- (3)
- obtain the optimal vibration suppression modification strategy for BEV high-speed gear transmission under full working conditions, and further extend the application to other gear transmissions.

## 2. Mathematical Model

#### 2.1. Dynamic Model of the Gear Transmission

#### 2.2. Excitation

_{t}is as follows:

_{k}(P) represents the loaded deformation at the kth meshing position under the nominal load P, and its value can be obtained by the gear LTCA procedure.

#### 2.3. Tooth Surface Modification

_{a}is the coordinate system established on the normal tooth profile of the tool, moving together with the tool; S

_{b}is the coordinate system established on the normal tooth surface of the tool, moving together with the tool; S

_{c}is the coordinate system established at the midpoint of the tooth width and pitch line of the transverse tooth surface of the rack tool; S

_{d}is the follower coordinate system established on the pinion; S

_{e}is a fixed coordinate system built on the pinion; d

_{0}is one-half of the normal tooth width; a is the parabolic coefficient of profile modification when the vertex is at O

_{a}; α

_{n}is the normal pressure angle; u is the distance from the point on the tool’s normal tooth profile to the parabolic vertex. The coordinates of the normal tooth profile of the rack tool in the coordinate system S

_{a}are expressed as follows:

_{b}and the coordinate system S

_{c}along the direction of the ${\mathrm{O}}_{\mathrm{b}}{\mathrm{z}}_{\mathrm{b}}$ axis; and ${\mathrm{r}}_{1}$ is the radius of reference circle of pinion (${\mathrm{d}}_{2}={\mathsf{\phi}}_{2}{\mathrm{r}}_{1}$).

_{f}is an auxiliary coordinate system established on the pinion tooth profile; S

_{g}is the coordinate system established on the pinion tooth profile along any helix angle direction. In the coordinate system S

_{g},

_{f},

#### 2.4. The Global Optimization Model

## 3. Experimental Verification and Analysis

#### 3.1. Experimental Verification of LTE Calculation Accuracy after Tooth Surface Modification

^{−2}and 1.840 × 10

^{−3}, respectively. The experimental results are shown in Figure 11 and Figure 12. The calculation result of the simulation method in this study is shown in Figure 13.

#### 3.2. Optimal Tooth Surface Modification Scheme

## 4. Conclusions

- (1)
- The bending-torsion-axis-swing coupling dynamics model of the helical gear system considering the stiffness excitation was established, the TVMS of the gear pair was calculated based on LTCA, and the calculation accuracy was verified by experiment.
- (2)
- Tooth profile and axial modifications were realized in TCA. The optimal tooth profile and axial modification parameters for the full working condition were obtained by using a genetic algorithm with the minimum root mean square of vibration acceleration as the optimization objective.
- (3)
- The vibration reduction effects of the optimal modification under a specific load and the full working condition were significantly different; the optimal modified tooth surface under full working condition had a better vibration suppression effect in the whole working load range. Therefore, more attention should be paid to the tooth surface modification under the full working conditions in BEV high-speed gear transmissions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Bend-torque-shaft-swing coupling model for high-speed stage helical gear transmission system.

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

a | Parabolic coefficient of profile modification |

b | Parabolic coefficient of axial modification |

c_{ix}, c_{iy}, c_{iz} | Equivalent support damping of driving and driven gears in Xx, y, and z directions (i = 1,2) |

c_{m} | Meshing damping |

c_{ijx}, c_{ijy} | Damping of the driving and driven gears corresponding to the torsional swing degrees of freedom (j = 1,2) |

${\mathrm{c}}_{1},{\mathrm{c}}_{2},{\mathrm{c}}_{3}$ | Constants |

d_{0} | One-half of the normal tooth width |

d_{1} | Distance between the coordinate system S_{b} and the coordinate system S_{c} along the direction of ${\mathrm{O}}_{\mathrm{b}}{\mathrm{z}}_{\mathrm{b}}$ axis |

F | Dynamic meshing force in the direction of the meshing line |

i_{12} | Transmission ratio of the gear pair |

I_{i} | Rotational inertia of the driving and driven gears (i = 1,2) |

k_{ix}, k_{iy}, k_{iz} | Equivalent support stiffness of driving and driven gears in x, y, and z directions (i = 1,2) |

k_{m} | Time-varying meshing stiffness |

k_{ijx}, k_{ijy} | Stiffness of the driving and driven gears corresponding to the torsional swing degrees of freedom (j = 1,2) |

K_{t} | Meshing stiffness |

m_{e} | Equivalent torsional mass of the gear pair |

M_{de}, M_{ec}, M_{cb}, M_{ba} | Coordinate transformation matrices |

M | Number of meshing points in one meshing period |

N | Number of the root mean square of the vibration acceleration in the working load range |

O_{1}, O_{2} | Geometric centers of driving and driven gears |

P | Force or moment |

q | Relative angular displacement |

R_{i} | Radius of the base circle of the driving and driven gears (i = 1,2) |

S_{a} | Coordinate system established on the normal tooth profile of the tool, moving together with the tool |

