# Modelling and Test of an Integrated Magnetic Spring-Eddy Current Damper for Space Applications

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−4}, even operating at high temperatures.

## 2. Mechanical and Electromagnetic Design and Analysis

#### 2.1. Mechanical Design

#### 2.2. Simulation of Stiffness—k

_{p}is the permanent magnetization µ

_{0}is a constant of permeability of vacuum and µ

_{r}corresponds to the relative permeability. The solver obtains the magnetic field distribution produced by a spatial distribution of objects with permanent magnetization [38] and a combination of DC current densities. All the simulations were run on a computer with Intel Core i4-4690 and 8 Gb of RAM memory.

#### 2.3. Simulation of Eddy Current Damping—c_{ed}

_{L}is damping force (Lorentz force), and v is the velocity of the moving suspended mass. By linking mechanical losses and eddy current losses, we can demonstrate that:

_{ed}, depending on the electromagnetic behavior of the magnet:

#### 2.4. Estimation of Coulomb Friction Damping Coefficient—c_{Cou}

_{f}, dissipates energy as ${W}_{Cou\text{}}=4\text{}\xb7{F}_{f}\xb7X$, where X is the displacement of the moving element. The energy dissipated by the friction damping force can be linked with an equivalent viscous damping coefficient for Coulomb friction, c

_{Cou}, using the oscillatory angular speed ω as [39]:

_{Cou}, we need to estimate friction force.

#### 2.5. Dynamical Model

_{0}is the amplitude of the driving motion, and ω is the frequency of the sinusoidal driving motion.

_{0}, driving frequency ω, undamped angular frequency ω, and the damping ratio $\xi $.

## 3. Prototype Manufacturing and Testbench Set-Up

#### 3.1. Prototype Manufacturing and Assembly

#### 3.2. Measurement System Set-Up and Data Analysis Procedure

## 4. Test Results

#### 4.1. Stiffness Static Test Results

#### 4.2. Damping Dynamic Free Vibration Test Results

^{2}due to a voltage bias error. Once the measurement signals were filtered, they were integrated twice in order to obtain displacement versus time measurements. From the displacement curve, logarithmic decay and oscillation period can be derived. Using logarithmic decay and oscillation period, damping ratio, natural frequency can be calculated and therefore, we obtain stiffness and total damping coefficient (eddy current damping plus Coulomb friction damping). A similar procedure was applied for the 2- and 3-magnets configurations. The results are listed in Table 4.

