# Vibration Reduction of an Overhung Rotor Supported by an Active Magnetic Bearing Using a Decoupling Control System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modelling of an Overhung Active Magnetic Bearing Rotor System

_{x}and θ

_{y}are the angular displacements of the whirling axis about the x- and y-axes, respectively; x

_{m}, y

_{m}, x

_{b}, and y

_{b}represent the displacements of the rotor at the magnetic and ball bearing locations; n defines the direction corresponding to the x- and y-axes; and the electromagnetic force and ball bearing forces are expressed as f

_{mn}and f

_{bn}, respectively, which are the locations of action of each force; and f

_{un}is the unbalanced force.

_{r}and I

_{a}are the transverse and polar mass moment of the inertia of the rotor, respectively; and l

_{m}, l

_{b}, and l

_{u}represent distances to the AMBS, ball bearing, and unbalanced force from the center of mass, respectively.

_{0}denotes the nominal current, and i

_{x}and i

_{y}are control currents acting on the x- and y-axes, respectively. Following previous studies [3,11], the total nonlinear forces along the axis of attractive electromagnetism can be determined as:

_{m}= μ

_{0}AN

^{2}/2, k

_{i}= 4k

_{m}i

_{0}/n

_{0}

^{2}, and k

_{s}= 4k

_{m}i

_{0}

^{2}/n

_{0}

^{3}define the current, displacement, and stiffness parameters of the magnets, respectively. We conducted the linearization using a Taylor expansion at the equilibrium point. The coil acting on the x- and y-axes similarly circulates with nominal current (i

_{0}) via Equation (2). Since the nominal air gaps (n

_{0}= x

_{0}= y

_{0}) are equal along the x- and y-axes, the displacement and current stiffness parameters that are obtained from the x-axis are the same as those obtained for the y-axis.

_{m}, x

_{b}, y

_{m}, and y

_{b}) and geometry coordinates (x, y, θ

_{x}, and θ

_{y}) is as shown in Equation (4). Similarly, an additional transformation from the plane of the sensor coordinates (x

_{ms}, x

_{bs}, y

_{ms}, and y

_{bs}) to the origin O can be described as

_{s}I

_{r}is larger than l

_{ms}I

_{a}and l

_{bs}I

_{a}. The two supports in the same direction, creating a gyroscopic effect, and coupling can be ignored. Simultaneously, |k

_{s}a

_{1}+ ka

_{3}| >> |k

_{s}a

_{2}+ ka

_{4}| and |ca

_{3}| >> |ca

_{4}| for the magnetic location sensor, and |k

_{s}a

_{6}+ ka

_{8}| >> |k

_{s}a

_{5}+ ka

_{7}| and |ca

_{8}| >> |ca

_{7}| for the ball bearing location sensor are satisfied. In the same axis, the coupling between two supports was completely ignored. Therefore, Equation (5) was simplified by reduction with a decentralized force estimator for the disturbance force. Only part of the dynamic motion correlated with a suspended magnet can be described by

_{x}and u

_{y}are the summation of the disturbance forces, including bearing forces, force of gravity, and unbalanced forces for both the x- and y-directions, respectively. The transfer function G

_{p}(s) is the quotient between the transformed displacement and input current of the decoupling rotor model:

_{a}) when connected to a coil:

_{s}) is the range of the pulse width modulation (PWM) power signal, where V

_{s}is the supply voltage. The equivalent of the voltage across the H-bridge load is related to the duty-cycle (δ) of its input control voltage v

_{n}. Thus, the control voltage is related to the control signal. Conventional controls are available for closing the feedback loop by showing that the coil current i

_{n}is appropriate for the control law [18]. The negative of pole in an open-loop AMBS is unstable when represented by a

_{1}k

_{s}. The most intuitive approach for a conditional control law for an AMB rotor obtained using Equation (8) is implementing a conventional PID control locally for each axis. A general stabilizing controller is designed with a proportion, integrator, and derivative gains as follows:

_{p}, g

_{d}, and g

_{i}are the gains of PID controller.

## 3. Decentralized Control of a Harmonic Disturbance Compensator

_{n}, as shown in Figure 3. The unknowns are the phase and amplitude of u

_{n}. The signal of a reference phase and rotation frequency Ω are assumed to be available. Normally, rotation speed Ω is constant or varies slowly. This motion is completely dependent on u

_{n}, which is an unknown unbalance described by Equation (7). The techniques for unbalance compensation related to the generalized feedback of the notch may narrow the stability margin of the closed-loop system. The negative phase of the notch was used to describe the characteristics of the transfer function that lead to instability below the natural frequency of a rigid mode. Especially for large scales and low speeds, the rotor may experience significant vibrations, caused by an unbalance. Herzog et al. designed a controller to ensure closed-loop stability using the insertion of a notch filter structure. Unlike the conventional structure of the generalized notch filter, we modified the phase-shift angle to substitute the transformation. The stable closed loop of an AMBS can be preserved only by adapting the improvement phase. The internal feedback structure of the HDC’s component notch is replaced by H(s) with a phase shift η

_{n}, as shown in Figure 3. Let n

_{ms}and c

_{n}denote the feedback components of input and output fault signals, respectively. The feedback components of dynamic can be described as

_{n}. Figure 3 depicts an unbalance vibration compensation with a PID controller. The transfer function from n

_{ms}to e

_{n}is estimated by evaluating Equation (13) at the frequency of Ω, as follows:

_{n}, which confirms its notch feedback characteristics. Each direction, without the gyroscopic effect and the coupling between the two-radius AMBS in the same axis, can be separated in two DOF in the x- and y-axes (radial direction). In the n-direction, to simplify the representation, the closed-loop stability with a PID controller was clarified. The other properties behave similarly. The transfer function from u

_{n}to e

_{n}is rearranged as.

