# Study of an Energy-Harvesting Damper Based on Magnetic Interaction

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Approach

_{2}= (E/R) flowing along the circuit, and the absorbed power should be W = IE. This absorbed energy is provided by the kinetics energy of the suspension system and should give rise to dissipated heat and regenerative braking through the force exerted by the magnetic field produced by I

_{2}on the magnet. Therefore, the maximum absorbed power takes place at the position of the secondary coil for which (dϕ/dt)) becomes maximum.

- Only in a few cases can the field produced by the magnet in the points forming the surface of the secondary coil be analytically expressed.
- The calculation of the flux would require knowing the analytical expression of the field for any point of the coil surface.
- During the magnet motion, the magnetic field in those exceptional positions admitting an analytical formulation loses this capability when changing position.
- Moreover, the oscillation of the magnet far from harmonic becomes stochastic in amplitude and frequency, making the calculation of the time derivative impossible.

_{1,2}. The calculation of M

_{1,2}leads to a solution formed by elliptical functions that only can be calculated by numerical methods. This case coincides with the case of a magnetic suspension, in which the magnet magnetic moment lies along the z-axis and the normal direction of the secondary coil is also oriented along the same z-axis and is placed coaxially with the magnet (Figure 1). Notice that outside the magnet the magnetic field geometry is analogous to that produced by a coil located at the magnet position and with the same magnetic moment.

_{1}and area s

_{1}. The component of the magnetic field produced by the primary coil along the normal direction to the secondary coil can be expressed at any point of s

_{2}as B = μ

_{0}I

_{1}f (z,r), where f (z,r) accounts for the dependence of the magnetic field with the distance between the coils and the relative orientation. Let us suppose that the primary coil (or the magnet) fixed to the suspension system oscillates around the point z* with amplitude z

_{0}and frequency ω

_{2}. Therefore, z = z*+ z

_{0}cos ω

_{2}t,

_{0}I

_{1}f (z,r) when I

_{1}is constant, the change of the magnetic field component at any point of the surface of the secondary coil could be only due to changes in the relative position of the two coils, that is, changes of z or r. Consider that, because of the cylindrical symmetry, the magnetic field component contributing to the flux through the secondary coil does not change under rotation around the z-axis, i.e., it is independent of the ϕ angle. The flux crossing the secondary coil becomes

_{0}If (z,r), but z and r are fixed; then, the flux through the secondary is given by:

## 3. Experimental Method

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A

_{φ}created by a circular coil with radius r

_{0}carrying a current of intensity I

_{0}, which at any point is located with respect to the coil at r and z in cylindrical coordinates, is given by

_{φ}

_{0}cos ω

_{2}t, a comparison with relation (2) yields

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**Figure 12.**(

**a**) Gradients Test 4.1; (

**b**) Gradients Test 4.2; (

**c**) Gradients Test 4.3; (

**d**) Gradients Test 4.4; (

**e**) Gradients Test 4.5; (

**f**) Gradients Test 4.6.

CASE 1. AC COIL-COIL SYSTEM | ||
---|---|---|

$B={\mu}_{0}I\left(t\right)f\left(z,r\right)$ | $\varphi =N{\mu}_{0}{I}_{0}cos\left({\omega}_{1}t\right)g\left(z,r\right)$ | $FEM=N{\mu}_{0}{I}_{0}{\omega}_{1}sin{\omega}_{1}tg\left(z,r\right)$ |

CASE 2. DC COIL-COIL SYSTEM | ||
---|---|---|

$B={\mu}_{0}If\left(z\left(t\right),r\right)$ | $\varphi =N{\mu}_{0}Ig\left(z,r\right)$ | $FEM=N{\mu}_{0}I{\omega}_{2}{z}_{0}sin{\omega}_{2}t\frac{dg\left(z,r\right)}{dz}$ |

MAXIMUM GRADIENT OF EACH TEST | |||||
---|---|---|---|---|---|

TEST 4.1 | TEST 4.2 | TEST 4.3 | TEST 4.4 | TEST 4.5 | TEST 4.6 |

1844 | 13,326 | 11,678 | 9602 | 12,774 | 13,566 |

GRADIENTS TEST 6 | |||
---|---|---|---|

dV (mV) | dA (cm) | dV/dA | |

5458 | 0.5 | 10,916 | |

6783 | 0.5 | 13,566 | |

TEST 4.6. B = 15 cm; A = 18 cm | |||

C (cm) | CHANNEL ONE (mV) | CHANNEL TWO (°) | REFERENCE (kHz) |

0 | 2.03 | −101.08 | 1 |

0.5 | 7488 | 83.34 | 1 |

1 | 14,271 | 82.88 | 1 |

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**MDPI and ACS Style**

Aberturas, S.; Hernando, A.; Olazagoitia, J.L.; García, M.Á.
Study of an Energy-Harvesting Damper Based on Magnetic Interaction. *Sensors* **2022**, *22*, 7865.
https://doi.org/10.3390/s22207865

**AMA Style**

Aberturas S, Hernando A, Olazagoitia JL, García MÁ.
Study of an Energy-Harvesting Damper Based on Magnetic Interaction. *Sensors*. 2022; 22(20):7865.
https://doi.org/10.3390/s22207865

**Chicago/Turabian Style**

Aberturas, Susana, Antonio Hernando, José Luis Olazagoitia, and Miguel Ángel García.
2022. "Study of an Energy-Harvesting Damper Based on Magnetic Interaction" *Sensors* 22, no. 20: 7865.
https://doi.org/10.3390/s22207865