Suppression of Base Excitation of Rotors on Magnetic Bearings

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2007, Article ID 91276, 10 pagesdoi:10.1155/2007/91276

Research ArticleSuppression of Base Excitation of Rotors on Magnetic Bearings

Steven Marx1 and C. Nataraj2

1 Environmental Quality Systems, Carderock Division, Naval Surface Warfare Center, Philadelphia,PA 19112, USA

2 Department of Mechanical Engineering, Center for Nonlinear Dynamics & Control, Villanova University, Villanova,PA 19085, USA

Received 1 March 2006; Revised 26 October 2006; Accepted 25 December 2006

Recommended by Hyeong-Joon Ahn

This paper deals with rotor systems that suffer harmonic base excitation when supported on magnetic bearings. Magnetic bearingsusing conventional control techniques perform poorly in such situations mainly due to their highly nonlinear characteristics. Thecompensation method presented here is a novel optimal control procedure with a combination of conventional, proportional,and differential feedback control. A four-degree-of-freedom model is used for the rotor system, and the bearings are modeled bynonlinear expressions. Each disturbance frequency is expected to produce a multiharmonic system response, a characteristic ofnonlinear systems. We apply optimal control choosing to minimize a performance index, which leads to the optimization of thetrigonometric coefficients in the correction current function. Results show that the control technique suppresses rotor vibration toamplitudes that were significantly smaller than the disturbance amplitudes for the entire range of disturbance frequencies applied.The control technique explored in this paper is a promising step towards the successful application of magnetic bearings to systemsmounted on moving platforms.

Copyright © 2007 S. Marx and C. Nataraj. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

Magnetic bearings suspend a rotor between electromagnetsand thus are noncontacting. They can significantly reducemachinery wear because the bearings themselves accumu-late nearly no wear; they can actively minimize rotor vibra-tion and, if widely used, they would reduce the amount ofoily waste generated. However, magnetic bearings’ poor per-formance when exposed to disturbances has prevented theiruse in many applications, particularly in transportation sys-tems. Magnetic bearings are naturally unstable and requirefeedback control. Conventional control techniques work wellwhen the bearings are exposed only to small disturbances,but quite poorly when exposed to more significant distur-bances from rotor unbalance or base motion. The distur-bances that may be easily handled in other similar applica-tions by control techniques fail when applied to magneticbearings. Magnetic bearings are very nonlinear and compen-sation techniques that are based on linearized system mod-els fail when disturbances cause the rotor to deviate too farfrom its centered position. This fact has hence generally lim-ited their application to machinery that is not exposed tosignificant disturbance motion.

The aviation industry in particular has been interested inapplying magnetic bearings to gas turbine engines. Storace etal. [1] provided an overview of the possible benefits of adapt-ing magnetic bearings to gas turbines. Eventually, magneticbearings could substantially improve the power-to-weightratio of gas turbines. Significant weight reductions wouldbe realized by the removal of the lube oil system, which in-cludes lube oil tanks, pumps, coolers, and a variety of tubing,valves, and fittings. Reduced life cycle costs would accom-pany the removal of the lube oil system due to increased bear-ing service life and reduced maintenance. Magnetic bear-ings would also enable new engine designs that spin fasterand operate at higher temperatures to achieve greater ther-mal efficiency, further increasing the power-to-weight ratio[2].

The US Navy also had an interest in magnetic bearings asa way to attenuate machinery noise. One study by the Navy[3] explored a feedback noise control method for a pumpwith magnetic bearings. The bearings would actively movethe pump impeller to suppress fluidborne noise. Nataraj[4, 5]; Nataraj and Calvert [6] studied the dynamic responseof rotating machinery with magnetic bearings mountedon a moving platform. Each identified significant technical

2 International Journal of Rotating Machinery

challenges that must be addressed prior to a magnetic bear-ing’s application on board Navy ships.

There are several sources of of nonlinearity; the magneticforce is inversely proportional to the square of gap length,and it is proportional to the square of coil current. Other, lessimportant, sources of nonlinearity are electromagnetic hys-teresis, eddy currents, and skewed force lines from geometricforce coupling between degrees of freedom [7]. Skewed forcelines become a concern only when air gaps are enlarged as apossible way to limit the effects of vibration [8].

