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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Unbalance in magnetically levitated rotor (MLR) can cause undesirable synchronous vibrations and lead to the saturation of the magnetic actuator. Dynamic balancing is an important way to solve these problems. However, the traditional balancing methods, using rotor displacement to estimate a rotor's unbalance, requiring several trial-runs, are neither precise nor efficient. This paper presents a new balancing method for an MLR without trial weights. In this method, the rotor is forced to rotate around its geometric axis. The coil currents of magnetic bearing, rather than rotor displacement, are employed to calculate the correction masses. This method provides two benefits when the MLR's rotation axis coincides with the geometric axis: one is that unbalanced centrifugal force/torque equals the synchronous magnetic force/torque, and the other is that the magnetic force is proportional to the control current. These make calculation of the correction masses by measuring coil current with only a single start-up precise. An unbalance compensation control (UCC) method, using a general band-pass filter (GPF) to make the MLR spin around its geometric axis is also discussed. Experimental results show that the novel balancing method can remove more than 92.7% of the rotor unbalance and a balancing accuracy of 0.024 g mm kg^{−1} is achieved.

Active magnetic bearings (AMBs) have several advantages over traditional bearings, such as low power losses, very long life, the elimination of the oil supply, vibration control, and diagnostic requirements, hence, AMBs have been widely used in the fields of energy-storing flywheels, turbo machinery and machine tools [

Due to material non-homogeneity and manufacturing errors, a rotor's inertia axis always misaligns with the geometric axis. This will inevitably result in rotor unbalance and produce centrifugal forces while the rotor is spinning. These centrifugal forces then transfer to the motor casing and generate vibration noises, which reduce the life of the machinery [

Off-line balancing is a widely used method to eliminate rotor unbalance [

The field balancing method employs correction masses to correct a rotor's mass distribution. It makes the rotor's inertia axis align with its geometric axis, removing the unbalance disturbance from the source [

In recent years, various field balancing methods have been developed. These methods fall into two major categories: influence coefficient methods and modal balancing methods [

Displacement sensor and angular-position sensor permit field balancing for MLR can be implemented without any additional instrumentation. More importantly, AMBs have the ability to actively control the rotor. Nevertheless, there is little research on field balancing for MLR in particular. Li

Based on an AMB rotor's active control property, this paper proposes a novel field balancing method. The method can simultaneously meet the requirements of high-efficiency and high-accuracy. It requires neither trial runs nor the MLR's precise model. In this method, the control current rather than rotor displacement is employed to calculate the correction masses, and no influence coefficients are required. After analysing the models of an unbalanced MLR, we find that the coincidence of a rotor's rotation axis with its geometric axis will bring about two benefits. One is that the unbalanced centrifugal force/torque equals the synchronous magnetic force/torque generated by the control current, which enables computation of correction masses using the control current with only a single start-up. The other is that the magnetic force is proportional to the control current, which makes the balancing highly accurate. The unbalance compensation control (UCC) method using a general band-pass filter (GPF), which enables the MLR to spin around its geometric axis, is also discussed.

As the first bending-critical speed (650 Hz) is far beyond the balancing speed (80 Hz), the rotor can be regard as peer rigid. All models in this study are based on this assumption.

The main structure of the magnetic levitation motor (MLM) is a motor which drives a rigid rotor supported by three AMBs, one thrust bearing and two radial bearings (see the computer-aided design diagram in

The AMB works in differential mode. Let _{ax}_{0} is the bias current, _{c}_{0} the normal air gap,

To intuitively model the unbalanced rotor, an unbalanced MLR supported by two radial AMBs is discussed here (_{axis}, _{axis} and _{axis} are the rotor's inertia axis, geometric axis, and rotation axis.

We establish the ground reference frame _{sa}_{sb}_{ma}_{mb}_{ax}_{ay}_{bx}_{by}

The unbalanced force is generated by the deviation of the inertia axis and rotation axis, which is divided into two parts. One is the deviation of the mass center and the rotation center, resulting in static unbalance. The other is the angle deviation of the inertia axis and rotation axis, resulting in dynamic unbalance.

