# Analysis of Magnetic Field Characteristics of a Giant Magnetostrictive Actuator with a Semi-Closed Magnetic Circuit

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Principles and Methods

#### 2.1. Basic Theory of Model Solving

^{2}; ρ is the charge density, and its unit is C/m

^{3}; B is the magnetic flux density vector, and its unit is Wb/m

^{2}; E is the electric field intensity vector, for which the unit is V/m; H is the magnetic density vector, unit A/m; J is the current density vector, unit A/m

^{2}; $\nabla $ is the Hamiltonian operator, in the three-dimensional axisymmetric problem $\nabla =\left(\frac{\partial}{\partial r},\frac{\partial}{\partial \phi},\frac{\partial}{\partial z}\right)$.

#### 2.2. COMSOL Multiphysics Finite Element Model Establishment

#### 2.2.1. Finite Element Model

_{r}is the relative magnetic permeability of the material, and the relative magnetic permeability of the disc spring can be ignored [19]. The u

_{r}value of the displacement conversion mechanism and the actuator shell (composed of preload bolt, outer shell, upper-end cover, and lower-end cover) were changed, respectively, and the influence of the material properties of the two on the distribution of the axial magnetic density of the GMM cylinder was analyzed and discussed.

#### 2.2.2. Setting Boundary Conditions and Loads

#### 2.2.3. Meshing

#### 2.2.4. Model Solution and Postprocessing

#### 2.3. Establishing a Coordinate System

#### 2.4. Solving the Cross-Sectional Magnetic Density

_{i}, and the magnetic density component is H

_{i}.

## 3. Results and Discussion

#### 3.1. Simulation Verification

#### 3.1.1. Influence of Displacement Conversion Mechanism on Magnetic Field Distribution Characteristic

_{r}of the shell material was set to 2000. The relative magnetic permeability of the displacement conversion mechanism was changed, the magnetic density of each section was obtained, respectively, and the data fitting was performed on the obtained results to obtain the axial magnetic density distribution of the GMM cylinder shown in Figure 10. Figure 11 shows the change in the magnetic density of the GMM cylinder near the output end, the middle part, and the near base end with the relative permeability of the displacement conversion mechanism.

_{r}, the magnetic density of the GMM cylinder near the output end, the middle part, and the near base end first increased and then decreased: the magnetic density near the output end reached the maximum value of 5.355 × 10

^{4}A/m around u

_{r}= 30; the magnetic density at the middle reached the maximum value of 4.553 × 10

^{4}A/m near u

_{r}= 20; the magnetic density near the base end reached the maximum value of 2.93 × 10

^{4}A/m near u

_{r}= 40. Through simulation analysis, it was found that when the displacement conversion mechanism u

_{r}< 100, the magnetic density on the cylinder changed greatly. In the range of 0 to 100, the variation amplitude of magnetic density could reach 1.0 × 10

^{4}A/m; when u

_{r}≥ 100, the magnetic density changed relatively gently, which is because the displacement conversion mechanism is nested with the GMM cylinder, and the relative permeability of the GMM cylinder is small (u

_{r}= 20). Therefore, when the relative magnetic permeability of the displacement conversion mechanism changed around 20, the influence on the distribution of magnetic density was more significant, and when the relative permeability of the displacement conversion mechanism exceeded the critical point (u

_{r}= 100), the influence on the distribution of the magnetic density on the cylinder weakened.

_{A}represents the magnetic density value of the middle section of the GMM cylinder; H

_{max}and H

_{min}represent the maximum magnetic density value near the output end of the GMM cylinder and the minimum magnetic density value near the base end, respectively. p

_{1}represents the degree of deviation of the magnetic density near the output end, and p

_{2}represents the degree of deviation of the magnetic density near the base end. The changes in the two with the relative permeability of the displacement conversion mechanism are shown in Figure 12. Through simulation, it was found that with the increase in u

_{r}, the magnetic density deviation at the base end first decreased and then increased, and the minimum value was 0.3026 when u

_{r}= 90, and the change range was large; the magnetic density deviation near the output end increased monotonically, and the increase was small, which is related to the displacement conversion mechanism and the magnetic permeability of the material at both ends of the cylinder. When u

_{r}= 20, the average deviation of the magnetic density between the two achieved the minimum value of 0.28345.