_{sn}is a displacement sensor gain. Then, the closed-loop stability is determined by roots.

## 4. Experiment and Results

_{s}= 24 VDC). The characteristics of the linearization zone for the relationship between the air gap and the obtainable control current of an AMBS on the x- and y-axes are shown in Figure 5.

#### 4.1. Frequency Boundary Detection with Impact Testing

#### 4.2. Performance Comparison

_{n}was used to choose an appropriate value to satisfy the stability criteria in Equation (19).

_{n}= 0 and ε = 500, and a rotor speed lower than 65 Hz under maximum speed, to relate to the peak of FRF in Figure 6. In the response optimization, the maximum overshoots the step response by 5%. Rise time and settling time are less than 0.5 and 1 s, respectively. The controller parameters were optimized to be g

_{p}= 0.01, g

_{d}= 0.0001, and g

_{i}= 0.5. The two sensors were defined to measure displacement on the x- and y-axes. The rotor’s orbital path was stabilized at the origin point, and was controlled by sustained air to whirl from the PID controller. The representation of rotor vibrations was predominated by rotational speed. The rotational speed of the rotor affected the amplitude. The operations of rotational accuracy depend on the whirling over a range of rotational speeds. The vibrations are created by the unbalanced rotor. The stabilization in the orbital origin can be monitored by the rotor with PID and HDC. The plot contains 10,000 successive samples (in the period at a steady state of 5 s). The improvement in rotation at a constant speed is clear. In the orbit plot, 95% of the operation occurred within the inside of a circular space with a diameter 20 µm, and centered on the orbital origin, as shown in Figure 7.

**■**) and Mid-Dum (

**♠**), respectively. The Mid-Bear (

**♦**) and End-Bear (

**●**) labels represent the position of the three-axial acceleration sensors that were used for contact measurement in the middle and end of the shafts, respectively.

#### 4.3. Diagnostic of Orbit Shape and Fatigue-Bearing Life Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Relationship between air gap and obtainable control current of an AMBS on the x- and y-axes.

**Figure 10.**Comparison of the orbits measured at a rotating speed of 23.5 Hz, with PID with HDC (

**a**) and without HDC (

**b**) for each of the sensing points.

**Figure 11.**Comparison of the orbits measured at a rotating speed of 43.6 Hz, with PID with HDC (

**a**) and without HDC (

**b**) for each of the sensing points.

**Figure 12.**Comparison of the orbits measured at a rotating speed of 50.3 Hz, with PID with HDC (

**a**) and without HDC (

**b**) for each of the sensing points.

**Figure 13.**Comparison of the orbits measured at a rotating speed of 60 Hz, with PID with HDC (

**a**) and without HDC (

**b**) for each of the sensing points.

Description | Parameter | Value (Unit) |
---|---|---|

Air gaps | x_{0} = y_{0} | 1 (mm) |

Pole face area | A | 4.025 × 10^{−6} (m^{2}) |

Winding number per coil | N | 60 (rev) |

Bias current | i_{0} | 2 (A) |

Range of current to control | i_{n} | 0–10 (A) |

Magnetic permeability | μ_{0} | 4π × 10^{−7} (Vs/Am) |

Amplifier gain | g_{a} | 48.3 (A/V) |

Displacement sensor gain | g_{ns} | 1.18 ^{I}, 1.32 ^{II} (V/mm) |

Current stiffness | k_{i} | 26.09 (N/A) |

Displacement stiffness | k_{n} | 198.24 (N/mm) |

^{I}x-axis,

^{II}y-axis.

FRF Indicator (Hz) | Damping (Hz) | Critical Damping (%) | ||
---|---|---|---|---|

Horizontal | Vertical | Horizontal | Vertical | |

23.5 | 0.756 | 9.537 | 3.210 | 2.300 |

43.6 | 0.445 | n/a | 1.030 | n/a |

50.3 | 0.777 | 1.490 | 1.590 | 3.090 |

69.0 | 0.845 | 0.988 | 1.230 | 1.430 |

Frequency (Hz) | End-Mag (■) | Mid-Dum (♠) | Mid-Bear (♦) | End-Bear (●) | Average |
---|---|---|---|---|---|

23.5 | 65.4 | 28.6 | 62.5 | 67.2 | 55.9 |

43.6 | 80.9 | 36.4 | 32.6 | 33.3 | 45.8 |

50.3 | 81.7 | 33.3 | 27.9 | 28.2 | 42.8 |

60.0 | 83.2 | 37.5 | 60.2 | 6.6 | 46.9 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Numanoy, N.; Srisertpol, J. Vibration Reduction of an Overhung Rotor Supported by an Active Magnetic Bearing Using a Decoupling Control System. *Machines* **2019**, *7*, 73.
https://doi.org/10.3390/machines7040073

**AMA Style**

Numanoy N, Srisertpol J. Vibration Reduction of an Overhung Rotor Supported by an Active Magnetic Bearing Using a Decoupling Control System. *Machines*. 2019; 7(4):73.
https://doi.org/10.3390/machines7040073

**Chicago/Turabian Style**

Numanoy, Nitisak, and Jiraphon Srisertpol. 2019. "Vibration Reduction of an Overhung Rotor Supported by an Active Magnetic Bearing Using a Decoupling Control System" *Machines* 7, no. 4: 73.
https://doi.org/10.3390/machines7040073