Reduction of Magnetically Induced Vibration of a Spoke ...prem.hanyang.ac.kr/down/Reduction of Magnetically Induced Vibration of a Spoke-Type IPM...Reduction of Magnetically Induced

IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 9, SEPTEMBER 2013 5097

Reduction of Magnetically Induced Vibration of a Spoke-Type IPM MotorUsing Magnetomechanical Coupled Analysis and Optimization

D. Y. Kim, J. K. Nam, and G. H. Jang

PREM, Department of Mechanical Engineering, Hanyang University, Seoul 133-791, Korea

We present an optimization methodology to reduce magnetically induced vibrations of a spoke-type interior permanent magnet (IPM)motor that we developed by performing magnetic and structural finite element analyses and optimization. The magnetic forces acting onthe teeth of the stator were calculated by magnetic finite element analysis and the Maxwell stress tensor method. The natural frequenciesand mode shapes of the stator were calculated by structural finite element analysis and verified by modal testing. The vibration of themotor due to the rotating magnetic force was calculated by the mode superposition method, and it was compared with the measuredvibration. Finally, two optimization problems were formulated and solved to reduce magnetically induced vibration: minimization ofmagnetic force and minimization of acceleration. We showed that minimization of acceleration was more effective than minimizationof magnetic force at reducing magnetically induced vibrations, because the former method effectively decreased the amplitudes of theexcitation frequencies of magnetic force by considering the transfer function of the motor.

Index Terms—IPM motor, magnetic forces, magnetically induced vibration, optimization methods.

I. INTRODUCTION

P ERMANENT MAGNET (PM) brushless dc (BLDC) mo-tors are widely used in many industry applications such as

home appliances and electric vehicles because they have highefficiency and easy controllability over a wide range of oper-ating speeds. These motors can be classified into two groups:surface-mounted PM (SPM) motors and interior PM (IPM) mo-tors. In SPM motors, the PMs are mounted on the surface of therotor, while in IPM motors, the PMs are in the interior of therotor core. Generally, IPM motors have higher power densityand efficiency than SPM motors because IPM motors utilize re-luctance torque as well as electromagnetic torque. Fig. 1 showsa spoke-type IPM motor that has high magnetic flux density inthe air gap due to placement of PMs on both sides of the polesof the rotor. However, concentration of the magnetic flux den-sity in the air gap distorts the back electromagnetic motive force(BEMF) [1], [2]. The high power density and distortion of theBEMF generate magnetically induced vibrations and high levelsof acoustic noise.Several researchers have investigated the characteristics of

vibration sources and magnetically-induced vibrations in mo-tors. Jang and Lieu investigated the effects of magnet geom-etry on the vibration of an electric motor [3]. They predictedthe magnetic force and dynamic reaction force at the mountingpoints of the motor according to variation of the geometry ofa PM. He et al. and Shin et al. investigated the radial and tan-gential magnetic forces of a PM motor through analytical cal-culations of electromagnetic field [4], [5]. Wu et al. presentedan analytical model of unbalanced magnetic force (UMF) in afractional-slot SPM motor [6]. However, they did not considerthe structural vibrations of the motor excited by magnetic force.Kim et al. investigated the UMF and dynamic response of the

Manuscript received December 31, 2012; revised February 21, 2013;accepted March 21, 2013. Date of publication April 03, 2013; date of cur-rent version August 21, 2013. Corresponding author: G. H. Jang (e-mail:[email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMAG.2013.2255307

Fig. 1. Structure of a spoke-type IPM motor.

rotors used in IPM and SPM motors [7]. They showed that IPMmotors have worse vibration characteristics than SPMmotors inthe presence of rotor eccentricity, but they investigated only thedynamic response of the rotor with bearings. They did not con-sider structural vibrations of a motor due to the magnetic forcesacting on the teeth of the stator, even though this is the dominantvibration source in a motor. Kim et al. investigated the vibra-tion of the stator due to magnetic force [8]. They predicted themagnetic force using the equivalent magnetizing current (EMC)method and the structural vibration of the stator by finite-ele-ment (FE) method. However, they assumed that the magneticforce acted on the center of a tooth, and they only took the ra-dial components of the magnetic force into account. Sun et al.investigated the effect of pole and slot combination on noiseand vibration in PM motors [9]. They predicted the magneticforce using the EMC and the FE methods, and they showed thecharacteristics of both magnetic force and vibration accordingto variation of pole and slot combination. Yim et al. investigatedthe vibration of an IPM motor due to magnetic force [10]. Theyshowed the magnetically induced vibrations of a stator resultfrom the dominant harmonics of the magnetic force. However,prior researchers [7]–[10] did not propose a concrete method-ology to reduce magnetically induced vibrations. Jung et al. in-vestigated the optimal design reducing magnetic forces of an