Some studies tried to define more generalized charac-teristics of magnetic bearings. Mohamed and Fawzi [9] ex-amined magnetic bearings’ response characteristics due tothe nonlinearity of the electromagnetic force. The authorsconstructed a five-degree-of-freedom model with a magneticthrust bearing and two radial magnetic bearings. They con-trolled the model with a linear feedback control system tostudy the nonlinear oscillations that occur due to gyroscopiceffects. The authors found that the model exhibited bifur-cation effects and was unstable around critical speeds. Theyalso proposed a nonlinear feedback control method. Chintaand Palazzolo [10] characterized the nonlinear responses ofa two-degree-of-freedom model with PD control in whichthe rotor speed, an excitation force, the static load, and theamount of coupling between the two axes were varied. Theysimulated the system using the trigonometric collocationmethod and were able to identify the conditions in which bi-furcation and changes in stability would occur. Bifurcationsin some conditions produced superharmonic response fre-quencies of one half and one quarter the forcing frequency.

A rotor’s responses to vibrations due to rotor unbalanceand base motion have attracted the most attention. Of thetwo kinds of disturbances, more papers explored methodsintended to manage the effects of rotor unbalance. Petelaand Botros [11] considered both linear and nonlinear modelsfor their method using multiaccess displacement and veloc-ity feedback to suppress vibration from a flexible shaft rotor.Results for the nonlinear model, however, were inconclusive,and they acknowledged that ranges of parameters, such asrotor unbalance and rotation speed, which would have noeffect on conventional bearings would cause instability withmagnetic bearings.

Lindlau and Knopse [12] applied feedback linearization,which linearizes the system at all operating points ratherthan just one neutral point. The response remained stableon a single-degree-of-freedom model, even when exposed toa disturbance with an amplitude of 90 percent of the gapwidth, and the simulated responses were similar to thosemeasured from a test rig. An approach was developed byKnopse et al. [13] to suppress vibration in a linear systemdue to unbalance-combined adaptive open-loop control withfeedback control. The adaptive open loop control system ad-justed the control signal’s magnitude and phase periodicallyin response to changes in the rotor’s operating conditions.The controller optimized the signal at each update by calcu-lating the minimum-expected rotor response. The controllerused an estimated influence coefficient matrix for the calcu-lation, which was estimated offline, prior to implementation,

by applying a known disturbance to the system and mea-suring the rotor’s response. The authors demonstrated thatthe method could suppress vibrations due to unbalance. Inanother approach, Cole et al. [14] employed PID feedbackcontrol combined with H∞ control, which used weights thatwere selected by optimizing a linearized model. The resultsdemonstrated that the control method suppressed vibrationmore effectively than using PID control alone; however, thepaper did not indicate the relative magnitude of the applieddisturbance.

Beale et al. [15, 16], Hisatani and Koizumi [17], andMizumo and Higuchi [18] considered methods that estimatethe forces of unbalance so that counter forces can be gen-erated by the magnetic bearing to induce the rotor to ro-tate about its center of mass. None of the techniques used anonlinear model. Beale et al. and Hisatani and Koizumi usedadaptive filtering to estimate the disturbance and to gener-ate counter force control signals. Beale et al. [16] noted somedisturbance amplitude and bandwidth limitations, outside ofwhich the technique will not be able to suppress vibration.

There has also been some interest in considering basemotion of a rotor supported on a magnetic bearing. Cole etal. [14] was among a few efforts that studied techniques thatcould limit the effects of base motion. They employed PIDcontrol in parallel with H∞ control of bearings in systemssubject to horizontal base impact (in addition to their ex-periments with rotor vibration mentioned above). The con-trol technique enabled the system to recover from the distur-bance when the control function weights were optimized forthe base motion disturbance and rotor vibration. Kasarda etal. [19] tried a variety of gains in a PID controller on a one-degree-of-freedom test stand mounted on a shaker table thatgenerated a sinusoidal base motion. They ran tests designedto find the limits of stability for several combinations of con-troller gains. They also found that some stable responses werelinear and others were nonlinear. Nataraj [5]; Nataraj andCalvert [6] demonstrated that a feedforward control methodthat would work well with a linearized model would notwork for the nonlinear model subject to a sinusoidal basemotion. Somewhat similar techniques were used by Knopseet al. [13] and Saunders et al. [20], who used a hybrid controlapproach. In their paper, the feedback gain, optimized with alinear quadratic Gaussian regulator, was added to a feedfor-ward signal designed to cancel the effects of steady, oscillatingdisturbances.

