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

Transcription

1 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 9, SEPTEMBER Reduction of Magnetically Induced Vibration of a Spoke-Type IPM Motor Using Magnetomechanical Coupled Analysis and Optimization D. Y. Kim, J. K. Nam, and G. H. Jang PREM, Department of Mechanical Engineering, Hanyang University, Seoul , 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 on the teeth of the stator were calculated by magnetic finite element analysis and the Maxwell stress tensor method. The natural frequencies and mode shapes of the stator were calculated by structural finite element analysis and verified by modal testing. The vibration of the motor due to the rotating magnetic force was calculated by the mode superposition method, and it was compared with the measured vibration. Finally, two optimization problems were formulated and solved to reduce magnetically induced vibration: minimization of magnetic force and minimization of acceleration. We showed that minimization of acceleration was more effective than minimization of magnetic force at reducing magnetically induced vibrations, because the former method effectively decreased the amplitudes of the excitation 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) motors are widely used in many industry applications such as home appliances and electric vehicles because they have high efficiency and easy controllability over a wide range of operating speeds. These motors can be classified into two groups: surface-mounted PM (SPM) motors and interior PM (IPM) motors. In SPM motors, the PMs are mounted on the surface of the rotor, while in IPM motors, the PMs are in the interior of the rotor core. Generally, IPM motors have higher power density and efficiency than SPM motors because IPM motors utilize reluctance torque as well as electromagnetic torque. Fig. 1 shows a spoke-type IPM motor that has high magnetic flux density in the air gap due to placement of PMs on both sides of the poles of the rotor. However, concentration of the magnetic flux density in the air gap distorts the back electromagnetic motive force (BEMF) [1], [2]. The high power density and distortion of the BEMF generate magnetically induced vibrations and high levels of acoustic noise. Several researchers have investigated the characteristics of vibration sources and magnetically-induced vibrations in motors. Jang and Lieu investigated the effects of magnet geometry on the vibration of an electric motor [3]. They predicted the magnetic force and dynamic reaction force at the mounting points of the motor according to variation of the geometry of apm.heet al. and Shin et al. investigated the radial and tangential magnetic forces of a PM motor through analytical calculations of electromagnetic field[4],[5].wuet al. presented an analytical model of unbalanced magnetic force (UMF) in a fractional-slot SPM motor [6]. However, they did not consider the 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, Date of publication April 03, 2013; date of current version August 21, Corresponding author: G. H. Jang ( ghjang@hanyang.ac.kr). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TMAG Fig. 1. Structure of a spoke-type IPM motor. rotors used in IPM and SPM motors [7]. They showed that IPM motors have worse vibration characteristics than SPM motors in the presence of rotor eccentricity, but they investigated only the dynamic response of the rotor with bearings. They did not consider structural vibrations of a motor due to the magnetic forces acting on the teeth of the stator, even though this is the dominant vibration source in a motor. Kim et al. investigated the vibration of the stator due to magnetic force [8]. They predicted the magnetic force using the equivalent magnetizing current (EMC) method and the structural vibration of the stator by finite-element (FE) method. However, they assumed that the magnetic force acted on the center of a tooth, and they only took the radial components of the magnetic force into account. Sun et al. investigated the effect of pole and slot combination on noise and vibration in PM motors [9]. They predicted the magnetic force using the EMC and the FE methods, and they showed the characteristics of both magnetic force and vibration according to variation of pole and slot combination. Yim et al. investigated the vibration of an IPM motor due to magnetic force [10]. They showed the magnetically induced vibrations of a stator result from the dominant harmonics of the magnetic force. However, prior researchers [7] [10] did not propose a concrete methodology to reduce magnetically induced vibrations. Jung et al. investigated the optimal design reducing magnetic forces of an IEEE

