unbalancesuppressionforambrotorsystemusing apf … · 2020. 1. 20. · apf-srfalgorithm yuanpingxu...

Research ArticleUnbalance Suppression for AMB Rotor System UsingAPF-SRF Algorithm

Yuanping Xu ,1 Haitong Wu,2 and Xudong Guan2

1National Key Laboratory of Science and Technology on Helicopter Transmission,Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China2College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Correspondence should be addressed to Yuanping Xu; [email protected]

Received 28 July 2019; Revised 18 December 2019; Accepted 24 December 2019; Published 20 January 2020

Academic Editor: Marco Tarabini

Copyright © 2020 Yuanping Xu et al. *is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

With the great capability in levitating rotors, active magnetic bearings (AMB) have been increasingly applied in high-speedrotating machinery applications. Typically, for the practical application of AMB rotor systems, synchronous unbalance vibrationhas long been a critical issue. *us, exploring a facile yet effective approach to address this issue has attracted tremendous interestfrom both industrial and academic perspectives. Herein, in this work, we propose a novel strategy to suppress the unbalancevibration through a compensation control algorithm method. *e compensation control algorithm which is established based onthe combination of first-order all-pass filter (APF) and synchronous rotating frame (SRF) algorithm demonstrates to be efficient ingreatly reducing the unbalance vibration and resulting in excellent system stability according to both simulation and experimentalresults. *is work not only confirms the applicability of APF-SRF algorithm for the AMB rotor systems but also opens up a newparadigm to modulate and minimize the unbalance vibration of AMB rotor systems.

1. Introduction

With themain function of reducing friction betweenmovingparts, bearing acts as one of the most important componentsin rotating machinery. Among the various bearings such asconventional ball and air and sliding bearings, activemagnetic bearings possess the inherent merits of avoidingmechanical frictions and eliminating lubrication, whichmake them highly desirable for constructing high-speedrotating machinery including blower, compressor, bear-ingless motor, and turbomachinery [1–6].

However, for the practical application of AMB rotorsystems, the synchronous unbalance vibration phenomenonhas long been a critical issue. Typically, the unbalance vi-bration induced by the mass unbalance of rotor woulddeteriorate the stabilization of the bearings system and causeserious damage effect. *us, exploring an effective approachto suppress the unbalance vibration of AMB rotor systemshas triggered enormous interest from both academic andindustrial communities.

To date, in the last three decades, a large number ofunbalance vibration suppression control techniques havebeen developed. For instance, Herzog et al. [7] presented ageneralized narrow-band notch filter for the AMB rotorsystem and provided industrial unbalance compensationapplication results. Zhu et al. [8, 9] proposed the iterativeseeking algorithm to produce additional forces to reduce therotor vibration which is suitable for both fix speed andvarying speed conditions. Lum et al. [10] presented anadaptive auto-centering unbalance compensation method,which is frequency independent and suitable for varyingspeed conditions. Based on the filtered-x LMS adaptive filter,Shi et al. [11] proposed an adaptive feedforward unbalancesuppression algorithm for AMB rotor systems with un-known mass imbalance. Cui et al. [12] proposed an im-proved robust odd repetitive control method to suppress theperiodic vibration. Peng et al. [13] developed a multipleresonant control strategy for AMB rotor systems to suppressthe multifrequency current harmonics during the varyingspeed conditions. Chen et al. [14] designed a fuzzy PID

HindawiShock and VibrationVolume 2020, Article ID 2606178, 10 pageshttps://doi.org/10.1155/2020/2606178

mailto:[email protected]://orcid.org/0000-0002-5264-2932https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/2606178

controller, thus reducing the unbalance vibration. Chen et al.[15] applied the adaptive immersion and invariance controlmethod to compensate the unbalance force for a strongnonlinearity three-pole AMB system. Noshadi and Zolfa-gharian [16] demonstrated an unbalance suppressionmethod through varying speed conditions by introducing aninner-loop repetitive disturbance observer-based controller.

