# Research on Magnetic Characteristics and Fuzzy PID Control of Electromagnetic Suspension

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Structure and Principle

#### 2.1. Design of the Electromagnetic Actuator

_{9}(3

^{2}× 3

^{2}) size optimization simulations of the actuator by the orthogonal experiment method are shown in Table 2.

_{9}(3

^{2}× 3

^{2}) size) is shown in Table 3. As can be observed, the combination (A3B3C2) is the optimal parameters, which are the height of large/small PM rings (12.5 mm), the radial length of large/small PM rings (12 mm), and the height of coils (9.5 mm).

#### 2.2. Sizes and Principles of the Electromagnetic Actuator

## 3. Model and Analysis of the Magnetic Field

#### 3.1. Magnetic Flux Density Model of a Single PM Ring

_{y}

_{11}and B

_{y}

_{12}generated by the surface currents 1 and 2 can be expressed as (2) and (3) [41,42,43]:

#### 3.2. Magnetic Flux Density Model of the Magnet Assembly

_{y}in the air gap can be expressed as (4) [43]:

#### 3.3. Theoretical Analysis of Magnetic Flux Density

_{y}is shown as regular fluctuations, and its value varies between 51.73 and 788.49 mT.

## 4. FEM Analysis of the Magnetic Field

#### 4.1. FEM Analysis Model

#### 4.2. FEM Analysis on the Magnetic Field

^{−3}T to 2.500 T. From the magnetic flux density results in the radial direction, it can be seen that the magnetic flux density in the air gap between the large PM rings and the small PM rings is about 550 mT, and the magnetic flux density at both ends is about 670 mT. The magnetic flux density of the large and small heat-dissipated rings is about 1.9 T. The magnetic flux density of the heat-dissipated rings in the middle of the same-size PM rings in the axial direction is relatively large.

^{−3}T to 2.500 T, which is the same as in Figure 6a. The results show that the magnetic flux between the large PM rings and the small PM rings is perpendicular to the axial direction of the electromagnetic actuator, and the directions of the multiple magnetic fluxes are parallel to each other. There will be divergence at the edge of each PM ring. Therefore, a magnetic field will be generated at the heat-dissipated rings. In theoretical calculations, the edge effects of the PM rings are assumed to be 0 mT, and this is reflected in the FEM simulation.

^{−4}T to 2.024 × 10

^{−4}T. The results show that the parallel magnetic field between the large PM rings and the small PM rings is visible, and the magnetic flux in the large heat-dissipated rings and the small heat-dissipated rings is repulsive; in the simulation boundary, due to the inevitable phenomenon of magnetic leakage, the magnetic flux must exist outside the electromagnetic actuator.

#### 4.3. FEM Analysis of the Magnetic Flux Density

#### 4.4. FEM Analysis on Magnetic Force

## 5. Vehicle Simulation Based on the Fuzzy PID Algorithm

#### 5.1. Vehicle Dynamics Model

_{iA}, F

_{iB}, F

_{iC}, and F

_{iD}are generated.

#### 5.2. Fuzzy PID Control Algorithm

^{2}, and the input of the fuzzy controller is the deviation e and the variation of the deviation ec. Based on fuzzy control rules, PID parameters are adjusted online to meet different requirements for PID parameters with different deviations e and incremental deviations ec. The simulation employs a variable-step solver with a maximum sampling step of 0.01 s, utilizing the Ode45 solver algorithm. The simulation is set to run for a duration of 5 s. The actives of the simulation are centroid acceleration, acceleration of the roll angle, acceleration of the pitch angle, dynamic deflections for four wheels, and dynamic loads for four wheels.

#### 5.3. Vehicle Simulation Results Based on Fuzzy PID under C-Grade Surface

_{p}= 1, k

_{i}= 20, and k

_{d}= 0.01.

^{2}less than that of the passive suspension. The acceleration of the pitch angle curve is shown in Figure 15. At 1.96 s, the maximum value of the electromagnetic suspension is 0.153 rad/s

^{2}less than that of the passive suspension.

#### 5.4. Vehicle Simulation Results Based on Fuzzy PID under a Deceleration Strip Surface

_{p}= 0.8, k

_{i}= 50, and k

_{d}= 0.001. This simulation analysis can provide a detailed analysis of the pitch and roll characteristics to showcase the advantages of the electromagnetic suspension.