2 5098 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 9, SEPTEMBER 2013 integrated starter and generator by using the response surface method [11]. Lee et al. investigated methods to reduce acoustic noise generated by an IPM motor [12]. They predicted the magnetic force using the EMC method and proposed an optimal design to reduce the acoustic noise of an IPM motor. However, they separated the optimal design into two components: a mechanical design component to increase the stiffness of the stator and an electromagnetic design component to reduce magnetic force. Jung et al. and Lee et al. [11], [12] did not consider the dynamic characteristics of the mechanical system in their optimal design to reduce structural vibrations and acoustic noise caused by magnetic force. Therefore, no prior studies have considered the effect of both magnetic and mechanical systems simultaneously on reducing magnetically induced vibrations. In this study, we present an optimization methodology to reduce magnetically induced vibrations of a spoke-type IPM motor that we developed by performing magnetic and structural FE analyses and optimization. The magnetic force acting on the teeth of a stator was calculated by magnetic FE analysis and the Maxwell stress tensor method. The natural frequencies and mode shapes of the stator were calculated by structural FE analysis and verified by modal testing. The vibration of the motor due to the rotating magnetic force was calculated by the mode superposition method, and the results were compared with the measured vibration of the motor. Finally, two optimization problems relating to minimization of magnetic force and minimization of acceleration were formulated and solved to determine if they could effectively reduce magnetically induced vibrations. Fig. 2. Measured,and phase currents using current probes. Fig. 3. Vector diagram in -axis reference frame. TABLE I SPECIFICATIONS OF THE MOTOR II. MAGNETIC FINITE ELEMENT ANALYSIS AND MAGNETIC FORCE A. Magnetic Finite-Element Analysis and Experimental Verification The phase currents applied in the windings were measured for a spoke-type IPM motor, as shown in Fig. 1 by using an oscilloscope and current probes. The measured phase currents and are shown in Fig. 2. The measured phase currents were decomposed by the Fourier series shown in (1) for application in a magnetic FE model of a spoke-type IPM motor: where and are the Fourier coefficient, the number of pole pairs, the rotating speed, and the phase angle between the a phase current and the -axis current, respectively. Equation (2) shows the relationship of,the -axis current, and the -axis current, as shown in Fig. 3, and (3) shows the relationship of the phase currents, -axis, and -axis current [13]. The and phase currents have phase delays of 120 and 240, respectively, with respect to the phase current: (1) (2) (3) We developed a two-dimensional FE model for a spoke-type IPM motor with 8 poles and 12 slots (Fig. 1) to calculate the magnetic field of the motor. The resulting FE model had elements. The FE model was fully modeled to describe rotor eccentricity (the geometric center of the rotor rotates on a rotational center with constant eccentricity), and the air gap of the FE model had three mesh layers to improve the numerical accuracy of the magnetic force. The elements in the air gap were uniformly divided such that the circumferential length of an element was equal to the rotational angle corresponding to the time step for FE analysis to avoid distortion of the elements in the moving mesh algorithm. Table I shows the major design specifications of the spoke-type IPM motor. The arrow in Fig. 1 indicates the direction of magnetization of the PM. The governing equation of the magnetic fieldisasfollows: (4)

3 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. 6. Comparison of simulated BEMF with measured BEMF at 1000 RPM. Fig. 5. Comparison of simulated surface magnetic flux density with measured surface magnetic flux density. where,and are the permeability, the magnetic vector potential, the current density in windings from external voltage sources, and the equivalent magnetization current density caused by the PM, respectively. The flow of magnetic flux calculated from (4) using the FE method is shown in 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 was measured every 0.5 during one revolution using a Gauss meter. Fig. 5 shows the simulated and measured surface magnetic flux density; the simulated magnetic flux density matched the measured flux density well. Furthermore, the BEMF was measured by using a spin-down test in which the BEMF was measured instantaneously once the electric power was turned off. Fig. 6 shows the simulated and measured BEMF at 1000 r/min; again, the simulated BEMF and measured BEMF values were consistent with one another. The RMS values of the simulated and measured BEMF were 43.8 and 43.1 Vrms, respectively. B. Characteristics of Magnetic Force Magnetic force was calculated from the magnetic flux density of the air gap using a Maxwell stress tensor and the following equation: (5) Fig. 7. Variation of magnetic force acting on a tooth of the stator: (a) Normal magnetic force and (b) tangential magnetic force. where, and are the Maxwell stress tensor, the magnetic flux density in the -direction, and Kronecker delta, respectively. The following normal and tangential force densities in the cylindrical coordinate were defined with both the Maxwell stress tensor and the relationship [14]: where and are the magnetic flux densities in the normal and tangential directions, respectively. Fig. 7 shows the variation in magnetic force acting on a tooth of a spoke-type IPM motor running at r/min with a phase current of 2.51 and a phase angle of 72 as the rotor (6) (7)

