# Electromechanical Coupling Dynamic Characteristics of the Dual-Motor Electric Drive System of Hybrid Electric Vehicles

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## Abstract

**:**

## 1. Introduction

## 2. Electromechanical Coupling Dynamics Model of the DEDS of an HEV

#### 2.1. Model of a PMSM

_{d}and u

_{q}denote the stator voltage of the d–q axis, respectively; i

_{d}and i

_{q}denote the stator current of the d–q axis, respectively; R denotes the stator resistance; ${\omega}_{e}$ denotes the electric angular velocity; L

_{d}and L

_{q}denote the inductance components of the d–q axis, respectively; ${\phi}_{f}$ denotes the flux linkage of permanent magnets; and P

_{n}denotes the number of pole pairs in the motor. The parameters of the PMSM are shown in Table 1.

#### 2.2. Nonlinear Model of the Inverter

_{d}to protect the transistor before the drive signal. The same two bridge arm transistors are in the off-state time, called dead time:

_{d}denotes the dead time; t

_{d}denotes the delay time; t

_{on}denotes the time required for the transistor to turn on; and t

_{off}denotes the time required for the transistor to turn off.

_{t}denotes the on-state voltage drop of the switching tube; v

_{d}denotes the off-state voltage drop of the current-continuing diode.

#### 2.3. Implementation of the SVPWM Algorithm

_{c}denotes carrier frequency and w

_{r}denotes modulated wave frequency.

_{c}, and its side band has a current frequency. The format is presented as |a f

_{e1}± f

_{c}| (a = 1, 2, …).

#### 2.4. Vector Control Model for PMSM

_{s}* denotes the target current; i

_{d}* and i

_{q}* denote the current instructions of d and q axis, respectively; U

_{d}* and U

_{q}* denote the voltage instructions of d and q axis, respectively; U

_{α}* and U

_{β}* denote the voltage instructions of SVPWM.

#### 2.5. Model of Gear System Dynamic of the DEDS

#### 2.5.1. Model of Gear Pair Torsional Vibration

_{1}and θ

_{2}denote the rotation angles of the main and driven gears, respectively; R

_{1}and R

_{2}denote the gear index circle radii; and e denotes a meshing error.

#### 2.5.2. Dynamic Model of Transmission System

_{M}

_{1}, J

_{M}

_{2}, and J

_{LN}denote the rotational inertia of motor 1, motor 2, and load, respectively; J

_{i}(i = 1, 2, …, 7) denotes the rotational inertia of the gear; gears 1 and 2 constitute the first gear pair; gears 3 and 4 constitute the second gear pair; gears 5 and 6 constitute the third gear pair; gears 4 and 7 constitute the fourth gear pair; θ

_{m}

_{1}, θ

_{m}

_{2}, and θ

_{L}denote the rotational inertia of motor 1, motor 2, and load, respectively; θ

_{i}(i = 1, 2, …, 7) denotes the angle of rotation of the i-th gear; k

_{j}and c

_{j}(j = 1, 2, …, 5) denote the stiffness and damping of the j-th shaft, respectively; k

_{mj}and c

_{mj}(j = 1, 2, 3, 4) denote the meshing stiffness and meshing damping of the j-th gear pair, respectively.

_{i}(i = 1, 2, …, 7) denotes the radius of the i-th gear base circle and F

_{a}(a = 1, 2, 3, 4) denotes the meshing force of the a-th gear pair.

#### 2.6. Electromechanical Coupling Model of the DEDS

_{LN}denotes the load torque of the system; and T

_{M}denotes the electromagnetic torque of motor.

#### 2.7. Analysis of the Inherent Torsional Vibration Characteristics of the DEDS

## 3. Analysis of Electromechanical Coupling Dynamics of the DEDS under Steady-State Conditions

_{et}(t = 1, 2) denotes the current frequencies of motor 1 and motor 2, respectively; f

_{mg}(g = 1, 2, 3, 4) denotes the gear meshing frequency of each stage.

