# Linearizing Control of a Distributed Actuation Magnetic Bearing for Thin-Walled Rotor Systems

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## Abstract

**:**

## 1. Introduction

## 2. DAMB Design and Force Model

## 3. Exact Linearizing Control

#### 3.1. Gradient-Based Numerical Solution

#### 3.2. Linear Approximation Solution

## 4. Experiments

#### 4.1. Thin-Walled Rotor and DAMB Test System

#### 4.2. DAMB Simulations

#### 4.3. Practical Implementation

#### 4.4. DAMB Force Control Evaluations

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Rotor-AMB control scheme: (

**a**) with feedback linearization (inverse AMB model) (

**b**) equivalent dynamics.

**Figure 8.**Simulations of DAMB with lumped-mass rotor for step-change in position demand (

**a**) step input in X direction (

**b**) step input in Y direction.

**Figure 9.**Bearing force computation (for ${F}_{x}$) with gradient-based root finding algorithm (

**a**) Bearing force between 0.06 to 0.12 second (

**b**) Bearing force solution updates at each time step (

**c**) Bearing force error updates at each time step.

**Figure 12.**Displacement response of the non-rotating thin-walled rotor due to (

**a**) step change in X demand (

**b**) step change in Y demand.

**Figure 13.**Displacement response of the non-rotating thin-walled rotor due to (

**a**) sinusoidal demand input in X direction (

**b**) sinusoidal demand input in Y direction.

**Figure 14.**Steady-state behavior of test system at rotational speed of 31.2 rad/s (5 Hz) with (

**a**) exact linearizing control with Harmonic Vibration Control implementation (

**b**) linear approximation control with synchronous (unbalance) control implementation.

**Figure 15.**Fourier transforms of displacement and current signals at rotational speed of 31.2 rad/s (5 Hz). Data is shown for x-axis components from each case in Figure 14.

Coefficient | Formula * | Coefficient | Formula * |
---|---|---|---|

${a}_{1},{a}_{2},{a}_{3},{a}_{4}$ | ${\mathsf{\mu}}_{0}{A}_{p}\sum \frac{2{N}_{1,j}{N}_{2,j}}{{\left({l}_{0}-2{u}_{j}\right)}^{2}}cos{\theta}_{j}$ | ${b}_{1},{b}_{2},{b}_{3},{b}_{4}$ | ${\mathsf{\mu}}_{0}{A}_{p}\sum \frac{2{N}_{1,j}{N}_{2,j}}{{\left({l}_{0}-2{u}_{j}\right)}^{2}}sin{\theta}_{j}$ |

${a}_{5},{a}_{6}$ | ${\mathsf{\mu}}_{0}{A}_{p}\sum \frac{{N}_{1,j}^{2}}{{\left({l}_{0}-2{u}_{j}\right)}^{2}}cos{\theta}_{j}$ | ${b}_{5},{b}_{6}$ | ${\mathsf{\mu}}_{0}{A}_{p}\sum \frac{{N}_{1,j}^{2}}{{\left({l}_{0}-2{u}_{j}\right)}^{2}}sin{\theta}_{j}$ |

${a}_{7},{a}_{8}$ | ${\mathsf{\mu}}_{0}{A}_{p}\sum \frac{{N}_{2,j}^{2}}{{\left({l}_{0}-2{u}_{j}\right)}^{2}}cos{\theta}_{j}$ | ${b}_{7},{b}_{8}$ | ${\mathsf{\mu}}_{0}{A}_{p}\sum \frac{{N}_{2,j}^{2}}{{\left({l}_{0}-2{u}_{j}\right)}^{2}}sin{\theta}_{j}$ |

Parameter | Symbol | Value | Units |
---|---|---|---|

Length | L | $0.8$ | m |

Radius | R | $0.0815$ | m |

Wall thickness | h | $0.00306$ | m |

Density | $\rho $ | 7850 | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

Parameter | Symbol | Value | Units |
---|---|---|---|

Pole face area | ${A}_{p}$ | $55.8$ | mm${}^{2}$ |

Permeability of free space | ${\mathsf{\mu}}_{0}$ | $4\pi \times {10}^{-7}$ | H/m${}^{-1}$ |

Maximum number of coil-turns | ${N}_{0}$ | 100 | |

Bearing A | |||

Gap size | ${s}_{0A}$ | $0.35$ | mm |

Effective flux path length | ${l}_{0A}$ | $1.1$ | mm |

Bearing B | |||

Gap size | ${s}_{0B}$ | $0.2$ | mm |

Effective flux path length | ${l}_{0B}$ | $0.8$ | mm |

**Table 4.**Unbalance estimation results from nonlinear force model: actual unbalance was $0.0074$∠${0}^{\circ}$.

Rotational Frequency (rad/s) | Estimated Force (N∠deg) | Estimated Unbalance (kg·m∠deg) |
---|---|---|

$19.0$ | $2.57\angle {2.3}^{\circ}$ | $0.0071\angle {2.3}^{\circ}$ |

$25.1$ | $4.68\angle {0.3}^{\circ}$ | $0.0074\angle {0.3}^{\circ}$ |

$31.2$ | $7.38\angle {-1.7}^{\circ}$ | $0.0076\angle {-1.7}^{\circ}$ |

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**MDPI and ACS Style**

Chamroon, C.; Cole, M.O.T.; Fakkaew, W. Linearizing Control of a Distributed Actuation Magnetic Bearing for Thin-Walled Rotor Systems. *Actuators* **2020**, *9*, 99.
https://doi.org/10.3390/act9040099

**AMA Style**

Chamroon C, Cole MOT, Fakkaew W. Linearizing Control of a Distributed Actuation Magnetic Bearing for Thin-Walled Rotor Systems. *Actuators*. 2020; 9(4):99.
https://doi.org/10.3390/act9040099

**Chicago/Turabian Style**

Chamroon, Chakkapong, Matthew O.T. Cole, and Wichaphon Fakkaew. 2020. "Linearizing Control of a Distributed Actuation Magnetic Bearing for Thin-Walled Rotor Systems" *Actuators* 9, no. 4: 99.
https://doi.org/10.3390/act9040099