0018-9464 © 2013 IEEE

5098 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 9, SEPTEMBER 2013

integrated starter and generator by using the response surfacemethod [11]. Lee et al. investigated methods to reduce acousticnoise generated by an IPM motor [12]. They predicted the mag-netic force using the EMC method and proposed an optimal de-sign to reduce the acoustic noise of an IPM motor. However,they separated the optimal design into two components: a me-chanical design component to increase the stiffness of the statorand an electromagnetic design component to reduce magneticforce. Jung et al. and Lee et al. [11], [12] did not consider the dy-namic characteristics of the mechanical system in their optimaldesign to reduce structural vibrations and acoustic noise causedby magnetic force. Therefore, no prior studies have consideredthe effect of both magnetic and mechanical systems simultane-ously on reducing magnetically induced vibrations.In this study, we present an optimization methodology to

reduce magnetically induced vibrations of a spoke-type IPMmotor that we developed by performing magnetic and structuralFE analyses and optimization. The magnetic force acting on theteeth of a stator was calculated by magnetic FE analysis and theMaxwell stress tensormethod.Thenatural frequencies andmodeshapes of the stator were calculated by structural FE analysis andverified by modal testing. The vibration of the motor due to therotating magnetic force was calculated by the mode superposi-tion method, and the results were compared with the measuredvibration of the motor. Finally, two optimization problemsrelating to minimization of magnetic force and minimization ofacceleration were formulated and solved to determine if theycould effectively reduce magnetically induced vibrations.

II. MAGNETIC FINITE ELEMENT ANALYSIS ANDMAGNETIC FORCE

A. Magnetic Finite-Element Analysis and ExperimentalVerification

The phase currents applied in the windings were measured fora spoke-type IPM motor, as shown in Fig. 1 by using an oscil-loscope and current probes. The measured phase currentsand are shown in Fig. 2. The measured phase currents weredecomposed by the Fourier series shown in (1) for applicationin a magnetic FE model of a spoke-type IPM motor:

(1)

where and are the Fourier coefficient, the number ofpole pairs, the rotating speed, and the phase angle between the aphase current and the -axis current, respectively. Equation (2)shows the relationship of , the -axis current, and the -axiscurrent, as shown in Fig. 3, and (3) shows the relationship of thephase currents, -axis, and -axis current [13]. The and phasecurrents have phase delays of 120 and 240 , respectively, withrespect to the phase current:

(2)

(3)

Fig. 2. Measured , and phase currents using current probes.

Fig. 3. Vector diagram in -axis reference frame.

TABLE ISPECIFICATIONS OF THE MOTOR

We developed a two-dimensional FE model for a spoke-typeIPM motor with 8 poles and 12 slots (Fig. 1) to calculate themagnetic field of the motor. The resulting FE model had 98 643elements. The FE model was fully modeled to describe rotoreccentricity (the geometric center of the rotor rotates on a rota-tional center with constant eccentricity), and the air gap of theFE model had three mesh layers to improve the numerical ac-curacy of the magnetic force. The elements in the air gap wereuniformly divided such that the circumferential length of an ele-ment was equal to the rotational angle corresponding to the timestep for FE analysis to avoid distortion of the elements in themoving mesh algorithm. Table I shows the major design speci-fications of the spoke-type IPM motor. The arrow in Fig. 1 in-dicates the direction of magnetization of the PM.The governing equation of the magnetic field is as follows:

(4)

KIM et al.: REDUCTION OF MAGNETICALLY INDUCED VIBRATION OF A SPOKE-TYPE IPM MOTOR 5099

Fig. 4. Simulated magnetic flux flow of a spoke-type IPM motor.

Fig. 5. Comparison of simulated surface magnetic flux density with measuredsurface magnetic flux density.

where , and are the permeability, the magneticvector potential, the current density in windings from ex-ternal voltage sources, and the equivalent magnetization cur-rent density caused by the PM, respectively. The flow of mag-netic flux calculated from (4) using the FE method is shownin Fig. 4.The surface magnetic flux density and BEMF were measured

to validate the accuracy of the developed magnetic FE model.The magnetic flux density along the surface of the rotor wasmeasured every 0.5 during one revolution using a Gauss meter.Fig. 5 shows the simulated and measured surface magnetic fluxdensity; the simulated magnetic flux density matched the mea-sured flux density well. Furthermore, the BEMF was measuredby using a spin-down test in which the BEMF was measuredinstantaneously once the electric power was turned off. Fig. 6shows the simulated and measured BEMF at 1000 r/min; again,the simulated BEMF and measured BEMF values were consis-tent with one another. The RMS values of the simulated andmeasured BEMF were 43.8 and 43.1 Vrms, respectively.

