thestabilityofmagneticlevitationmillingsystembasedon...

Research ArticleThe Stability of Magnetic Levitation Milling System Based onModal Decoupling Control

Xiaoli Qiao and Xiaoping Tang

Shaoxing University Yuanpei College Shaoxing 312000 China

Correspondence should be addressed to Xiaoli Qiao qiaoxiaoli168163com

Received 6 July 2019 Revised 17 September 2019 Accepted 18 September 2019 Published 19 May 2020

Academic Editor Chao Tao

Copyright copy 2020 Xiaoli Qiao and Xiaoping Tang is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

Milling stability not only reduces the surface quality of the workpiece but also seriously restricts the high-speed development ofCNCmachine tools e electric spindle rotor system with the active magnetic bearing has a strong gyro coupling effect and withthe increasing rotor speed it will become a major unfavorable factor for the stability of the system during high-speed milling estrong gyro coupling effect makes the stability region narrow at the time of high-speed milling So a modal decoupling controlmethod that can reduce the effects of the gyro effect on the magnetic levitation milling system under high-speed milling isproposed e effects of the gyro coupling of the magnetic bearing rotor on the milling stability region before and after thedecoupling control are studied which show that the modal decoupling control technology can reduce the effects of the gyro effecton the magnetic levitation milling system

1 Introduction

e high-speed milling is the only way to improve millingefficiency and the magnetic suspension support technologyis one of the worldrsquos recognized high technology and ofcourse the first choice for high-speed milling machines ehigh-speed magnetic suspension electric spindle rotor sys-tem has a strong gyro coupling effect at high speed whichwill greatly affect the stability of the high-speed machiningsystem and then affect the efficiency of milling processingand product quality When the electric spindle rotor systemis treated as the rigid rotor system the modes of rotation andtranslation motion will occur When the rotor of the electricspindle rotates at high speed the rotational mode can bedivided into positive precession mode and opposite pre-cession mode under the action of the strong gyro couplingeffect of the rotor In the actual magnetic suspension electricspindle rotor system the bandwidth of the power amplifierand the sensor is limited which will lead to a time delay ofthe electromagnetic force and the force produced by themagnetic bearing will put energy into the electric spindlerotor the energy of the opposite precession mode will be put

into the positive feedback and be accumulated which willlead to a unstable milling system In addition the dampingeffect of the control force will also decrease as the precessionmodal frequency decreases especially when the controllerincludes the integral part the phase advance of the lowfrequency band is difficult to guarantee When the preces-sion mode frequency finally enters the range in which theintegral parameter acts the precession mode will also causethe milling system to be unstable [1]

In order to reduce the gyro coupling effect of the high-speed magnetic suspension electric spindle rotor system onmilling stability it is necessary to suppress the positiveprecession mode and the opposite precession mode gen-erated by the gyro coupling effect In this regard varioussolutions have been proposed mainly including controlmethods based on modern control theory such as slidingmode control [2] μ synthesis [3] gain-scheduled Hinfincontrol [4] Cholesky decomposition reduction [5] and LQRcontrol [6] Although these control methods had played arole in suppressing the gyro effect these algorithms arecomplex and computationally intensive and are difficult toimplement in engineering due to hardware conditions

HindawiShock and VibrationVolume 2020 Article ID 7839070 9 pageshttpsdoiorg10115520207839070

ere is also a cross-feedback control method based on thetraditional PD controller in which the cross-feedback can bedivided into displacement crossover [7] velocity crossovervelocity and displacement crossover [8] and displacementcrossover combined with electromagnetic force [9] e ad-vantage of cross-feedback especially the speed cross-feedbackmethod is that the structure is simple and is easy to beimplemented e disadvantage is that there is still no effectivecross-feedback parameter designmethod In addition when thetraditional PD control is used the characteristics of each modeare difficult to be independently adjusted because there is thestrong coupling between the rotational mode and the trans-lational mode of the magnetic electric spindle rotor ereforethe control methods based on this have certain limitations

In view of this based on the mathematical model of theactive magnetic bearing electric spindle rotor system a modaldecoupling control method is proposed which can decouplethe rotating mode and the translational mode of the activemagnetic bearing electric spindle rotor system in order tocontrol the stiffness and damping of the translationalmode andthe rotation mode independently which can reduce effectivelythe influence of the gyro effect on the high-speed millingstability area and then improve the stability area of the high-speed magnetic suspension electric spindle milling system

2 Mathematical Model of Four-Degree-of-Freedom Magnetic Suspension ElectricSpindle Rotor System

e structure of the magnetic bearing rigid electric spindlerotor system is shown in Figure 1 Ideally the axis of therotor coincides with the center of the two radial bearings Inorder to describe the mutual position between the rotor thesensor and the electromagnetic bearing the main coordi-nate system oxyz is set up in which the coordinate origin isat the centroid point c of the rotor and e z-axis coincideswith the center line of the two radial bearings e distancefrom the center of the left and right radial electromagneticbearings A and B to point c is respectively lmA and lmB andlmA is a negative number while lmB is a positive numberedistances of the left and right sensors A and B to c arerespectively lsA and lsB lsA is negative and lsB is positive Inorder to make the analysis easy three coordinate systems ofradial planes are established which are the sensor coordinatesystem the active magnetic bearing coordinate system andthe centroid coordinate system the coordinate origin is seton the center line of the two radial bearings When the rotorrotates the coordinates at the center of mass of the rotor arex y θx and θye coordinates at the left and right sensorsare xsA xsB ysA and ysB and the coordinates of the left andright active magnetic bearings are xmA xmB ymA and ymBrespectively e ignorance of the sensor and magneticbearing installation position error canmake xsA xmA xaxsB xmB xb ysA ymA ya ysB ymB yblsA lmA la lsB lmB lb and l lsA minus lsB When thedisplacement of the rotor is xa ya xb and yb in the xa yaand yb in the y α β and directions the displacement of therotor geometric center O is

x lb

Lxa minus

la

Lxb

y lb

Lya minus

la

Lyb

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(1)

e angle at which the rotor rotates counterclockwiseabout the x-axis is

θx yb minus ya

L (2)

e angle at which the rotor rotates counterclockwisearound the y-axis is

θy xa minus xb

L (3)