#### 4.3. Transmissibility Curve

_{0}is 4.98 Hz (31.4 rad/s) common for all configurations. This can be explained since load mass was adjusted to set a separation distance of 10 mm. Damping ratios are different depending on the configuration. Damping ratios, $\xi $, are calculated as 0.03, 0.08, and 0.12 for 2, 3, and 4 magnets configuration, respectively.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Hu, S.D.; Li, H.; Tzou, H.S. Precision Microscopic Actuations of Parabolic Cylindrical Shell Reflectors. J. Vib. Acoust.
**2015**, 137, 011013. [Google Scholar] [CrossRef] - Liu, L.; Cao, D.; Wei, J.; Tan, X.; Yu, T. Rigid-Flexible Coupling Dynamic Modeling and Vibration Control for a Three-Axis Stabilized Spacecraft. J. Vib. Acoust.
**2017**, 139, 041006. [Google Scholar] [CrossRef] - Mainenti-Lopes, I.; Souza, L.; De Sousa, F. Design of a Nonlinear Controller for a Rigid-Flexible Satellite Using Multi-Objective Generalized Extremal Optimization with Real Codification. Shock. Vib.
**2012**, 19, 947–956. [Google Scholar] [CrossRef] - Diez-Jimenez, E.; Musolino, A.; Raugi, M.; Rizzo, R.; Sani, L. A Magneto-Rheological Brake Excited by Permanent Magnets. Appl. Comput. Electromagn. Soc. J.
**2019**, 34, 186–191. [Google Scholar] - Mohamed, K.T.; Ata, A.A.; El-Souhily, B.M. Dynamic Analysis Algorithm for a Micro-Robot for Surgical Ap-plications. Int. J. Mech. Mater. Des.
**2011**, 7, 17–28. [Google Scholar] [CrossRef] - Yu, D. The Dynamic Stress of Stranded-Wire Helical Spring and Its Useful Life. J. Nanjing Univ. Sci. Technol.
**1994**, 75, 24–29. [Google Scholar] - Wang, S.; Li, X.; Lei, S.; Zhou, J.; Yang, Y. Research on torsional fretting wear behaviors and damage mechanisms of stranded-wire helical spring. J. Mech. Sci. Technol.
**2011**, 25, 2137–2147. [Google Scholar] [CrossRef] - Diez-Jimenez, E.; Perez-Diaz, J.; Ferdeghini, C.; Canepa, F.; Bernini, C.; Cristache, C.; Sanchez-Garcia-Casarrubios, J.; Valiente-Blanco, I.; Ruiz-Navas, E.; Martínez-Rojas, J. Magnetic and morphological characterization of Nd2Fe14B magnets with different quality grades at low temperature 5–300 K. J. Magn. Magn. Mater.
**2018**, 451, 549–553. [Google Scholar] [CrossRef] - Xu, Y.; Zhou, J.; Jin, C. Identification of dynamic stiffness and damping in active magnetic bearings using transfer functions of electrical control system. J. Mech. Sci. Technol.
**2019**, 33, 571–577. [Google Scholar] [CrossRef] - Esnoz-Larraya, J.; Valiente-Blanco, I.; Cristache, C.; Sánchez-García-Casarrubios, J.; Diez-Jimenez, E.; Perez-Diaz, J.L. OPTIMAGDRIVE: High-performance magnetic gears development for space applications. In Proceedings of the 17th European Space Mechanisms and Tribology Symposium, Hatfield, UK, 20–22 September 2017. [Google Scholar]
- Cristache, C.; Diez-Jimenez, E.; Valiente-Blanco, I.; Sanchez-Garcia-Casarrubios, J.; Perez-Diaz, J.L. Aeronautical Magnetic Torque Limiter for Passive Protection against Overloads. Machines
**2016**, 4, 17. [Google Scholar] [CrossRef] [Green Version] - Valiente-Blanco, I.; Cristache, C.; Sanchez-Garcia-Casarrubios, J.; Rodriguez-Celis, F.; Perez-Diaz, J.L. Mechanical Impedance Matching Using a Magnetic Linear Gear. Shock. Vib.
**2017**, 2017, 1–9. [Google Scholar] [CrossRef] [Green Version] - Perez-Diaz, J.L.; Valiente-Blanco, I.; Cristache, C.; Sanchez-García-Casarubios, J.; Rodriguez, F.; Esnoz, J.; Diez-Jimenez, E.; Sanchez-García-Casarrubios, J.; Larraya, J.E. A novel high temperature eddy current damper with enhanced performance by means of impedance matching. Smart Mater. Struct.
**2019**, 28, 025034. [Google Scholar] [CrossRef] - Valiente-Blanco, I.; Diez-Jimenez, E.; Cristache, C.; Álvarez-Valenzuela, M.A.; Perez-Diaz, J.L. Characterization and Improvement of Axial and Radial Stiffness of Contactless Thrust Superconducting Magnetic Bearings. Tribol. Lett.
**2013**, 54, 213–220. [Google Scholar] [CrossRef] - Perez-Diaz, J.L.; Diez-Jimenez, E.; Valiente-Blanco, I.; Herrero-De-Vicente, J. Stable thrust on a finite-sized magnet above a Meissner superconducting torus. J. Appl. Phys.
**2013**, 113, 63907. [Google Scholar] [CrossRef] - Perez-Diaz, J.L.; Diez-Jimenez, E.; Valiente-Blanco, I.; Cristache, C.; Alvarez-Valenzuela, M.-A.; Sanchez-Garcia-Casarrubios, J.; Ferdeghini, C.; Canepa, F.; Hornig, W.; Carbone, G.; et al. Performance of Magnetic-Superconductor Non-Contact Harmonic Drive for Cryogenic Space Applications. Machines
**2015**, 3, 138–156. [Google Scholar] [CrossRef] [Green Version] - Diez-Jimenez, E.; Sander, B.; Timm, L.; Perez-Diaz, J.L. Tailoring of the flip effect in the orientation of a magnet levitating over a superconducting torus: Geometrical dependencies. Phys. C Supercond.