^{2}smaller than when using passive suspension. In the simulation results of the acceleration of the roll angle, there is a slight delay, with a delay time of 0.04 s. The peak values of the accelerations of the roll angle and the pitch angle decreased by 0.319 rad/s

^{2}and 0.015 rad/s

^{2}, respectively.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Structure of the electromagnetic suspension system. (

**a**) 1/4 MacPherson-type independent suspension. (

**b**) Electromagnetic shock absorber.

**Figure 3.**Dimensional drawing. (

**a**) Sectional view in the radial direction. (

**b**) Sectional view in the axial direction.

**Figure 6.**FEM analysis of magnetic field for half of the actuator. (

**a**) The magnetic flux density. (

**b**) The direction of magnetic flux. (

**c**) The trend of magnetic flux.

**Figure 9.**Simulation results of magnetic force characteristics. (

**a**) The current–magnetic force curve. (

**b**) The displacement–magnetic force curve.

Level | Height of Large/Small PM Rings (mm) A | Radial Length of Large/Small PM Rings (mm) B | Height of Coils (mm) C |
---|---|---|---|

1 | 10.5 | 10 | 5.5 |

2 | 11.5 | 11 | 9.5 |

3 | 12.5 | 12 | 12.5 |

**Table 2.**L

_{9}(3

^{2}× 3

^{2}) size optimization simulations of the actuator by the orthogonal experiment method.

Number | Height of Large/Small PM Rings (mm) | Radial Length of Large/Small PM Rings (mm) | Height of Coils (mm) | Magnetic Force (N) |
---|---|---|---|---|

1 | A1 | B1 | C1 | 156 |

2 | A1 | B2 | C2 | 171 |

3 | A1 | B3 | C3 | 156 |

4 | A2 | B1 | C2 | 135 |

5 | A2 | B2 | C3 | 159 |

6 | A2 | B3 | C1 | 171 |

7 | A3 | B1 | C3 | 165 |

8 | A3 | B2 | C1 | 150 |

9 | A3 | B3 | C2 | 180 |

Value Name | Height of Coil_{2}(mm) | Thickness of Coil_{2}(mm) | Height of Soft Iron Ring_{1}(mm) |
---|---|---|---|

K_{j}_{1} | 543 | 516 | 537 |

K_{j}_{2} | 525 | 540 | 546 |

K_{j}_{3} | 555 | 567 | 540 |

K_{jp}_{1} | 181 | 172 | 179 |

K_{jp}_{2} | 175 | 180 | 182 |

K_{jp}_{3} | 185 | 189 | 180 |

R_{j} | 10 | 17 | 3 |

Primary and secondary order | B > A > C | ||

Optimal levels | A3 | B3 | C2 |

Optimal combination | A3B3C2 |

_{jk}(k = 1, 2, 3) is the sum of the simulation results with the same level k in the j

^{th}column. K

_{jpk}is the mean value of the simulation results with the same level k in the j

^{th}column. R

_{j}is the range of K

_{jpk}.