When themotion equation of the four-degree-of-freedomelectric spindle rotor system with active electromagneticbearing is established since the bearing air gap is small it canbe assumed that (1) the rotor is an axisymmetric rigid rotorthat is the moments of inertia around the x and y axes areequal (2) e structure and parameters of the radial four-degree-of-freedom are exactly the same (3) the magnetic fieldin the electromagnet is evenly distributed the magnetic fluxleakage is ignored and the magnetic material does not exhibitsaturation characteristics (4) the influence of the magneticresistance and loss of the core material are ignored Withoutconsidering the influence of external damping factors it iseasy to establish the motion equation of the axially symmetricactive magnetic bearing electric spindle rotor systemaccording to the rotor dynamics theory [10]

m eurox FmxA + FmxB

m euroy FmyA + FmyB

Jeuroθy minus Jzω _θx FmxAla minus FmxBlb

Jeuroθx minus Jzω _θy minus FmyAla + FmyBlb

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(4)

c

Oz

y

x

Act

ive m

agne

tic b

earin

g B

Disp

lace

men

t sen

sor B

Elec

tric

spin

dle r

otor

Wor

kpie

ce

Act

ive m

agne

tic b

earin

g A

Disp

lace

men

t sen

sor A

lmAlsA lsB

lmB

L

lcw

Figure 1 Active magnetic bearing-supported motorized spindlemilling system

2 Shock and Vibration

where m is the electric spindle rotor quality Jx Jy J isthe rotor moment inertia around x and y axis and yz is therotor moment inertia around z axisFmxA FmxB FmyA andFmyB are respectively the electro-magnetic force of the end A and end B in the direction of x

and y of the active electromagnetic bearingsEquation (4) is transformed into a matrix

Meuroq + G _q LfF (5)

HereM

J 0 0 00 m 0 00 0 J 00 0 0 m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is the mass matrix of the rotor

system G

0 0 JzΩ 00 0 0 0

minus JzΩ 0 0 00 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is the gyro effect matrix of

the rotor system Lf

la minus lb 0 01 1 0 00 0 minus la lb0 0 1 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is force arm coef-

ficient matrix F (FmxA FmxB FmyA FmyB)T is the elec-tromagnetic force vector of the active magnetic bearing andq (θy x θx y)T is the coordinate of the centroid of therigid electric spindle

e electromagnetic force generated by the activemagnetic bearing is a quadratic function of the coil currentand the air gap between the stator and rotor [10] as follows

Fm μ0AaN2

4i2

δ2 (6)

where μ0 is air permeability Aa is the cross-sectional area ofthe iron core and air gap N is the number of turns i is thecoil current and δ is the length of the air gap

When the structure parameter is constant and the rotormoves in a small range near its static equilibrium positionequation (6) can be expanded as Taylor at the static equi-librium point and the high-order small amount is omittede electromagnetic force can be expressed as a linearfunction of the current stiffness coefficient and the dis-placement stiffness coefficient [12] Since the rotor is placedvertically the bias currents on all the poles can be the samee electromagnetic force can be expressed as

F

FmxA

fmxB

fmyA

fmyB

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ks 0 0 0

0 ks 0 0

0 0 ks 0

0 0 0 ks

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xa

xb

ya

yb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

ki 0 0 0

0 ki 0 0

0 0 ki 0

0 0 0 ki

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

imxa

imxb

imya

imyb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ksqs + Kii

(7)

Here qs (xaxbyayb)T is the displacement of theactive magnetic bearing relative to the equilibrium point i

(imxaimxbimyaimyb)T is control current Ks diag(ks

ks ks ks) is the matrix of electromagnetic bearing dis-placement stiffness coefficient and Ki diag(ki ki ki ki) isthe matrix of the current stiffness coefficient of the elec-tromagnetic bearing

After the structural parameters of the electromagneticbearing the current and the gap at the operating point aredetermined the current stiffness coefficient and the dis-placement stiffness coefficient of the electromagnetic bearingare both constant

Putting equation (7) into equation (5) the followingequation is obtained

Meuroq + G _q Lf Ksqs + Kii( 1113857 (8)

In order to unify the coordinate system the sensorcoordinate system qs on the right side of equation (8) isconverted into a centroid coordinate system q e coor-dinate conversion relationship can be obtained as follows

xa

xb

ya

yb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

la 1 0 0

minus lb 1 0 0

0 0 minus la 1

0 0 lb 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

θy

x

θx

y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

at isqs Tbq (10)

Here Tb

la 1 0 0minus lb 1 0 00 0 minus la 10 0 lb 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is the transformation matrix

of the electromagnetic bearing B coordinate to the centroidcoordinate

e matrix Tb is the transpose of the matrix Lf

Tb LTf (11)

Putting equations (9) and (11) into equation (8) thefollowing equation is obtained

Meuroq + G _q LfKsLTfq + LfKii (12)

en Kss LfKsLTf and the following equation can be

obtained

Meuroq + G _q minus Kssq LfKii (13)

Here

Kss

ks l2a + l2b( 1113857 ks la minus lb( 1113857 0 0

ks la minus lb( 1113857 2ks 0 0

0 0 ks l2a + l2b( 1113857 ks minus la + lb( 1113857

0 0 ks minus la + lb( 1113857 2ks

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

en KTss (LfKsL

Tf)T LfKsL

Tf Kss

3 Modal Decoupling Principles

e purpose of the modal decoupling controller is to achievethe independent control between the rotational mode andthe translational modal of the electromagnetic suspensionspindle system Firstly the displacement signal at the foursensors of the input modal decoupling controller is con-verted into the translational displacement signal and therotational angle signal at the center of mass of the rotorat

Shock and Vibration 3

is the coordinates of the sensor coordinate system xa xb yaand yb are converted into the coordinates of the centroidcoordinate system x y θx and θy In this way the rotormodes can be controlled independently e PD controlalgorithm is still used in the modal decoupling controller toproduce control signal