Static unbalance causes an unbalance force. The relative positions of the rotation center, geometric center and mass center when the rotor spins are shown in

The positions of the geometric center and mass center in

The rotor's dynamic unbalance causes an unbalance torque. _{axis}, _{axis} and _{axis}, where _{axis}_{axis}_{rg}_{rg}_{ri}_{ri}_{axis} and _{axis} in _{axis}_{gi}_{gi}_{axis} in _{axis} in _{axis}

For the purposes of this paper, the rotor is modeled as rigid. The control block diagram of the MLR system is shown in ^{T}_{rg}_{rg}Y β_{rg}^{T}_{gi}_{gi}_{gi}_{gi}^{T}

The generalized mass matrix _{r}_{z}_{x}_{y}_{x}_{y}

The AMBs only provide forces at the locations indicated in _{m}_{ax}f_{bx}f_{ay}f_{by}^{T}. _{m}_{ax}_{bx}_{ay}_{by}

We denote _{m}_{ax}x_{bx}x_{ay}x_{by}^{T}_{ax}i_{bx}i_{ay}i_{by}^{T}_{m}

According to _{rg}

Substituting

When we describe the magnetic force produced by rotor vibration, it is necessary to transform from the gravity-center coordinates to the bearing coordinates since the rotor dynamics are formulated in the gravity-center coordinate system. The matrix _{αm}_{_}_{q}_{rg}_{m}

The sensors are non-collocated with the bearing center of action. To account for this, an additional transform is required to move from the sensors to the gravity center coordinate system.

For traditional supported rotors, knowing the rotor unbalance _{rg}_{rg}

For a MLR, however, we can obtain the expression of

When we use _{rg}_{m}

As the MLR's control diagram shown in _{s}_{αs}_{_}_{q}q_{rg}_{amp}

We know that the value of _{s} will reduce to be zero. _{rg}

Because the transformation from _{m} to _{s} is linear (shown in _{m}^{T} at zero-displacement status. Substituting _{m}_{m} = [_{ax}_{bx}_{ay}_{by}^{T} can be simplified to:
_{i}_{0}/s_{0}^{2}_{m}_{s}

Theoretically, any unbalance distribution in a rigid rotor can be balanced in two different planes [_{1}, _{2} and _{3}, and _{1} + _{2} + _{3} =

Assuming that the correction masses _{ca}_{cb}_{ma}_{mb}

According to the double-plane balancing method, if _{c} can balance the rigid rotor completely, the following formulation will be obtained:

For the slender rotor considered in this study, _{r}_{z}

The correction masses can be solved from _{1}, _{2}, _{3} can be obtained precisely from the mechanical layout, rotor speed Ω can be measured by the Hall sensor. According to _{c}_{i}_{m}, _{rg}, _{ma} and _{mb}, and the magnetic force is the linear function of the rotor vibrations.

The correction masses on the two balancing planes can be solved further as:

Field balancing under the zero-displacement status greatly improves calculation accuracy and reduces the computation process, requiring only a single start-up. In addition, rigid balancing just needs to be implemented at a low speed. The magnetic force is small in this case, and the AMB saturation will not take place, ensuring the magnetic force is in proportion to the control current.

The advantages of field balancing under the zero-displacement status have been discussed in detail in the section above. In this section, we will discuss how to control the rotor to rotate under this status.

It is known that the object of active vibration control is to make the rotor spin around its inertia axis, which requires the synchronous magnetic force between rotor and stator to be infinitesimal (equivalent to the synchronous support-stiffness being infinitesimal). Under the zero-displacement status, the rotor is forced to spin around its geometric axis. Its synchronous displacement is infinitesimal whereas the synchronous support-stiffness is infinite, a situation also called unbalance compensation control method.