4 5100 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 9, SEPTEMBER 2013 Fig. 9. Variation of amplitudes of 1X, 2X, 8X, 10X, and 16X of the normal magnetic 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 the tangential magnetic force along the circumference of the airgap. Torque was determined from the following equation [15]: where and aretheresultanttorqueandtheradiusoftherotor, respectively. Iron loss from the stator core was calculated by the following equation [16]: (8) (9) Fig. 8. Frequency spectrum of normal magnetic force on the center of a stator tooth according to rotor eccentricity: (a) Without rotor eccentricity; (b) with rotor eccentricity of 0.1 mm; and (c) with rotor eccentricity of 0.2 mm. rotates 90. The rotor eccentricity was assumed to be 25 m based on consideration of the internal clearance of the ball bearings (tolerance is from 4 to 11 m) and tolerance between the housing of the motor and the bearing (tolerance for clearance fit is 15 to 20 m). The normal magnetic force is distributed along the tooth, while the tangential magnetic force is concentrated on the edge of the tooth. Both normal and tangential magnetic forces repeat every 45 so that their frequency components are the 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 rotor eccentricity is shown in Fig. 8; it is clear that rotor eccentricity generates additional harmonics. Fig. 9 shows the amplitude variation of major harmonics of the normal magnetic force acting on the center of a tooth due to rotor eccentricity. The amplitudes of 1X, 8X, and 16X increased linearly, while the amplitudes of 2X and 10X increased proportionally to the square of rotor eccentricity. where,and are the coefficient of the hysteresis loss, peak value of the magnetic flux density, driving frequency, conductivity of the material, thickness of lamination, and the coefficient of excess losses, respectively. III. STRUCTURAL FINITE ELEMENT ANALYSIS AND EXPERIMENTAL VERIFICATION The three-dimensional structural FE model shown in Fig. 10 was developed to simulate the vibrations induced by magnetic forces acting on the teeth of the stator. The FE model had elements consisting of tetrahedral elements, brick elements, and 247 beam and rigid-link elements. A. Free Vibrational Analysis of a Stator and Experimental Verification The stator shown in Fig. 11 has a laminated structure consisting of stacked thin plates to reduce eddy current loss due to changes in magnetic flux. These laminations are generally fixed by bolting, welding, or caulking. Therefore, the elastic modulus and shear modulus of stator lamination in the -axis are much lower than that of an isotropic structure [17]. The stator was modeled with orthotropic material to account for the mechanical property of stator lamination. The shear modulus was determined from the following equation [18]: (10)

5 KIM et al.: REDUCTION OF MAGNETICALLY INDUCED VIBRATION OF A SPOKE-TYPE IPM MOTOR 5101 Fig. 12. Equivalent magnetic nodal force acting on a tooth of the stator. Fig. 10. Fig. 11. Finite element model composed of motor housing, shaft, and stator. Finite-element model of a stator. B. Forced Vibrational Analysis and Experimental Verification Forced vibration of a spoke-type IPM motor excited by a rotating magnetic force can be represented by the following equation: (11) where and are the mass matrix, the damping matrix, and the stiffness matrix, respectively. and are the equivalent nodal force vector and nodal displacement vector, respectively. is determined from the modal damping ratios measured experimentally. Normal and tangential magnetic forces were applied to 15 nodal points on every tooth, as shown in Fig. 12. Axial magnetic force was not included because the housing was made of aluminum and there was no overhang between the stator and the rotor to generate axial magnetic force. The magnetic force acting on the teeth of the stator was calculated from the surface integral of the magnetic force density. The calculated magnetic forces were as follows: TABLE II COMPARISON OF NATURAL FREQUENCIES AND MODE SHAPES OBTAINED BY FE ANALYSIS WITH THOSE OBTAINED EXPERIMENTALLY (12) (13) The equivalent nodal force vector was calculated from the following equation: (14) where and are the elastic modulus, shear modulus, and Poisson s ratio, respectively. The developed FE model of the stator was validated by comparing simulated natural frequencies and mode shapes with measured ones through modal testing, as shown in Table II. The simulated natural frequencies and mode shapes matched well with the measured ones (within 7% error). where and are the shape function and the magnetic force vector, respectively, calculated from the magnetic FE analysis. The vibration of the motor due to the rotating magnetic force was calculated by the mode superposition method. Nodal displacements can be expressed as the linear superposition of multiplying the mode vector by the modal displacement with the following equation [19]: (15) where, and are the th mode shape vector, the modal displacement, and the number of mode shapes used in

