# Experimental Analysis and Full Prediction Model of a 5-DOF Motorized Spindle

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generations of Mathematical Models

#### 2.1. ClassicalFull Model

#### 2.1.1. Classical Full Nonlinear Model

_{0}is the permeability in vacuum. In this model, one part of the parameters (i.e., the structure and electromagnetic parameters with subscript (1) belongs to the AC 2-DOF HMB. F

_{cn}

_{1j}is the classical nonlinear model of the radial suspension forces of the AC 2-DOF HMB; x

_{1}and y

_{1}are positive displacements of the rotor in the x- and y-directions; φ

_{1j}is the radial resultant magnetic fluxes in the air-gap corresponding to each pole; S

_{r1}is the face area of the radial magnetic pole; δ

_{r1}is the length of the uniform air-gap without rotor eccentricity; F

_{m1}is the magnetomotive force provided to the outer circuit of the AC 2-DOF HMB; i

_{j}

_{1}is the radial three-phase control current; j = A, B, C; N

_{r1}is the number of turns that the radial control coils. The other part of the parameters (i.e., the structure and electromagnetic parameters with subscript (2) belongs to the AC–DC 3-DOF HMB. F

_{cn}

_{2j}and F

_{cnz}are the classical nonlinear model of the radial and axial suspension forces of the AC–DC 3-DOF HMB; x

_{2}and y

_{2}are positive displacements in the x- and y-directions and z is right displacement of the rotor; φ

_{2j}is the radial resultant magnetic fluxes in the air-gap corresponding to each pole; F

_{m2}is the magnetomotive force provided to the outer circuit of the AC–DC 3-DOF HMB; i

_{z}is the axial control current and i

_{j}

_{2}is the radial three-phase control current; j = A, B, C. F

_{z}

_{1}is the right magnetic suspension force acting on the rotor; F

_{z}

_{2}is the left magnetic suspension force acting on the rotor; N

_{r2}is the number of turns that the radial control coils; N

_{z}is the number of turns that the axial control coils; φ

_{z}

_{1}and φ

_{z}

_{2}are the axial resultant magnetic-biased fluxes generated by the permanent magnet and axial control coils in the right and left axial air-gaps, respectively; G

_{z1}and G

_{z2}describe the right and the left air-gap permeances, respectively; G

_{A}, G

_{B}, and G

_{C}describe the radial air-gap permeances.

#### 2.1.2. Classical Full Linear Model

_{cl}

_{1x}, F

_{cl}

_{1y}, F

_{cl}

_{2x}, F

_{cl}

_{2y}and F

_{clz}) can be written as follows:

_{x}

_{1}, i

_{y}

_{1}are the current components in the x- and y-axis transformed from the three-phased currents by Clark coordinate transformation; k

_{cl}

_{1xy}is the radial force-displacement coefficient; k

_{cl}

_{1ir}is the radial force-current coefficient. The other part of the parameters with subscript2 belongs to the AC–DC 3-DOF HMB: i

_{x}

_{2}, i

_{y}

_{2}are the current components in the x- and y-axis transformed from the three-phased currents by Clark coordinate transformation; k

_{cl}

_{2xy}is the radial force-displacement coefficient; k

_{cl}

_{2ir}is the radial force-current coefficient; k

_{clz}is the axial force-displacement coefficient, and k

_{cliz}is the axial force-current coefficient.

_{r2}is the face area of the radial magnetic pole; δ

_{r2}is the length of the uniform air-gap without rotor eccentricity and δ

_{z}is the axial air-gap length of the AC–DC 3-DOF HMB; S

_{z}is the face area of the axial magnetic pole of the AC–DC 3-DOF HMB; and S

_{mag}is the average area of the inner and outer faces of the permanent magnet.