Size Parameter | Description | Value (Unit: mm) |
---|---|---|

H_{c} | Axial length of the coils | 9.5 |

H_{m} | Axial length of the PM ring | 12.5 |

H_{hd} | Axial length of the heat-dissipated ring | 3 |

H_{whole} | Axial length of the actuator | 96 |

D_{whole} | Outside diameter of the actuator | 80.6 |

D_{cmax} | Outside diameter of the coils | 62.3 |

D_{cmin} | Inside diameter of the coils | 56.6 |

D_{smax} | Outside diameter of the small PM rings | 52.6 |

D_{smin} | Inside diameter of the small PM rings | 40.6 |

D_{bmax} | Outside diameter of the large PM rings | 76.6 |

D_{bmin} | Inside diameter of the large PM rings | 64.6 |

Variable | Description |
---|---|

Q(a, b) | Horizontal and vertical coordinates of any current source |

P(x, y) | Horizontal and vertical coordinates of any point in a plane |

j | Ring number from left to right, j = 1–6 |

θ_{1}, θ_{2} | Angle between the upper and lower points of surface current 1 |

θ | Angle between the line of point Q and point P and the horizontal line |

r | Distance between point Q and point P |

B_{yj}_{1} | Magnetic flux density of surface current 1 |

B_{yj}_{2} | Magnetic flux density of surface current 2 |

B_{y}_{1} | Magnetic flux density of large PM rings |

B_{y}_{2} | Magnetic flux density of small PM rings |

B_{y} | Magnetic flux density of the PM ring array |

Variable | Description |
---|---|

x_{b} | Displacement of body centroid |

m_{b} | Mass of body centroid |

θ | Pitch angle |

φ | Roll angle |

a | Distance from the center of mass to the front axle |

b | Distance from the center of mass to the rear axle |

t_{f} | 1/2 front track |

t_{r} | 1/2 rear track |

m_{wA} | Mass of the left front wheel |

m_{wB} | Mass of the right front wheel |

m_{wC} | Mass of the left rear wheel |

m_{wD} | Mass of the right rear wheel |

x_{wA} | Displacement of the left front wheel |

x_{wB} | Displacement of the right front wheel |

x_{wC} | Displacement of the left rear wheel |

x_{wD} | Displacement of the right rear wheel |

k_{tA} | Stiffness of the left front wheel |

k_{tB} | Stiffness of the right front wheel |

k_{tC} | Stiffness of the left rear wheel |

k_{tD} | Stiffness of the right rear wheel |

k_{sA} | Stiffness of the left front wheel spring |

k_{sB} | Stiffness of the right front wheel spring |

k_{sC} | Stiffness of the left rear wheel |

k_{sD} | Stiffness of the right rear wheel |

c_{sA} | Left front wheel damping coefficient |

c_{sB} | Right front wheel damping coefficient |

c_{sC} | Left rear wheel damping coefficient |

c_{sD} | Right rear wheel damping coefficient |

x_{gA} | Left front wheel road displacement |

x_{gB} | Right front wheel road displacement |

x_{gC} | Left rear wheel road displacement |

x_{gD} | Right rear wheel road displacement |

x_{bA} | Left front wheel spring mass displacement |

x_{bB} | Right front wheel spring mass displacement |

x_{bC} | Left rear wheel sprung mass displacement |

x_{bD} | Right rear wheel sprung mass displacement |

Parameters (Unit) | Value | Parameters (Unit) | Value |
---|---|---|---|

m_{s} (kg) | 1836 | c_{sA} (N/(m/s)) | 1800 |

m_{wA} (kg) | 50 | c_{sB} (N/(m/s)) | 1800 |

m_{wB} (kg) | 50 | c_{sC} (N/(m/s)) | 2000 |

m_{wC} (kg) | 50 | c_{sD} (N/(m/s)) | 2000 |

m_{wD} (kg) | 50 | k_{tA} (N/m) | 230,000 |

I_{p} (kg·m^{2}) | 3411 | k_{tB} (N/m) | 230,000 |

I_{r} (kg·m^{2}) | 676 | k_{tC} (N/m) | 230,000 |

k_{sA} (N/m) | 57,000 | k_{tD} (N/m) | 230,000 |

k_{sB} (N/m) | 57,000 | a (m) | 1.455 |

k_{sC} (N/m) | 64,000 | b (m) | 1.514 |

k_{sD} (N/m) | 64,000 | t_{l} (m)/t_{r} (m) | 0.805 |

U | Ec | |||||||
---|---|---|---|---|---|---|---|---|

NB | NM | NS | Z | PS | PM | PB | ||

E | NB | PB | PB | PM | PM | PS | Z | Z |

NM | PB | PB | PM | PS | PS | Z | NS | |

NS | PM | PM | PS | PS | Z | NS | NS | |

Z | PM | PM | PS | Z | NS | NM | NM | |

PS | PS | PS | Z | NS | NS | NM | NM | |

PM | PS | Z | NS | NM | NM | NM | NB | |

PB | Z | Z | NM | NM | NM | NB | NB |

U | Ec | |||||||
---|---|---|---|---|---|---|---|---|

NB | NM | NS | Z | PS | PM | PB | ||

E | NB | PB | PB | PM | PM | PS | Z | Z |

NM | PB | PB | PM | PS | PS | Z | NS | |

NS | PM | PM | PS | PS | Z | NS | NS | |

Z | PM | PM | PS | Z | NS | NM | NM | |

PS | PS | PS | Z | NS | NS | NM | NM | |

PM | PS | Z | NS | NS | NM | NB | NB | |

PB | Z | Z | NM | NM | NM | NB | NB |

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## Share and Cite

**MDPI and ACS Style**

Wei, W.; Yu, S.; Li, B.
Research on Magnetic Characteristics and Fuzzy PID Control of Electromagnetic Suspension. *Actuators* **2023**, *12*, 203.
https://doi.org/10.3390/act12050203

**AMA Style**

Wei W, Yu S, Li B.
Research on Magnetic Characteristics and Fuzzy PID Control of Electromagnetic Suspension. *Actuators*. 2023; 12(5):203.
https://doi.org/10.3390/act12050203

**Chicago/Turabian Style**

Wei, Wei, Songjian Yu, and Baozuo Li.
2023. "Research on Magnetic Characteristics and Fuzzy PID Control of Electromagnetic Suspension" *Actuators* 12, no. 5: 203.
https://doi.org/10.3390/act12050203