#### 3.1. Single-Motor Drive Mode

_{e}

_{1}, and the harmonic frequencies 5f

_{e}

_{1}and 7f

_{e}

_{1}caused by the inverter’s nonlinear characteristics of in the single-motor drive mode, the harmonics caused by the nonlinearity of the inverter are marked in red in the figure. Because of the role of low-pass filtering of the motor circuit and the large inertia of the gear system, resulting in a small amplitude of the higher harmonics, the higher harmonics are not listed in this paper. In addition, the current spectrum contains mechanical system vibration frequencies, presented as |a f

_{e}

_{1}± b f

_{mg}|(a,b = 1, 2, …), which is due to the direct coupling of the electrical and mechanical parts through the motor shaft. The current’s fundamental frequency is modulated by the meshing frequency of gears, so that the current exhibits complex frequency characteristics, indicating that the operation of the gear system can be monitored by the current.

#### 3.2. Dual-Motor Drive Mode

_{m}

_{1}in the torque PSD of motor 2. The results show that there is a strong coupling effect between the two motors in the dual-motor drive mode.

## 4. Analysis of Electromechanical Coupling Dynamics of the DEDS under Impact Conditions

#### 4.1. Single-Motor Drive Mode

_{e}

_{1}± f

_{N}

_{1}| appear on both sides of the power supply frequency with the system’s first-order natural frequency as the interval, indicating that the motor stator current has a certain ability to monitor the transient torsional vibration of the DEDS.

#### 4.2. Dual-Motor Drive Mode

_{e}denotes the ratio factor of the speed of the two motors.

## 5. Conclusions

_{e1}± b f

_{mg}| (a,b = 1, 2, …); the stator current can be used as the monitoring signal of the steady-state healthy operation of the gear transmission system. The fluctuations of the electromagnetic torque and the dynamic meshing force of the gear pair are primarily excited by the meshing frequency of the gear pair at each level, and the harmonic frequency generated by the nonlinearity of the inverter; the electromagnetic torque spectrum of one motor in the dual-motor drive mode contains the harmonic components of the other motor, and the dynamic meshing force of the gear pair contains harmonic components of both motors in the frequency domain. There is an obvious electromechanical coupling effect between the electrical system and gear system of the DEDS. There is also a significant coupling effect between the two motors in the dual-motor drive mode.

_{e1}± f

_{N1}| in the stator current’s signal. This indicates that the stator current has a feedback effect on the torsional vibration of the system caused by the change in the external load; appropriate damping of the motor shaft and reducing its stiffness will reduce the torsional vibration of the gear system caused by impact load. The different torsional amplitude values of the gearing system can be reflected in the motor stator current’s frequency signal. Moreover, compared with the single-motor drive mode, the speed synchronization error of the dual-motor drive mode will aggravate the torsional vibration amplitude of the gear system under impact conditions. The impact energy caused by an external impact load on the gear system can be suppressed by reducing the speed synchronization error with appropriate control measures.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

HEV | Hybrid electric vehicle | ||

DEDS | Dual-motor electric drive system | ||

PMSM | Permanent magnet synchronous motor | ||

DC | Direct current | ||

IGBT | Insulated gate bipolar transistor | ||

SVPWM | Space vector pulse width modulation | ||

PWM | Pulse width modulation | ||

PSD | Power spectral density | ||

PI | Proportional integral | ||

Formula Symbols | |||

${u}_{d}$ | Stator voltage of the d axis | ${u}_{q}$ | Stator voltage of the q axis |

${i}_{d}$ | Stator current of the d axis | ${i}_{q}$ | Stator current of the q axis |

${\phi}_{d}$ | Flux linkage of the d axis | ${\phi}_{q}$ | Flux linkage of the q axis |

${\omega}_{e}$ | Electric angular velocity | $R$ | Stator resistance |

$P$ | Power of PMSM | $n$ | Rated speed of PMSM |

${T}_{e}$ | Rated Torque of PMSM | ${\phi}_{f}$ | Flux linkage of permanent magnets |

${L}_{d}$ | Inductance component of the d axis | ${L}_{q}$ | Inductance component of the q axis |

