# Study on Beat Vibration of a High Temperature Superconducting EDS Maglev Vehicle at Low Speed

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

#### 2.1. Definition of Two Coordinate Systems

#### 2.2. Dynamic Equations of the Vehicle

#### 2.3. Levitation Force Based on Dynamic Circuitry Theory

**Figure 4.**Levitation force on each SCM (${F}_{\mathrm{z}\_\mathrm{left}}^{i}(t)={F}_{\mathrm{z}\_\mathrm{right}}^{i}(t)={F}_{\mathrm{z}}^{i}(t)$).

## 3. Theory Vertical Equilibrium Point

## 4. Multi-Body Dynamic Model of the EDS Vehicle

- (1)
- Decoupling of Levitation forces

- (2)
- Model elements in multi-body dynamic model

- (3)
- Simulation conditions

- (4)
- Solver setting

## 5. Results and Discussion

#### 5.1. Simulation Results

#### 5.2. Vertical Displacement and Pitch Angle Performance under Different Conditions

- (1)
- The mean value of the vertical displacement ${z}_{\mathrm{mean}}$, which indicates that the vertical balance point is defined as:

- (2)
- The amplitude of the vertical displacement ${z}_{\mathrm{amp}}$ is defined as:

- (3)
- The mean value of the pitch angle of the vehicle ${\gamma}_{\mathrm{mean}}$, which indicates that the mean pitch angle is defined as:

- (4)
- The amplitude of the pitch angle of the vehicle is defined as ${\gamma}_{\mathrm{amp}:}$

- (1)
- Influence of initial vertical displacement $z(0)$

- (2)
- Influence of velocity of vehicle $v$

- (3)
- Influence of MMF of SCM

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The 200 m straight EDS maglev test line with high-temperature superconducting (HTS) at the CRRC.

**Figure 2.**Experimental running results when $v=13.7\hspace{0.33em}\mathrm{mps}$: (

**a**) vertical displacement acceleration $\ddot{z}(t)$; (

**b**) pitch angle $\gamma (t)$.

**Figure 3.**Sketch and force diagram of test line with SCM coils and ground coils at the CRRC: (

**a**) cross section; (

**b**) vertical view; (

**c**) eight levitation forces.

**Figure 5.**Levitation force on left SCMs when $v=13\hspace{0.33em}\mathrm{mps}$ and $MMF=300\hspace{0.33em}\mathrm{kAt}$: (

**a**) levitation force ${F}_{z}^{1}(x)$; (

**b**) levitation force ${F}_{z}^{2}(x)$; (

**c**) levitation force ${F}_{z}^{3}(x)$; (

**d**) levitation force ${F}_{z}^{4}(x)$.

**Figure 6.**Decomposition of levitation force: (

**a**) i-th levitation force ${F}_{z}^{i}(x,z)$; (

**b**) decomposition of i-th levitation force; (

**c**) force diagram of vehicle with $\gamma (t)=0$; (

**d**) decomposition of force diagram of vehicle with $\gamma (t)=0$.

**Figure 7.**Theory equilibrium point: (

**a**) ${F}_{\mathrm{z}\_\mathrm{all}}(z)$ when $v\in [13,\hspace{0.17em}\hspace{0.17em}14,\hspace{0.17em}\hspace{0.17em}15]\hspace{0.33em}\mathrm{mps}$ and $MMF=300\hspace{0.33em}\mathrm{kAt}$; (

**b**) theory vertical equilibrium point ${z}_{0}$ at $MMF\in [300,310,320]\hspace{0.33em}\mathrm{kAt}$ and $v\in [13,\hspace{0.17em}\hspace{0.17em}14,\hspace{0.17em}\hspace{0.17em}15]\hspace{0.33em}\mathrm{mps}$.

**Figure 9.**The (

**a**) $z(t)$ and (

**b**) $\gamma (t)$ of the vehicle based on multi-body dynamics when $v=14\hspace{0.33em}\mathrm{mps}$, $MMF=310\hspace{0.33em}\mathrm{kAt}$, and $z(0)=40\hspace{0.33em}\mathrm{mm}$.