**2011**, 471, 229–232. [Google Scholar] [CrossRef] - Xu, L.; Zhu, X. Natural Frequencies and Vibrating Modes for a Magnetic Planetary Gear Drive. Shock. Vib.
**2012**, 19, 1385–1401. [Google Scholar] [CrossRef] - Hao, X.-H.; Zhu, X.-J. Forced Responses of the Parametric Vibration System for the Electromechanical Integrated Magnetic Gear. Shock. Vib.
**2015**, 2015, 1–17. [Google Scholar] [CrossRef] [Green Version] - Hao, X.-H.; Zhu, H.-Q.; Pan, D. Nonlinear Resonance Responses of Electromechanical Integrated Magnetic Gear System. Shock. Vib.
**2018**, 2018, 1–16. [Google Scholar] [CrossRef] [Green Version] - Diez-Jimenez, E.; Montero, R.S.; Muñoz, M.M. Towards Miniaturization of Magnetic Gears: Torque Performance Assessment. Micromachines
**2017**, 9, 16. [Google Scholar] [CrossRef] [Green Version] - Paden, B.; Groom, N.; Antaki, J.F. Design Formulas for Permanent-Magnet Bearings. J. Mech. Des.
**2003**, 125, 734–738. [Google Scholar] [CrossRef] - Salauddin, M.; A Halim, M.; Park, J.Y. A magnetic-spring-based, low-frequency-vibration energy harvester comprising a dual Halbach array. Smart Mater. Struct.
**2016**, 25, 095017. [Google Scholar] [CrossRef] - Wang, T.; Zhu, Z.; Zhu, S. Comparison of vibration energy harvesters with fixed and unfixed magnetic springs. Electron. Lett.
**2018**, 54, 646–647. [Google Scholar] [CrossRef] - Qian, K.-X.; Zeng, P.; Ru, W.-M.; Yuan, H.-Y. Novel magnetic spring and magnetic bearing. IEEE Trans. Magn.
**2003**, 39, 559–561. [Google Scholar] [CrossRef] - Otake, Y. Development of a Horizontal Component Seismometer Using a Magnetic Spring. Rev. Sci. Instrum.
**2000**, 71, 4576–4581. [Google Scholar] [CrossRef] - Sun, F.; Zhang, M.; Jin, J.; Duan, Z.; Jin, J.; Zhang, X. Mechanical analysis of a three-degree of freedom same-stiffness permanent magnetic spring. Int. J. Appl. Electromagn. Mech.
**2016**, 52, 667–675. [Google Scholar] [CrossRef] - Robertson, W.; Cazzolato, B.S.; Zander, A. A multipole array magnetic spring. IEEE Trans. Magn.
**2005**, 41, 3826–3828. [Google Scholar] [CrossRef] [Green Version] - Zheng, Y.; Li, Q.; Yan, B.; Luo, Y.; Zhang, X. A Stewart isolator with high-static-low-dynamic stiffness struts based on negative stiffness magnetic springs. J. Sound Vib.
**2018**, 422, 390–408. [Google Scholar] [CrossRef] - Yao, H.; Wang, T.; Wen, B.; Qiu, B. A tunable dynamic vibration absorber for unbalanced rotor system. J. Mech. Sci. Technol.
**2018**, 32, 1519–1528. [Google Scholar] [CrossRef] - Li, Q.; Zhu, Y.; Xu, D.; Hu, J.; Min, W.; Pang, L. A negative stiffness vibration isolator using magnetic spring combined with rubber membrane. J. Mech. Sci. Technol.
**2013**, 27, 813–824. [Google Scholar] [CrossRef] - Poojary, U.R.; Gangadharan, K. Integer and Fractional Order-Based Viscoelastic Constitutive Modeling to Predict the Frequency and Magnetic Field-Induced Properties of Magnetorheological Elastomer. J. Vib. Acoust.
**2018**, 140. [Google Scholar] [CrossRef] - Zhang, X.; Xia, X.; Xiang, Z.; You, Y.; Li, B. An Online Active Balancing Method Using Magnetorheological Effect of Magnetic Fluid. J. Vib. Acoust.
**2018**, 141, 011008. [Google Scholar] [CrossRef] - Michaud, S.; Vedovati, F.; Catalan, J.; Zahnd, B.; Herrscher, M.; Omiciuolo, M.; Patti, S. Sentinel-4 Scanner Sub-system. In Proceedings of the 17th European Space Mechanisms and Tribology Symposium, Hatfield, UK, 20–22 September 2017; pp. 20–22. [Google Scholar]
- Liebold, F.; Allegranza, C.; Seiler, R.; Junge, A. Modelling and Simulation of Electromagnetic Effects. In Proceedings of the 16th European Space Mechanisms and Tribology Symposium, Bilbao, Spain, 23–25 September 2015; pp. 23–25. [Google Scholar]
- Bae, J.-S.; Hwang, J.-H.; Park, J.-S.; Kwag, D.-G. Modeling and experiments on eddy current damping caused by a permanent magnet in a conductive tube. J. Mech. Sci. Technol.
**2009**, 23, 3024–3035. [Google Scholar] [CrossRef] - Kim, J.-H.; Lee, Y.-G.; Kim, C.-G. An experimental study on a new air-eddy current damper for application in low-frequency accelerometers. J. Mech. Sci. Technol.
**2015**, 29, 3617–3625. [Google Scholar] [CrossRef] - Ansoft Ansys Maxwell V15—Help Assistant; ANSYS Inc.: Canonsburg, PA, USA, 2018.
- Stutts, D.S. Equivalent Viscous Damping; Missouri University of Sciences and Technology: Rolla, MO, USA, 2009. [Google Scholar]

