# Dynamic Vibration Absorbing Performance of 5-DoF Magnetically Suspended Momentum Wheel Based on Damping Regulation

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## Abstract

**:**

## 1. Introduction

## 2. Force Modelling of 5-DoF MSMW

#### 2.1. System View of 5-DoF MSMW

_{a}, and the span distance of the axial displacement sensor is l

_{s}.

#### 2.2. Displacement and Force Coordinates of 5-DoF MSMW

_{uz}

_{1}, f

_{uz}

_{2}, f

_{uz}

_{3}, f

_{uz}

_{4}, f

_{lz}

_{1}, f

_{lz}

_{2}, f

_{lz}

_{3}, f

_{lz}

_{4}] could control the axial displacement of the flywheel rotor in the z-axis. Moreover, the deflection torques

**T**

_{a}= [T

_{α}, T

_{β}] are generated to control the torsions of the flywheel rotor around the y-axis and the x-axis, so the deflection torques could be expressed as

#### 2.3. Translational Stiffness and Damping Characteristics of 5-DoF MSMW

_{s}. The closed-loop transfer function from the reference displacement to the outputted displacement could be written to

#### 2.4. Torsional Stiffness and Damping Characteristics of 5-DoF MSMW

## 3. Translational Vibration Modelling of 5-DoF MSMW

#### 3.1. Translational Vibration Model of 5-DoF MSMW

_{s}, and the axial displacement of the flywheel rotor is defined as z

_{r}. For the radial displacement of the 5-DoF MSMW, the radial displacement of the flywheel rotor is defined as x

_{r}during the torsional state, and the torsional angle of the flywheel rotor is chosen as β. For the stiffness and damping coefficients of the whole system, the stiffness and damping parameters of the connection joint between the stator base of the 5-DoF MSMW and the shacking table are defined as K

_{j}and C

_{j}. For the axial 3-DoF AMB unit, the translational stiffness and damping coefficients are K

_{a}and C

_{a}, respectively. The torsional stiffness and damping coefficients unit are K

_{ta}and C

_{at}, separately.

_{s}is the mass of the stator base, and m

_{r}is the mass of the flywheel rotor. f

_{s}is the excitation force acting on the stator base, and it is usually expressed as ${f}_{s}={F}_{0}\mathrm{sin}\omega t$.

_{r}and Z

_{s}are both the complex numbers indicating the displacement variations.

_{r}and Z

_{s}together, there are

_{r}of the FW rotor, we get the following function

_{m}is a constant value, by simplifying the transfer functions of the translational vibration models, (18) and (19) could be further written to

#### 3.2. Translational Vibration Model of 5-DoF MSMW with Active Controllable Stiffness and Damping

## 4. Torsional Vibration Modelling of 5-DoF MSMW

#### 4.1. Torsional Vibration Model of 5-DoF MSMW

_{s}, and the torsional angle of the flywheel rotor around the x-axis is β. For the deflection torque generated by the axial 3-DoF AMB to control the torsional motion of the flywheel rotor, the stiffness coefficient could be defined as K

_{ta}, and the damping coefficient is C

_{ta}. For the connection joint between the 5-DoF MSMW and the shacking table, the torsional stiffness coefficient is K

_{tj}, and the torsional damping coefficient is C

_{tj}. Therefore, the torsional vibration models of the flywheel rotor and the stator base could be written to

_{j}is a fixed value, the torsional functions in (31) could be further simplified to

#### 4.2. Torsional Vibration Model of 5-DoF MSMW with Active Controllable Stiffness and Damping

## 5. Numerical Simulation

#### 5.1. Translational Vibration of 5-DoF MSMW

_{a}= 500, the response magnitudes of the translational vibration with different damping coefficients are plotted in Figure 6a. In detail, as shown by the red line, the maximum magnitude of the translational vibration is about 26.08 dB at the frequency 87.25 Hz when the damping coefficient is C