Typically, the reported various techniques includingthese mentioned above can be mainly classified into twocategories according to the different mechanisms. *e basicmechanism for the first category is to provide magneticbearings with higher magnetic control stiffness to force therotor spin around its axis of geometry compulsorily [17, 18].*is strategy proves to be applicable for those situationsrequiring high rotating precision, i.e., spindle. However, onthe other hand, using this approach may give rise to thesaturation of power amplifier. *e core mechanism of thesecond category lies in making the rotor to rotate around itsinertial axis to reduce unbalance vibration [7, 19]. Moreover,there are also further unbalance control strategies to improvethe system stability while passing bending critical speeds [20].

In this work, we present a novel approach based on thecombination of first-order all-pass filter (APF) and syn-chronous rotating frame (SRF) algorithm to minimize theunbalance vibration of AMB rotor systems. For the firsttime, the SRF method which has been widely adopted inelectric and electronic systems [21–25] was applied in theAMB rotor system for unbalance compensation [26].Making use of the SRF-based method, one can realize thegood stability of AMB rotor systems by only adjusting thecompensate angle. It also should be pointed out here thataccording to the hypothesis of previous work, the rotordisplacement measurements in the two orthogonal axespossess the same amplitude with a phase difference of π/2.However, in the real AMB rotor system, even though we canuse the same control parameters to improve the isotropy, theanisotropy for x and y direction is still inevitable and thus therotor displacement orbit is not always an ideal circle (seeFigure 1) due to the discrepancy in manufacturing processand electrical component performance. Such phenomenonmay greatly limit the SRF-based controller performance.Herein, to address this problem, we propose a more practicalalgorithm for AMB rotor unbalance compensation. *eprinciple of APF-SRF algorithm is given, and the stabilityanalysis of the AMB rotor system using this proposed APF-SRF algorithm is performed. Both simulation and experi-mental results confirm the efficiency of this novel APF-SRFalgorithm in suppressing unbalance vibration of the AMBrotor system and meanwhile ensuring its good stability.

*e remainder of paper is organized as follows. First,Section 2 describes the principles of the proposed APF-SRFunbalance suppression algorithm. Next, we present thesimulation and experimental results in Section 3. Last,conclusions are drawn in Section 4.

2. Principles

2.1. SRFAlgorithm. *e SRF algorithm is usually adopted toextract the harmonics by using Park transformation, which

is also known as the d-q method [22, 27]. *e Park trans-formation matrix is described as follows:

T(ωt) �cos(ωt) sin(ωt)

− sin(ωt) cos(ωt) . (1)

*e block diagram of the SRF algorithm is pictured inFigure 2. *e [xα, xβ] denote the two orthogonal periodicdiscrete-time signals which have the same amplitude andfrequency in a stationary frame.*erefore, using the discreteFourier series, they can be written as follows:

xα

xβ

⎡⎣ ⎤⎦ �

∞

n�1An cos nωt + θn(

∞

n�1An sin nωt + θn(

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (2)

Using Park transformation, (3) could be obtained bymultiplying (1) with (2) and simplified using the trigono-metric formula as follows:

xd

xq

⎡⎣ ⎤⎦ � T(ωt)xα

xβ

⎡⎣ ⎤⎦ �

∞

n�1An cos (n − 1)ωt + θn

∞

n�1An sin (n − 1)ωt + θn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (3)

where xd and xq are the outputs of the Park transformation inthe synchronous frame. From (3), it is clear that after beingtransformed by the Park transformation, the fundamentaland harmonics components can be separated easily. For theAMB rotor system, the high-frequency harmonic compo-nents will induce high-frequency vibration, which can befiltered out by applying a low-pass filer (LPF), and thefundamental components can be described as follows:

xd

xq

⎡⎣ ⎤⎦ �A1 cos θ1A1 sin θ1

. (4)

Theory

Real

20

20

10

10

0

0

–10

–10–20

–20

y (µm

)

x (µm)

Figure 1: Rotor displacement orbit due to the performance dis-crepancy in x and y directions.