**Figure 10.**The (

**a**) $\ddot{z}(t)$ and (

**b**) $\ddot{\gamma}(t)$ response based on multi-body dynamics when $v=14\hspace{0.33em}\mathrm{mps}$, $MMF=310\hspace{0.33em}\mathrm{kAt}$, and $z(0)=40\hspace{0.33em}\mathrm{mm}$.

**Figure 11.**FFT transformation of vertical displacement acceleration and pitch angle acceleration for Figure 10: (

**a**) FFT transformation of $\ddot{z}$; (

**b**) FFT transformation of $\ddot{\gamma}$.

**Figure 12.**Simulation results when $MMF=300\hspace{0.33em}\mathrm{kAt}$: (

**a**) ${z}_{\mathrm{mean}}$; (

**b**) ${z}_{\mathrm{amp}}$ (

**c**) ${\gamma}_{\mathrm{mean}}$; (

**d**) ${\gamma}_{\mathrm{amp}}$.

**Figure 13.**Simulation results when $MMF=310\hspace{0.33em}\mathrm{kAt}$: (

**a**) ${z}_{\mathrm{mean}}$; (

**b**) ${z}_{\mathrm{amp}}$ (

**c**) ${\gamma}_{\mathrm{mean}}$; (

**d**) ${\gamma}_{\mathrm{amp}}$.

**Figure 14.**Simulation results when $MMF=320\hspace{0.33em}\mathrm{kAt}$: (

**a**) ${z}_{\mathrm{mean}}$; (

**b**) ${z}_{\mathrm{amp}}$; (

**c**) ${\gamma}_{\mathrm{mean}}$; (

**d**) ${\gamma}_{\mathrm{amp}}$.

Mass m/kg | Moment of Inertia ${\mathit{J}}_{\mathit{X}\mathit{X}}/\mathbf{kg}{\mathbf{m}}^{2}$ | Moment of Inertia ${\mathit{J}}_{\mathit{Y}\mathit{Y}}/\mathbf{kg}{\mathbf{m}}^{2}$ | Moment of Inertia ${\mathit{J}}_{\mathit{Z}\mathit{Z}}/\mathbf{kg}{\mathbf{m}}^{2}$ |
---|---|---|---|

2935 | 2536 | 4581 | 6353 |

Scheme | $\mathit{X}$/m | $\mathit{Y}$/m | $\mathit{Z}$/m |
---|---|---|---|

1st left SCM | 1.35 | −1.033 | 0 |

2nd left SCM | 0.81 | −1.033 | 0 |

3rd left SCM | −0.81. | −1.033 | 0 |

4th left SCM | 1.35 | −1.033 | 0 |

1st right SCM | 1.35 | 1.033 | 0 |

2nd right SCM | 0.81 | 1.033 | 0 |

3rd right SCM | −0.81 | 1.033 | 0 |

4th right SCM | −1.35 | 1.033 | 0 |

Factor | Value | Description |
---|---|---|

$z(0)$ | $[30,31,\cdots ,50]\hspace{0.33em}\mathrm{mm}$ | Initial vertical displacement, level number is 21. |

$v$ | $[13,14,15]\hspace{0.33em}\mathrm{mps}$ | Velocity of vehicle, level number is 3. |

$MMF$ | $[300,310,320]\hspace{0.33em}\mathrm{kAt}$ | MMF of SCM onboard, level number is 3. |

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**MDPI and ACS Style**

Yu, Q.-S.; Wang, M.; Yao, G.-F.; Zhang, S.-X.; Yang, J.; Shao, N.
Study on Beat Vibration of a High Temperature Superconducting EDS Maglev Vehicle at Low Speed. *Appl. Sci.* **2023**, *13*, 3131.
https://doi.org/10.3390/app13053131

**AMA Style**

Yu Q-S, Wang M, Yao G-F, Zhang S-X, Yang J, Shao N.
Study on Beat Vibration of a High Temperature Superconducting EDS Maglev Vehicle at Low Speed. *Applied Sciences*. 2023; 13(5):3131.
https://doi.org/10.3390/app13053131

**Chicago/Turabian Style**

Yu, Qing-Song, Min Wang, Guo-Feng Yao, Shi-Xuan Zhang, Jing Yang, and Nan Shao.
2023. "Study on Beat Vibration of a High Temperature Superconducting EDS Maglev Vehicle at Low Speed" *Applied Sciences* 13, no. 5: 3131.
https://doi.org/10.3390/app13053131