This paper examines a novel method designed to sup-press rotor motion due to base excitation with an empha-sis on the nonlinear model for the magnetic bearings. Themethod combines PD feedback with feedforward optimalcontrol, where a known base motion is used to select a set ofpredetermined frequencies and their amplitudes for a controlsignal optimized to suppress the rotor response.

2. MATHEMATICAL MODEL

A typical radial magnetic bearing arrangement (see Figure 1)is four electromagnets with two opposing electromagnetsaligned along each radial axis where each electromagnet has

S. Marx and C. Nataraj 3

Controlcurrent (Ic)

Biascurrent (Ib)

++ +

−+ −

Rotor

+

++

−+−

Controlcurrent (Ic)

Biascurrent (Ib)

Figure 1: An 8-pole radial magnetic bearing.

z

WΓ

y

Vβ

Fmz1

L1

Fmy1 L2Fmz2

Fmy2

Figure 2: The four-degree-of-freedom model.

two poles. We consider a four-degree-of-freedom model ofa rigid rotor that is supported at each end by such a mag-netic bearing. The rotor is modeled as a shaft with a saddle-mounted disk perpendicular to the shaft (see Figure 2). Eachend is free to move radially with the bearing forces being ap-plied along the y and z axes. Base motion and rotor unbal-ance disturbance forces could also be applied to both direc-tions at each end. This model also includes the gyroscopiceffect. The gyroscopic moments applied to each direction ofeach bearing depend on the angular position and velocity ofthe disk in all three directions, which couples the xy and xzmotion.

Using standard techniques from analytical dynamics withEuler angles and Lagrange equations, and assuming smalldisplacements, the equations of motion can be derived. The

term Frc represents the forces and moments applied to the ro-tor at its center of gravity, which include the magnetic bear-ing forces (Fm) and disturbance forces such as unbalance.The resulting equation of motion is

⎡⎢⎢⎢⎣

mr 0 0 00 mr 0 00 0 Jd 00 0 0 Jd

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

VWβΓ

⎤⎥⎥⎥⎦−Ωr

⎡⎢⎢⎢⎣

0 0 0 00 0 0 00 0 0 −Jp0 0 Jp 0

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

VWβΓ

⎤⎥⎥⎥⎦ = Frc,

(1)

where Jd and Jp are the diametral and polar mass momentsof inertia. The equation also can be represented as

Mr p −ΩrGr p = Frc, (2)

where p is a displacement vector of the rotor’s positions andangles at the rotor’s center.

These equations can be then converted to the horizontaland vertical positions at each bearing, q = Tqp, using thetransformation matrix

Tq =

⎡⎢⎢⎢⎣

1 0 0 −L1

0 1 L1 01 0 0 L2

0 1 −L2 0

⎤⎥⎥⎥⎦ . (3)

The terms L1 and L2 are the distances from the center of grav-ity to the bearings. Equation (2) then becomes

MrT−1q q −ΩrGrT

−1q q = Fr , (4)

where Fr are the forces applied at the bearings and p is thevector of state variables of the rotor’s positions, and theirderivatives, at each bearing. When nondimensionalized, (4)can be restated as

Amx − Agx = fr . (5)

The equation is then rearranged into the form

[I 00 Am

][xx

]=[

0 I0 Ag

][xx

]+

[c0Fm

]+

[0−Fs

](6)

in which Fr is separated into the magnetic bearing forces (Fm)and the static load on the rotor (Fs). Note that these forcesare the sum of the the individual forces at the bearings trans-formed appropriately. The next section discusses the forcemodel for each axis at each bearing.

2.1. Magnetic bearing model

The magnetic bearing forces can be shown to be the follow-ing; detailed derivations can be found in Schweitzer et al. [21]and Nataraj [4]. We directly present nondimensional forceshere:

fm = fm0

(i21

4g21

)− fm0

(i22

4g22

), (7)


Referencecurrent PD

Correctioncurrent (icc)

Correction currentfunction selector

Controlcurrent (ic)

Base motion (x f )

Bearing and rotorRotor position (x)

Figure 3: Control scheme.

where the magnetic bearing constant ( fm0) is a function ofthe bearing and rotor parameters. The currents applied (i)and the rotor gaps (g) are

i1 = 1 + is − ic, i2 = 1− is + ic,

g1 = 1− (x − u), g2 = 1 + (x − u).(8)

The magnetic bearing force is then reduced to

fm = fm0

4

[( (1 + is − ic

)2

(1− (x − u)

)2

)−( (

1− is + ic)2

(1 + (x − u)

)2

)], (9)

where x is the appropriate displacement component, u(t) isthe base excitation in that direction, and ic is the control cur-rent.