Field balancing described in this study is realized at a fixed speed, without guaranteeing the stability of the closed loop over the full-speed range, but requiring a high-accuracy algorithm. Thus, referring to the GNF discussed by Hezorg

As the gyroscopic coupling of the slender rotor is weak, an SISO controller woke well. And the stability analysis is based on SISO controller. In

The GPF is briefly described as follows. First, the cross-correlation coefficients between the control error signal

Then; the synchronous component of the error signal is reconstructed:

The synchronous component passes through integrators until it vanishes, and a constant synchronous control output is achieved through GPF channel. The control output cancels out the unbalance forces.

To ensure the stability margin of the GNF, a transform matrix was strung behind the integral part [_{ω}

Herzog utilized the root locus to analysis the GNF's stability [

Following the analysis of GNF, the stability of the GPF will be discussed here. However, unlike the GNF, the stable region of the GPF will be solved exactly in this study. First, the AMB control system described in _{r}

Using the small gain theorem [

That is:

It is found from

Substituting system parameters and the expression of _{C} = _{ω}_{S} = _{ω}_{p} and _{d} are proportionality and differential coefficient.

Solving the above equations, _{c} and _{s} are obtained as:

The expression of ^{2} can be gained from

Substituting _{c}, _{s} and Ω can be solved. Parameters _{c} and _{s} that satisfy the critical stability under the different rotor speeds are shown in

_{ω}_{ω}

Any pole falling to the right-half plane will make the AMB control system unstable. There are two poles in _{ω}

The GPF will make the rotor displacement converge to zero as long as _{S} and _{C} lie in the stable region in _{ω} is set 140°, and

As discussed in Section 3, the procedure of computing the correction masses will be simplified greatly when the rotor is controlled under the zero-displacement status. Nevertheless, control current and current stiffness are indispensable, and the two factors will directly influence the computational precision. Accurate acquisition of the above two factors will be discussed in this section.

Due to manufacturing and assembling errors, the current stiffness is usually not the same as its design value, and balancing accuracy will decrease when using the design value. Because the geometric axis aligns with the AMBs centerline at zero-displacement status, the current stiffness loss produced by eddy current can be ignored in low-speed filed balancing. Here, torque equilibrium method, a simple but useful method, is employed to test the current stiffness.

As shown in

From

Because unbalance is a rotor's feature, the unbalance disturbance force is synchronized with the rotation rotor. Thus, the correction masses in _{z} is the size of the sine/cosine tables,

The rated speed (1,000 Hz) of the rotor in the study is higher than its first bending frequency (650 Hz), so the rotor is flexible. It is necessary to implement flexible balancing. However, to make the flexible balancing more precise, the beforehand rigid balancing is indispensable. Rigid balancing is mainly discussed in this study. The rotor is rigid when it rotates below 50% of the first bending critical speed _{bending}, weakly rigid when above 50% and below 70% of _{bending}, and flexible when above 70% [

To evaluate the effectiveness of the proposed balancing method, experiments were carried out on an MLM, which was fixed vertically on a metal base (as shown in

The inertia axis and the rotation axis of the rotor are not aligned when rotating because of the residual unbalance. To reduce the unbalance, correction masses should be added to the two balancing planes at appropriate positions. There are 12 thread holes meanly distributing in every balancing plane. Screws matching the correction masses are fixed in the thread holes that match the correction angles. If the correction angle mismatches any thread hole, two screws that equivalent the correction masses can be placed in the two nearby thread holes.

Before assembled in the casing, the rotor has been balanced on the balancing machine. The GPF is added to the AMB control system at 0.7 s, and changes of the rotor displacement and coil current are shown in

The rotor's synchronous displacement rapidly decreases to zero (the high-frequency residuals of

According to the method discussed in Section 5, the measured current stiffness are obtained and listed in

According to ^{−1} and 1.962 g mm kg^{−1}), far from the standard of balance grade for a motor [^{−1} and 0.024 g mm kg^{−1}), superior to the standard of the highest balance grade in ISO1940–2003 [

The rotor spins at frequency 80 Hz under the PD controller, and a comparison of the rotor displacements before and after balancing is given in

