# Dynamic Modeling and Vibration Characteristics Analysis of Transmission Process for Dual-Motor Coupling Drive System

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Dynamic Model of DCDS with TMM

#### 2.1. Establishment of the Model for the Whole DCDS

_{p}can be generated. The planet carrier ${Z}_{c}$ which supports the planets is rotated around the center line. Furthermore, the differential bevel gear is promoted by the assisted gear ${Z}_{3}$, which is engaged with the planet carrier. During the driving process, the sun gear ${Z}_{s}$ is promoted by motor 1 at a rated speed to improve the energy efficiency. The gear ring ${Z}_{r}$ is coordinately rotated by assistant motor 2 to adjust the rotational speed. The electric vehicle can be driven by the differential, which is composed of two pairs of bevel gears.

_{jL}and Z

_{jD}are the input drive torque and output load torque of the jth substructure, and j = 1, 2, …, 9. T

_{j}is the transmission matrix between the two adjacent substructures. Then, the whole transmission matrix of the DCDS based on TMM can be expressed as

_{W}is constructed, owing to the difference of the location for each substructure and the coupling relationship between two transmission components. The states of each substructure are transmitted simultaneously with the overall state vector Z.

_{1}, E

_{2}, E

_{3}, and E

_{4}are 5 order, 10 order, 15 order, and 20 order unit matrices, respectively; T

_{I}, T

_{II}, T

_{III}, T

_{IV}, and T

_{V}represent the transfer matrix of Shafts I–V at certain transmission processes; and ${T}_{11}^{{C}_{n}}$, ${T}_{12}^{{C}_{n}}$, ${T}_{21}^{{C}_{n}}$, and ${T}_{22}^{{C}_{n}}$ are the elements of the coupling transfer matrix for the engagement of the individual gear and planetary gears.

#### 2.2. Modeling of Meshing Process for an Individual Pair of Gears

_{1D}and T

_{2D}are the input drive torques of the driving gear and the driven gear, respectively. T

_{1L}and T

_{2L}are the output load torques of the driving gear and the driven gear, respectively. R

_{b1}and R

_{b2}are the base radii. e(t) is the geometric transmission error along with the meshing line. k

_{m}is the meshing stiffness, and c is the damp.

#### 2.3. Modeling of the Transmission Process for the Planetary Gear

_{i}(i = 1, 2, 3), the carrier as c, and the gear ring as r. The rotational angles of the sun gear, planet gear, and carrier are labeled as ${\theta}_{s}$, ${\theta}_{pi}$, and ${\theta}_{c}$. The meshing stiffness between the sun gear and the ith planet gear is k

_{spi}. The base radii of the sun gear, planet gear, and gear ring are R

_{bs}, R

_{bp}, and R

_{br}. Several sets of coordinates can be used to describe the geometrical relationship of the DCDS. One set is given by the absolute coordinate, which is established based on the fixed center $O\left(x,y,z\right)$. The second set is given by the actual coordinate of the sun gear, which is designated as {S

_{s}-X

_{s}Y

_{s}Z

_{s}}. The planet gears are given by the equivalent coordinates {S

_{p}-X

_{p}Y

_{p}Z

_{p}}, which orbit around the sun gear with a variable angle velocity w

_{z}.

_{D}and T

_{L}. Then, the vibration differential equation can be established as

_{s}, E

_{pi}, and E

_{r}. ${\phi}_{s}$ is the angle between E

_{s}and the external meshing line of the ith planet gear. ${\delta}_{s}$ is the angle between E

_{s}and the x axis. ${\gamma}_{pi}$ and ${\phi}_{pi}$ are the angles between E

_{pi}and the internal and external meshing lines, respectively. ${\delta}_{pi}$ is the angle between E

_{pi}and the x

_{pi}axis. ${\phi}_{r}$ is the angle between E

_{r}and the internal meshing line of the ith planet gear. ${\delta}_{r}$ is the angle between E

_{r}and the x axis. From Figure 7 and Figure 8, the following geometric relationship can be obtained:

_{s}and E

_{pi}on the external meshing line can be expressed as

_{pi}and E

_{r}on the internal meshing line can be expressed as

#### 2.4. Modeling of the Transmission Process for the Differential Bevel Gear

_{i}and I

_{si}are the length and rotational inertia of the ith shaft segment, and k

_{s}is the torsional stiffness.