B. Characteristics of Magnetic Force

Magnetic force was calculated from the magnetic flux densityof the air gap using a Maxwell stress tensor and the followingequation:

(5)

Fig. 6. Comparison of simulated BEMF with measured BEMF at 1000 RPM.

Fig. 7. Variation of magnetic force acting on a tooth of the stator: (a) Normalmagnetic force and (b) tangential magnetic force.

where , and are the Maxwell stress tensor,the magnetic flux density in the -direction, and Kro-necker–delta, respectively. The following normal and tangen-tial force densities in the cylindrical coordinate were definedwith both the Maxwell stress tensor and the relationship

[14]:

(6)

(7)

where and are the magnetic flux densities in the normaland tangential directions, respectively.Fig. 7 shows the variation in magnetic force acting on a tooth

of a spoke-type IPMmotor running at 15 246 r/min with a phasecurrent of 2.51 and a phase angle of 72 as the rotor


Fig. 8. Frequency spectrum of normal magnetic force on the center of a statortooth according to rotor eccentricity: (a) Without rotor eccentricity; (b) withrotor eccentricity of 0.1 mm; and (c) with rotor eccentricity of 0.2 mm.

rotates 90 . The rotor eccentricity was assumed to be 25 mbased on consideration of the internal clearance of the ball bear-ings (tolerance is from 4 to 11 m) and tolerance between thehousing of the motor and the bearing (tolerance for clearance fitis 15 to 20 m). The normal magnetic force is distributed alongthe tooth, while the tangential magnetic force is concentratedon the edge of the tooth. Both normal and tangential magneticforces repeat every 45 so that their frequency components arethe eighth harmonics of the number of poles.The frequency spectrum of the normal magnetic force acting

on the center of a tooth according to the variations in rotoreccentricity is shown in Fig. 8; it is clear that rotor eccentricitygenerates additional harmonics. Fig. 9 shows the amplitudevariation of major harmonics of the normal magnetic forceacting on the center of a tooth due to rotor eccentricity. Theamplitudes of 1X, 8X, and 16X increased linearly, while theamplitudes of 2X and 10X increased proportionally to thesquare of rotor eccentricity.

Fig. 9. Variation of amplitudes of 1X, 2X, 8X, 10X, and 16X of the normalmagnetic force on the center of a tooth according to rotor eccentricity.

C. Calculation of Torque and Iron Loss

The torque of a motor can be calculated by integrating thetangential magnetic force along the circumference of the air-gap. Torque was determined from the following equation [15]:

(8)

where and are the resultant torque and the radius of the rotor,respectively.Iron loss from the stator core was calculated by the following

equation [16]:

(9)

where , and are the coefficient of the hys-teresis loss, peak value of the magnetic flux density, driving fre-quency, conductivity of the material, thickness of lamination,and the coefficient of excess losses, respectively.

III. STRUCTURAL FINITE ELEMENT ANALYSIS ANDEXPERIMENTAL VERIFICATION

The three-dimensional structural FE model shown in Fig. 10was developed to simulate the vibrations induced by magneticforces acting on the teeth of the stator. The FE model had233 786 elements consisting of 92 701 tetrahedral elements,140 838 brick elements, and 247 beam and rigid-link elements.

A. Free Vibrational Analysis of a Stator and ExperimentalVerification

The stator shown in Fig. 11 has a laminated structure con-sisting of stacked thin plates to reduce eddy current loss due tochanges in magnetic flux. These laminations are generally fixedby bolting, welding, or caulking. Therefore, the elastic modulusand shear modulus of stator lamination in the -axis are muchlower than that of an isotropic structure [17]. The stator wasmodeled with orthotropic material to account for the mechan-ical property of stator lamination. The shear modulus was de-termined from the following equation [18]:

(10)


Fig. 10. Finite element model composed of motor housing, shaft, and stator.

Fig. 11. Finite-element model of a stator.

TABLE IICOMPARISON OF NATURAL FREQUENCIES AND MODE SHAPES OBTAINED BY

FE ANALYSIS WITH THOSE OBTAINED EXPERIMENTALLY

where and are the elastic modulus, shear modulus, andPoisson’s ratio, respectively.The developed FE model of the stator was validated by

comparing simulated natural frequencies and mode shapes withmeasured ones through modal testing, as shown in Table II.The simulated natural frequencies and mode shapes matchedwell with the measured ones (within 7% error).