**Figure 1.**3D CAD model of the device in four magnets configuration. (1)—top spherical end, (2) NdFeB N38H magnets, (3) conductive aluminum inner rod, (4) linking hollowed rod, and (5) bottom spherical end.

**Figure 3.**2D axisymmetric FEM results for the magnetic field in four magnets configuration (50 mm outer diameter).

**Figure 4.**Simulation results of force/volume ratio for different magnet outer diameters (inner diameter and magnet thickness were fixed parameters).

**Figure 5.**(

**a**) Load capacity and (

**b**) stiffness calculation for different magnets separation distances and different magnets configurations.

**Figure 6.**Maximum force simulation for different numbers of magnets with alternative magnetization directions.

**Figure 10.**Parts and assembled prototype. (1)—adapted top end, (2) NdFeB N38H magnets, (3) conductive aluminium inner rod, (4) threaded M4 holes for sensors and (5) bottom end.

**Figure 11.**Test bench: (1)—prototype in 4 magnets configuration, (2) baseplate, (3) vertical direction accelerometer, (4) current source, (5) data acquisition card, (6) personal computer and (7) 2 kg load mass.

**Figure 15.**Acceleration measurements treatment procedure. (

**a**) Raw direct acceleration measurements; (

**b**) Low pass filtered signal.

Part | Main Dimensions (mm) | Material | Weight (g) |
---|---|---|---|

Top and bottom ends | Aluminum 7075 | 107 | |

Permanent Magnet | NdFeB N38H | 49 | |

Inner conductive rod | Aluminum 7075 | 47 | |

Linking hollowed rod | Aluminum 7075 | 60 | |

TOTAL for 2 magnets configuration | 419 | ||

TOTAL for 3 magnets configuration | 468 | ||

TOTAL for 4 magnets configuration | 517 |

Common Parameters | |||
---|---|---|---|

Rotation angle | 0.55° | Friction Coeff. | 0.05 |

Pair of forces arm | 5 mm | X displacement | ±2 mm |

Configuration | 2 Magnets | 3 Magnets | 4 Magnets |

Torque magnet 1 (mNm) | 3.6 | 3.5 | 5.8 |

Torque magnet 2 (mNm) | - | 2.9 | 2.2 |

Torque magnet 3 (mNm) | - | - | 7.5 |

Total Normal force (N) | 0.72 | 1.28 | 3.1 |

Total axial Friction force (N) | 0.050 | 0.08 | 0.217 |

Natural radian frequency (rad/s) | 31.30 | 31.30 | 31.30 |

Damping coefficient (Ns/m) | 1.02 | 1.821 | 4.41 |

Configuration/Topology | Load (N) | k—Stiffness (N/m) | c_{ed}—Eddy Current Damping(Ns/m) | c_{cou}—Coulomb Friction Damping(Ns/m) |
---|---|---|---|---|

2 Magnets Configuration | 22.36 | 2236 | 5.15 | 1.02 |

3 Magnets Configuration | 18.51 | 1851 | 10.30 | 1.82 |

4 Magnets Configuration | 19.46 | 1946 | 15.45 | 4.41 |

Configuration | Load at 10 mm Distance (N) | k—Stiffness (N/m) | c—Damping Coefficient (Ns/m) | |||
---|---|---|---|---|---|---|

Measure. | Simul. | Measure. | Simul. | Measure. | Simul. | |

2 Magnets | 29.5 | 22.36 | 2410 | 2236 | 5.45 | 6.17 |

3 Magnets | 22.5 | 18.51 | 2050 | 1851 | 10.52 | 12.12 |

4 Magnets | 22.9 | 19.46 | 2090 | 1946 | 17.25 | 19.86 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Diez-Jimenez, E.; Alén-Cordero, C.; Alcover-Sánchez, R.; Corral-Abad, E.
Modelling and Test of an Integrated Magnetic Spring-Eddy Current Damper for Space Applications. *Actuators* **2021**, *10*, 8.
https://doi.org/10.3390/act10010008

**AMA Style**

Diez-Jimenez E, Alén-Cordero C, Alcover-Sánchez R, Corral-Abad E.
Modelling and Test of an Integrated Magnetic Spring-Eddy Current Damper for Space Applications. *Actuators*. 2021; 10(1):8.
https://doi.org/10.3390/act10010008

**Chicago/Turabian Style**

Diez-Jimenez, Efren, Cristina Alén-Cordero, Roberto Alcover-Sánchez, and Eduardo Corral-Abad.
2021. "Modelling and Test of an Integrated Magnetic Spring-Eddy Current Damper for Space Applications" *Actuators* 10, no. 1: 8.
https://doi.org/10.3390/act10010008