Since the action point of control signal is at the masscenter of the rotor the actual electromagnetic bearing islocated at both ends of the rotor it is also necessary toconvert the control signal at the center of mass of therotor into a control signal of the electromagnetic force atthe left and right electromagnetic bearings at is thecoordinates of the centroid coordinate system are con-verted into the coordinates of the bearing coordinatesystem Finally the obtained electromagnetic forcecontrol signal together with the negative stiffness com-pensation signal is added to the power amplifier whichwill generate the control current to control the stabilityof the milling process [13]

e control flow is shown in Figure 2e specific process of the modal decoupling controlled

and its mathematical expression is designed As shown inFigure 1 if lmA ne minus lmB the distance between the A and Belectromagnetic bearings to the origin of the coordinates isdifferent and the negative stiffness matrix Kss is not a di-agonal matrix which will weaken the effect of modaldecoupling After the translation of the centroid of the rotoran electromagnetic force of the same magnitude is generatedat the bearing positions If it is an asymmetrical bearing anonzero torque is generated with respect to the centroid cwhich will cause the rotor to rotate In order to make theproportional and differential parameters of the PD con-troller act more directly on the modal stiffness the negativestiffness Kss must be compensated

If the current signal input to the power amplifier isdivided into a current ic of the modal decoupling controllerand a negative stiffness compensation current ik the currentsignal i which will be input to the power amplifier by thecontroller is

i ic + ik (15)

Putting equation (15) into equation (13) the followingequation can be obtained

Meuroq + G _q + Kssq LfKiic + LfKiik (16)

As shown in equation (16) in order to make equationKssq LfKiik established it is necessary to compensate thenegative stiffness Putting equation (10) into Kssq LfKiikik can be expressed as

ik Kminus 1i B

minus 1KssL

minus 1s qs (17)

What will be discussed is the expression of the currentsignal ic of the modal decoupling controller For the input ofthe controller the output current signal by the conventionaldistributed PD controller is

ic minus Pqs minus D _qs (18)

HereP diag(pxA pxB pyA pyB) is the proportionalcoefficient matrix and D diag(dxA dxB dyA dyB) is adifferential coefficient matrix

It can be known from equation (15) that the input changeof any one control channels of the controller will cause thechange in the rotation angle and the displacement of thecenter of mass of the rotor which makes the rotational modeand the displacement mode coupled to each other at theinput end erefore the coordinates xsA xsB ysA and ysB

of the sensor coordinate system should be converted intothe coordinates of the centroid coordinate system x y θxand θy that is the matrix qs should be converted into amatrix q e previous deduction leads to q (LT

f)minus 1qswhich means a coordinate transformation link with a re-lation matrix (LT

f)minus 1 is added before the sensor and the PDcontroller In the physical sense the direct adjustment of themodal offset signals by the PD control algorithm isaccomplished

For the output the expression LfKi is

LfKi

KilbA KilbB 0 0

Ki Ki 0 0

0 0 KilbA KilbB

0 0 Ki Ki

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)

It can be seen that since LfKi is not a diagonal matrixthere is a coupling between the rotational mode and thetranslational mode In order to decouple the rotational modeand the translational mode at the output it is also necessaryto set a transformation matrix T to make LfKiT convertedinto a diagonal matrix ere are many different expressionsfor this kind of diagonal matrix If LfKiT I the matrix T isobtained

T Kminus 1i L

minus 1f

1ki lbA minus lbB( 1113857

minus 1 lmB 0 0

1 minus lmA 0 0

0 0 minus 1 lmA

0 0 1 minus lmA

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

From the physical point of view modal decouplingcontrol algorithm can directly adjust the modal of the rotorafter the decoupling of the input end but the output controlsignal at this time for on the center of mass of the rotor isthe signal required for the stability of the rotor not theelectromagnetic force signal required for the two radialbearings erefore a conversion link must be provided toconvert the torque signal into the electromagnetic forcesignal of the electromagnetic bearing in which way thedecoupling between the input and output of the controller isrealized and finally the current command signal of the PDcontroller is obtained as follows

ic minus Kminus 1i L

minus 1f (P + D) L

Tf1113872 1113873

minus 1qs (21)

Combining equations (15) (18) and (21) the followingequation is obtained


i ic + ik minus Kminus 1i L

minus 1f P + D minus kss( 1113857 L

Tf1113872 1113873

minus 1qs (22)

Putting equations (19) and (22) into equation (8) then

Meuroq + G _q + Pq + D _q 0 (23)

P diag pr pp pr pp1113872 1113873 (24)

D diag dr dp dr dp1113872 1113873 (25)

where pr and dr are respectively the proportional anddifferential coefficients in the rotational mode controlchannel while pp and dp are respectively the proportionaland differential coefficients in the translational mode controlchannel

It can be seen from equations (24) and (25) that thetranslational mode and the rotational mode control channelare independent of each other so that the stiffness anddamping of each mode can be adjusted by changing pr ppdr and dp In addition the gyro matrix G only has acoupling effect on the rotatingmodal control channel and noinfluence on the translational modal control channel andthe gyro coupling effect is significantly reduced

4 Influence of Cross-Coupling Effect of ActiveMagnetic Bearing Gyro on Grinding Stability

41 Dynamic Milling Force Model e dynamic millingprocess with four cutter cylindrical end mills is shown inFigure 3 [11]

e feed direction of the tool is defined as the x directionand the coordinate direction is shown in Figure 4 e ϕj

shows the angular position of the tooth e tangential andradial cutting forces of the tooth j are expressed as Ftj andFnj the angle between the radial cutting force and the vectorcutting force is expressed as β e transient cutting force ofthe jth tooth of the end mill can be expressed as follows [11]

Fxj minus Ktbg ϕj1113872 1113873Δx sin ϕj1113872 1113873cos ϕj1113872 1113873 + Knsin2 ϕj1113872 11138731113872 1113873

+Δy Kn sin ϕj1113872 1113873cos ϕj1113872 1113873 + cos2 ϕj1113872 11138731113872 1113873

⎛⎝ ⎞⎠

(26)

Fxj Ktbg ϕj1113872 1113873Δx minus Kncos ϕj1113872 1113873sin ϕj1113872 1113873 + sin2 ϕj1113872 11138731113872 1113873

+Δy sin ϕj1113872 1113873cos ϕj1113872 1113873 minus Kncos2 ϕj1113872 11138731113872 1113873

⎛⎝ ⎞⎠

(27)

e cutting force vector FAx and FAy of each of thecutter end mills with Nt teeth in the x and y directions is