Finally, is, the current to compensate for a static load, isdetermined from the relation is = fs/ fm0, which is derivedby assuming that the sum of the magnetic bearing forces( f1 and f2) equals the static load ( fs), or

fs = fm0

4

((1 + is

)2 − (1− is)2)

. (10)

3. CONTROL

The control system configuration proposed in this paper hasa feedback loop and a feedforward loop (see Figure 3). Thefeedback loop is a conventional PD controller with propor-tional and derivative gain (Kp and Kd) that yields a currenti f b = Kpx + Kdx.

The feedforward loop is based on the fundamental ob-servation that nonlinear systems exhibit sub and super har-monics when subjected to harmonic excitation. Hence, wepostulate a control scheme that compensates for these fre-quency terms by supplying correction currents at the appro-priate frequencies

icc =N∑

j=1

aj cos jωbt +N∑

j=1

bj sin jωbt, (11)

where the frequencies of the correction current are multiplesand fractions of the base motion frequency (ωb). The final

control current is then the sum of the feedback current andthe feedforward correction current ic = i f b + icc.

We next seek to minimize a performance index

J = φ(t f)

+∫ t f

0

(xTQx + i

TccRicc

)dt, (12)

where Q and R are weighting matrices. We add a final statepenalty that was a function of the final time of the simulationperiod just completed to ensure convergence to the optimalsolution; this is a Bolza type of cost function [22]. If the sys-tem were linear, we would be able to use conventional linearoptimal control theory. In addition, since the linear responsewould be harmonic, it is possible to derive simpler versionsof the Riccati equation [23]. In fact, that is the procedure weused with the linearized versions of the above equations toprovide a starting solution for the nonlinear optimal controlprocess.

Being nonlinear, the response of a system suspended bymagnetic bearings due to any particular disturbance is notpossible to predict analytically and a single optimal solutioncannot be derived. Hence, we implement the scheme numer-ically. In order to make the problem more insightful, andsince we are only dealing with harmonic disturbances, we cannow choose to optimally determine the coefficients in the se-ries (11), rather than elaborate functions of time as is donein conventional optimal control.

The intended application of the results is then as follows.A table of optimal parameters is obtained prior to imple-menting the control method by measuring the responses ofa simulated magnetic bearing system to a range of expecteddisturbances. A correction current function selector is thenused to select the optimal control current parameters from alookup table, based on the measured disturbance. Note that,in many practical situations (such as ship-board machinerywhich was the motivation for this paper), the disturbance fre-quencies are often known a priori to some extent.

4. NUMERICAL RESULTS

In order to generate an initial guess of the correction currentcoefficients, we used a linearized system model with optimal


control. A valid guess had to produce a stable solution, oth-erwise the iterative selection of coefficients while trying tofind an optimized set would not produce meaningful solu-tions. To start with, guesses were based on a solution usinglinear optimal control (with linearized magnetic bearing ex-pressions). The details of the linear optimal control are notshown here in the interest of brevity. The optimized coef-ficients found for the correction current to compensate fora base motion that is near the base motion for which coef-ficients are being searched, are used as the initial guess forsubsequent runs.

In this case, a least-squares-fit curve-fitting techniquewas used to identify the trigonometric series coefficients forthe correction current. The first guess would have coefficientsfor only one correction current frequency (the base motionfrequency) although many more are needed for the nonlinearclosed-loop model.

The optimization process was iterative, repeating thesimulation of the nonlinear closed-loop system. The processwas started using the initial guess coefficients, generating aresponse curve and calculating the initial performance index(12) for the response curve. The coefficients were then al-tered for all three frequencies prior to the next iteration, theresponse was found again, and the performance index wasrecalculated. The coefficients were altered after each iterationusing the simplex search method to select successive sets ofvariables to find values that approach a minimum. The pro-cess was repeated until the performance indices converged tosome minimum value within a specified tolerance. Note thatthe feedback control parameters are not part of the optimiza-tion process; although we have investigated that aspect (andfound it to be not so efficacious), it is not included in thispaper as the focus of the paper is on the feedforward con-troller. Also note that the transient solutions are completelydropped from this analysis as the steady solution is the mainfocus of this paper. We do believe that the transient solutionis important especially for situations dealing with shock, butcould be the subject of future research.