_{a}= 0.04. When the damping coefficient is increased to C

_{a}= 0.10, the maximum magnitude of the translational vibration is reduced to 9.34 dB. Therefore, the damping coefficient of the axial 3-DoF AMB unit could suppress the response magnitude of the translational vibration. Moreover, when the damping coefficient of the axial 3-DoF AMB unit is chosen as the fixed value C

_{a}= 0.06, the response magnitudes of the translational vibration at different stiffness coefficients are shown in Figure 6b. The response magnitude is marked by the red line when the stiffness coefficient is K

_{a}= 200, and the maximum response magnitude is 18.31 dB when the response frequency is at 86.5 Hz. The maximum value of the response magnitude is about 24 dB at 84.5 Hz when the stiffness coefficient is increased to K

_{a}= 800. According to the response curves, the variation of the stiffness coefficient causes little influence on the response magnitude, but it could change the natural response frequency of the translational vibration.

#### 5.2. Torsional Vibration of 5-DoF MSMW

_{jt}= 58, and the torsional damping coefficient is chosen as C

_{jt}= 0.025. For the flywheel rotor, the response curves of the torsional vibration are plotted in Figure 7 when the stiffness coefficient and the damping coefficient of the axial 3-DoF AMB unit are chosen as different values. As shown in Figure 7a, the stiffness coefficient of the axial 3-DoF AMB unit is defined as Kat = 5, and the vibration magnitude of the torsional vibration is −33.52 dB at the 475 Hz when the damping coefficient is chosen as C

_{at}= 0.4. The vibration magnitude of the torsional vibration is reduced to 41.1 dB when the damping coefficient is increased to C

_{at}= 1.0. Thus, the response magnitude of the torsional vibration could be mitigated by the damping coefficient of the axial 3-DoF AMB unit. The frequency of the torsional vibration could also be regulated by the stiffness coefficient, and the resonant vibration of the torsion could be avoidable. As illustrated in Figure 7b, the vibration magnitude of the torsional vibration is 8.58 dB when the stiffness coefficient is defined as K

_{at}= 4, and it is changed to 26.9 dB when the stiffness coefficient is increased to 32. Therefore, for the torsional vibration of the 5-DoF MSMW, the damping coefficient of the axial 3-DoF AMB unit could effectively control the vibration magnitude, and then the stiffness coefficient could change the natural frequency of the torsional vibration.

## 6. Experimental Validation

#### 6.1. Vibration Measurement System of 5-DoF MSMW

^{2}, and the polar moment of inertia 0.15 kgm

^{2}. For the axial 3-DoF AMB unit, the current stiffness is 128 N/A, and the displacement stiffness is −132.7 N/mm. For the radial 2-DoF AMB unit, the current stiffness is 132.26 N/A, and the displacement stiffness is −352 N/mm.

#### 6.2. Translational Vibration Experiment of 5-DoF MSMW

_{a}= 0.06. The displacement deflection plotted by the red line is reduced to −0.14 μm when the damping coefficient is increased to C

_{a}= 0.10. Moreover, the random disturbance is imposed on the 5-DoF MSMW along the axial direction, and the displacement curves of the flywheel rotor are plotted in Figure 9b. The root mean square (RMS) value is used to evaluate the axial displacement of the flywheel rotor. As plotted by the green line, the RMS is 0.063 μm when the damping coefficient is C

_{a}= 0.06. The RMS of axial displacement is mitigated to 0.015 μm when the damping coefficient is increased to C

_{a}= 0.10.

_{a}= 0.06. When the damping coefficient is increased to 0.10, the maximum deflection of the axial displacement shown by the red line is 0.06 μm.

#### 6.3. Torsional Vibration Experiment of 5-DoF MSMW

_{at}= 0.6 and it is reduced to −0.23°, shown by the red line when the torsional damping is increased to C

_{at}= 1.0. In addition, the torsional angles of the flywheel rotor suffering the random disturbance are illustrated in Figure 11b, the root mean square (RMS) value of the torsional angle is 0.060° with the torsional damping C

_{at}= 0.6, and then the RMS value of the flywheel rotor’s torsional angle is reduced to 0.032° when the torsional damping is increased to C

_{at}= 1.0.