2 Shock and Vibration

*e filtered outputs of the Park transformation arecompared with their references values, and the PI controlleris employed to avoid steady state error. *e PI controller hasan infinite gain for the DC component. When the systemreaches the steady state, the gain of the PI controller could bedenoted as K. *us, we can obtain the equation described asfollows:

id

iq

⎡⎣ ⎤⎦ �KA1 cos θ1KA1 sin θ1

. (5)

Finally, using the inverse Park transformation, the sta-tionary frame fundamental component at ω can be obtainedas follows:

iα

iβ

⎡⎣ ⎤⎦ � Tinv(ωt)id

iq

⎡⎣ ⎤⎦, (6)

where

Tinv(ωt) �cos(ωt + φ) − sin(ωt + φ)

sin(ωt + φ) cos(ωt + φ) , (7)

where φ represents the compensation phase angle and can beused to ensure the close-loop system stability. *erefore, (6)can be written as

iα

iβ

⎡⎣ ⎤⎦ �KA1 cos ωt + θ1 + φ(

KA1 sin ωt + θ1 + φ( . (8)

From (8), it is clear that the control signals iα and iβ havethe same frequency, but with a phase lead φ.

2.2. First-Order All-Pass Filter. As described above, thetransformation from the stationary frame to the synchro-nous frame using Park transformation requires two or-thogonal components which possess the same amplitudeand frequency. However, in the real AMB rotor system, thediscrepancy in manufacturing process and electrical com-ponents performance would result in the anisotropy in x andy directions, giving rise to limited SRF-based controllerperformance. To overcome the limitation, the first-order all-pass filter (APF) is employed in this work to generate twoquadrature signals. APF is widely applied in the signalprocessing, and it could shift the phase of an input signalfrom 0 to π while keeping the amplitude constant.

Consequently, it can be used in the instrumentation andcommunication systems such as multiphase oscillators[28–30]. *e first-order APF transfer function is given by

GAPF(s) �ω0 − sω0 + s

, (9)

where ω0 represents the corner frequency of the system ofthe first-order APF. *is transfer function has unity gain forall frequencies and a phase lag:

ϕ � − 2 arctanωω0

, (10)

where phase lag ϕ varies from 0 to − 180 degrees as ω goesfrom zero to infinity. Bode plots from transfer function of(10) are shown in Figure 3. It can be concluded from Figure 3that the shifting phase is zero while ω is DC and − 90 degreewhen ω�ω0. *erefore, for the input signal with frequencyof ω0, the first-order APF can shift its phase with 90 degrees.If the ω0 is a variable and equals to the rotor displacementsynchronous component, the first-order APF output is al-ways 90 degrees lagged from input rotor displacement signal.*erefore, one axis rotor displacement signal could generatetwo quadrature signals, which will satisfy the SRF algorithmrequirement.

2.3. APF-SRF Unbalance Suppression. Combining the first-order APF and SRF together, the closed-loop AMB rotorsystem control block diagram involving the APF-SRF un-balance suppression controller is shown in Figure 4. Here,H(s) is the unbalance suppression controller, C(s) is theoriginal levitation controller, A(s) represents the amplifier,P(s) represents the controlled plant, and Ks denotes thedisplacement sensor.

*e unbalance suppression controller H(s) is connectedin parallel with the original system levitation controller C(s),and the detected displacement deviation is the input. *eunbalance suppression signals are added to the originalsystem controller output. Figure 5 gives the specific structureof unbalance suppression controller H(s), which contains afirst-order APF and a SRF.*e rotor displacement signal x istransformed to two signals [xα, xβ], which have the sameamplitude but 90 degrees lagged using first-order APF.*en,the two signals [xα, xβ] are transformed to DC component[xd, xq] with Park transformation. For the rotor bearingssystem, considering the periodic forced excitation withfrequency ω, the unbalance response is typical with thefrequency of excitation coinciding with rotor speed, i.e.,ω�Ω. *erefore, the ω could be obtained from the speedsensor. To filter the high-frequency components, a first-order low-pass filter is employed in this work and its transferfunction is

Gf(s) �1

τs + 1, (11)

where τ � 1/(2πfc) and, in this paper, fc � 10Hz. *en, the PIcontroller guarantees no steady state error for the filteredsignals and the PI controller transfer function is

LPFPI

PI

xα

iα

xd xd

id

iq

xq xq

αβdq

αβ

dq

xβ

iβ

Figure 2: SRF algorithm block diagram.