${u}_{dc}$ | Battery direct current voltage | ${P}_{n}$ | Number of pole pairs |

${T}_{M1}$ | Electromagnetic torque of PMSM 1 | ${T}_{M2}$ | Electromagnetic torque of PMSM 2 |

${T}_{d}$ | Dead time of the inverter | ${t}_{d}$ | Delay time of the inverter |

${t}_{on}$ | IGBT turn-on time | ${t}_{off}$ | IGBT turn-off time |

$\Delta u$ | Average error voltage of A-phase bridge | ${v}_{d}$ | Conduction voltage drops of freewheeling diode |

${T}_{\mathrm{PWM}}$ | Pulse width modulation cycle | ${v}_{t}$ | Voltage drops of IGBT switch |

${w}_{r}$ | Modulated wave frequency | ${f}_{c}$ | Modulation carrier frequency |

${\omega}_{e}{}^{*}$ | Target speed of the motor | ${i}_{s}{}^{*}$ | Target current |

${i}_{d}{}^{*}$ | Current instruction of the d axis | ${i}_{q}{}^{*}$ | Current instruction of the q axis |

${U}_{d}{}^{*}$ | Voltage instruction of the d axis | ${U}_{q}{}^{*}$ | Voltage instruction of the d axis |

${U}_{\alpha}{}^{*}$ | Voltage instruction of the SVPWM | ${U}_{\beta}{}^{*}$ | Voltage instruction of the SVPWM |

${w}_{c}$ | Carrier frequency | $\delta $ | Meshing displacement of the gear |

${k}_{v}$ | Gear pair meshing stiffness | ${c}_{v}$ | Gear pair meshing damping |

$e$ | Gear meshing error | b | Gear pair clearance |

${J}_{M1}$ | Rotational inertia of motor 1 | ${k}_{1}$ | Stiffness of shaft 1 |

${J}_{M2}$ | Rotational inertia of motor 2 | ${k}_{2}$ | Stiffness of shaft 2 |

${J}_{1}$ | Rotational inertia of gear 1 | ${k}_{3}$ | Stiffness of shaft 3 |

${J}_{2}$ | Rotational inertia of gear 2 | ${k}_{4}$ | Stiffness of shaft 4 |

${J}_{3}$ | Rotational inertia of gear 3 | ${k}_{5}$ | Stiffness of shaft 5 |

${J}_{4}$ | Rotational inertia of gear 4 | ${F}_{1}$ | Meshing force of gear pair 1 |

${J}_{5}$ | Rotational inertia of gear 5 | ${F}_{2}$ | Meshing force of gear pair 2 |

${J}_{6}$ | Rotational inertia of gear 6 | ${F}_{3}$ | Meshing force of gear pair 3 |

${J}_{7}$ | Rotational inertia of gear 7 | ${F}_{4}$ | Meshing force of gear pair 4 |

${J}_{LN}$ | Rotational inertia of load | ${\theta}_{1}$ | Rotation angle of gear 1 |

${c}_{1}$ | Damping of shaft 1 | ${\theta}_{2}$ | Rotation angle of gear 2 |

${c}_{2}$ | Damping of shaft 2 | ${\theta}_{3}$ | Rotation angle of gear 3 |

${c}_{3}$ | Damping of shaft 3 | ${\theta}_{4}$ | Rotation angle of gear 4 |

${c}_{4}$ | Damping of shaft 4 | ${\theta}_{5}$ | Rotation angle of gear 5 |

${c}_{5}$ | Damping of shaft 5 | ${\theta}_{6}$ | Rotation angle of gear 6 |

${R}_{1}$ | Radius of gear 1 | ${\theta}_{7}$ | Rotation angle of gear 7 |

${R}_{2}$ | Radius of gear 2 | ${\theta}_{M1}$ | Rotation angle of motor 1 |

${R}_{3}$ | Radius of gear 3 | ${\theta}_{M2}$ | Rotation angle of motor 2 |

${R}_{4}$ | Radius of gear 4 | ${T}_{LN}$ | Load torque of system |

${R}_{5}$ | Radius of gear 5 | ${T}_{LN1}$ | Load torque of motor 1 |

${R}_{6}$ | Radius of gear 6 | ${T}_{LN1}$ | Load torque of motor 1 |

${R}_{7}$ | Radius of gear 7 | $\theta $ | Angular displacement matrix of system |