In order to illustrate the efficacy of the method, we use anexample here of a rotor in a standard U.S. Navy fire pump.The air gap length is a typical one for magnetic bearings. Itis saddle mounted, and can reasonably assumed to be a rigidrotor: shaft weight= 76 lbs; disk weight= 30 lbs; shaft diam-eter = 2.75 in; disk diameter = 20.9 in; shaft length = 45 in;disk position = 26 in; air gap = 0.020 in; fm0 = 10; nomi-nal rotational speed = 30, 000 rpm. The selection of controlparameters for the model was based primarily on combina-tions of feedback gains that were selected based on the stan-dard control techniques applied to the linearized system andpredicted low responses from numerical simulations for thenonlinear system. The parameters selected were Kp = 1.1,and Kd = 0.0200166. The differential feedback gain producesa damping ratio of ζ = 0.1.

All mechanical and control parameters were nondimen-sionalized so that the solution methods used here could beapplied to a variety of rotor designs. We used nondimen-sional base motion frequencies (Ωb) in the range of 0.1 to3.0. The base motion amplitude used for each model and for

0 5 10 15 20 25 30 35

Time

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Dis

plac

emen

t

W1 w/ FF compensationW1 w/o FF compensation

V1 w/ FF compensationV1 w/o FF compensation

Figure 4: Effect of optimal FF controller for Ωb = 0.6.

all base motion frequencies was 0.9, or ninety percent of theair gap length.

The same three relative frequencies of one, two, and fourtimes the base motion frequency were used for the correc-tion current for each case and for all base motion frequencies.The correction current frequencies were selected by identify-ing what the dominant response frequencies were, generally,for the entire range of base motion frequencies prior to op-timization. The use of three correction current frequenciesrequired at least six subharmonics. Six subharmonics couldhave multiples of two, three, four, five, and six times the basemotion frequency. Two superharmonic frequencies, one halfand one third of the base motion frequency, were used be-cause either one may contribute to a response.

First we compare results from the optimal feedforwardcontrolled system with a conventionally controlled system inorder to demonstrate the need for our approach. Examplesare shown at two speeds, (see Figures 4 and 5). Note thatthe response with the conventional PD control is too largeto be shown in its entirety on the same graph as the optimalfeedforward controlled system. Response at other speeds issimilar.

All the vertical motion response plots resulting from theapplication of optimized correction current coefficients hadamplitudes that were significantly less than the amplitudeof the base motion. In fact, the response amplitudes whenjust the correction current was optimized for the linearizedmodel (labeled as “w/o opt. coeff.”) were typically less thanten percent of the amplitude of the base motion. It should benoted that the comparison in all the subsequent figures is be-tween an optimal linear system and an optimal nonlinear sys-tem both with the feedforward controller proposed in this pa-per, and investigates and emphasizes the need for the nonlinearmodel. Response without the FF controller is at least an orderof magnitude worse and is not compared any more.


0 5 10 15

Time

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

Dis

plac

emen

t

W1 w/ FF compensationW1 w/o FF compensation

V1 w/ FF compensationV1 w/o FF compensation

Figure 5: Effect of optimal FF controller for Ωb = 1.3.

0 50 100 150 200

Time

−0.08

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

Dis

plac

emen

t

W1 (vert.) w/ opt. coeff.W1 (vert.) w/o opt. coeff.

V1 (hor.) w/ opt. coeff.V1 (hor.) w/o opt. coeff.

Figure 6: Response plots at Ωb = 0.1, bearing 1.

Figures 6, 7, 8, 9, 10, and 11 are sample plots of responsesof the four-degree-of-freedom model. In fact, because twomost dominant response frequencies (see Figures 12 and 13)were usually one and two times the base motion frequency, itis possible that using only those frequencies as the correctioncurrent frequencies would have been sufficient to suppressmost of the motion.