6 5102 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 9, SEPTEMBER 2013 Fig. 14. Four design variables of the design optimization problem. TABLE III LOWER AND UPPER LIMIT OF THE FOUR DESIGN VARIABLES USED IN THE DESIGN OPTIMIZATION PROBLEMS Fig. 13. Frequency response of (a) simulated acceleration and (b) measured acceleration at point A of a stator. the mode superposition method, respectively. The number of superposed modes was set to 30 after ensuring that acceleration was converged. The simulated acceleration was compared with the measured acceleration to verify the accuracy of the forced vibration analysis of a spoke-type IPM motor. The simulated and measured acceleration of the upper center point of the stator marked by A in Fig. 10 is shown in Fig. 13. Major vibration sources are centrifugal force and gyroscopic moment due to the unbalanced mass of the rotor, which generate the first harmonic, but the simulation model did not include these centrifugal forces or the gyroscopic moment, therefore the first harmonic was not observed in the simulated acceleration results shown in Fig. 13(a). The rotor is supported by ball bearings, and the bearing frequency of 783 Hz shown in Fig. 13(b) originated from defects of the outer race of the ball bearings [20], [21]. Most simulated frequency components and their amplitudes matched well with the measured ones with the exception of the first harmonic and 783 Hz.Furthermore,thefirst harmonic did not disappear in the frequency response even when the electric power was turned off, which implies that the first harmonic was not caused by electromagnetic sources, but by mechanical sources. IV. DESIGN OPTIMIZATION TO REDUCE MAGNETICALLY-INDUCED VIBRATION Two optimization problems were formulated and solved to reduce magnetically-induced vibration: minimization of the magnetic force and minimization of acceleration. A. Minimization of the Magnetic Force The optimization problem to minimize the magnetic force acting 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 objective function is the sum of the normal force and tangential force on a tooth of the stator as shown in Fig. 12. The magnetic force was calculated from (12) (13). Torque was constrained to be equal or larger than that of the initial design, and iron loss was constrained to be equal or smaller than that of the initial design. The volume of the PM was also constrained to be equal to that in the initial design. Fig. 14 shows the four design variables of the design optimization problem: the pole angle of the rotor, the length of the PM, the length of a tooth, and the thickness of the tooth shoe. Their lower and upper limits are specified in Table III. Fig. 15 shows the procedure used to minimize magnetic force. It took approximately 13 hours using a computer with a Quad core CPU (2.93 GHz) and 16.0 GB RAM to complete one process. Therefore, the Kriging metamodel was constructed from FE analysis results for 36 experimental points with full factorial design (FFD) that had two levels for the pole angle of the rotor and the length of the PM and three levels for the thickness

7 KIM et al.: REDUCTION OF MAGNETICALLY INDUCED VIBRATION OF A SPOKE-TYPE IPM MOTOR 5103 Fig. 16. Optimization procedure to minimize the acceleration of the motor. Fig. 15. Optimization procedure to minimize the magnetic force. TABLE IV COMPARISON OF THE OPTIMAL DESIGN TO MINIMIZE THE MAGNETIC FORCE WITH THE INITIAL DESIGN force decreased by 4.8%, iron loss decreased by 9.9%, while the torque remained the same as in the initial design. B. Analysis of Magnetically-Induced Vibrations We formulated the optimization problem to minimize the motor acceleration induced by magnetic force excitation as follows: (18) TABLE V DESIGN VARIABLES OF THE OPTIMAL DESIGN TO MINIMIZE MAGNETIC FORCES (19) of the tooth shoe and the length of a tooth [22]. In optimization of nonlinear problems, all the searching method for optimal solution has the same averaged performance over all the problems according to the no free lunch theorem [23] and previous research for comparison of searching methods [24]. In this research, the progressive quadratic response surface method (PRQSM) was chosen to search for the optimal solution [25]. The PQRSM does not have the same disadvantage as gradient-based optimization method of converging on local optima. The computational time required by the metamodel to search for the optimal point was about 1 minute, and the number of iterations required for convergence 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 where is the acceleration at the th point on stator. The objective function is the sum of radial acceleration (RMS) of 18 points on the motor, as shown in Fig. 10. Acceleration was calculated by solving (11). The constraints were the same as those used in the optimization problem to minimize magnetic force in (17). Fig. 16 shows the optimization procedure to minimize the acceleration of the motor. It took about 16 hours to complete one process using the same computer as that used to minimize the magnetic force. The same metamodel, design variables, and optimization method were used to solve the optimization problem to minimize acceleration. The results of the optimization problem are shown in Table VI, and the optimal design variables are shown in Table VII. The acceleration decreased by 7.0%, while the torque remained at the same level and iron loss decreased by 5.1% compared with the initial design.