Negative stiffness compensation

Centroid coordinatesconverted to

magnetic bearing

coordinates Pow

er am

plifi

er Magnetic bearing

coordinatesconverted to

centroidcoordinates

Elec

tric

spin

dle r

otor

syste

mx

y

θx

θy

PD

PD

PD

PD

xa

xb

ya

yb

xlowast

ylowast

θlowast

x

θlowast

y

ixixa

iya

iyb

ixbiy

iθx

iθy

Figure 2 Modal decoupling control system

β

Fnj

Fj

Ftj

Fjx

y

j ndash 1

j

n

ϕj

Figure 3 Dynamic milling process


FAx 1113944

Nt

j1Fxj

FAy 1113944

Nt

j1Fyj

(28)

By putting equations (26) and (27) into equation (28)the cutting force expression shown in the following equationcan be obtained

FAx 12

bKt cxxΔxi + cxyΔyi1113872 1113873

FAx 12

bKt cyxΔxi + cyyΔyi1113872 1113873

(29)

where

cxx 12cos(2φ) minus 2Knφ + KnS1113858 1113859

1113868111386811138681113868φe

φs

cxy minus12sin(2φ) + 2φ minus Kn cos(2φ)1113858 1113859

1113868111386811138681113868φe

φs

cyx minus12sin(2φ) minus 2φ minus Kn cos(2φ)1113858 1113859

1113868111386811138681113868φe

φs

cyy 12cos(2φ) + 2Knφ + Kn sin(2φ)1113858 1113859

1113868111386811138681113868φe

φs

(30)

where Kt is the tangential milling stiffness Kn is the radialmilling stiffness b is the milling width and ΔxiΔyi is thedynamic depth of cut in the x and y directions and the startand end angles of the tooth action are expressed as ϕs and ϕerespectively

42 Stability Analysis When the stability of the activemagnetic bearing cutting system is analyzed it is mainly

about the dynamic displacement change of the workpieceend however there is the gap between the workpiecemounting position and the position of the electromagneticbearing B and the control force of the electromagneticbearing is acted on the position of the electromagneticbearing In order to reduce stability analysis error thedisplacement at the workpiece is corrected by the correctionfactor kw that is xcw kwxmB erefore the first step is touse the displacement of the electromagnetic bearing to in-dicate the following

q

θy

x

θx

y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1L

minus1L

0 0

lb

L

la

L0 0

0 0 minus1L

1L

0 0lb

L

la

L

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

middot

xbA

xbB

ybA

ybB

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tsqs (31)

Putting equation (31) into equation (23) the motionequation of the magnetic suspension rigid electric spindlerotor system can be obtained in magnetic bearing B coor-dinates as

MTs euroqs +(G + D)Ts _qs + TsPqs 0 (32)

If Δe [kwΔxmBΔymB]T is the dynamic cutting depthof milling equation (35) can be obtained by putting equation(29) into equation (32)

MTs Δeuroe +(G + D)Ts Δ _e + TsP Δe 12

bKt

Nt

2πΗΔe

(33)

0 02 04 06 08 1ndash04

ndash03

ndash02

ndash01

0

01

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)

ndash04 ndash02 0 02 04ndash04

ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)

Figure 4e displacement at the workpiece position based on the conventional PID (a) displacement diagram in the x and y direction (b)running track at the position of the workpiece


e dimension of the matrix H is 2 times 2 and the pro-portional coefficient is cxx cxy cyx cyy

In order to analyze the effect of the gyro effect of theactive magnetic bearing on the region of the milled stabilityconvert equation (33) as follows [14]

minus MTsω2c +(G + D)Tsωc + TsP1113966 1113967 Δe iωc( 11138571113864 1113865

Nt

4πbKtH

middot 1 minus eminus iωcT

1113872 1113873 Δe iωc( 11138571113864 1113865

(34)

If Γ(iωc) 1 minus MTsω2c + (G + D)Tsωc + TsP1113864 1113865 then

equation (34) is arranged out to obtain the closed-loopdynamic milling characteristic equation

I minusNt

4πbKt 1 minus e

minus iωcT1113872 1113873[B]Γ iωc( 11138571113882 1113883 Δe iωc( 11138571113864 1113865 0 (35)

If ξ minus (Nt4π)bKt(1 minus eminus iωcT) the eigenvalues ofequation (35) are

ξ ξRe + iξIm (36)

When the real part of all the eigenvalues of equation (35)is less than zero the milling process is stable Otherwise themilling process is unstable

Equation (36) and eminus iωcT cos(ωcT) minus i sin(ωcT) areput into ξ minus (Nt4π)bKt(1 minus eminus iωcT) and the criticalmilling width blim is obtained

blim minus4π

NtKt

ξRe + iξIm1113864 1113865 +ξIm sin ωcT( 1113857 minus iξRe + ξIm1113864 1113865

1 minus cos ωcT( 11138571113896 1113897

(37)

Make the imaginary part zero

ζ ξImξRe

sin ωcT( 1113857

1 minus cos ωcT( 1113857 (38)

Equations (38) and (37) are arranged

blim minus2π

NtKt

ξRe 1 + ξ21113872 1113873 (39)

where T (π minus 2tanminus 1ξ + 2Nπ)ωc is the number of ripplesbetween the two teeth and ωc chatter frequency

5 Milling System Stability Area Analyses

e influence of the traditional PID controller and modaldecoupling control on the displacement of the workpiece atthe milling position is analyzed by simulation e influenceof the gyro effect on the milling stability region is analyzede parameters of the electric spindle rotor system used in

Table 1 Electric spindle parameters

m Rotor mass 258 kgla Radial electromagnet center distance 150mmJx Jy Moment of inertia of the rotor around the x-axis 02251 kgm2

Jz Moment of inertia of the rotor around the z-axis 03388 kgm2

δ Radial electromagnetic bearing ideal air gap 04times10minus 3mRs Spindle radius 50mmki Current stiffness coefficient 377NAks Displacement stiffness coefficient 1508e4Nm

0 02 04 06 08 1ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)


ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)

Figure 5 e displacement at the workpiece position based on modal decoupling stability area of milling (a) without control and (b) withmodal decoupling control


the simulation are shown in Table 1 and the milling pa-rameters are N4 4 and Kt 2173Nmm2