Shock and Vibration 3

GPI(s) � kp +ki

s, (12)

where kp is the proportional coefficient and ki is the integralcoefficient. In this paper, kp � 0.5 and ki � 3.*e PI controlleroutput signals [id, iq] are transformed to control signals [iα,iβ] by applying the inverse Park transformation, and thesecontrol signals are added to the original system controller tosuppress the same frequency vibration.

2.4. Stability Analysis. *e transfer function of the first-order low-pass filter and PI controller in the stationary framecan be analyzed in the frequency domain [26] to obtain

Gf(s)GPI(s) � Gf(s − jω)GPI(s − jω)e

jφ

�1

τ(s − jω) + 1kp +

ki

s − jω e

jφ.

(13)

*us, the transfer function of APF-SRF unbalancesuppression controller can be written as

H(s) �iα

x�

kp(s − jω) + ki ejφ

(τ(s − jω) + 1)(s − jω). (14)

*erefore, as shown in Figure 4, the characteristicequation of the closed-loop system is

1 +(C(s) + H(s))A(s)P(s)Ks � 0. (15)(15) can be rewritten as

1 +H(s)A(s)P(s)Ks

1 + C(s)A(s)P(s)Ks� 0. (16)

As shown in Figure 4, without unbalance suppressioncontroller H(s), the closed-loop transfer function fromamplifier input u to sensor output x could be written as

R(s) �A(s)P(s)Ks

1 + C(s)A(s)P(s)Ks. (17)

*erefore, substituting (14) and (17) into (16) yields

ς s, ki( � (s − jω) (τ(s − jω) + 1) + kpejφ

R(s)

+ kiejφ

R(s) � 0.(18)

*e derivative of (18) with respect to ki can be obtainedwhen ki � 0 is as follows:

zs ki(

zki

s�jω,ki�0� −

zς s, ki( /zkizς s, ki( /zs

� −ejφR(jω)

1 + kpejφR(jω).

(19)

*e AMB rotor system is stable before the APF-SRF isadded; therefore, R(s) is stable. After plunging the APF-SRFcontroller, in order to satisfy the stability requirements to ensurethe motion of root locus to left, we can conclude from (19) that

Re −ejφR(jω)

1 + kpejφR(jω) < 0,

Im −ejφR(jω)

1 + kpejφR(jω) � 0.

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(20)

*estability condition of (20) is equivalent to the following:

−π2< arg −

ejφR(jω)1 + kpejφR(jω)

<π2

. (21)

From (21), we can conclude that the system stability isrelated to the PI parameters, the compensation phase angleφ, and R(s). With the appropriate PI parameters, the stabilityof the closed-loop system can be guaranteed by tuning thecomplement phase angle.

3. Simulation and Experimental Results

3.1. Experimental System. Figure 6 shows the mechanicalstructure of the experiential AMB rotor employed in this

100 101 102 103 104

1

0.5

0

–0.5

–10

–40

–90

–135

–180

Mag

nitu

de (d

B)Ph

ase (

deg)

Frequency (Hz)

ω = ω0

Figure 3: Bode plot of first-order APF.

+

–

+ref

x

u

Ks

H(s)

C(s) A(s) P(s)

Figure 4: Block diagram of the closed-loop control system in-volving the APF-SRF unbalance suppression controller.

LPF

SRF

APF

PI

PI

xd xd

xq xqαβ

dq

xα = x

xβ

id

iq αβdq

iα

iβx

ωt ωt + φω = Ω ∫

Figure 5: Structure of the APF-SRF unbalance suppressioncontroller.


work.*e length of the rotor is around 1m, and the weight isaround 14.5 kg. *ree AMB (AMB1, AMB2, and AMB3) areassembled in this test rig, and in this work, only AMB1 andAMB3 are activated for rotor levitation. Another radial AMBlocated around the center of the rotor (AMB2) is designed asthe noncontact electromagnetic exciter to apply onlineelectromagnetic force during the operation. A 5 kW in-duction motor provides rotating power for the rotor. *e airgaps for radial and thrust AMB are 0.25mm and 0.6mm,respectively. Table 1 lists the other parameters of the test rig.Figure 7 presents the radial and thrust AMB adopted in thiswork.

