# An Integrated Vibration Elimination System with Mechanical-Electrical-Magnetic Coupling Effects for In-Wheel-Motor-Driven Electric Vehicles

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## Abstract

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## 1. Introduction

- A novel IVES was developed, containing a practical DVAS equipped between the IWM and the suspension, as well as a robust ASS based on the delay-dependent $H\infty $ controller. The UEMF was considered in this system and the mechanical-electrical-magnetic coupling effects of IWMD EVs were observed.
- A frequency-compatible tire (FCT) model integrating the RRM and FRM was developed to ensure adaptability to different frequency ranges. It also improved the accuracy of vertical tire forces, which were further inputted as an external excitation to the IVES (instead of the road roughness), further improving the accuracy.
- A novel virtual prototype was designed, combining CATIA, ADAMS, and MATLAB/Simulink environment to establish a high-fidelity multi-body model for the IVES. Particularly, the IVES structure was developed, aiming to maximize its integration ability and minimize the impact on the original chassis structure.

## 2. The Mathematical Model of SIWMS

#### 2.1. The UEMF Model in Driving Conditions

#### 2.2. The FCT Model

#### 2.3. DVAS-ASS Integrated Model

## 3. The IVES Optimization Control

#### 3.1. The UEMF Influence on the Dynamic Performances of the Vehicle

#### 3.2. Active Suspension Control

- (1)
- Without external perturbations, the closed-loop system shown in Equation (30) is asymptotically stable.
- (2)
- The performance ${\u2225{\mathbf{z}}_{\mathbf{1}}\left(t\right)\u2225}_{\infty}^{}\le \gamma w{\left(t\right)}_{\infty}^{}$ is minimized subject to Equation (30), where $\gamma $ is the bounded $H\infty $ parametrization.
- (3)
- The time-domain constraint $\left|{\mathbf{z}}_{\mathbf{2}}\left(t\right)\right|\le 1$ must be satisfied.

**P**> 0,

**Q**> 0, and

**R**> 0 exist, and a general matrix

**K${}_{t}$**satisfying the following linear matrix equations (LMIs) is:

**X**exists, making the following inequalities feasible under LMIs:

**K${}_{t}$**can be given by

**K${}_{t}$ = WX${}^{-1}$**

**Proof.**

**X**> 0, matrices (or scalars)

**M**and

**N**with compatible dimensions, the following inequality holds:

**P**,

**Q**, and

**R**being symmetric positive definite matrices. The derivative of ${\mathbf{V}}_{1}\left(t\right)$ is shown next:

**Lemma 1**, that gets:

## 4. Simulation and Analyses

#### 4.1. Performance Comparison of Different Structures

**P**,

**Q**, and

**R**. Thus, the program can be solved via the solver-mincx within the LMI toolbox. Through LMI algorithms, the control gain matrix K${}_{t}$ of the ASS controller (considering the time delay) is obtained with the minimum guaranteed closed-loop $H\infty $ performance index $\gamma $${}_{t}$ = 8.82. Such output implies that for any time-delay satisfying $0\le \tau \le 40\mathrm{ms}$, the controller can stabilize the system with the $H\infty $ performance.

**K**${}_{c}$ of the Controller with the minimum guaranteed closed-loop $H\infty $ performance index $\gamma $${}_{c}$ = 6.28 is obtained as:

- Case 1—equipped with the IWM and passive suspension;
- Case 2—equipped with the DVAS in IWM and passive suspension;
- Case 3—equipped with the DVAS in IWM and the ASS using the control gain matrix K${}_{c}$;
- Case 4—equipped with the DVAS in IWM and the ASS using the control gain matrix K${}_{t}$.

