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

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## 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

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**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 |

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**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