*e rotor is modeled using the finite element methodand Nelson–Timoshenko beam is adopted. *e rotor is splitinto 69 elements, and Figure 8 shows the theoretical free-freeundamped mode shapes of the rotor. Considering thebearing stiffness and gyroscopic effect, a Campbell diagramis drawn at 106N/m support stiffness, as illustrated inFigure 9. With the gyroscopic effects, the natural frequenciesare the functions of the rotor speed and splitting into for-ward and backward modes.

*e control program is implemented on a dSPACE1202MicroLabBox, and the sample rate was set as 20 kHz. *edecentralized PID control strategy is employed to stabilizethe AMB rotor system, and its PID coefficients are kpc � 1.7,kic � 1, and kdc � 0.003. *e whole system is pictured inFigure 10.

3.2. Simulation. For the AMB rotor system, the system isopen-loop unstable and the feedback closed-loop control isindispensable due to the magnetic force property. *erefore,stability is the primary concern for the AMB rotor system.Using the system parameters mentioned above, Figure 11presents the closed-loop system-dominant root locus in thecomplex plane depending on the phase compensation angleφ. It is clear that without phase compensation (φ� 0), theroot loci (red color) move to the right direction with in-creasing speed. At the speed of 35Hz, the root loci cross theimage axis and then enter the right-half plane with in-creasing rotating speed.*e root loci located at the right-halfplane mean that the system loses stability with the proposedunbalance compensation method when φ� 0. To stabilizethe system, phase-frequency characteristic of the R(s)function is adopted as phase compensation angle and theblue dot is the root loci with introducing the phase com-pensation angle. It is clear that the root loci locate at theentire left-half plane over the operation speed range indi-cating the system remains stable.

AMB1AMB2 Motor AMB3

Rotor Control system

Figure 6: Structure of the AMB rotor.

Table 1: Basic parameters of the test rig.

Symbol Quantity Valuesm Rotor mass 14.5 kgKi Force/current factor 91.5N/AKx Force/displacement factor 7.3×105N/mKamp Amplifier gain 0.4A/Vfw Amplifier cutoff frequency 1224HzKs Sensor gain 20000V/m

(a) (b)

Figure 7: *e radial and thrust AMB employed in this paper. (a)Radial AMB. (b) *rust AMB.

0 1

1st bending mode2nd bending mode3rd bending mode

1

1.5

0.5

0

–0.5

–1

–1.5

Relat

ive d

ispla

cem

ent

0.2 0.4 0.6 0.8AMB rotor length (m)

Figure 8: *eoretical free-free mode shapes of the rotor.

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

1st bending mode

2nd bending mode

3rd bending mode

Conical modeParallel mode

Freq

uenc

y (H

z)

Speed (Hz)

1X

Figure 9: Campbell diagram showing splitting of natural fre-quencies at 106N/m support stiffness.


To simulate the system performance, the rotor unbalanceresponse is calculated. From t� 0 s to t� 0.5 s, the APF-SRFunbalance suppression controller is disabled. Two phasecompensation situation, φ� 0 and φ� π/2 are tested.Figure 12(a) shows the simulated displacement vibration at35Hz with φ� 0. When the unbalance suppression con-troller is activated, the displacement vibration becomesdivergence and the filtered DC output becomes unstable(Figure 12(b)), which means the AMB rotor system is out ofcontrol. Figure 12(c) shows the simulated displacementvibration at 35Hz with φ� π/2. When the unbalance sup-pression controller is activated, the displacement vibrationcan be minimized and the filtered DC output is convergent,as indicated in Figure 12(d).

*e simulation results indicate that with the appropriatecompensation phase angle φ to ensure the system stability,the APF-SRF algorithm is an effectivemethod for AMB rotorsystem unbalance suppression.

3.3. Experimental Results. *e control program is imple-mented on a dSPACE1202 MicroLabBox, and the samplerate was set as 20 kHz. *e decentralized PID control

strategy is employed to stabilize the AMB rotor system. Inthe test, the phase compensation angle is acquired from thetransfer function R(s) using the sine sweeping test from thereal test rig in the levitation condition.