#### 4.2. Virtual Prototype Validation for the IVES

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The IWM characteristics: (

**a**) ${T}_{\mathrm{mean}}$ versus turning angle; (

**b**) ${T}_{\mathrm{std}}$ versus turning angle; (

**c**) single-phase UEMF of SRM; (

**d**) the magic formula.

**Figure 7.**Quarter vehicle models equipped with the IWM: (

**a**) passive suspension, (

**b**) DVAS, (

**c**) ASS and DVAS.

**Figure 9.**Coupling effects between the SMA, the ECC, and the UEMF: (

**a**) the SMA time response; (

**b**) the SMA frequency response; (

**c**) the ECC time response; (

**d**) the ECC frequency response.

**Figure 11.**Responses of four cases under the road excitation: (

**a**) SMA time response; (

**b**) SMA frequency response; (

**c**) ECC time response; (

**d**) ECC frequency response.

**Figure 13.**Responses of the MM and the VP: (

**a**) time response of the SMA; (

**b**) frequency response of the SMA; (

**c**) time response of the ECC; (

**d**) frequency response of the ECC.

Parameters | Value | Parameters | Value |
---|---|---|---|

DVAS-ASS Integrated Model | Passive Suspension System | ||

k${}_{s}$ | 3.2 × 10${}^{4}$ N/m | m${}_{ams}$ | 34.5 kg |

c${}_{s}$ | 1.8 × 10${}^{3}$ N·s/m | Driving conditions | |

k${}_{b}$ | 2.08 × 10${}^{7}$ N/m | R${}_{t}$ | 0.3160 m |

k${}_{d}$ | 5.3 × 10${}^{4}$ N/m | $\mu $ | 0.0066 |

c${}_{d}$ | 1.9 × 10${}^{3}$ N·s/m | I${}_{t}$ | 0.546 kg·m${}^{2}$ |

k${}_{t}$ | 1.8 × 10${}^{6}$ N/m | FTC model | |

c${}_{t}$ | 510 × N·s/m | q${}_{V}$${}_{1}$ | 8.5352 × 10${}^{-8}$ m s${}^{2}$ |

m${}_{s}$ | 332 kg | q${}_{V}$${}_{2}$ | 8.81 × 10${}^{4}$ s |

m${}_{ax}$ | 25 kg | q${}_{Fz}$${}_{1}$ | 1.4389 × 10${}^{5}$ N/m |

m${}_{ms}$ | 9.5 kg | q${}_{Fz}$${}_{2}$ | 4.5090 × 10${}^{6}$ N/m${}^{2}$ |

m${}_{mr}$ | 22.5 kg | ||

m${}_{t}$ | 6.15 kg | ||

$\Sigma $m | 389 kg |

RMS | SMA (m/s^{2})- Decrement (%) | ECC (10^{−5} m)-Decrement (%) | RS (10^{−3} m)-Decrement (%) | TD (10^{−4} m)-Decrement (%) |
---|---|---|---|---|

Case 1 | 0.9941- | 2.365- | 4.229- | 4.328- |

Case 2 | 0.8001-↓19.5% | 1.673-↓29.2% | 4.221-↓0.2% | 4.325-↓0.07% |

Case 3 | 0.4148-↓58.2% | 1.625-↓31.2% | 4.223-↓0.1% | 4.322-↓0.1% |

Case 4 | 0.3217-↓67.6% | 1.444-↓38.9% | 4.222-↓0.2% | 4.318-↓0.2% |

Case | Time Delay | SMA (m/s^{2})-Deteriorate | ECC (10^{−5} m)-Deteriorate |
---|---|---|---|

Case 3 | $\tau $ = 0 ms | 0.4848 | 1.625 |

$\tau $ = 15 ms | 0.6279-↑29.5% | 1.667-↑2.5% | |

controller | $\tau $ = 30 ms | 0.7634-↑57.5% | 1.694-↑4.4% |

Case 4 | $\tau $ = 0 ms | 0.3217 | 1.444 |

$\tau $ = 15 ms | 0.3467-↑7.7% | 1.449-↑0.3% | |

controller | $\tau $ = 30 ms | 0.3545-↑10.1% | 1.447-↑0.2% |

RMS | SMA (m/s^{2}) | ECC (10^{−5} m) | RS (10^{−3} m) | TD (10^{−3} mm) |
---|---|---|---|---|

MM | 0.3217 | 1.444 | 4.222 | 4.318 |

VP | 0.3089 | 1.323 | 4.142 | 4.039 |

Error $\eta $ | 4.12% | 9.12% | 1.9% | 6.9% |

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## Share and Cite

**MDPI and ACS Style**

Zhao, Z.; Gu, L.; Wu, J.; Zhang, X.; Yang, H.
An Integrated Vibration Elimination System with Mechanical-Electrical-Magnetic Coupling Effects for In-Wheel-Motor-Driven Electric Vehicles. *Electronics* **2023**, *12*, 1117.
https://doi.org/10.3390/electronics12051117

**AMA Style**

Zhao Z, Gu L, Wu J, Zhang X, Yang H.
An Integrated Vibration Elimination System with Mechanical-Electrical-Magnetic Coupling Effects for In-Wheel-Motor-Driven Electric Vehicles. *Electronics*. 2023; 12(5):1117.
https://doi.org/10.3390/electronics12051117

**Chicago/Turabian Style**

Zhao, Ze, Liang Gu, Jianyang Wu, Xinyang Zhang, and Haixu Yang.
2023. "An Integrated Vibration Elimination System with Mechanical-Electrical-Magnetic Coupling Effects for In-Wheel-Motor-Driven Electric Vehicles" *Electronics* 12, no. 5: 1117.
https://doi.org/10.3390/electronics12051117