_{r}= 1, the magnetic permeability of the material was the same as that of air; when u

_{r}> 1, the displacement conversion mechanism was nested with other components of the GMA, affecting the distribution of the magnetic density.

_{r}≈ 10, the magnetic density on the GMM cylinder was larger, and the distribution uniformity was good. Based on this conclusion, when discussing the influence of the shell u

_{r}on the magnetic density of the GMM cylinder, the displacement conversion mechanism material u

_{r}was set to 10 in the COMSOL Multiphysics simulation software.

#### 3.1.2. Influence of Shell on Magnetic Field Distribution Characteristic

_{r}< 45, the magnetic density near the output end was smaller than that in the middle part; when u

_{r}≥ 45, the magnetic density near the output end exceeded the middle part; the magnetic density near the base end was obviously lower than that of the middle part and the magnetic density near the output end.

_{r}of the shell, the magnetic density on the GMM cylinder increased accordingly. This is because the torsional effect of the shell on the magnetic field lines is continuously strengthened, and more magnetic field lines are closed in the device loop. In particular, when u

_{r}< 90, the magnetic density on the cylinder increased greatly: the magnetic density near the output increased from 6.678 × 10

^{3}A/m to 4.673 × 10

^{4}A/m; the magnetic density in the middle increased from 2.431 × 10

^{4}A/m to 4.237 × 10

^{4}A/m; the magnetic density near the base increased from 1.045 × 10

^{4}A/m to 2.284 × 10

^{4}A/m.

_{r}on the deviation of the axial magnetic density of the GMM cylinder is described by referring to the definition in Equation (8). However, because under different shell u

_{r}conditions, the maximum values near the output end were not all larger than the magnetic density at the middle part, so the deviation of the axial magnetic density is defined as

_{1}and p

_{2}are the deviations of the magnetic density near the output end and the near base end, respectively, and the changes in the two with the relative permeability of the shell are shown in Figure 15. It can be seen from the simulation results that when u

_{r}< 5, the deviation of the magnetic density at both ends of the GMM cylinder was larger, and the deviation of the magnetic density at the near output end was greater than that at the near base end. As u

_{r}increased, the magnetic density at both ends decreased rapidly, and the uniformity of the magnetic field was enhanced. As the relative magnetic permeability of the displacement conversion mechanism was small, and the near-base end of the GMM cylinder was in contact with the head of the “T”-type plunger, the magnetic density near the base end was small. Therefore, when the shell had u

_{r}≥ 5, the deviation of the magnetic density near the base end was much larger than that near the output end. The deviation of the average magnetic density at both ends had the minimum value of 0.26922 when u

_{r}= 2000.

_{r}= 1, the magnetic permeability of the material was the same as that of air; it is worth noting that the internal medium of the device was connected with the external air medium; when u

_{r}> 1, the shell worked with other components of GMA to separate the inner space of GMA from the external air medium, which affected the distribution of the magnetic density on the GMM cylinder.

#### 3.1.3. Influence of Air Gap Geometry on Axial Magnetic Density of GMM Cylinder

^{4}A/m to 2.402 × 10

^{4}A/m.

#### 3.2. Experimental Verification

#### 3.2.1. Experimental System

#### 3.2.2. Comparison between Simulation Results and Experimental Results

_{H}.

_{S}is the simulation result, and H

_{E}is the experimental test result. Through calculation, it was found that the maximum relative error appeared at the fourth measuring point, which was about 8.5%, and the relative errors of other measuring points were all less than 5%. Therefore, it can be considered that the simulation analysis carried out in this paper can accurately reflect the real physical phenomena and verify the correctness of the established simulation model.