A new method of field dynamic balancing for an MLR has been proposed that requires no trial weights or rotor-bearing model, but which can make balancing simultaneously highly efficient and highly accurate. First, the MLR dynamics including static unbalance and dynamic unbalance was modeled. According to the models, we found that making the rotor's rotation axis align with the geometric axis would bring two benefits. One was that the unbalanced centrifugal force/torque equaled the synchronous magnetic force/torque generated by the control current; the other is that the magnetic force was proportional to the control current, assisting in making field balancing highly accurate. Second, using the UCC method with a GPF enabled the MLR to spin around its geometric axis. Finally, the correction masses was calculated using control current rather than rotor displacement, which required only a single startup, and particularly without the need for influence coefficients. The experimental results showed that this novel balancing method increased the balance grade by two orders of magnitude based on off-line balancing, achieving 0.024 g mm kg^{−1}. Though the proposed balancing method performs well in the experiment, it still has many limitations as follows:

It is effective only for rigid rotors;

The off-line balancing is indispensable. The proposed method requires that the magnetic force should be able to counterbalance the unbalance centrifugal force, so it is not effective for rotors that have large unbalances;

The rotor's mass distribution should change little in operation. Otherwise, an active vibration control method should be used in conjunction with the propose method.

This work is supported in part by the National Nature Science Funds of China (No. 61203203), and in part by the National Major Project for the Development and Application of Scientific Instrument Equipment of China (No. 2012YQ040235), Aviation science fund of China (No. 2012zb51019).

The authors declare no conflict of interest.

Structure diagram of a MLM.

Schematic of an unbalanced rotor supported by two radial AMBs.

MLR's rotation center, geometric center and mass center in ground and rotor-fixed reference frame.

MLR's rotation axis, geometric axis and inertia axis in the ground and rotor-fixed angular coordinate system.

The control block diagram of the MLR.

Force-current/displacement characteristic of the radial AMB.

The control block diagram of the MLR system.

Schematic of the double-plane balancing method.

AMB control system including the GPF.

Amplitude-frequency response characteristic of the controller (GPF + PD).

The simplified control diagram and its transformation of the MLR.

Parameters _{C}_{S}

Poles and zeros of the AMB system with GPF at speed of 80 Hz.

Signals before and after adding the GPF (simulation performed with Simulink).

Schematic of current stiffness measurement.

Photograph of the 4 kW MLM, AMB control box and motor control box.

Sensor outputs before and after adding the GPF at 80 Hz.

Amplitude and phase of synchronous displacement before and after adding the GPF.

Amplitude and phase of synchronous current before and after adding the GPF.

Rotor displacement at 80 Hz.

Amplitude of synchronous displacement and current of AX-channel from 80 Hz to 0 before (bold curve) and after (thin curve) balancing at 80 Hz.

Amplitude of rotor's synchronous displacement after field balancing with the rotation speed decreasing from 500 Hz to 0.

Design parameters of the 4 kW MLM.

_{r} |
Length of the rotor | 554 | mm |

Mass of the rotor | 6.85 | kg | |

Power | 4 | kW | |

_{bending} |
The first bending frequency | 650 | Hz |

_{r} |
The transverse moment of inertia | 0.1147 | kg m^{2} |

_{z} |
The polar moment of inertia | 0.002529 | kg m^{2} |

_{0} |
Bias current | 1.2 | A |

_{i} |
Current stiffness | 43 | N A^{−1} |

_{s} |
Negative position stiffness | −0.21 × 10^{6} |
N m^{−1} |

_{0} |
Radial protective clearance | 0.2 | mm |

_{1} |
The distance from balancing plane-A to AMB- |
166 | mm |

_{2} |
The distance of the two AMBs | 193 | mm |

_{3} |
The distance from AMB- |
168.5 | mm |

_{1} |
The correction radius of balancing plane- |
20 | mm |

_{2} |
The correction radius of balancing plane- |
15 | mm |

The measured results of the current stiffness.

AX | 33.45 | N A^{−1} |

AY | 33.99 | N A^{−1} |

BX | 37.77 | N A^{−1} |

BY | 36.06 | N A^{−1} |