Figure 4 shows the displacement track at the workpieceposition under the PID controller It can be seen fromFigure 4 that the rotor of the electric spindle cannot be stablysuspended at the operation starts e PID controller isturned on at 002 s and the electric spindle rotor graduallystabilizes under the action of the controller e displace-ment in the x and y directions of the workpiece positiongradually decreases but there is fluctuation which is due tothe influence of the gyro cross-effect of the electric spindlerotor system e modal of the electric spindle rotor isdecoupled on the basis of the PID controller and the ro-tational mode and the translational modal of the rotorsystem of the electric spindle are independently controlled Itcan be seen from Figure 5 that the modal decoupling controlcan make the displacement of the workpiece position rapidlyreduced and stable Figure 5(a) shows when the modaldecoupling control is also turned on at 002 s the dis-placement at the workpiece position quickly approacheszero (stable) and the fluctuations existing in the conven-tional PID control are overcome

Figure 6 shows the effect of gyro effect on the stability ofmilling at different speeds n 6000 rmin and n 15000 rmin the influence of gyro effect on the stability of milling isanalyzed

It can be seen from Figure 6(a) that the gyro effect willshift the milling stability to the right and move it slightlyupward e higher the speed the more the right shiftswhich indicate that the gyro effect makes the frequency ofthe critical stability zone of milling frequency value increaseAnd the higher the speed of the electric spindle rotor thehigher the frequency of the critical stability zone of themilling is is because the gyro effect is related to the speedof the electric spindle and as the rotor speed of the electricspindle increases the gyro effect becomes more obviousemodal decoupling control can greatly reduce the gyro effect

on the critical stability zone of milling which is shown inFigure 6(b)

6 Conclusions

Based on the mathematical model of electric spindle rotorsystem with the active magnetic bearing the influence of thetraditional PID controller and the modal decoupling controlon the displacement of the milling workpiece position areanalyzed e displacement at the workpiece positionquickly approaches zero (stable) with the modal decouplingcontrolling and the fluctuations existing in the traditionalPID control are overcome

e effect of the gyro effect on the stability of milling atdifferent rotor speeds of the electric spindle is analyzed esimulation results show that the gyro effect can shift thestability area of themilling to the right And this effect will bemore obvious as the rotor speed of the electric spindle in-creases e proposed modal decoupling control can reducethe influence of gyro effect on the stability of milling byseparately adjusting the parameters of the translationalmode and the rotational mode

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by the National Natural ScienceFoundation of China (51505296) and Zhejiang ProvincialPublic Foundation of China (LGG19E050014)

0 05 1 15 2 25times 104

0

02

04

06

08

1 times 10ndash3

Spindle speed (rmin)

Dep

th o

f mill

ing

blim

(m) Without gyroscopic

effect

With gyroscopiceffect

(a)

times 104

times 10ndash3

0 05 1 15 2 250

02

04

06

08

1


Dep

th o

f mill

ing

blim

(m)

Without gyroscopiceffect

With modaldecoupling control


(b)

Figure 6 Effect of modal decoupling control on the stability of milling stability area of milling (a) without control and (b) with modaldecoupling control


References

[1] C Yue H Gao X Liu S Y Liang and L Wang ldquoA review ofchatter vibration research in millingrdquo Chinese Journal ofAeronautics vol 32 no 2 pp 215ndash242

[2] S Sivrioglu and K Nonami ldquoSliding mode control with time-varying hyperplane for AMB systemsrdquo IEEEASME Trans-actions on Mechatronics vol 3 no 1 pp 51ndash59 1998

[3] A Mystkowski ldquoSensitivity and stability analysis of mu-synthesis AMB flexible rotorrdquo Solid State Phenomenavol 164 no 133 pp 313ndash318 2010

[4] S Sivrioglu and K Nonami ldquoAn experimental evaluation ofrobust gain-scheduled controllers for AMB system with gy-roscopic rotorrdquo in Proceedings of the 6th InternationalSymposium on Magnetic Bearings pp 352ndash361 CambridgeMA USA August 1998

[5] Y C Zhang G J Sun and Y J Zhang ldquoExperimental ver-ification for zero power control of 05 kwh class flywheelsystem using magnetic bearing with gyroscopic effectrdquo inProceedings of the 1st International Conference on MachineLearning and Cybernetics pp 2059ndash2062 Beijing ChinaNovember 2002

[6] P Efrain B Douglas and K Edson ldquoVibration control usingactive magnetic actuators with the LQR control techniquerdquo inProceedings of the 7th Brazilian Conference on DynamicsControl and Applications pp 1ndash6 Presidente Prudente BrazilMay 2008

[7] K Zhang and X Dai ldquoDynamic analysis and control of anenergy storage flywheel rotor with active magnetic bearingsrdquoin Proceedings of the 2010 International Conference on DigitalManufacturing amp Automation pp 573ndash576 ChangshaChina 2010

[8] B Wang and Z Deng ldquoAnalysis of cross feedback control forthe magnetically suspended flywheel rotorrdquo in Proceedings ofthe 12th International Symposium on Magnetic Bearingspp 567ndash572 Nanjing China 2010

[9] K Zheng L Zhao and H Zhao ldquoApplication of magneticforce lead control on a flywheel suspended by AMSrdquo ChineseJournal of Mechanical Engineering vol 40 no 7 pp 175ndash1792004

[10] K Jiang ldquoActive magnetic bearingmdashrotor vibration controltechnologyrdquo PhD Zhejiang University Zhejiang China2011

[11] X Qiao and C Zhu ldquoActive control of milling chatter basedon the built-in force actuatorrdquo Journal of Mechanical Engi-neering vol 1 no 48 pp 187ndash192 2012

[12] R Siegwart R Larsouneur and A Traxler ldquoDesign andperformance of a high speed milling spindle in digital con-trolled active magnetic bearingsrdquo in Proceedings of the 2ndInternational Symposium on Magnetic Bearings University ofTokyo Tokyo Japan pp 197ndash204 1990

[13] Q Zhang ldquoResearch on the suppression of gyroscopic effect ofactive magnetic bearing-flywheel energy storage systemrdquo2012

[14] Z Li and Q Liu ldquoModeling and simulation of chatter stabilityfor circular millingrdquo Journal of Mechanical Engineeringvol 46 no 7 pp 181ndash186 2010


ere is also a cross-feedback control method based on thetraditional PD controller in which the cross-feedback can bedivided into displacement crossover [7] velocity crossovervelocity and displacement crossover [8] and displacementcrossover combined with electromagnetic force [9] e ad-vantage of cross-feedback especially the speed cross-feedbackmethod is that the structure is simple and is easy to beimplemented e disadvantage is that there is still no effectivecross-feedback parameter designmethod In addition when thetraditional PD control is used the characteristics of each modeare difficult to be independently adjusted because there is thestrong coupling between the rotational mode and the trans-lational mode of the magnetic electric spindle rotor ereforethe control methods based on this have certain limitations