For horizontal motion, however, the dominant frequen-cies were more variable. Figures 14 and 15 have magni-tude response profiles where one of the two most domi-nant frequencies were three times the base motion frequen-cies. Therefore, the indirect effects on horizontal motion bythe gyroscopic effects of the four-degree-of-freedom model

0 5 10 15 20 25 30 35

Time

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Dis

plac

emen

t




0 5 10 15 20

Time

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Dis

plac

emen

t




could require the addition of three times the vertical motionfrequency to the horizontal motion correction current.

There is one obvious overall trend, which is a general de-cline in the overall magnitudes of the responses with increas-ing base motion frequency (see Figure 16). The one excep-tion is the root mean square plot of the solutions from thenumerical method for the two-degree-of-freedom model.When the base motion frequency of Ωb = 1.6 was applied,the initial simulation failed when starting with the optimizedcontrol current coefficients found for Ωb = 1.5. Therefore,the optimization cycle had to start with the optimized co-efficients for the linearized model. Because the coefficientsfor the linearized model generated responses with higher


0 5 10 15

Time

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Dis

plac

emen

t




0 2 4 6 8 10

Time

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

Dis

plac

emen

t




magnitudes, the RMS plot jumped up at Ω = 1.6. This maybe an indication of the model’s sensitivity to deviations fromthe expected base motion frequency. This sensitivity shouldbe explored because high sensitivity would require smallerfrequency intervals in a feedforward control lookup table.

Another noteworthy feature of the vertical response plotsare the static offsets, with the neutral points for the oscilla-tions occurring around some point below the bearings’ cen-ters. This is likely due to the effect of the static load onthe nonlinear system, even though a compensating current(is) was applied. Flowers et al. [24] explored this problemand proposed an integrally augmented state feedback con-trol method as a remedy. In none of the cases, however, was

0 1 2 3 4 5 6 7

Time

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Dis

plac

emen

t




0 5 10 15

Response frequencies (niΩb)

10−8

10−6

10−4

10−2

100

Mag

nit

ude

w/ opt. coeff.w/o opt. coeff.

Figure 12: Vert. response mag. profiles at Ωb = 1.0, bearing 2.

the static offset any larger than ten percent of the nominalgap length.

The optimal control algorithm actually added amplitudeto the horizontal motion. The response plot model wouldshow no motion in the undisturbed horizontal directionprior to optimization, but as the optimization method itera-tively tried different sets of control current coefficients, mo-tion was added even though the performance index was re-duced. Without a disturbance applied, the optimal solutionfor horizontal motion was the correction current coefficientsfound for the linearized model. Figures 17 and 18 are plotswhere the horizontal motion increased by a small amount.More details of all the studies can be found in Marx [25].


0 5 10 15


10−8

10−6

10−4

10−2

Mag

nit

ude


Figure 13: Vert. response mag. profiles at Ωb = 2.2, bearing 2.

0 5 10 15


10−12

10−10

10−8

10−6

10−4

10−2

Mag

nit

ude


Figure 14: Horiz. response mag. profiles at Ωb = 1.4, bearing 1.

5. CONCLUSION

This paper developed a combination of PD feedback and anoptimal feedforward control procedure that can effectivelylimit the deleterious influence of harmonic base motion vi-bration on a rotor suspended by magnetic bearings. We con-sidered a four-degree-of-freedom rigid rotor model with twomagnetic bearings, and derived a procedure to compute cor-rection currents based on the sub and super harmonic re-sponses of nonlinear systems.

Three correction current frequencies were used for allbase motion frequencies applied, which were one, two, andfour times the base motion frequency. The same three relative

0 5 10 15


10−8

10−6

10−4

10−2

Mag

nit

ude



0 0.5 1 1.5 2 2.5 3

Ωb

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Mag

nit

ude

V1 (hor.)W1 (vert.)

V2 (hor.)W2 (vert.)

Figure 16: RMS plots for 4 DOF model.

correction current frequencies were used exclusively so thattheir coefficients could be used to initiate optimization of co-efficients for nearby base motion frequencies. Although thecombination of correction current frequencies was not nec-essarily the dominant response frequencies for all base mo-tion frequencies, they were sufficient to consistently producesatisfactory results.

In summary, the response with the proposed optimalfeedforward controller is orders of magnitude better than asystem without one. This is so even when the response is inthe nonlinear regime. While the results from this work weresatisfactory, we believe that much needs to be done beforemagnetic bearings can be used successfully and reliably in


0 5 10 15


10−10

10−8

10−6

10−4

10−2

Mag

nit

ude



0 5 10 15


10−10

10−8

10−6

10−4

10−2

Mag

nit

ude



very demanding scenarios. Some initial thoughts in this re-gard follow.