_{at}= 0.6. The maximum deflection of the radial displacement is mitigated to 0.05 μm at the damping coefficient C

_{at}= 1.0. In addition, as shown in Figure 12b, the power spectrum density of the radial displacement is also reduced by increasing the damping coefficients.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Mahfouz, A.; Pritykin, D.; Biggs, J. Hybrid Attitude Control for Nano-Spacecraft: Reaction Wheel Failure and Singularity Handling. J. Guid. Control. Dyn.
**2021**, 44, 548–558. [Google Scholar] [CrossRef] - Ismail, Z.; Varatharajoo, R. A study of reaction wheel configurations for a 3-axis satellite attitude control. Adv. Space Res.
**2009**, 45, 750–759. [Google Scholar] [CrossRef] - Kumar, K.D.; Godard; Abreu, N.; Sinha, M. Fault-tolerant attitude control of miniature satellites using reaction wheels. Acta Astronaut.
**2018**, 151, 206–216. [Google Scholar] [CrossRef] - Abd-Elhay, A.-E.R.; Murtada, W.A.; Yosof, M.I. A high accuracy modeling scheme for dynamic systems: Spacecraft reaction wheel model. J. Eng. Appl. Sci.
**2022**, 69, 1–22. [Google Scholar] [CrossRef] - Fu, C.; Sinou, J.-J.; Zhu, W.; Lu, K.; Yang, Y. A state-of-the-art review on uncertainty analysis of rotor systems. Mech. Syst. Signal Process.
**2023**, 183, 109619. [Google Scholar] [CrossRef] - Jia, Q.; Li, H.; Chen, X.; Zhang, Y. Observer-based reaction wheel fault reconstruction for spacecraft attitude control systems. Aircr. Eng. Aerosp. Technol.
**2019**, 91, 1268–1277. [Google Scholar] [CrossRef] - Tuysuz, A.; Achtnich, T.; Zwyssig, C.; Kolar, J.W. A 300 000-r/min Magnetically Levitated Reaction Wheel Demonstrator. IEEE Trans. Ind. Electron.
**2018**, 66, 6404–6407. [Google Scholar] [CrossRef] - Xiang, B.; Liu, H.; Yu, Y. Gimbal effect of magnetically suspended flywheel with active deflection of Lorentz-force magnetic bearing. Mech. Syst. Signal Process.
**2022**, 173, 109081. [Google Scholar] [CrossRef] - Zhai, L.; Han, B.; Liu, X.; Zhao, J. Losses estimation, thermal-structure coupled simulation analysis of a magnetic-bearing reaction wheel. Int. J. Appl. Electromagn. Mech.
**2019**, 60, 33–53. [Google Scholar] [CrossRef] - Xiang, B.; Wen, T.; Liu, Z. Vibration analysis, measurement and balancing of flywheel rotor suspended by active magnetic bearing. Measurement
**2022**, 197, 111305. [Google Scholar] [CrossRef] - Dagnaes-Hansen, N.A.; Santos, I.F. Magnetically suspended flywheel in gimbal mount-Test bench design and experimental validation. J. Sound Vib.
**2019**, 448, 197–210. [Google Scholar] [CrossRef] - Dagnaes-Hansen, N.A.; Santos, I.F. Magnetically suspended flywheel in gimbal mount–Nonlinear modelling and simulation. J. Sound Vib.
**2018**, 432, 327–350. [Google Scholar] [CrossRef] - Saeed, N.A.; El-Shourbagy, S.M.; Kamel, M.; Raslan, K.R.; Awrejcewicz, J.; Gepreel, K.A. On the Resonant Vibrations Control of the Nonlinear Rotor Active Magnetic Bearing Systems. Appl. Sci.