${r}_{1}$ | Motor 1 to load transmission ratio | ${r}_{2}$ | Motor 2 to load transmission ratio |

$J$ | Inertia matrix of system | $K$ | Stiffness matrix of system |

$C$ | Damping matrix of system | ${c}_{m}$ | Meshing damping of gears |

${f}_{Ni}$ | Natural frequency of motor 1 drive mode of system | ${f}_{Ni}\prime $ | Natural frequency of motor 2 drive mode of system |

${f}_{e1}$ | Current frequencies of motor 1 | ${f}_{e2}$ | Current frequencies of motor 2 |

${f}_{m1}$ | Meshing frequency of gear pair 1 | ${f}_{m2}$ | Meshing frequency of gear pair 2 |

${f}_{m3}$ | Meshing frequency of gear pair 3 | ${f}_{m4}$ | Meshing frequency of gear pair 4 |

${\omega}_{1}$ | Speed of motors 1 | ${\omega}_{2}$ | Speed of motors 2 |

${\omega}_{mean}$ | Average speed of two motors | ${K}_{e}$ | Ratio factor of the speed of the two motors |

$E$ | Speed synchronization error | $S$ | Impact energy |

$F\left(t\right)$ | Actual value of dynamic meshing force | ${F}^{\prime}\left(t\right)$ | Dynamic meshing force steady-state target value |

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**Figure 2.**Inverter A-phase bridge arm circuit and output voltage waveform: (

**a**) inverter A-phase bridge arm structure diagram; (

**b**) waveform of A-phase bridge arm signal when i

_{a}$>$ 0.

**Figure 4.**SVPWM algorithm implementation diagram: (

**a**) voltage vector space diagram; (

**b**) schematic diagram of PWM pulse signal generation.

**Figure 7.**Torsional vibration model of helical gear pair: (

**a**) stress diagram; (

**b**) transmission diagram.

**Figure 10.**Electrical system vibration response: (

**a**) current time domain; (

**b**) current frequency domain; (

**c**) electromagnetic torque time domain; (

**d**) electromagnetic torque PSD.

**Figure 11.**The meshing force of the first gear pair in the single-motor drive mode: (

**a**) time domain; (

**b**) PSD.

**Figure 14.**The meshing force of the first gear pair in the dual-motor drive mode: (

**a**) time domain; (

**b**) PSD.

**Figure 17.**The meshing force of the first gear pair: (

**a**) time domain; (

**b**) time-frequency domain (0-80 Hz).

**Figure 20.**Time-frequency domain of the stator current of motor 1 with different values of motor shaft damping: (

**a**) case 5; (

**b**) case 3; (

**b**) case 4.

**Figure 22.**Time-frequency domain of the stator current of motor 1 with different values of motor shaft stiffness: (

**a**) case 3; (

**b**) case 2; (

**c**) case 1.

**Figure 25.**Motor 1 speed and synchronization error under different combinations of PI parameters: (

**a**) motor 1 speed; (

**b**) synchronization error.

**Figure 26.**The meshing force of the first gear pair: (

**a**) meshing force of the first gear pair at time-varying meshing stiffness; (

**b**) root mean square value of the meshing force of the first gear pair; (

**c**) meshing force decay process.

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

Motor 1 | Motor 2 | |

Power P (kW) Number of pole pairs P _{n}Rated speed n (rpm) Rated torque T _{e} (Nm)Stator resistance R ($\mathsf{\Omega}$) D axis inductance L _{d} (mH)Q axis inductance L _{q} (mH) | 90 5 4000 214 0.012 0.196 0.149 | 80 5 4800 153 0.012 0.101 0.296 |