## 4. Conclusions

- (1)
- By increasing the relative permeability of the transfer mechanism, the magnetic field intensity on the GMM cylinder first increased and then decreased, whereas the deviation of the average magnetic field intensity at both ends first decreased and then increased. In order to make the axial magnetic field intensity of the GMM cylinder larger and more uniform, the displacement conversion mechanism should be made of materials with low magnetic permeability.
- (2)
- When the relative permeability of the shell increased, the magnetic field density on the GMM cylinder increased monotonously, and the deviation of magnetic field density at both ends decreased monotonously. In order to make the axial magnetic field intensity of the GMM cylinder large and uniform, the device shell should be made of high-permeability material.
- (3)
- With the increase in air gap size, its constraint on magnetic field lines weakened, the magnetic field density on the GMM cylinder decreased, and the deviation degree of magnetic field density increased. Therefore, in order to improve the utilization efficiency of the excitation magnetic field, the geometric size of the air gap should be reduced as much as possible under the condition of meeting the pretightening force requirements.
- (4)
- The experimental test results showed that the established simulation model can correctly reflect the actual physical characteristics of GMA, and it was also verified that it is feasible to use the averaging method based on magnetic circuit theory to solve the magnetic field intensity of GMM cylinder axial section.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 7.**(

**a**) The magnetic field intensity on an axis of the cylinder taken as the research object; (

**b**) the magnetic field intensity of a section on the cylinder taken as the research object.

**Figure 8.**Radial magnetic field distribution of the cylinder section at 50% of the total length of the GMM cylinder.

**Figure 10.**The axial magnetic density distribution of the GMM cylinder by changing the relative permeability of the displacement conversion mechanism. (

**a**) u

_{r}=1, 5, 10, 20, 40, 100, (

**b**) u

_{r}=100, 200, 500, 1000, 2000.

**Figure 11.**Change in magnetic density of GMM cylinder by changing the relative permeability of the displacement conversion mechanism.

**Figure 12.**Influence of relative permeability of displacement conversion mechanism on deviation of axial magnetic density of GMM cylinder.

**Figure 13.**The axial magnetic density distribution of the GMM cylinder by changing the relative permeability of the shell. (

**a**) u

_{r}=1, 5, 10, 20, 40, 100, (

**b**) u

_{r}=100, 200, 500, 1000, 2000.

**Figure 14.**Changes in the magnetic density of the GMM cylinder by changing the relative permeability of the shell.

**Figure 15.**Influence of relative permeability of shell on deviation of axial magnetic density of GMM cylinder.

**Figure 16.**The axial magnetic density distribution of the GMM cylinder by changing the air gap size.

**Figure 23.**Comparison of magnetic field intensity between simulation and experiment: (

**a**) Z = 0 mm; (

**b**) Z = 10 mm; (

**c**) Z = 20 mm; (

**d**) Z = 30 mm; (

**e**) Z = 40 mm; (

**f**) Z varied from 0 mm to 40 mm.

NO. | Structure Name | Material u_{r} |
---|---|---|

1 | Preload bolt | 2000 |

2 | Upper-end cover | 2000 |

3 | Disc spring | — |

4 | Outer shell | 2000 |

5 | Excitation coil | 1 |

6 | Coil skeleton | 1 |

7 | Air gap | 1 |

8 | GMM cylinder | 20 |

9 | “T” plunger | 10 |

10 | Lower end cover | 2000 |

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**MDPI and ACS Style**

Zhou, Z.; He, Z.; Xue, G.; Zhou, J.; Rong, C.; Liu, G.
Analysis of Magnetic Field Characteristics of a Giant Magnetostrictive Actuator with a Semi-Closed Magnetic Circuit. *Actuators* **2022**, *11*, 108.
https://doi.org/10.3390/act11040108

**AMA Style**

Zhou Z, He Z, Xue G, Zhou J, Rong C, Liu G.
Analysis of Magnetic Field Characteristics of a Giant Magnetostrictive Actuator with a Semi-Closed Magnetic Circuit. *Actuators*. 2022; 11(4):108.
https://doi.org/10.3390/act11040108

**Chicago/Turabian Style**

Zhou, Zhaoqi, Zhongbo He, Guangming Xue, Jingtao Zhou, Ce Rong, and Guoping Liu.
2022. "Analysis of Magnetic Field Characteristics of a Giant Magnetostrictive Actuator with a Semi-Closed Magnetic Circuit" *Actuators* 11, no. 4: 108.
https://doi.org/10.3390/act11040108