In view of this based on the mathematical model of theactive magnetic bearing electric spindle rotor system a modaldecoupling control method is proposed which can decouplethe rotating mode and the translational mode of the activemagnetic bearing electric spindle rotor system in order tocontrol the stiffness and damping of the translationalmode andthe rotation mode independently which can reduce effectivelythe influence of the gyro effect on the high-speed millingstability area and then improve the stability area of the high-speed magnetic suspension electric spindle milling system

2 Mathematical Model of Four-Degree-of-Freedom Magnetic Suspension ElectricSpindle Rotor System

e structure of the magnetic bearing rigid electric spindlerotor system is shown in Figure 1 Ideally the axis of therotor coincides with the center of the two radial bearings Inorder to describe the mutual position between the rotor thesensor and the electromagnetic bearing the main coordi-nate system oxyz is set up in which the coordinate origin isat the centroid point c of the rotor and e z-axis coincideswith the center line of the two radial bearings e distancefrom the center of the left and right radial electromagneticbearings A and B to point c is respectively lmA and lmB andlmA is a negative number while lmB is a positive numberedistances of the left and right sensors A and B to c arerespectively lsA and lsB lsA is negative and lsB is positive Inorder to make the analysis easy three coordinate systems ofradial planes are established which are the sensor coordinatesystem the active magnetic bearing coordinate system andthe centroid coordinate system the coordinate origin is seton the center line of the two radial bearings When the rotorrotates the coordinates at the center of mass of the rotor arex y θx and θye coordinates at the left and right sensorsare xsA xsB ysA and ysB and the coordinates of the left andright active magnetic bearings are xmA xmB ymA and ymBrespectively e ignorance of the sensor and magneticbearing installation position error canmake xsA xmA xaxsB xmB xb ysA ymA ya ysB ymB yblsA lmA la lsB lmB lb and l lsA minus lsB When thedisplacement of the rotor is xa ya xb and yb in the xa yaand yb in the y α β and directions the displacement of therotor geometric center O is

x lb

Lxa minus

la

Lxb

y lb

Lya minus

la

Lyb

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(1)

e angle at which the rotor rotates counterclockwiseabout the x-axis is

θx yb minus ya

L (2)

e angle at which the rotor rotates counterclockwisearound the y-axis is

θy xa minus xb

L (3)

When themotion equation of the four-degree-of-freedomelectric spindle rotor system with active electromagneticbearing is established since the bearing air gap is small it canbe assumed that (1) the rotor is an axisymmetric rigid rotorthat is the moments of inertia around the x and y axes areequal (2) e structure and parameters of the radial four-degree-of-freedom are exactly the same (3) the magnetic fieldin the electromagnet is evenly distributed the magnetic fluxleakage is ignored and the magnetic material does not exhibitsaturation characteristics (4) the influence of the magneticresistance and loss of the core material are ignored Withoutconsidering the influence of external damping factors it iseasy to establish the motion equation of the axially symmetricactive magnetic bearing electric spindle rotor systemaccording to the rotor dynamics theory [10]

m eurox FmxA + FmxB

m euroy FmyA + FmyB

Jeuroθy minus Jzω _θx FmxAla minus FmxBlb

Jeuroθx minus Jzω _θy minus FmyAla + FmyBlb

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(4)

c

Oz

y

x

Act

ive m

agne

tic b

earin

g B

Disp

lace

men

t sen

sor B

Elec

tric

spin

dle r

otor

Wor

kpie

ce

Act

ive m

agne

tic b

earin

g A

Disp

lace

men

t sen

sor A

lmAlsA lsB

lmB

L

lcw

Figure 1 Active magnetic bearing-supported motorized spindlemilling system





HereM

J 0 0 00 m 0 00 0 J 00 0 0 m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


system G

0 0 JzΩ 00 0 0 0

minus JzΩ 0 0 00 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


the rotor system Lf


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣




Fm μ0AaN2

4i2

δ2 (6)



F

FmxA

fmxB

fmyA

fmyB

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ks 0 0 0

0 ks 0 0

0 0 ks 0

0 0 0 ks

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xa

xb

ya

yb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

ki 0 0 0

0 ki 0 0

0 0 ki 0

0 0 0 ki

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

imxa

imxb

imya

imyb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ksqs + Kii

(7)








xa

xb

ya

yb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

la 1 0 0

minus lb 1 0 0

0 0 minus la 1

0 0 lb 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

θy

x

θx

y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

at isqs Tbq (10)

Here Tb


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣




Tb LTf (11)




obtained


Here

Kss




0 0 ks minus la + lb( 1113857 2ks

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

en KTss (LfKsL

Tf)T LfKsL

Tf Kss









i ic + ik (15)




ik Kminus 1i B

minus 1KssL

minus 1s qs (17)









LfKi

KilbA KilbB 0 0

Ki Ki 0 0

0 0 KilbA KilbB

0 0 Ki Ki

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


T Kminus 1i L

minus 1f


minus 1 lmB 0 0

1 minus lmA 0 0

0 0 minus 1 lmA

0 0 1 minus lmA

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)



minus 1f (P + D) L

Tf1113872 1113873

minus 1qs (21)





Tf1113872 1113873

minus 1qs (22)


Meuroq + G _q + Pq + D _q 0 (23)

P diag pr pp pr pp1113872 1113873 (24)

D diag dr dp dr dp1113872 1113873 (25)









⎛⎝ ⎞⎠

(26)



⎛⎝ ⎞⎠

(27)




magnetic bearing

coordinates Pow

er am

plifi

er Magnetic bearing


centroidcoordinates

Elec

tric

spin

dle r

otor

syste

mx

y

θx

θy

PD

PD

PD

PD

xa

xb

ya

yb

xlowast

ylowast

θlowast

x

θlowast

y

ixixa

iya

iyb

ixbiy

iθx

iθy


β

Fnj

Fj

Ftj

Fjx

y

j ndash 1

j

n

ϕj



FAx 1113944

Nt

j1Fxj

FAy 1113944

Nt

j1Fyj

(28)