Fig. 12. Equivalent magnetic nodal force acting on a tooth of the stator.

B. Forced Vibrational Analysis and Experimental Verification

Forced vibration of a spoke-type IPM motor excited by arotating magnetic force can be represented by the followingequation:

(11)

where and are the mass matrix, the damping ma-trix, and the stiffness matrix, respectively. and arethe equivalent nodal force vector and nodal displacement vector,respectively. is determined from the modal damping ratiosmeasured experimentally.Normal and tangential magnetic forces were applied to 15

nodal points on every tooth, as shown in Fig. 12. Axial mag-netic force was not included because the housing was made ofaluminum and there was no overhang between the stator andthe rotor to generate axial magnetic force. The magnetic forceacting on the teeth of the stator was calculated from the surfaceintegral of the magnetic force density. The calculated magneticforces were as follows:

(12)

(13)

The equivalent nodal force vector was calculated from thefollowing equation:

(14)

where and are the shape function and the magneticforce vector, respectively, calculated from the magnetic FEanalysis.The vibration of the motor due to the rotating magnetic force

was calculated by the mode superposition method. Nodal dis-placements can be expressed as the linear superposition of mul-tiplying the mode vector by the modal displacement with thefollowing equation [19]:

(15)

where , and are the th mode shape vector, themodal displacement, and the number of mode shapes used in


Fig. 13. Frequency response of (a) simulated acceleration and (b) measuredacceleration at point A of a stator.

the mode superposition method, respectively. The number ofsuperposed modes was set to 30 after ensuring that accelerationwas converged.The simulated acceleration was compared with the measured

acceleration to verify the accuracy of the forced vibration anal-ysis of a spoke-type IPM motor. The simulated and measuredacceleration of the upper center point of the stator marked byA in Fig. 10 is shown in Fig. 13. Major vibration sources arecentrifugal force and gyroscopic moment due to the unbalancedmass of the rotor, which generate the first harmonic, but the sim-ulation model did not include these centrifugal forces or the gy-roscopic moment, therefore the first harmonic was not observedin the simulated acceleration results shown in Fig. 13(a). Therotor is supported by ball bearings, and the bearing frequencyof 783 Hz shown in Fig. 13(b) originated from defects of theouter race of the ball bearings [20], [21]. Most simulated fre-quency components and their amplitudes matched well with themeasured ones with the exception of the first harmonic and 783Hz. Furthermore, the first harmonic did not disappear in the fre-quency response even when the electric power was turned off,which implies that the first harmonic was not caused by electro-magnetic sources, but by mechanical sources.

IV. DESIGN OPTIMIZATION TO REDUCEMAGNETICALLY-INDUCED VIBRATION

Two optimization problemswere formulated and solved to re-duce magnetically-induced vibration: minimization of the mag-netic force and minimization of acceleration.

Fig. 14. Four design variables of the design optimization problem.

TABLE IIILOWER AND UPPER LIMIT OF THE FOUR DESIGN VARIABLES USED IN THE

DESIGN OPTIMIZATION PROBLEMS

A. Minimization of the Magnetic Force

The optimization problem to minimize the magnetic forceacting on the teeth of the stator was formulated as follows:

(16)

(17)

where and are the normal and tangential magnetic force,respectively, at the th point on a tooth of the stator. The ob-jective function is the sum of the normal force and tangentialforce on a tooth of the stator as shown in Fig. 12. The magneticforce was calculated from (12)–(13). Torque was constrained tobe equal or larger than that of the initial design, and iron losswas constrained to be equal or smaller than that of the initial de-sign. The volume of the PM was also constrained to be equalto that in the initial design. Fig. 14 shows the four design vari-ables of the design optimization problem: the pole angle of therotor, the length of the PM, the length of a tooth, and the thick-ness of the tooth shoe. Their lower and upper limits are specifiedin Table III. Fig. 15 shows the procedure used to minimize mag-netic force. It took approximately 13 hours using a computerwitha Quad core CPU (2.93 GHz) and 16.0 GB RAM to completeone process. Therefore, the Kriging metamodel was constructedfrom FE analysis results for 36 experimental points with full fac-torial design (FFD) that had two levels for the pole angle of therotor and the length of the PM and three levels for the thickness


Fig. 15. Optimization procedure to minimize the magnetic force.