#### 2.2. Improved Full Model

#### 2.2.1. Improved Full Nonlinear Model

_{in}

_{1j}is the improved full nonlinear model of the radial suspension forces generated by the resultant magnetic fluxes of each radial air-gap; l is the equivalent length of rotor; r is radius of rotor; dS

_{r1}is per unit area; and θ is the dimensional mechanical angle. With the rotor eccentricity at time t, the expression of the magnetic flux density in air-gap B (θ, t) is determined by the permanent magnet and the control current. The other part of the parameters with subscript2 belongs to the AC–DC 3-DOF HMB: F

_{in}

_{2j}is the improved full nonlinear model of the radial suspension forces generated by the resultant magnetic fluxes of each radial air-gap; F

_{inz}is the improved full nonlinear model of the axial suspension force; G

_{z1}and G

_{z2}describe the right and the left air-gap permeances, respectively; G

_{A}, G

_{B}, and G

_{C}describe the radial air-gap permeances; G

_{mag}describes the permanent magnetic permeance; φ

_{cz}is the axial control magnetic fluxes generated by the control coils; φ

_{pz}

_{1}and φ

_{pz}

_{2}are the axial magnetic biased fluxes generated by the permanent magnet in right and left axial air-gaps, respectively.

#### 2.2.2. Improved Full Linear Model

_{il}

_{1x}, F

_{il}

_{1y}, F

_{il}

_{2x}, F

_{il}

_{2y}and F

_{ilz}) can be written as follows:

_{mag}is the average area of the inner and outer faces of the permanent magnet; and δ

_{mag}is the vertical distance from the outer surface to inner surface of the permanent magnet.

_{il}

_{1xy}is the radial force-displacement coefficient; k

_{il}

_{1ir}is the radial force-current coefficient. The other part of the parameters with subscript2 belongs to the AC–DC 3-DOF HMB: k

_{il}

_{2xy}is the radial force-displacement coefficient; k

_{il}

_{2ir}is the radial force-current coefficient; k

_{ilz}is the axial force-displacement coefficient; and k

_{iliz}is the axial force-current coefficient.

#### 2.3. SwitchingFullModel

## 3. Switching Full Model Analysis Based on Experiments

_{xx}

_{1}, F

_{yy}

_{1}, F

_{xx}

_{2}, F

_{yy}

_{2}and F

_{z}) using the force-current transformation modules. Notably, the radial and axial force-current transformation modules reflect the efforts of the mathematical suspension force models. Therefore, the mathematical models of the suspension forces are important to obtain precise control of these magnetic bearings. The key mathematical expressions (i.e., mathematical models of the suspension forces) in the algorithm of the control system of the 5-DOF motorized spindle are as mentioned above.

_{xx}

_{1}, F

_{yy}

_{1}, F

_{xx}

_{2}, F

_{yy}

_{2}, and F

_{zz}) into radial and axial control current reference signals (i

_{A1}*, i

_{B1}*, i

_{C1}*, i

_{A2}*, i

_{B2}*, i

_{C2}*, and i

_{z}*) for the 5-DOF HMBs. The control current reference signals function as the reference signals of the internal closed loop current control link. Moreover, the exciting currents can be obtained using the current-controlled voltage-source inverter and axial switching power amplifier. Through this process, control currents are adjusted using the control system with a negative position and current feedbacks to readjust flux distribution. Thus, the accuracy of a suspension forces model directly affects the effectiveness of the control system.

#### 3.1. Start-of-Suspension Response Experiment

#### 3.2. Suspension Experiment

#### 3.3. Disturbance Response Experiment

## 4. Full Prediction Model of Suspension Force according to Operating State

#### 4.1. Comparative Experiment

#### 4.2. Verification Experiment

#### 4.3. Stiffness Tests and AccuracyAnalysis of the Model

_{x}and radial suspension force F

_{x}, and the relationship between the displacement x and suspension force F

_{x}by the calculated results of a nonlinear model. The force modeling errors between the three full model curves and the experiment results can be derived from Figure 12. Because the composition of full prediction model is the same as the switching full model only with differences in freedom, the accuracy of full prediction model is in accord with switching full model.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Prototype of the five-degrees-of-freedom (5-DOF) AC hybrid magnetic bearings (HMB)-supported motorized spindle.