S_{b} | Coordinate system established on the normal tooth surface of the tool, moving together with the tool |

S_{c} | Coordinate system established at the midpoint of the tooth width and pitch line of the transverse tooth surface of the rack tool |

S_{d} | Follower coordinate system established on the pinion |

S_{e} | Fixed coordinate system built on the pinion |

S_{f} | Auxiliary coordinate system established on the pinion tooth profile |

S_{g} | Coordinate system established on the pinion tooth profile along any helix angle direction |

T_{1}, T_{2} | Driving and driven gear torque |

u | Distance from the point on the tool’s normal tooth profile to the parabolic vertex |

x_{i}, y_{i} | Bending vibration displacement of driving and driven gears (i = 1,2) |

X | Root mean square of the vibration acceleration at a certain working load under full working conditions |

Y | Vibration displacement in the direction of the meshing line |

z_{i} | Axial vibration displacement of the driving and driven gears (i = 1,2) |

Z | Line displacement or angular displacement deformation |

β | Helical gear helix angle |

${\mathsf{\delta}}_{1}$ | Geometric transmission error |

${\mathsf{\delta}}_{2}$ | Tooth bending deformation |

${\mathsf{\delta}}_{3}$ | Contact deformation |

Δ${\mathsf{\phi}}_{1}$, Δ${\mathsf{\phi}}_{2}$ | Actual angle of the pinion and gear measured by the circular grating |

θ_{x}, θ_{y} | Swings around the x- and y-axes |

θ_{z} | Torsional deformation around the z-axis |

ξ | Damping ratio |

${\mathsf{\phi}}_{1}$ | Angle between the meshing plane and the positive direction of the y-axis |

${\mathsf{\phi}}_{2}$ | Rotation angle of the pinion during gear machining |

Parameters | Pinion | Gear |
---|---|---|

Number of teeth | 22 | 59 |

Spiral direction | RH | LH |

Normal module (mm) | 2 | |

Normal pressure angle (°) | 18.5 | |

Helix angle (°) | 32 | |

Face width (mm) | 33 | 31.5 |

Profile shift coefficient (mm) | 0.4578 | −0.31 |

Elastic modulus (GPa) | 210 |

Modification Scheme | Tooth Profile Modification Coefficient: a | Axial Modification Coefficient: b |
---|---|---|

Scheme 1 (Optimum modification at a load of 100 N·m) | 1.048 × 10^{−2} | 1.840 × 10^{−3} |

Scheme 2 (Optimum modification at a load of 300 N·m) | 8.600 × 10^{−3} | 3.950 × 10^{−3} |

Scheme 3 (Optimum modification at a load of 400 N·m) | 8.540 × 10^{−3} | 4.890 × 10^{−3} |

Scheme 4 (Optimum modification under full working conditions) | 7.450 × 10^{−3} | 2.540 × 10^{−3} |

**Table 4.**Root mean square value of vibration acceleration in the direction of the meshing line under different working loads (m/s

^{2}).

Working Load | Scheme 1 | Scheme 2 | Scheme 3 | Scheme 4 | Unmodified |
---|---|---|---|---|---|

100 N·m | 10.1806 | 40.0622 | 40.3820 | 12.6791 | 12.8866 |

150 N·m | 8.2216 | 18.0759 | 34.7084 | 6.6689 | 15.2156 |

200 N·m | 10.9045 | 7.8333 | 14.0231 | 5.9006 | 13.6982 |

250 N·m | 10.9960 | 5.6143 | 7.9189 | 1.3355 | 10.7765 |

300 N·m | 9.4872 | 2.2351 | 7.1550 | 3.6751 | 10.1485 |

350 N·m | 11.3276 | 3.6645 | 4.7836 | 6.5724 | 10.7539 |

400 N·m | 11.5764 | 6.9905 | 5.6930 | 10.7864 | 12.6968 |

Average | 10.3849 | 12.0680 | 16.3806 | 6.8026 | 12.3109 |

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**MDPI and ACS Style**

Du, J.; Hu, L.; Mao, J.; Zhang, Y.
Optimal Vibration Suppression Modification Method for High-Speed Helical Gear Transmission of Battery Electric Vehicles under Full Working Conditions. *Machines* **2021**, *9*, 226.
https://doi.org/10.3390/machines9100226

**AMA Style**

Du J, Hu L, Mao J, Zhang Y.
Optimal Vibration Suppression Modification Method for High-Speed Helical Gear Transmission of Battery Electric Vehicles under Full Working Conditions. *Machines*. 2021; 9(10):226.
https://doi.org/10.3390/machines9100226

**Chicago/Turabian Style**

Du, Jinfu, Liang Hu, Jin Mao, and Yanchao Zhang.
2021. "Optimal Vibration Suppression Modification Method for High-Speed Helical Gear Transmission of Battery Electric Vehicles under Full Working Conditions" *Machines* 9, no. 10: 226.
https://doi.org/10.3390/machines9100226