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

Battery DC voltage u_{dc} (V)Modulation carrier period T _{PWM} (us)Dead time t _{d} (us)IGBT turn-on time t _{on} (us)IGBT turn-on time t _{off} (us)The conduction voltage drops of freewheeling diode v _{d} (V) | 450 100 4 1 2 2 |

Voltage drops of IGBT switch v_{t} (V)Modulation carrier frequency f _{c} (kHz) | 3 10 |

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

Stiffness (Nm/rad) | k_{1} = k_{5} = 1.2 × 10^{6}; k_{2} = k_{3} = 1 × 10^{7}; k_{4} = 8 × 10^{3} |

Damping (Nm $\xb7$ s/rad) | c_{1}= c_{5} = 4; c_{2}= c_{3}= c_{4} = 1.7 |

Inertia (kg $\xb7$ m^{2}) | J_{M}_{1} = 6.15 × 10^{−3}; J_{M}_{2} = 5.36 × 10^{−3}; J_{LN} = 0.36; J_{1} = 1.51 × 10^{−4};J _{2} = 1.32 × 10^{−3}; J_{3} = 1.67 × 10^{−4}; J_{4} = 7.2 × 10^{−3}; J_{5} = 1.92 × 10^{−4};J _{6} = 1.2 × 10^{−2}; J_{7} = 1.69 × 10^{−4} |

Meshing damping c_{m} | 100 |

Transmission ratio r_{1} | 10.5 |

Transmission ratio r_{2} | 8.75 |

Order | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

${f}_{Ni}$(Hz) | 0 | 22.1 | 56.4 | 159.2 | 255.3 | 734.8 |

${f}_{Ni}\prime $(Hz) | 0 | 23.3 | 58.6 | 159.2 | 255.3 | 734.8 |

Working Mode | System Load Torque (Nm) | Motor Speed (rpm) | |
---|---|---|---|

Motor 1 | Motor 2 | ||

Single-motor drive mode | 1155 | 3000 | / |

Dual-motor drive mode | 1155 | 3000 | 2500 |

Working Mode | System Load Torque (Nm) | Motor Speed (rpm) | ||
---|---|---|---|---|

Before Impact | After Impact | Motor 1 | Motor 2 | |

Single-motor drive mode | 1155 | 2310 | 3000 | / |

Dual-motor drive mode | 1155 | 2310 | 3000 | 2500 |

Case Name | Damping (Nm∙s/rad) | Stiffness (Nm/rad) |
---|---|---|

Case 1 Case 2 Case 3 Case 4 | 4 4 4 40 | 10^{3}10 ^{4}10 ^{5}10 ^{5} |

Case 5 | 400 | 10^{5} |

**Table 8.**Calculation results of the impact energy of the transmission system with different PI parameters.

PI Parameter Combinations | $\mathbf{Impact}\text{}\mathbf{Energy}\text{}(\times $$10{2}^{\text{}}\mathbf{N}\mathbf{\xb7}\mathbf{s})$ |
---|---|

Case 1 Case 2 Case 3 | 8.4432 6.0279 4.8503 |

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## Share and Cite

**MDPI and ACS Style**

Ge, S.; Hou, S.; Yao, M. Electromechanical Coupling Dynamic Characteristics of the Dual-Motor Electric Drive System of Hybrid Electric Vehicles. *Energies* **2023**, *16*, 3190.
https://doi.org/10.3390/en16073190

**AMA Style**

Ge S, Hou S, Yao M. Electromechanical Coupling Dynamic Characteristics of the Dual-Motor Electric Drive System of Hybrid Electric Vehicles. *Energies*. 2023; 16(7):3190.
https://doi.org/10.3390/en16073190

**Chicago/Turabian Style**

Ge, Shuaishuai, Shuang Hou, and Mingyao Yao. 2023. "Electromechanical Coupling Dynamic Characteristics of the Dual-Motor Electric Drive System of Hybrid Electric Vehicles" *Energies* 16, no. 7: 3190.
https://doi.org/10.3390/en16073190