# Stability Analysis of Electromechanical Coupling Torsional Vibration for Wheel-Side Direct-Driven Transmission System under Transmission Clearance and Motor Excitation

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. System Electromechanical Coupling Dynamics Modelling

_{a}, i

_{b}, and i

_{c}represent the three-phase stator currents of the PMSM, respectively; u

_{a}, u

_{b}, and u

_{c}represent the three-phase stator input voltages of the PMSM, respectively; θ

_{1}and θ

_{2}are the rotation angle of the PMSM and the angle of the wheel hub; J

_{1}and J

_{2}show the equivalent moment of inertia on the PMSM shaft and the moment of inertia of the wheel hub; K and C represent the torsional stiffness and damping coefficient of the system transmission shaft, respectively; e

_{c}indicates the system transmission clearance; T

_{m}is the electromagnetic torque of the PMSM; and T

_{l}is the load torque of the wheel hub. Furthermore, the meanings of the symbols used in this paper are listed in Table A1 in Appendix A. The following assumptions are made: (1) the self-inductance and the mutual-inductance of each phase winding of the PMSM are sinusoidal, by ignoring the magnetic saturation and magnetic flux leakage; (2) the influences of the temperature and the frequency change on the PMSM parameters are ignored; and (3) the rotor of the PMSM has no eccentricity and the air gap is uniform.

_{1}, e

_{2}, and e

_{3}), which are the three-phase stator electric charge of the PMSM, and the mechanical subsystem includes two parts (θ

_{1}and θ

_{2}), which are the output angles of the PMSM and the wheel hub. The generalized coordinates of the WDTS are shown in Table 1.

_{m}

_{1}) and the magnetic energy generated by the interaction of the flux linkage, caused by the permanent magnet rotor, and the stator current (W

_{m}

_{2}), which can be shown as:

_{a}, L

_{b}, and L

_{c}are the self-inductance coefficients of the three-phase stator winding of the PMSM; H represents the mutual-inductance coefficient of the three-phase stator winding of the PMSM; and ${\psi}_{f}$ is the flux linkage of the permanent magnet rotor for the PMSM.

_{a}, R

_{b}, and R

_{c}are the three-phase stator winding resistances of the PMSM, respectively.

_{4}). The generalized force corresponding to the angle of the wheel hub is the load torque (Q

_{5}), and the generalized forces of the mechanical part can be represented as:

_{g}is the Lagrange operator. For the WDTS, L

_{g}can be shown as:

_{1}= e

_{1}, the voltage equation can be expressed as:

_{D}and I

_{D}are the stator direct-axis voltage and current of the PMSM, respectively. u

_{Q}and I

_{Q}are the stator quadrature-axis voltage and current of the PMSM, respectively. w

_{e}is the electric angular speed of the PMSM (${\omega}_{e}=\mathfrak{M}{\omega}_{1}$) and $\mathfrak{M}$ is the pole-pair of the PMSM. L

_{D}and L

_{Q}are the D–Q axis inductance component. The PMSM used in our paper is the surface-mounted PMSM and L

_{D}= L

_{Q}.

_{c}) and less than the negative critical value (−e

_{c}), the system dynamics behaviors are similar. Therefore, the dynamic characteristics of the WDTS are considered when the angle difference of the transmission shaft is greater than the positive critical value (e

_{c}). Furthermore, by defining ${\omega}_{1}={\dot{\theta}}_{1}$, ${\omega}_{2}={\dot{\theta}}_{2}$, $\vartheta ={\theta}_{1}-{\theta}_{2}$, and L

_{D}= L

_{Q}= L, the dynamic model with electromagnetic parameters decoupled of the WDTS of the electric bus can be expressed as:

## 3. System Model Analysis and Verification

_{f}= 0.125 Wb, J

_{1}= 25.28 Kg·m

^{2}, J

_{2}= 1200 Kg·m

^{2}, C = 100 N·m·s/rad, K = 10

^{6}N·m/rad, and $\mathfrak{M}=4$.