**Figure 6.**Trajectories and displacement waves of the rotor adopting “switching model” at the start-of-suspension state. (

**a**) Mass center orbit of the rotor under the support of the AC 2-DOF HMB; (

**b**) Mass center orbit of the rotor under the support of the AC–DC 3-DOF HMB; (

**c**) Displacement waveforms of the rotor under the support of the AC–DC 3-DOF HMB.

**Figure 7.**Radial trajectories of the rotor at the stable suspension state. (

**a**) Mass center orbit of the suspended rotor under the support of the AC 2-DOF HMB; (

**b**) Mass center orbit of the suspended rotor under the support of the AC–DC 3-DOF HMB; (

**c**) Displacement waveforms of the rotor under the support of the AC–DC 3-DOF HMB.

**Figure 10.**Trajectories and displacement waves of the rotor adopting “full prediction model” at the start-of=suspension state. (

**a**) Mass center orbit of the rotor under the support of the AC 2-DOF HMB; (

**b**) Mass center orbit of the rotor under the support of the AC–DC 3-DOF HMB; (

**c**) Displacement waveforms of the rotor under the support of the AC–DC 3-DOF HMB.

**Figure 12.**Comparison between the measurement result and three calculation results. (

**a**) Relationship between the control current i

_{x}and radial suspension force F

_{x}; (

**b**) Relationship between the displacement x and suspension force F

_{x}.

“Full operation-domain model” | Model | Equation | Current | Displacement | Multi-Zone |

Model (1): Improved full nonlinear model | Equations (5) and (6) | 0–1 (A) & 1–1.5 (A) | 0–0.25 (mm) & 0.25–0.49 (mm) | Zone (1) | |

Model (2): Classical full nonlinear model | Equations (1) and (2) | 1.5–2 (A) | 0.25–0.49(mm) | Zone (2) | |

Model (3):- ①
- Improved full nonlinear model (if the last model is Model (1))
- ②
- Classical full nonlinear model (if the last model is Model (2) or Model (4))
| - ①
- Equations (5) and (6)
- ②
- Equations (1) and (2)
| 1–1.5 (A) | 0.4–0.49 (mm) | Zone (3) | |

Model (4): Classical full nonlinear model | Equations (1) and (2) | 1.5–2 (A) | 0.4–0.49 (mm) | Zone (4) |

**Table 2.**The “full prediction model” of suspension force according to operating state selection table.

“Full prediction model” of suspension force according to operating state | Operating State | Initial Model | Position | Model Variation | Switching Point |

start-of-suspension | Model (1) | Initial position: 0–0.25 (mm) | Changeless Model (1) | none | |

Model (3) | Initial position: 0.25–0.44 (mm) | Model (3)→Model (1) | When the displacement enters into 0–0.25 (mm) | ||

Model (4) | Initial position: 0.44–0.49 (mm) | Model (4)→Model (1) | When the displacement enters into 0–0.25 (mm) | ||

suspension | Model (1) | Operating position: 0–0.25 (mm) | Changeless Model (1) | none | |

Model (2) | Operating position: 0.25–0.49 (mm) | Model (2)→Model (1) | When the displacement enters into 0–0.25 (mm) | ||

disturbed state | Model (1) | Maximum deviation postion: 0–0.25 (mm) | Changeless Model (1) | none | |

Model (1) | Maximum deviation postion: 0.25–0.49 (mm) | Model (1)→Model (2–4)→Model (1) | When the displacement enters into 0–0.25 (mm), twice |

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**MDPI and ACS Style**

Zhang, W.; Zhu, H.; Yang, H.; Chen, T. Experimental Analysis and Full Prediction Model of a 5-DOF Motorized Spindle. *Energies* **2017**, *10*, 75.
https://doi.org/10.3390/en10010075

**AMA Style**

Zhang W, Zhu H, Yang H, Chen T. Experimental Analysis and Full Prediction Model of a 5-DOF Motorized Spindle. *Energies*. 2017; 10(1):75.
https://doi.org/10.3390/en10010075

**Chicago/Turabian Style**

Zhang, Weiyu, Huangqiu Zhu, Hengkun Yang, and Tao Chen. 2017. "Experimental Analysis and Full Prediction Model of a 5-DOF Motorized Spindle" *Energies* 10, no. 1: 75.
https://doi.org/10.3390/en10010075