**2022**, 12, 8300. [Google Scholar] [CrossRef] - Soni, T.; Dutt, J.K.; Das, A. Dynamic behavior and stability of energy efficient electro-magnetic suspension of rotors involving time delay. Energy
**2021**, 231, 120906. [Google Scholar] [CrossRef] - Zheng, S.; Wang, C. Rotor Balancing for Magnetically Levitated TMPs Integrated With Vibration Self-Sensing of Magnetic Bearings. IEEE/ASME Trans. Mechatron.
**2021**, 26, 3031–3039. [Google Scholar] [CrossRef] - Li, X.; Dietz, D.; An, J.; Erd, N.; Gemeinder, Y.; Binder, A. Manufacture and Testing of a Magnetically Suspended 0.5 kWh-Flywheel Energy Storage System. IEEE Trans. Ind. Appl.
**2022**, 58, 1–11. [Google Scholar] [CrossRef] - Li, X.; Palazzolo, A.; Wang, Z. A Combination 5-DOF Active Magnetic Bearing for Energy Storage Flywheels. IEEE Trans. Transp. Electrif.
**2021**, 7, 2344–2355. [Google Scholar] [CrossRef] - Cole, M.O.T.; Fakkaew, W. An Active Magnetic Bearing for Thin-Walled Rotors: Vibrational Dynamics and Stabilizing Control. IEEE/ASME Trans. Mechatron.
**2018**, 23, 2859–2869. [Google Scholar] [CrossRef] - Xiang, B.; Wong, W. Electromagnetic vibration absorber for torsional vibration in high speed rotational machine. Mech. Syst. Signal Process.
**2020**, 140, 106639. [Google Scholar] [CrossRef] - Lusty, C.; Keogh, P. Active Vibration Control of a Flexible Rotor by Flexibly Mounted Internal-Stator Magnetic Actuators. IEEE/ASME Trans. Mechatron.
**2018**, 23, 2870–2880. [Google Scholar] [CrossRef] - Hutterer, M.; Wimmer, D.; Schrödl, M. Control of magnetically levitated rotors using stabilizing effects of gyroscopes. Mech. Syst. Signal Process.
**2021**, 166, 108431. [Google Scholar] [CrossRef] - Numanoy, N.; Srisertpol, J. Vibration Reduction of an Overhung Rotor Supported by an Active Magnetic Bearing Using a Decoupling Control System. Machines
**2019**, 7, 73. [Google Scholar] [CrossRef] [Green Version] - Xiang, B.; Wong, W. Decoupling control of magnetically suspended motor rotor with heavy self-weight and great moment of inertia based on internal model control. J. Vib. Control
**2021**, 28, 1591–1604. [Google Scholar] [CrossRef] - Gallego, G.B.; Rossini, L.; Achtnich, T.; Araujo, D.M.; Perriard, Y. Novel Generalized Notch Filter for Harmonic Vibration Suppression in Magnetic Bearing Systems. IEEE Trans. Ind. Appl.
**2021**, 57, 6977–6987. [Google Scholar] [CrossRef] - Peng, C.; He, J.; Deng, Z.; Liu, Q. Parallel mode notch filters for vibration control of magnetically suspended flywheel in the full speed range. IET Electr. Power Appl.
**2020**, 14, 1672–1678. [Google Scholar] [CrossRef] - Gong, L.; Zhu, C. Synchronous Vibration Control for Magnetically Suspended Rotor System Using a Variable Angle Compensation Algorithm. IEEE Trans. Ind. Electron.
**2020**, 68, 6547–6559. [Google Scholar] [CrossRef]

