# Characterization of LCR Parallel-Type Electromagnetic Shunt Damper for Superconducting Magnetic Levitation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Studies

#### 2.1. Modeling, Mechanism, and Governing Equations

_{0}= Acosωt on a shaking table, where A and w are the amplitude and the angular frequency of excitation, respectively. As the superconductor is cooled with liquid nitrogen, the magnet is held at a height z

_{0}from the surface of the superconductor, and the magnet, after being released, is balanced with its own weight at the initial levitation height z

_{st}. Let z be the vertical displacement of the magnet from that initial levitation position, and x be its relative displacement, as seen from the shaking table. k

_{1}denotes the spring constant obtained from a linear approximation of the electromagnetic force acting on the magnet. The mass and the magnetic moment of the magnet are denoted by m and M

_{m}, respectively, and the magnetic permeability of the vacuum is denoted by μ

_{0}. ϕ is an important parameter in this study, denoting the electromechanical coupling coefficient between the magnet and the electric circuit of the shunt damper. Figure 2 shows two types of LCR shunt circuits, series and parallel. i denotes the current flowing in the circuit; V

_{emf}is the electromotive force induced in the coil. L, R

_{L}, and C denote the inductance of the coil, the internal resistance, and the electric capacity of the capacitor, respectively. The external resistance is denoted by R

_{0}. In the optimum design of these circuit constants, the existence of this external resistance cannot be ignored in an actual system and can have a great influence.

_{1})

^{1/2}, A, and A(k

_{1}/L)

^{1/2}, respectively. The dimensionless governing equations for x and the current i can be written as follows for the series-type and parallel-type dampers [6].

#### 2.2. Numerical Integrations

#### 2.3. Some Discussions

## 3. Experimental Verification of Primary Resonance Suppression

#### 3.1. Experimental Setup and Measurement Method

#### 3.2. Experimental Results

_{o}= 0.15 Ω and R

_{o}= 99.4 Ω and when there is no damper. According to Figure 5, the reduction rate of the maximum amplitude with the use of the series-type damper compared to that without the damper is 74.3% with R

_{o}= 0.15 Ω, but the amplitude does not decrease with R

_{o}= 99.4 Ω. On the other hand, Figure 6 shows that the reduction rate of the maximum amplitude with the use of the parallel-type damper is 69.6% with R

_{o}= 0.15 Ω, and 80% even with R

_{o}= 99.4 Ω.

## 4. Numerical Prediction of Nonlinear Resonance Suppression

_{o}= 0.15 Ω, respectively. In both figures, (a) and (b) show the time history of the motion and its frequency analysis results, respectively. There is a marked difference between the time histories of the series type shown in Figure 7a and the parallel type shown in Figure 8a. According to the frequency analysis results in (b) of both figures, the latter shows a vibration waveform with a single frequency of 1.6, which is the excitation frequency, while the former consists of two vibration components, one with an excitation frequency of 1.6 and the other with a frequency of 8, which is half of the excitation frequency of 1.6. The vibration component at one half of the excitation frequency is caused by the nonlinearity of the electromagnetic force due to superconductivity. This is a type of nonlinear resonance called a subharmonic resonance. Thus, with a larger excitation amplitude, the nonlinear resonance can be confirmed near $\nu $ = 1.6 with the series-type damper. However, these nonlinear resonances of the subharmonic component were not found with the parallel-type damper. These numerical predictions imply a possible amplitude reduction in nonlinear resonance with the parallel-type damper.

## 5. Conclusions

- (1)
- With the series-type damper, the reduction rate of the resonance amplitude reached 74.3% with an external resistance ${R}_{o}$ = 0.15 Ω, while the reduction effect was not obtained with ${R}_{o}$ = 99.4 Ω.
- (2)
- With the parallel-type damper, the reduction rate of the resonance amplitude reached 69.6% with an external resistance value ${R}_{o}$ = 0.15 Ω, and a reduction rate of up to 80% was obtained even with R
_{O}= 99.4 Ω. Unlike the series-type damper, the experiments confirmed that the parallel-type damper can maintain the effect of reducing the resonance amplitude even if the external resistance of the circuit is larger. - (3)
- It was confirmed by numerical calculation that the parallel-type shunt damper can also be expected to reduce amplitude at resonances caused by the nonlinearity of the magnetic force. It was also confirmed by numerical calculations that the damping effect on subharmonic resonance when the external resistance ${R}_{o}$ is increased is reduced in the series type but maintained in the parallel type.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Analytical model of superconducting levitation system with an electromagnetic shunt damper.

**Figure 3.**Numerical result of relationship between nondimensional maximum amplitude X and external resistance value ${R}_{o}$, obtained with each of the series-type and parallel-type dampers and without damper.

**Figure 5.**Frequency response plots of the vibration amplitude of the permanent magnet obtained by experiments with the series type shunt damper.

**Figure 6.**Frequency response plots of the vibration amplitude of the permanent magnet obtained by experiments with the parallel-type shunt damper.

**Figure 7.**Numerical results of the motion of the magnet above the superconductor excited with the amplitude $A=1.0$ mm and frequency $\nu =1.6$ using the series-type shunt damper, showing subharmonic resonance. (

**a**) Time history; (

**b**) FFT.

**Figure 8.**Numerical results of the motion of the magnet above the superconductor excited with the amplitude $A=1.0$ mm and frequency $\nu =1.6$ using the parallel-type shunt damper, showing no subharmonic resonance. (

**a**) Time history; (

**b**) FFT.

**Figure 9.**Numerical result of the relationship between maximum amplitude of the vibration component at half the excitation frequency and external resistance value R

_{o}obtained with each of the series-type and parallel-type dampers.

m [kg] | ${c}_{M}\text{}$[N·s/m] | ${M}_{m}\text{}$[Wb·m] | $A$ [mm] |

0.0335 | 0.423 | $1.01\times {10}^{-6}$ | 0.1 |

${z}_{0}$ [mm] | ${z}_{st}$ [mm] | $L$ [H] | ${R}_{L}$$\text{}\left[\mathsf{\Omega}\right]$ |

$10.0$ | $9.45$ | $1.50\times {10}^{-3}$ | 0.500 |

$C$ [mF] | ${\mu}_{0}$ [H/m] | $\varphi $$\text{}$[V·s/m] | |

$20.8$ | $1.26\times {10}^{-6}$ | 1.18 |

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**MDPI and ACS Style**

Fujita, K.; Sugiura, T.
Characterization of LCR Parallel-Type Electromagnetic Shunt Damper for Superconducting Magnetic Levitation. *Actuators* **2022**, *11*, 216.
https://doi.org/10.3390/act11080216

**AMA Style**

Fujita K, Sugiura T.
Characterization of LCR Parallel-Type Electromagnetic Shunt Damper for Superconducting Magnetic Levitation. *Actuators*. 2022; 11(8):216.
https://doi.org/10.3390/act11080216

**Chicago/Turabian Style**

Fujita, Kentaro, and Toshihiko Sugiura.
2022. "Characterization of LCR Parallel-Type Electromagnetic Shunt Damper for Superconducting Magnetic Levitation" *Actuators* 11, no. 8: 216.
https://doi.org/10.3390/act11080216