8 5104 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 9, SEPTEMBER 2013 TABLE VI COMPARISON OF THE OPTIMAL DESIGN TO MINIMIZE ACCELERATION WITH THE INITIAL DESIGN TABLE VII DESIGN VARIABLES OF THE OPTIMAL DESIGN TO MINIMIZE ACCELERATION Fig. 17. Amplitudes of acceleration of the dominant harmonic component for the two optimization problems. TABLE VIII COMPARISON OF THE TWO OPTIMIZATION PROBLEMS even though the magnetic excitation force (0.436 N) in minimization of the acceleration is greater than that (0.415 N) in minimization of the magnetic force, because the optimization method to minimize acceleration included the transfer function of the motor. In addition, the effects of the direction of excitation on structural vibrations were included in the transfer function of the motor to minimize acceleration. The amplitudes of acceleration corresponding to 8X, 10X, and 16X for the two optimization methods are shown in Fig. 17. The amplitudes of accelerations corresponding to 10X and 16X decreased in the minimization of the magnetic force while the amplitude of 8X increased. In contrast, the amplitudes of acceleration corresponding to 8X and 16X, which are the first and the second dominant frequency components of acceleration, mainly decreased in minimization of the acceleration. The optimization method to minimize the acceleration reduced the magnetically-induced vibrations effectively by considering the mechanical transfer function of the motor in optimization because the structural vibration is the response of the transfer function excited by the magnetic force. V. CONCLUSION We developed an optimization methodology to reduce magnetically-induced vibration of a spoke-type IPM motor by performing magnetic and structural FE analyses and optimization. Simulated and experimentally measured magnetic flux density and BEMF values were compared to verify the magnetic FE model. Simulated natural frequencies, mode shapes, and the acceleration of the motor induced by excitation of the rotating magnetic force were compared to measured values to verify the structural FE model. Finally, two optimization problems were formulated and solved to reduce magnetically induced vibration: minimization of the magnetic force and minimization of the motor acceleration. We showed that minimization of acceleration was more effective at reducing magnetically induced vibration than minimization of the magnetic force, because the former method effectively decreased the amplitudes of the excitation frequencies of the magnetic force by taking the transfer function of the motor into consideration. Our proposed method can be effectively extended to other electric machines to reduce magnetically induced vibrations or to maximize the torque while maintaining structural vibrations. Our methodology can be applied in electromagnetic designs and structural designs to reduce structural vibrations and acoustic noise caused by magnetic force. ACKNOWLEDGMENT This research was supported by Samsung Electronics Company, Ltd. The results obtained for the two optimization problems are compared in Table VIII. Minimization of acceleration (0.622 m/s ) decreased magnetically-induced vibration more effectively than minimization of the magnetic force (0.658 m/s ), REFERENCES [1] Q. Chen, G. Liu, W. Gong, L. Qu, and W. Zhao, Design of a spoketype permanent-magnet motor with optimal winding configuration for electric vehicle applications, J. Appl. Phys., vol. 111, no. 7, 2012, 07E710 07E [2] K.Y.Hwang,S.B.Rhee,B.Y.Yang,andB.I.Kwon, Rotorpoledesign in spoke type BLDC motor by RSM, in Proc. IEEE Conf. Electromagn. Field Comput., 2006,p.425. [3] G. H. Jang and D. K. Lieu, The effect of magnet geometry on electric motor vibration, IEEE Trans. Magn., vol. 27, no. 6, pp , Nov