Additional sets of correction current coefficients shouldbe found for each base motion frequency, with each set se-lected to compensate for a different base motion magnitude,with the magnitudes separated by some fixed intervals. Alookup table should then be assembled. How well the coef-ficients suppress vibrations that have frequencies or ampli-tudes between those for which optimized coefficients werefound should be determined. And, the technique should beapplied to a variety of rotor configurations, including ro-tors with flexible shafts. The technique developed (and ap-plied for rigid rotors here) is quite general; however, it can beanticipated that flexible rotor models would introduce ad-

ditional complexities that would challenge the current algo-rithm.

Eventually, the control method could be enhanced by in-troducing algorithms that would estimate the actual distur-bance and periodically adjust the correction current param-eters to adapt them to changing rotor characteristics. How-ever, the control technique studied in this effort cannot beultimately regarded as successful until it is proven on a teststand and then on in-service machinery. Even if successful, itis still only one part of a more complete solution. We focusedon steady harmonic excitation and steady response. However,a magnetic bearing must also be able to limit the transientresponse due to step or impulse disturbances, which couldinclude severe momentary disturbances such as shock froma nearby underwater explosion or a hard aircraft landing. Forexample, an additional feedback loop can be added that useslinear quadratic regulator (LQR) control, feedback lineariza-tion, or H∞ control. The added feedback loop could be en-gaged temporarily, for at least as long as the expected tran-sient response, after being triggered by a signal from a plat-form mounted sensor. These techniques are being exploredin our current work.

The numerical method used here was indeed success-ful; however, with harmonic excitations, much better insightcan be gleaned from a seminumerical approach (such as thetrigonometric collocation method, [26]), and is also the sub-ject of our current work.

REFERENCES

[1] A. F. Storace, D. Sood, J. P. Lyons, and M. A. Preston, “Integra-tion of magnetic bearings in the design of advanced gas tur-bine engines,” Transactions of the ASME, Journal of Engineeringfor Gas Turbines and Power, vol. 117, no. 4, pp. 655–665, 1995.

[2] G. Montague, M. Jansen, R. Jansen, B. Ebihara, A. Provenza,and A. Palazzolo, “Experimental high temperature characteri-zation of a magnetic bearing for turbomachinery,” in Proceed-ings of the 59th Annual Forum of the American Helicopter Soci-ety, pp. 6–8, Phoenix, Arizona, May 2003.

[3] G. E. Piper, “Active feedback noise control of a magnetic bear-ing pump,” Noise Control Engineering Journal, vol. 45, no. 2,pp. 78–84, 1997.

[4] C. Nataraj, “Design of magnetic bearings for propulsion drive-shafts: a feasibility study,” Tech. Rep., Naval Surface WarfareCenter, Carderock Division, Annapolis, Md, USA, 1997.

[5] C. Nataraj, “Dynamic response of rotating machinery on mag-netic bearings,” Tech. Rep., Naval Surface Warfare Center,Carderock Division, Annapolis, Md, USA, 1998.

[6] C. Nataraj and T. E. Calvert, “Compensation of base motion inmagnetic-bearing supported rotors for Navy applications,” inProceedings of ASNE Intelligent Ships Symposium, Philadelphia,Pa, USA, May 2003.

[7] L. N. Virgin, T. F. Walsh, and J. D. Knight, “Nonlinear behaviorof a magnetic bearing system,” Transactions of the ASME: Jour-nal of Engineering for Gas Turbines and Power, vol. 117, no. 3,pp. 582–588, 1995.

[8] N. Motee and M. S. de Queiroz, “Control of active magneticbearings with a smart bias,” in Proceedings of the 41st IEEEConference on Decision and Control, vol. 1, pp. 860–865, LasVegas, Nev, USA, December 2002.


[9] A. M. Mohamed and P. E. Fawzi, “Nonlinear oscillations inmagnetic bearing systems,” IEEE Transactions on AutomaticControl, vol. 38, no. 8, pp. 1242–1245, 1993.

[10] M. Chinta and A. B. Palazzolo, “Stability and bifurcation ofrotor motion in a magnetic bearing,” Journal of Sound and Vi-bration, vol. 214, no. 5, pp. 793–803, 1998.