Firstly, a fixed rotating speed 9000 rpm is selected toevaluate the proposed APF-SRF unbalance compensationalgorithm. Figure 13 shows the changes of rotor displace-ment at 9000 rpmwhile the proposed unbalance suppressionmethod is activated. *e maximum displacement ampli-tudes for both AMB1 and AMB3 in two orthogonal di-rections are almost suppressed from 15 μm to 5 μm.Figure 14 shows the comparison of rotor displacement orbitsat 9000 rpm for both AMB1 and AMB3 with an unbalancecompensation controller or not, for which the same con-clusions can be obtained.

To evaluate the proposed method in the variable speedcondition, the rotor was decelerated from 11100 to 0 rpmwiththe proposed unbalance compensation algorithm. Figure 15gives the displacement amplitude from 11100 to 0 rpm usingthe proposed unbalance compensation algorithm. For thesake of comparison, the corresponding displacement am-plitude with the absence of proposed unbalance compensa-tion algorithm is also given, as illustrated in Figure 16.Obviously, the proposed method demonstrates to be effectivein suppressing the unbalance phenomenon and stablizing thesystem, particularly at high speed. It also should be noted herethat due to the different vibration conditions, the decelerationtime of the two figures exists as a weak discrepancy.

*e ISO14839-2:2004 (vibration of rotating machineryequipped with active magnetic bearings-part 2: evaluation ofvibration) standard is selected for evaluation. According toISO14839-2:2004, the vibration index is defined Dmax/Cmin,where Dmax is the maximum peak displacement which is

Dmax � max��

x2(t) + y2(t)

, (22)

RotorAMB

Speedsensor

PC

dSPACE

Amplifier

Frequencyconverter

Oscilloscope

Figure 10: AMB rotor system.

–200

0

200

400

600

800

1000

1200

1400

35Hz

Real axis

Without phase compensationWith phase compensation

Imag

e axi

s

–12 –10 –8 –6 –4 –2 0 2 4

Figure 11: Closed-loop-dominant root locus.

40

20

0

–20

–40

x (µ

m)

0 10.5t (s)

(a)

40

20

0

–20

–400 10.5

t (s)

x d (µ

m)

(b)

–20

0

20

x (µ

m)

0 10.5t (s)

(c)

–20

–10

0

10

x d (µ

m)

0 10.5t (s)

(d)

Figure 12: Simulation results of the AMB rotor system using APF-SRF unbalance compensation algorithm. (a) Rotor displacementwith φ� 0. (b) Filtered output with φ� 0. (c) Rotor displacementwith φ� π/2. (d) Filtered output with φ� π/2.


x 1 (µ

m)

20

10

0

–10

–200 1 2 3

Time (s)

(a)y 1

(µm

)

20

10

0

–10

–200 1 2 3

Time (s)

(b)

x 2 (µ

m) 10

0

–10

–20

20

0 1 2 3Time (s)

(c)

y 2 (µ

m)

20

10

0

–10

–200 1 2 3

Time (s)

(d)

Figure 13: *e changes of rotor displacement at 9000 rpm with the proposed APF-SRF unbalance compensation algorithm. (a) AMB1 xdirection. (b) AMB1 y direction. (c) AMB3 x direction. (d) AMB3 y direction.

y 1 (µ

m)

20

10

0

–10

–20

x1 (µm)–20 –10 0 10 20

(a)

y 1 (µ

m)

20

10

0

–10

–20

x1 (µm)–20 –10 0 10 20

(b)

x2 (µm)

y 2 (µ

m)

20

10

0

–10

–20–20 –10 0 10 20

(c)

y 2 (µ

m)

20

10

0

–10

–20

x2 (µm)–20 –10 0 10 20

(d)

Figure 14: Rotor displacement orbits at 9000 rpm. (a) AMB1 without unbalance compensation. (b) AMB1 with unbalance compensation.(c) AMB3 without unbalance compensation. (d) AMB3 with unbalance compensation.