9 KIM et al.: REDUCTION OF MAGNETICALLY INDUCED VIBRATION OF A SPOKE-TYPE IPM MOTOR 5105 [4] G. He, Z. Huang, and D. Chen, Two-dimensional field analysis on electromagnetic vibration-and-noise sources in permanent-magnet direct current commutator motors, IEEE Trans. Magn., vol. 47, no. 4, pp , Apr [5] H.J.Shin,J.Y.Choi,H.I.Park,andS.M.Jang, Vibrationanalysisand measurements through prediction of electromagnetic vibration sources of permanent magnet synchronous motor based on analytical magnetic field calculations, IEEE Trans. Magn., vol. 48, no. 11, pp , Nov [6] L. J. Wu, Z. Q. Zhu, J. T. Chen, and Z. P. Xia, An analytical model of unbalanced magnetic force in fractional-slot surface-mounted permanent magnet machines, IEEE Trans. Magn., vol. 46, no. 7, pp , Jul [7] T.J.Kim,S.M.Hwang,K.T.Kim,W.B.Jeong,andC.U.Kim, Comparison of dynamic response for IPM and SPM motors by considering mechanical and magnetic coupling, IEEE Trans. Magn., vol. 37, no. 4, pp , Jul [8] J.M.Kim,T.Sun,S.H.Lee,D.J.Kim,andJ.P.Hong, Evaluation and improved design about acoustic noise and vibration in IPMSM, in Proc.Int.Conf.Electr.Mach.Syst., 2010, pp [9] T. Sun, G. H. Lee, J. P. Hong, and M. R. Choi, Effect of pole and slot combination on noise and vibration in permanent magnet synchronous motor, IEEE Trans. Magn., vol. 47, no. 5, pp , May [10] K. H. Yim, J. W. Jang, G. H. Jang, M. K. Kim, and K. N. Kim, Forced vibration analysis of an IPM motor for electrical vehicles due to magnetic force, IEEE Trans. Magn., vol. 48, no. 11, pp , Nov [11] J. W. Jung, S. H. Lee, G. H. Lee, J. P. Hong, D. H. Lee, and K. N. Kim, Reduction design of vibration and noise in IPMSM type integrated starter and generator for HEV, IEEE Trans. Magn., vol. 47, no. 6, pp , Jun [12] S. H. Lee, J. P. Hong, S. H. Hwang, W. T. Lee, J. Y. Lee, and Y. K. Kim, Optimal design for noise reduction in interior permanent-magnet motor, IEEE Trans. Ind. Appl., vol. 45, no. 6, pp , [13] N. Bianchi and T. M. Jahns, Design, Analysis, and Control of Interior PM Synchronous Machines. Padova, Italy: Cleup, [14] S. J. Salon, Finite Element Analysis of Electrical Machines. Norwell, MA, USA: Kluwer, [15] G. H. Jang and J. W. Yoon, Torque and unbalanced magnetic force in a rotational unsymmetric brushless DC motors, IEEE Trans. Magn., vol. 32, no. 5, pp , Sep [16] F. Fiorillo and A. Novikov, An improved approach to power losses in magnetic laminations under nonsinusoidal induction waveform, IEEE Trans. Magn., vol. 26, no. 5, pp , Sep [17] S. D. Garvey, The vibrational behaviour of laminated components in electrical machines, in Proc. Int. Conf. Electr. Mach. Drives, 1989, pp [18] N. E. Dowling, The Mechanical Behavior of Materials, 3rded. Englewood Cliffs, NJ, USA: Prentice-Hall, [19] R. R. Craig, Structural Dynamics: An Introduction to Computer Methods. New York, NY, USA: Wiley, [20] G. H. Jang and S. W. Jeong, Vibration analysis of a rotating system due to the effect of ball bearing waviness, J. Sound Vib., vol. 269, pp , [21] T. A. Harris, Rolling Bearing Analysis, 4th ed. New York, NY, USA: Wiley-Interscience, [22] R. H. Myers and D. C. Montgomery, Response Surface Methodology Process and Product Optimization Using Designed Experiments. New York, NY, USA: Wiley, [23] D. H. Wolpert and W. G. Macready, No free lunch theorems for optimization, IEEE Trans. Evol. Comput., vol. 1, no. 1, pp , [24] F. R. Fulginei and A. Salvini, Comparative analysis between modern heuristics and hybrid algorithms, Int. J. Comput. Math. Electr. Electron. Eng., vol. 26, no. 2, pp , [25] K. J. Hong, M. S. Kim, and D. H. Choi, Efficient approximation method for constructing quadratic response surface model, J. Mech. Sci. Technol., vol. 15, pp , 2001.