# Design of a Noncontact Torsion Testing Device Using Magnetic Levitation Mechanism

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Structure and Working Principle

#### 2.1. Structure Overview

#### 2.2. Principle of Levitation

#### 2.3. Principle of Torque Application

## 3. Control of X and RY

#### 3.1. Plant Model

#### 3.1.1. State Space Model Derivation

_{1}, z

_{2}, z

_{3}, and z

_{4}are the four airgaps between the electromagnets and the iron plates. O is the centroid of the levitated object. M is the point on the surface of the iron plate at the same vertical height as the midpoint of z

_{1}and z

_{2}. Obviously, ${z}_{M}=\frac{{z}_{1}{+z}_{3}}{2}$. l is the vertical distance between the lower and upper attractive-type permanent magnetic bearings. h is the vertical distance between the lower and upper electromagnets. In Figure 6b, F

_{1}, F

_{2}, F

_{3}, and F

_{4}denote the attractive forces of the four electromagnets on the iron plates, respectively. F

_{P}and T

_{P}, respectively denote the resultant force and resultant torque of the four attractive-type permanent magnetic bearings on the levitated object.

_{1}, F

_{2}, F

_{3}, and F

_{4}are shown in Equation (1), where a and c are two coefficients depending on the shape and dimensions of the electromagnets and the iron plates, and i

_{1}, i

_{2}, i

_{3}, and i

_{4}are the currents of the four electromagnets, respectively. As shown in Equation (2), F

_{1}, F

_{2}, F

_{3}, and F

_{4}can be linearized at a working point (i

_{0}, z

_{0}), where i

_{0}and z

_{0}are a constant current and a constant airgap, respectively. In Equation (2), F

_{0}denotes the magnetic force when the current and airgap are just i

_{0}and z

_{0}, and $\mathsf{\Delta}{i}_{1}={i}_{1}-{i}_{0}$, $\mathsf{\Delta}{i}_{2}={i}_{2}-{i}_{0},\text{}\mathsf{\Delta}{i}_{3}={i}_{3}-{i}_{0},\mathsf{\Delta}{i}_{4}={i}_{4}-{i}_{0},{z}_{1}={z}_{1}-{z}_{0}$, $\mathsf{\Delta}{z}_{2}={z}_{2}-{z}_{0},\text{}\mathsf{\Delta}{z}_{3}={z}_{3}-{z}_{0},\mathsf{\Delta}{z}_{4}={z}_{4}-{z}_{0}$, in addition, k

_{i}and k

_{z}are the current coefficient and airgap coefficient, respectively. Their expressions are shown in Equation (3), which was derived by calculating the derivative of the magnetic force (Any of F

_{1}, F

_{2}, F

_{3,}and F

_{4}) with respect to the current (Any of i

_{1}, i

_{2}, i

_{3,}and i

_{4}) and the airgap (Any of z

_{1}, z

_{2}, z

_{3,}and z

_{4}) at the working point (i

_{0}, z

_{0}).

_{P}and T

_{P}can be regarded as two generalized forces with stiffness, so Equation (4) can be rewritten as Equation (5), where k

_{a}and k

_{r}are the stiffnesses of F

_{P}and T

_{P,}respectively.

#### 3.1.2. Determination of Plant Model Parameter

_{i}, k

_{z}, k

_{r}, and k

_{a}. Among them, m, J, h, and l can be easily calculated based on the densities, dimensions, and shapes of the solids; their values are 1.731 kg, 3.3 × 10

^{−3}kg·m

^{2}, 0.143 m, and 0.0774 m, respectively. k

_{i}and k

_{z}can be calculated using Equation (3), however, the calculation requires the values of a and c. Therefore, the attractive force of a single electromagnet to an iron plate was calculated at various combinations of current and airgap by FEM analysis. After the analysis, the result data were imported into the Curve Fitting Toolbox of MATLAB to obtain a fitting surface, which was generated in the form of $F=\frac{{\mathit{ai}}^{2}}{{(z+c)}^{2}}$, which is the general equation of electromagnetic force. The generated fitting surface is shown in Figure 7, where the black point is the original data. With the generation of the fitting surface, a and c were also calculated out by the toolbox; their values were 1.204 × 10

^{−5}Nm

^{2}A

^{−2}, 2.38 × 10

^{−3}m, respectively. Furthermore, i

_{0}and z

_{0}were set as 2 A and 3 mm, and then k

_{i}and k

_{z}were calculated using Equation (3); their values were 1.6639 N/A and 618.5426 N/m, respectively.

_{r}and k

_{a}, another FEM analysis was conducted on the four attractive-type permanent magnetic bearings. In the FEM analysis, the four permanent magnets in the bearing holder were set to be static, and the four permanent magnets embedded in the iron plate were set to be movable; then, F

_{P}and T

_{P}are calculated under various lateral displacements (z

_{M}-z

_{0}) and rotation angles (θ) of the four permanent magnets embedded in the iron plate. The results are shown in Figure 8. Figure 8a shows the result of F

_{P}, and Figure 8b shows the result of T

_{P}. It can be drawn from the slope of the red dashed line in both graphs that the stiffnesses k

_{a}and k

_{r}are approximately +12.5 N/mm and +0.133 N·m/° in the vicinity of the initial point (abscissa = 0). So far, all the plant model parameters have been caught, and the main parameter values are shown in Table 3.