0 20 40 60 80 100 120 140 160 180–20

0

20

Disp

lace

men

tam

plitu

de (µ

m)

Time (s)

(a)

0 20 40 60 80 100 120 140 160 180–20

0

20

Disp

lace

men

tam

plitu

de (µ

m)

Time (s)

(b)

0 20 40 60 80 100 120 140 160 180–20

0

20

Disp

lace

men

tam

plitu

de (µ

m)

Time (s)

(c)

0 20 40 60 80 100 120 140 160 180–20

0

20

Disp

lace

men

tam

plitu

de (µ

m)

Time (s)

(d)

Figure 15: Displacement amplitude from11100 to 0 rpm using the proposed unbalance compensation algorithm. (a) AMB1 x direction. (b)AMB1 y direction. (c) AMB3 x direction. (d) AMB3 y direction.


where Cmin is the minimum value of radial or axial clearancebetween the rotor and stator. Figure 17 shows the com-parison of the vibration index for AMB1 depending on withthe proposed method or without. In the entire speed range,the proposed method could stabilize the system and possesswell unbalance suppressionperformance. Figure 18 showsthe comparison of the vibration index for AMB3, and we canobtain the same conclusion.

0 20 40 60 80 100 120 140–20

0

20

Disp

lace

men

tam

plitu

de (µ

m)

Time (s)

(a)

0 20 40 60 80 100 120 140–20

0

20

Disp

lace

men

tam

plitu

de (µ

m)

Time (s)

(b)

0 20 40 60 80 100 120 140–20

0

20

Disp

lace

men

tam

plitu

de (µ

m)

Time (s)

(c)

0 20 40 60 80 100 120 140–20

0

20

Disp

lace

men

tam

plitu

de (µ

m)

Time (s)

(d)

Figure 16: Displacement amplitude from11100 to 0 rpm with the absence of proposed unbalance compensation algorithm. (a) AMB1 xdirection. (b) AMB1 y direction. (c) AMB3 x direction. (d) AMB3 y direction.

A�er compensationBefore compensation

0.3

0.2

0.1

00 2000 4000 6000 8000 10000 12000

Speed (rpm)

Dm

ax/C

min

Figure 17: *e comparison of the vibration index for AMB1.

A�er compensationBefore compensation

0.3

0.2

0.1

00 2000 4000 6000 8000 10000 12000

Speed (rpm)

Dmax

/Cmin

Figure 18: *e comparison of the vibration index for AMB3.

15

10

5

010000

5000

0 0100

200300

Disp

lace

men

t (µm

)

Spectrum (H

z)Speed (rpm)

Figure 19: Comparison rotor displacement waterfall diagram inAMB1 x direction without algorithm.


Figure 19 shows the rotor displacement waterfall diagram inAMB1 x direction without any suppression method over theentire speed range. We can find that a maximum synchronousdisplacement vibration is up to 9.3μm at 8340 rpm. Figure 20shows the rotor displacement waterfall diagram in AMB1 xdirection with the proposed suppression method over the entirespeed range, and it is clear that the proposed method is veryeffective especially at high speed.

4. Conclusion

Synchronous unbalance vibration has long been a criticalissue for the practical application of AMB. In this work, wepropose a feedforward compensation controller to suppressthe unbalance vibration, which is rooting in the combinationof APF and SRF transformation. *e proposed algorithmdemonstrates to be more effective over the traditional SRF-based control method by generating two orthogonal com-ponents using APF. Simply applying the APF-SRF algorithmto the AMB rotor system may lead to instability, and acompensation phase angle is introduced further to ensure agood stability. According to both simulation and experi-mental results, the APF-SRF algorithm proves to be efficientin suppressing unbalance vibration of AMB rotor system,meanwhile ensuring its good stability. *is work not onlydemonstrates the applicability of APF-SRF algorithm for theAMB rotor system but also opens up a new paradigm tomodulate and minimize the unbalance vibration of the AMBrotor system.

Data Availability

*e experimental figure data used to support the findings ofthis study are included within the article. *e Matlab sim-ulation, control program data used to support the findings ofthis study are available from the corresponding author.

Conflicts of Interest

*e authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

*is work was supported in part by the National NaturalScience Foundation of China (51905263), Key Laboratory ofVibration and Control of Aero-Propulsion System, Ministryof Education (VCAME201902), Jiangsu Natural ScienceYouth Fund (BK20190412), China Postdoctoral ScienceFoundation (2018M642245), Jiangsu Postdoctoral ResearchFunding (2019K188), and Fundamental Research Funds forthe Central Universities (NZ2018460).

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