_{z}is the stator voltage frequency of the PMSM. The three-phase stator voltage frequencies of the PMSM are given as 10 Hz, 15 Hz, and 20 Hz. Substituting the relevant physical parameters, the stable output speeds of the PMSM should be 150 r/min, 225 r/min, and 300 r/min, according to the calculation formula of the synchronous motor speed. On the virtual simulation platform of the WDTS, when the three-phase stator voltage frequencies of the PMSM are 10 Hz, 15 Hz, and 20 Hz, the output speed curves of the PMSM are obtained as Figure 3.

## 4. System Model Solving and Resonance Analysis

_{m}is represented as a cosine function for equivalently describing the non-stationary transition process [30]. Then the influences of the temperature and the frequency change on the PMSM dynamic characteristics are ignored [31,32,33]. And the output torque of the PMSM in the start-up stage is taken as:

_{0}is the zero-order approximate solution of Equation (20), x

_{1}is the first-order approximate solution of Equation (20).

_{k}, respectively.

_{1}p = 0 and ${D}_{1}\gamma =0$. By eliminating $\gamma $ from Equation (30), it is obtained that:

_{c}= 0, the response curve is basically symmetrical and the resonance frequency is in a very narrow band, corresponding to the linear forced vibration. However, when e

_{c}≠ 0, the response curve is bent, which corresponds to the nonlinear forced vibration. Therefore, when the transmission clearance is considered, the torsional vibration of the WDTS presents complex nonlinear characteristics.

_{c}= 0.005 rad as an example, when the initial disturbance frequency of the PMSM increases gradually, the path of the system amplitude frequency response curve is: A→B→C. When it reaches the critical point C, with the further increase of the initial disturbance frequency of the PMSM, the system torsional vibration amplitude response jumps from point C to point D. When the initial disturbance frequency of the PMSM decreases gradually, the path of the system amplitude frequency response curve is: D→E. When it reaches the critical saddle point E, with the further decrease of the initial disturbance frequency of the PMSM, the system torsional vibration amplitude response skips from point E to point B. This jumping phenomenon of the response amplitudes will cause the fatigue damage to the WDTS of the electric bus, which must be restrained in the practical application.

## 5. Conclusions

- (i)
- Under the steady-state operation, with the drive voltage signals of different frequencies applied, the numerical results of the PMSM speed of the constructed WDTS dynamic model are consistent with the theoretical calculation results of the synchronous motor speed. Then, the effectiveness of the constructed model is verified;
- (ii)
- For the non-stationary transition process in the start-up stage of the PMSM, a cosine function disturbance excitation is used to describe it. From the amplitude frequency characteristic curve of the system torsional vibration response, it is seen that with the increase of the PMSM torque disturbance amplitude, the main resonance response amplitude of the WDTS will increase, but it will not affect the resonance region of the system torsional vibration, and, in actual application, the PMSM torque disturbance amplitude should be suppressed; and
- (iii)
- With the increase of the transmission clearance, the system torsional vibration response shows complex nonlinear characteristics, which is manifested by the jump and bifurcation of the system torsional vibration response amplitudes. This phenomenon will cause fatigue damage to the WDTS, which should be effectively suppressed in the actual application. According to the analysis results of this paper, it can provide an important reference for the efficient utilization of the WDTS in the electric bus.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Symbols | Meanings |
---|---|