#### 3.2. Control Design

_{1}, D

_{1}, P

_{2}, and D

_{2}can be regarded as two PD-gains of two independent PD-controllers. If the feedback matrix is used, a closed-loop system matrix will be as follows.

_{1}and ζ

_{2}denote the damping ratios of the systems, ω

_{n}

_{1}and ω

_{n}

_{2}denote the undamped oscillation angular frequencies of the systems. Comparing Equations (19) and (20), the following equations can be derived.

_{1}, ζ

_{2}, ω

_{n}

_{1}, and ω

_{n}

_{2}can be obtained as follows:

_{1}, ζ

_{2}, ω

_{n}

_{1}, and ω

_{n}

_{2}. In addition, damping ratio (ζ

_{1}, ζ

_{2}) and undamped oscillation angular frequency (ω

_{n}

_{1}, ω

_{n}

_{2}) determine the dynamic performance of the system. In more detail, as shown in Equations (25) and (26), damping ratio determines overshoot (M

_{p}

_{1}, M

_{p}

_{2}) of step response, while undamped oscillation angular frequency jointly determines settling time (t

_{s}

_{1}, t

_{s}

_{2}) of step response.

_{1}and ζ

_{2}. As a result, the overshoots will approach 4.33%. In addition, 0.025 s is taken for both t

_{s}

_{1}and t

_{s}

_{2}, to realize that ω

_{n}

_{1}and ω

_{n}

_{2}should be 169.73 s

^{−1}. Then, substituting ζ

_{1}= ζ

_{2}= 0.707 and ω

_{n}

_{1}= ω

_{n}

_{2}= 169.73 s

^{−1}into Equations (23) and (24), PD-gains that realize the desired dynamics performance will be obtained as Table 4.

#### 3.3. Control Simulation

## 4. Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Consent for Publication

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**Figure 1.**Schematic diagram: (

**a**) Traditional torsion testing mechanism; (

**b**) Noncontact torsion testing mechanism.

**Figure 2.**Physical structure of the noncontact torsion testing device: (

**a**) Overall structure (

**b**) Partial exploded view.

**Figure 5.**FEM analysis results of the permanent magnetic gear: (

**a**) Torque (

**b**) Attractive force between the two permanent magnets in a vertical direction.

**Figure 6.**Illustration of the plant model: (

**a**) Illustration of the dimensions (

**b**) Illustration of the forces.

**Figure 8.**FEM analysis results of the four attractive-type permanent magnetic bearings: (

**a**) Analysis result of F

_{P}(

**b**) Analysis result of T

_{P}.

**Figure 11.**Experiment setup: (

**a**) A Photograph of the experiment setup (

**b**) Overall system structure diagram.

Name | Material/Model | Number |
---|---|---|

Torque sensor | Forsentek FT05 | 1 |

Servo motor | Maxon A-max 32 | 1 |

Displacement sensor | Panasonic GP-XC12ML | 2 |

Permanent magnetic gear | N40 | 1 |

Attractive-type permanent magnetic bearing | N40 | 4 |

Degree of Freedom | Control Type | Component That Implements Control |
---|---|---|

X | Active | The four electromagnets |

Y | Passive | The four attractive-type permanent magnetic bearings |

Z | Passive | The four attractive-type permanent magnetic bearings |

RX | Passive | The four attractive-type permanent magnetic bearings |

RY | Active | The four electromagnets |

RZ | Passive | The four attractive-type permanent magnetic bearings |

Parameter Name | Parameter Value |
---|---|

a (Nm^{2}A^{−2}) | 1.204 × 10^{−5} |

c (mm) | 2.38 |

i_{0} (A) | 2 |

z_{0} (mm) | 3 |

k_{i} (N/A) | 1.6639 |

k_{z} (N/m) | 618.5426 |

m (kg) | 1.731 |

J (kg·m^{2}) | 0.0033 |

h (m) | 0.143 |

l (m) | 0.0774 |

Parameter Name | Parameter Value |
---|---|

P_{1} (A/m) | 9886 |

D_{1} (A·s/m) | 64 |

P_{2} (A/m) | 3166 |

D_{2} (A·s/m) | 23 |

Parameter Name | Parameter Value |
---|---|

i_{0} (A) | 2 |

z_{0} (mm) | 3 |

P_{1} (A/m) | 10,000 |

I_{1} (A/m/s) | 1000 |

D_{1} (A·s/m) | 50 |

P_{2} (A/m) | 7000 |

I_{2} (A/m/s) | 1000 |

D_{2} (A·s/m) | 35 |

Property Name | Property Value |
---|---|

Material | Z-ASA PRO |

Length (mm) | 67 |

Width (mm) | 6 |

Thickness (mm) | 1.6 |

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**MDPI and ACS Style**

Ren, M.; Oka, K.
Design of a Noncontact Torsion Testing Device Using Magnetic Levitation Mechanism. *Actuators* **2023**, *12*, 174.
https://doi.org/10.3390/act12040174

**AMA Style**

Ren M, Oka K.
Design of a Noncontact Torsion Testing Device Using Magnetic Levitation Mechanism. *Actuators*. 2023; 12(4):174.
https://doi.org/10.3390/act12040174

**Chicago/Turabian Style**

Ren, Mengyi, and Koichi Oka.
2023. "Design of a Noncontact Torsion Testing Device Using Magnetic Levitation Mechanism" *Actuators* 12, no. 4: 174.
https://doi.org/10.3390/act12040174