FAx 12


FAx 12


(29)

where


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs

(30)




q

θy

x

θx

y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1L

minus1L

0 0

lb

L

la

L0 0

0 0 minus1L

1L

0 0lb

L

la

L

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

middot

xbA

xbB

ybA

ybB

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tsqs (31)





bKt

Nt

2πΗΔe

(33)

0 02 04 06 08 1ndash04

ndash03

ndash02

ndash01

0

01

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)


ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)






Nt

4πbKtH


1113872 1113873 Δe iωc( 11138571113864 1113865

(34)



I minusNt

4πbKt 1 minus e






blim minus4π

NtKt


1 minus cos ωcT( 11138571113896 1113897

(37)


ζ ξImξRe

sin ωcT( 1113857

1 minus cos ωcT( 1113857 (38)


blim minus2π

NtKt

ξRe 1 + ξ21113872 1113873 (39)








0 02 04 06 08 1ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)


ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)








6 Conclusions



Data Availability




Acknowledgments


0 05 1 15 2 25times 104

0

02

04

06

08

1 times 10ndash3


Dep

th o

f mill

ing

blim


effect


(a)

times 104

times 10ndash3

0 05 1 15 2 250

02

04

06

08

1


Dep

th o

f mill

ing

blim

(m)




(b)



References



















HereM

J 0 0 00 m 0 00 0 J 00 0 0 m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


system G

0 0 JzΩ 00 0 0 0

minus JzΩ 0 0 00 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


the rotor system Lf


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣




Fm μ0AaN2

4i2

δ2 (6)



F

FmxA

fmxB

fmyA

fmyB

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ks 0 0 0

0 ks 0 0

0 0 ks 0

0 0 0 ks

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xa

xb

ya

yb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

ki 0 0 0

0 ki 0 0

0 0 ki 0

0 0 0 ki

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

imxa

imxb

imya

imyb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ksqs + Kii

(7)








xa

xb

ya

yb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

la 1 0 0

minus lb 1 0 0

0 0 minus la 1

0 0 lb 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

θy

x

θx

y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

at isqs Tbq (10)

Here Tb


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣




Tb LTf (11)




obtained


Here

Kss




0 0 ks minus la + lb( 1113857 2ks

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

en KTss (LfKsL

Tf)T LfKsL

Tf Kss









i ic + ik (15)




ik Kminus 1i B

minus 1KssL

minus 1s qs (17)









LfKi

KilbA KilbB 0 0

Ki Ki 0 0

0 0 KilbA KilbB

0 0 Ki Ki

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


T Kminus 1i L

minus 1f


minus 1 lmB 0 0

1 minus lmA 0 0

0 0 minus 1 lmA

0 0 1 minus lmA

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)



minus 1f (P + D) L

Tf1113872 1113873

minus 1qs (21)





Tf1113872 1113873

minus 1qs (22)


Meuroq + G _q + Pq + D _q 0 (23)

P diag pr pp pr pp1113872 1113873 (24)

D diag dr dp dr dp1113872 1113873 (25)









⎛⎝ ⎞⎠

(26)



⎛⎝ ⎞⎠

(27)




magnetic bearing

coordinates Pow

er am

plifi

er Magnetic bearing


centroidcoordinates

Elec

tric

spin

dle r

otor

syste

mx

y

θx

θy

PD

PD

PD

PD

xa

xb

ya

yb

xlowast

ylowast

θlowast

x

θlowast

y

ixixa

iya

iyb

ixbiy

iθx

iθy


β

Fnj

Fj

Ftj

Fjx

y

j ndash 1

j

n

ϕj



FAx 1113944

Nt

j1Fxj

FAy 1113944

Nt

j1Fyj

(28)


FAx 12


FAx 12


(29)

where


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs

(30)




q

θy

x

θx

y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1L

minus1L

0 0

lb

L

la

L0 0

0 0 minus1L

1L

0 0lb

L

la

L

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

middot

xbA

xbB

ybA

ybB

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tsqs (31)





bKt

Nt

2πΗΔe

(33)

0 02 04 06 08 1ndash04

ndash03

ndash02

ndash01

0

01

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)


ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)






Nt

4πbKtH


1113872 1113873 Δe iωc( 11138571113864 1113865

(34)



I minusNt

4πbKt 1 minus e






blim minus4π

NtKt


1 minus cos ωcT( 11138571113896 1113897

(37)


ζ ξImξRe

sin ωcT( 1113857

1 minus cos ωcT( 1113857 (38)


blim minus2π

NtKt

ξRe 1 + ξ21113872 1113873 (39)








0 02 04 06 08 1ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)


ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)








6 Conclusions



Data Availability




Acknowledgments


0 05 1 15 2 25times 104

0

02

04

06

08

1 times 10ndash3


Dep

th o

f mill

ing

blim


effect


(a)

times 104

times 10ndash3

0 05 1 15 2 250

02

04

06

08

1


Dep

th o

f mill

ing

blim

(m)




(b)



References





















i ic + ik (15)




ik Kminus 1i B

minus 1KssL

minus 1s qs (17)









LfKi

KilbA KilbB 0 0

Ki Ki 0 0

0 0 KilbA KilbB

0 0 Ki Ki

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


T Kminus 1i L

minus 1f


minus 1 lmB 0 0

1 minus lmA 0 0

0 0 minus 1 lmA

0 0 1 minus lmA

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)



minus 1f (P + D) L

Tf1113872 1113873

minus 1qs (21)





Tf1113872 1113873

minus 1qs (22)


Meuroq + G _q + Pq + D _q 0 (23)

P diag pr pp pr pp1113872 1113873 (24)

D diag dr dp dr dp1113872 1113873 (25)









⎛⎝ ⎞⎠

(26)



⎛⎝ ⎞⎠

(27)




magnetic bearing

coordinates Pow

er am

plifi

er Magnetic bearing


centroidcoordinates

Elec

tric

spin

dle r

otor

syste

mx

y

θx

θy

PD

PD

PD

PD

xa

xb

ya

yb

xlowast

ylowast

θlowast

x

θlowast

y

ixixa

iya

iyb

ixbiy

iθx

iθy


β

Fnj

Fj

Ftj

Fjx

y

j ndash 1

j

n

ϕj



FAx 1113944

Nt

j1Fxj

FAy 1113944

Nt

j1Fyj

(28)