TABLE IVCOMPARISON OF THE OPTIMAL DESIGN TO MINIMIZE THE MAGNETIC FORCE

WITH THE INITIAL DESIGN

TABLE VDESIGN VARIABLES OF THE OPTIMAL DESIGN TO MINIMIZE MAGNETIC FORCES

of the tooth shoe and the length of a tooth [22]. In optimizationof nonlinear problems, all the searchingmethod for optimal solu-tion has the same averaged performance over all the problems ac-cording to the “no free lunch theorem” [23] andprevious researchfor comparison of searching methods [24]. In this research, theprogressive quadratic response surface method (PRQSM) waschosen to search for the optimal solution [25]. The PQRSMdoesnot have the same disadvantage as gradient-based optimizationmethod of converging on local optima. The computational timerequired by the metamodel to search for the optimal point wasabout 1 minute, and the number of iterations required for con-vergence was 37 for this optimization problem.The results of the optimization are shown in Table IV, and the

optimal design variables are shown in Table V. The magnetic

Fig. 16. Optimization procedure to minimize the acceleration of the motor.

force decreased by 4.8%, iron loss decreased by 9.9%, whilethe torque remained the same as in the initial design.

B. Analysis of Magnetically-Induced Vibrations

We formulated the optimization problem to minimize themotor acceleration induced by magnetic force excitation asfollows:

(18)

(19)

where is the acceleration at the th point on stator. The ob-jective function is the sum of radial acceleration (RMS) of 18points on the motor, as shown in Fig. 10. Acceleration was cal-culated by solving (11). The constraints were the same as thoseused in the optimization problem to minimize magnetic force in(17). Fig. 16 shows the optimization procedure to minimize theacceleration of the motor. It took about 16 hours to complete oneprocess using the same computer as that used to minimize themagnetic force. The same metamodel, design variables, and op-timization method were used to solve the optimization problemto minimize acceleration.The results of the optimization problem are shown in

Table VI, and the optimal design variables are shown inTable VII. The acceleration decreased by 7.0%, while thetorque remained at the same level and iron loss decreased by5.1% compared with the initial design.


TABLE VICOMPARISON OF THE OPTIMAL DESIGN TO MINIMIZE

ACCELERATION WITH THE INITIAL DESIGN

TABLE VIIDESIGN VARIABLES OF THE OPTIMAL DESIGN TO MINIMIZE ACCELERATION

Fig. 17. Amplitudes of acceleration of the dominant harmonic component forthe two optimization problems.

TABLE VIIICOMPARISON OF THE TWO OPTIMIZATION PROBLEMS

The results obtained for the two optimization problems arecompared in Table VIII. Minimization of acceleration (0.622m/s ) decreased magnetically-induced vibration more effec-tively than minimization of the magnetic force (0.658 m/s ),

even though the magnetic excitation force (0.436 N) in min-imization of the acceleration is greater than that (0.415 N) inminimization of the magnetic force, because the optimizationmethod to minimize acceleration included the transfer functionof themotor. In addition, the effects of the direction of excitationon structural vibrations were included in the transfer function ofthe motor to minimize acceleration. The amplitudes of accelera-tion corresponding to 8X, 10X, and 16X for the two optimizationmethods are shown in Fig. 17. The amplitudes of accelerationscorresponding to 10X and 16X decreased in the minimizationof the magnetic force while the amplitude of 8X increased. Incontrast, the amplitudes of acceleration corresponding to 8Xand 16X, which are the first and the second dominant frequencycomponents of acceleration, mainly decreased in minimiza-tion of the acceleration. The optimization method to minimizethe acceleration reduced the magnetically-induced vibrationseffectively by considering the mechanical transfer function ofthe motor in optimization because the structural vibration is theresponse of the transfer function excited by the magnetic force.

V. CONCLUSION

We developed an optimization methodology to reduce mag-netically-induced vibration of a spoke-type IPM motor by per-forming magnetic and structural FE analyses and optimization.Simulated and experimentally measured magnetic flux densityand BEMF values were compared to verify the magnetic FEmodel. Simulated natural frequencies, mode shapes, and the ac-celeration of the motor induced by excitation of the rotatingmagnetic force were compared to measured values to verify thestructural FE model. Finally, two optimization problems wereformulated and solved to reduce magnetically induced vibra-tion: minimization of the magnetic force and minimization ofthe motor acceleration. We showed that minimization of accel-eration was more effective at reducing magnetically induced vi-bration than minimization of the magnetic force, because theformer method effectively decreased the amplitudes of the ex-citation frequencies of the magnetic force by taking the transferfunction of the motor into consideration. Our proposed methodcan be effectively extended to other electric machines to re-duce magnetically induced vibrations or to maximize the torquewhile maintaining structural vibrations. Our methodology canbe applied in electromagnetic designs and structural designs toreduce structural vibrations and acoustic noise caused by mag-netic force.

ACKNOWLEDGMENT

This research was supported by Samsung Electronics Com-pany, Ltd.

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