**Figure 1.**The system view of the 5-DoF MSFW. (

**a**) The main components of the 5-DoF MSMW; (

**b**) the displacement terms of the 5-DoF MSMW.

**Figure 2.**The structure and force models of the 2−DoF radial AMB. (

**a**) The top view of the 2−DoF radial AMB; (

**b**) the force distributions of the 2−DoF radial AMB.

**Figure 3.**The structure and force models of the 3-DoF axial AMB. (

**a**) The upper end of the 3-DoF axial AMB; (

**b**) the force distributions of the 3-DoF axial AMB; (

**c**) the force distributions of the 3-DoF axial AMB.

**Figure 5.**The force models of the 5-DoF MSMW on the shacking table. (

**a**) The equivalent translational vibration model of the 5-DoF MSMW; (

**b**) the equivalent torsional vibration model of the 5-DoF MSMW.

**Figure 6.**The translational vibration of the flywheel rotor along the axial direction. (

**a**) The translational vibration amplitudes of the flywheel rotor with different damping coefficients; (

**b**) the translational vibration amplitudes of the flywheel rotor with different stiffness coefficients.

**Figure 7.**The torsional vibration of the flywheel rotor along the axial direction. (

**a**) The torsional vibration amplitudes of the flywheel rotor with different damping coefficients. (

**b**) The torsional vibration amplitudes of the flywheel rotor with different stiffness coefficients.

**Figure 8.**The vibration experimental systems of the 5-DoF MSMW. (

**a**) The translational vibration system of the 5-DoF MSMW. (

**b**) The torsional vibration system of the 5-DoF MSMW.

**Figure 9.**The translational displacements of the 5−DoF MSMW with different types of disturbances. (

**a**) The axial translational displacements of the flywheel rotor suffering the impulse disturbance with different damping coefficients. (

**b**) The axial translational displacements of the flywheel rotor suffering the random disturbance with different damping coefficients.

**Figure 11.**The torsional angles of the 5−DoF MSMW with different types of disturbances. (

**a**) The torsional angles of the flywheel rotor suffering the impulse disturbance with different damping coefficients; (

**b**) The torsional angles of the flywheel rotor suffering the random disturbance with different damping coefficients.

**Figure 12.**The vibration characteristics of the 5−DoF MSMW when the rotating speed is 2000 rpm. (

**a**) The radial displacement of the flywheel rotor at 2000 rpm. (

**b**) The power spectrum density of the radial displacement at 2000 rpm.

Symbol | Quantity | Value |
---|---|---|

m_{r} | Mass of flywheel rotor | 4.2 kg |

m_{s} | Mass of stator base | 12 kg |

J_{e} | Equatorial moment of inertia | 0.02865 kgm^{2} |

J_{p} | Polar moment of inertia | 0.01508 kgm^{2} |

k_{ix} | Current stiffness of radial AMB unit | 132.26 N/A |

k_{dx} | Displacement stiffness of radial AMB unit | −352 N/mm |

k_{iz} | Current stiffness of axial AMB unit | 128 N/A |

k_{dz} | Displacement stiffness of axial AMB unit | −132.7 N/mm |

k_{s} | Displacement sensitivity of eddy-current sensor | 4 V/mm |

k_{w} | Amplification coefficient of power system | 3.6 A/V |

K_{j} | Translational stiffness coefficient of connection joint | 600 |

C_{j} | Translational damping coefficient of connection joint | 0.02 |

K_{jt} | Torsional stiffness coefficient of connection joint | 58 |

C_{jt} | Torsional damping coefficient of connection joint | 0.025 |

Frequency | Vibration Amplitude | Speed | Imposing Axis |
---|---|---|---|

10–20 Hz | 10 mm | 2 octave/min | z-axis |

20–100 Hz | 16 g | 2 octave/min | z-axis |

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## Share and Cite

**MDPI and ACS Style**

Xiang, B.; Liu, H.
Dynamic Vibration Absorbing Performance of 5-DoF Magnetically Suspended Momentum Wheel Based on Damping Regulation. *Actuators* **2023**, *12*, 152.
https://doi.org/10.3390/act12040152

**AMA Style**

Xiang B, Liu H.
Dynamic Vibration Absorbing Performance of 5-DoF Magnetically Suspended Momentum Wheel Based on Damping Regulation. *Actuators*. 2023; 12(4):152.
https://doi.org/10.3390/act12040152

**Chicago/Turabian Style**

Xiang, Biao, and Hu Liu.
2023. "Dynamic Vibration Absorbing Performance of 5-DoF Magnetically Suspended Momentum Wheel Based on Damping Regulation" *Actuators* 12, no. 4: 152.
https://doi.org/10.3390/act12040152