[11] G. Petela and K. K. Botros, “Magnetic bearing controlof flexible shaft vibrations based on multi-access velocity-displacement feedback,” Transactions of the ASME, Journal ofEngineering for Gas Turbines and Power, vol. 117, no. 1, pp.188–197, 1995.

[12] J. D. Lindlau and C. R. Knopse, “Feedback linearization of anactive magnetic bearing with voltage control,” IEEE Transac-tions on Control Systems Technology, vol. 10, no. 1, pp. 21–31,2002.

[13] C. R. Knopse, R. W. Hope, S. M. Tamer, and S. J. Fedigan,“Robustness of adaptive unbalance control of rotors with mag-netic bearings,” Journal of Vibration and Control, vol. 2, no. 1,pp. 33–52, 1996.

[14] M. O. T. Cole, P. S. Kneogh, and C. R. Burrows, “Vibrationcontrol of a flexible rotor/magnetic bearing system subject todirect forcing and base motion disturbances,” Proceedings ofthe Institution of Mechanical Engineers—Part C: Journal of Me-chanical Engineering Science, vol. 212, no. 7, pp. 535–546, 1998.

[15] S. Beale, B. Shafai, P. LaRocca, and E. Cusson, “Adaptive forcedbalancing for magnetic bearing control systems,” in Proceed-ings of the 31st Conference on Decision and Control, pp. 3535–3539, Tucson, Ariz, USA, December 1992.

[16] S. Beale, B. Shafai, P. LaRocca, and E. Cusson, “Adaptiveforced balancing for multivariable systems,” Transactions of theASME, Journal of Dynamic Systems, Measurement, and Con-trol, vol. 117, no. 4, pp. 496–502, 1995.

[17] M. Hisatani and T. Koizumi, “Adaptive filtering for unbal-anced vibration suppression,” in Proceedings of the 4th Interna-tional Symposium on Magnetic Bearings (ISMB ’94), pp. 125–130, Zurich, Switzerland, August 1994.

[18] T. Mizumo and T. Higuchi, “Design of magnetic bearing con-trollers based on disturbance estimation,” in Proceedings of the2nd International Symposium on Magnetic Bearings, pp. 281–288, Tokyo, Japan, July 1990.

[19] M. E. Kasarda, J. Clements, A. L. Wicks, C. D. Hall, and R. G.Kirk, “Effect of sinusoidal base motion on a magnetic bear-ing,” in Proceedings of the IEEE International Conference onControl Applications, vol. 1, pp. 144–149, Anchorage, Alaska,USA, September 2000.

[20] W. R. Saunders, H. H. Robertshaw, and R. A. Burdisso, “A hy-brid structural control approach for narrow-band and impul-sive disturbance rejection,” Noise Control Engineering Journal,vol. 44, no. 1, pp. 11–21, 1996.

[21] G. Schweitzer, H. Bleuler, and A. Traxler, Active Magnetic Bear-ings, vdf Hochschulverlag an der ETH, Zurich, Germany, 1994.

[22] R. F. Stengel, Optimal Control and Estimation, Dover, NewYork, NY, USA, 1994.

[23] C. Nataraj and S. Marx, “An efficient linear optimal controlalgorithm for harmonic base excitation of flexible rotor sys-tems,” submitted to Journal of Vibration and Control.

[24] G. T. Flowers, G. Szasz, V. S. Trent, and M. E. Greene, “Astudy of integrally augmented state feedback control for an ac-tive magnetic bearing supported rotor system,” Transactions ofthe ASME, Journal of Engineering for Gas Turbines and Power,vol. 123, no. 2, pp. 377–382, 2001.

[25] S. Marx, “Compensation for base motion in magnetic bear-ing rotors,” Master’s thesis, Villanova University, Villanova, Pa,USA, 2005.

[26] C. Nataraj and H. D. Nelson, “Periodic oscillations in rotor dy-namic systems with nonlinear supports: a general approach,”Transactions of the ASME, Journal of Vibration, Acoustics Stressand Reliability in Design, vol. 111, pp. 187–193, 1989.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2010

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Suppression of Base Excitation of Rotors on Magnetic Bearings

Documents

Transcript of Suppression of Base Excitation of Rotors on Magnetic Bearings