FAx 12


FAx 12


(29)

where


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs

(30)




q

θy

x

θx

y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1L

minus1L

0 0

lb

L

la

L0 0

0 0 minus1L

1L

0 0lb

L

la

L

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

middot

xbA

xbB

ybA

ybB

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tsqs (31)





bKt

Nt

2πΗΔe

(33)

0 02 04 06 08 1ndash04

ndash03

ndash02

ndash01

0

01

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)


ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)






Nt

4πbKtH


1113872 1113873 Δe iωc( 11138571113864 1113865

(34)



I minusNt

4πbKt 1 minus e






blim minus4π

NtKt


1 minus cos ωcT( 11138571113896 1113897

(37)


ζ ξImξRe

sin ωcT( 1113857

1 minus cos ωcT( 1113857 (38)


blim minus2π

NtKt

ξRe 1 + ξ21113872 1113873 (39)








0 02 04 06 08 1ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)


ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)








6 Conclusions



Data Availability




Acknowledgments


0 05 1 15 2 25times 104

0

02

04

06

08

1 times 10ndash3


Dep

th o

f mill

ing

blim


effect


(a)

times 104

times 10ndash3

0 05 1 15 2 250

02

04

06

08

1


Dep

th o

f mill

ing

blim

(m)




(b)



References


















Tf1113872 1113873

minus 1qs (22)


Meuroq + G _q + Pq + D _q 0 (23)

P diag pr pp pr pp1113872 1113873 (24)

D diag dr dp dr dp1113872 1113873 (25)









⎛⎝ ⎞⎠

(26)



⎛⎝ ⎞⎠

(27)




magnetic bearing

coordinates Pow

er am

plifi

er Magnetic bearing


centroidcoordinates

Elec

tric

spin

dle r

otor

syste

mx

y

θx

θy

PD

PD

PD

PD

xa

xb

ya

yb

xlowast

ylowast

θlowast

x

θlowast

y

ixixa

iya

iyb

ixbiy

iθx

iθy


β

Fnj

Fj

Ftj

Fjx

y

j ndash 1

j

n

ϕj



FAx 1113944

Nt

j1Fxj

FAy 1113944

Nt

j1Fyj

(28)


FAx 12


FAx 12


(29)

where


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs

(30)




q

θy

x

θx

y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1L

minus1L

0 0

lb

L

la

L0 0

0 0 minus1L

1L

0 0lb

L

la

L

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

middot

xbA

xbB

ybA

ybB

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tsqs (31)





bKt

Nt

2πΗΔe

(33)

0 02 04 06 08 1ndash04

ndash03

ndash02

ndash01

0

01

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)


ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)






Nt

4πbKtH


1113872 1113873 Δe iωc( 11138571113864 1113865

(34)



I minusNt

4πbKt 1 minus e






blim minus4π

NtKt


1 minus cos ωcT( 11138571113896 1113897

(37)


ζ ξImξRe

sin ωcT( 1113857

1 minus cos ωcT( 1113857 (38)


blim minus2π

NtKt

ξRe 1 + ξ21113872 1113873 (39)








0 02 04 06 08 1ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)


ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)








6 Conclusions



Data Availability




Acknowledgments


0 05 1 15 2 25times 104

0

02

04

06

08

1 times 10ndash3


Dep

th o

f mill

ing

blim


effect


(a)

times 104

times 10ndash3

0 05 1 15 2 250

02

04

06

08

1


Dep

th o

f mill

ing

blim

(m)




(b)



References
















FAx 1113944

Nt

j1Fxj

FAy 1113944

Nt

j1Fyj

(28)


FAx 12


FAx 12


(29)

where


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs


1113868111386811138681113868φe

φs

(30)




q

θy

x

θx

y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1L

minus1L

0 0

lb

L

la

L0 0

0 0 minus1L

1L

0 0lb

L

la

L

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

middot

xbA

xbB

ybA

ybB

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tsqs (31)





bKt

Nt

2πΗΔe

(33)

0 02 04 06 08 1ndash04

ndash03

ndash02

ndash01

0

01

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)


ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)






Nt

4πbKtH


1113872 1113873 Δe iωc( 11138571113864 1113865

(34)



I minusNt

4πbKt 1 minus e






blim minus4π

NtKt


1 minus cos ωcT( 11138571113896 1113897

(37)


ζ ξImξRe

sin ωcT( 1113857

1 minus cos ωcT( 1113857 (38)


blim minus2π

NtKt

ξRe 1 + ξ21113872 1113873 (39)








0 02 04 06 08 1ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)


ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)








6 Conclusions



Data Availability




Acknowledgments


0 05 1 15 2 25times 104

0

02

04

06

08

1 times 10ndash3


Dep

th o

f mill

ing

blim


effect


(a)

times 104

times 10ndash3

0 05 1 15 2 250

02

04

06

08

1


Dep

th o

f mill

ing

blim

(m)




(b)



References



















Nt

4πbKtH


1113872 1113873 Δe iωc( 11138571113864 1113865

(34)



I minusNt

4πbKt 1 minus e






blim minus4π

NtKt


1 minus cos ωcT( 11138571113896 1113897

(37)


ζ ξImξRe

sin ωcT( 1113857

1 minus cos ωcT( 1113857 (38)


blim minus2π

NtKt

ξRe 1 + ξ21113872 1113873 (39)








0 02 04 06 08 1ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

t (s)

Xw (m

m)

Yw (m

m)

YwXw

(a)


ndash02

0

02

04

Xw (mm)

Yw (m

m)

(b)








6 Conclusions



Data Availability




Acknowledgments


0 05 1 15 2 25times 104

0

02

04

06

08

1 times 10ndash3


Dep

th o

f mill

ing

blim


effect


(a)

times 104

times 10ndash3

0 05 1 15 2 250

02

04

06

08

1


Dep

th o

f mill

ing

blim

(m)




(b)



References





















6 Conclusions



Data Availability




Acknowledgments


0 05 1 15 2 25times 104

0

02

04

06

08

1 times 10ndash3


Dep

th o

f mill

ing

blim


effect


(a)

times 104

times 10ndash3

0 05 1 15 2 250

02

04

06

08

1


Dep

th o

f mill

ing

blim

(m)




(b)



References
















References
















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