## 3. Dynamic Characteristics Analysis and Test of the Transmission Process for DCDS

#### 3.1. Dynamic Characteristics Analysis

^{11}Pa. The parameters of the planetary gear and differential bevel gear for the computational analysis are shown in Table 1.

_{0}is the initial distance of the contact surface, q is the actual motion displacement, and STEP is the step function. The simulated results are shown in Figure 9, Figure 10 and Figure 11. As the rotational speed increased from 1 to 50 r/s, the torsional, radial, and axial vibrations became more severe. The torsional and radial vibrations were almost the same when considering axial motion or not, which means that the axial motion made no difference to the dynamic characteristics. Moreover, Figure 11 shows that the difference of the axial vibration amplitudes in the z axis was several times lower than the torsional and radial vibrations. Each pair of meshing gears had a symmetrical structure, and the dynamic characteristics had periodic behavior. Figure 10 shows that the radial vibrations were very close in the x and y axes. Hence, the axial vibration of the gears could be neglected in the dynamic analysis of the DCDS, especially at low frequencies.

#### 3.2. Experiment Setup and Test

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 12.**(

**a**) Time domain dynamic response of DCDS driven by main motor 1; (

**b**) Frequency domain dynamic response of DCDS driven by main motor 1.

**Figure 13.**(

**a**) Time domain dynamic response of DCDS driven by main motor 2; (

**b**) Frequency domain dynamic response of DCDS driven by main motor 2.

**Figure 15.**The normalized model of the first four vibration modes with different meshing stiffness: (

**a**) The first mode shape varying from the axial length; (

**b**) the second mode shape varying from the axial length; (

**c**) the third mode shape varying from the axial length; (

**d**) the fourth mode shape varying from the axial length.

**Figure 19.**(

**a**) Experimental result of the dynamic response for the substructure 2; (

**b**) Experimental result of the dynamic response for the substructure 4; (

**c**) Experimental result of the dynamic response for the substructure 6; (

**d**) Experimental result of the dynamic response for the substructure 8.

Planetary Gear | Differential Bevel Gear | ||
---|---|---|---|

Number of teeth | 20 | Number of teeth | 35 |

Modulus | 1.5 | Modulus at main aspect | 1.5 |

Pressure angle | 20 | Pressure angle | 20 |

Length of shaft | 30 mm | Cone apex angle | 45 |

Helical angle | 12.5 |

**Table 2.**Comparison of the first four natural frequencies between the dynamic model and the experimental test.

Order | Natural Frequency of Dynamic Model (rad/s) | Natural Frequency of Experimental Test (rad/s) | Relative Error (%) |
---|---|---|---|

First | 219.604 | 211.008 | 4.07 |

Second | 500.219 | 482.932 | 3.58 |

Third | 1083.379 | 1005.428 | 7.75 |

Fourth | 1389.928 | 1354.596 | 2.61 |

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**MDPI and ACS Style**

Fan, W.; Yang, Y.; Su, X.
Dynamic Modeling and Vibration Characteristics Analysis of Transmission Process for Dual-Motor Coupling Drive System. *Symmetry* **2020**, *12*, 1171.
https://doi.org/10.3390/sym12071171

**AMA Style**

Fan W, Yang Y, Su X.
Dynamic Modeling and Vibration Characteristics Analysis of Transmission Process for Dual-Motor Coupling Drive System. *Symmetry*. 2020; 12(7):1171.
https://doi.org/10.3390/sym12071171

**Chicago/Turabian Style**

Fan, Wei, Yongfei Yang, and Xiangang Su.
2020. "Dynamic Modeling and Vibration Characteristics Analysis of Transmission Process for Dual-Motor Coupling Drive System" *Symmetry* 12, no. 7: 1171.
https://doi.org/10.3390/sym12071171