J_{1} | equivalent moment of inertia on the PMSM shaft |

J_{2} | moment of inertia of the wheel hub |

θ_{1} | rotation angle of the PMSM |

θ_{2} | angle of the wheel hub |

u_{a} | A-phase stator input voltage of the PMSM |

u_{b} | B-phase stator input voltage of the PMSM |

u_{c} | C-phase stator input voltage of the PMSM |

i_{a} | A-phase stator current of the PMSM |

i_{b} | B-phase stator current of the PMSM |

i_{c} | C-phase stator current of the PMSM |

R_{a} | A-phase stator winding resistance of the PMSM |

R_{b} | B-phase stator winding resistance of the PMSM |

R_{c} | C-phase stator winding resistance of the PMSM |

L_{a} | self-inductance coefficient of the a-phase stator winding |

L_{b} | self-inductance coefficient of the b-phase stator winding |

L_{c} | self-inductance coefficient of the c-phase stator winding |

${\psi}_{f}$ | magnetic potential of the permanent magnet rotor of the PMSM |

K | torsional stiffness of the system transmission shaft |

C | damping coefficient of the system transmission shaft |

e_{c} | system transmission clearance |

T_{m} | electromagnetic torque of the PMSM |

T_{l} | load torque of the wheel hub |

w_{e} | electric angular speed of the PMSM |

u_{D} | stator direct-axis voltage of the PMSM |

I_{D} | stator direct-axis current of the PMSM |

u_{Q} | stator quadrature-axis voltage of the PMSM |

I_{Q} | stator quadrature-axis current of the PMSM |

f_{z} | stator voltage frequency of the PMSM |

$\mathfrak{M}$ | pole-pair of the PMSM |

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**Figure 1.**Physical model of the WDTS: (

**a**) wheel-side direct-driven unit; and (

**b**) equivalent transmission model of the WDTS.

**Figure 2.**Virtual simulation platform of the WDTS: 1. Voltage amplitude; 2. Voltage transmitter module; 3. Coordinate positive transformation module (C

_{3}

_{s}

_{/2}

_{s}+ C

_{2s/2r}); 4. PMSM module; 5. Inverse coordinate transformation module; 6. Three-phase stator current display module; 7. Torque conversion module; 8. PMSM speed display module; 9. wheel hub speed display module; 10. Motion equation module; 11. Voltage frequency; 12. Voltage initial phase.

**Figure 3.**Output speeds waveform diagram of the PMSM under different voltage frequencies in the virtual simulation platform of the WDTS.

**Figure 4.**Three-phase stator current and quadrature-axis current curves of the PMSM in the start-up phase (f

_{z}= 20 Hz): (

**a**) three-phase stator current; and (

**b**) quadrature-axis current.

**Figure 5.**Influence of PMSM torque disturbance amplitude on the amplitude frequency response curve of the main resonance for the torsional vibration of the WDTS.

**Figure 6.**Influence of the transmission clearance on the amplitude frequency response curve of the main resonance for the torsional vibration of the WDTS: (

**a**) influence law of different transmission clearance; and (

**b**) amplitude jump phenomenon.

Generalized Coordinates | Electromagnetic Subsystem | Mechanical Subsystem | |||
---|---|---|---|---|---|

Stator of the PMSM | Output Angles of the PMSM | Output Angles of the Wheel Hub | |||

j = 1 | j = 2 | j = 3 | j = 4 | j = 5 | |

q_{j} | e_{1} | e_{2} | e_{3} | ${\theta}_{1}$ | ${\theta}_{2}$ |

${\dot{q}}_{j}$ | i_{a} | i_{b} | i_{c} | ${\dot{\theta}}_{1}$ | ${\dot{\theta}}_{2}$ |

${Q}_{j}$ | u_{a} | u_{b} | u_{c} | T_{m} | T_{l} |

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## Share and Cite

**MDPI and ACS Style**

Ju, J.; Liu, Y.; Zhang, C.
Stability Analysis of Electromechanical Coupling Torsional Vibration for Wheel-Side Direct-Driven Transmission System under Transmission Clearance and Motor Excitation. *World Electr. Veh. J.* **2022**, *13*, 46.
https://doi.org/10.3390/wevj13030046

**AMA Style**

Ju J, Liu Y, Zhang C.
Stability Analysis of Electromechanical Coupling Torsional Vibration for Wheel-Side Direct-Driven Transmission System under Transmission Clearance and Motor Excitation. *World Electric Vehicle Journal*. 2022; 13(3):46.
https://doi.org/10.3390/wevj13030046

**Chicago/Turabian Style**

Ju, Jinyong, Yufei Liu, and Chunrui Zhang.
2022. "Stability Analysis of Electromechanical Coupling Torsional Vibration for Wheel-Side Direct-Driven Transmission System under Transmission Clearance and Motor Excitation" *World Electric Vehicle Journal* 13, no. 3: 46.
https://doi.org/10.3390/wevj13030046