resonance characteristics of an electric power steering motor

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APSAEM12 Journal of the Japan Society of Applied Electromagnetics and Mechanics Vol.21, No.3 (2013)

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Resonance Characteristics of an Electric Power Steering Motor

Noboru NIGUCHI*1, Katsuhiro HIRATA*1 and Teruaki KONDO*1

This paper describes the resonance characteristics of two types of commonly-used electric power steering motors: the 8-pole-12-slot and the 10-pole-12-slot motor. The relationship between the excitation modes and characteristic vibrational modes is clarified by conducting a coupled magnetic field - structural analysis and carrying out a modal analysis on a prototype. Finally, the 8-pole-12-slot and 10-pole-12-slot motors are compared in terms of their gener-ated noise.

Keywords: resonance characteristics, excitation mode, characteristic vibrational mode, coupled analysis. (Received: 31 May 2012, Revised: 14 June 2013)

1. Introduction

Recently, the noise reduction of motors in automo-biles, especially electric power steering motors (EPS motors), have been gathering attention. The EPS system shown in Fig. 1 reduces fuel consumption because EPS motors operate only when the steering wheel is turned. In comparison, conventional hydraulic power steering systems constantly draws power from the engine. Therefore, vehicles installed with EPS systems are increasing all over the world. However, the close proximity in which the motor is installed to the driver means that the driver experiences an undesirable amount of noise � especially nowadays where hybrid and electric vehicles that employ motors instead of engines are increasing.

One of the causes of the noise is electromagnetic vi-bration. The stator vibrates due to the radial electromag-netic force caused by the attraction and repulsion force between permanent magnets and electromagnets [1-3]. In particular, the electromagnetic vibration increases at characteristic frequencies. However, it is difficult to avoid operation at characteristic frequencies using design technique because the EPS motors operates continuously from low speeds to high speeds.

In a previous study on resonance characteristics, the behavior of the stator of an induction motor was clari-fied by vibrating the stator with shakers [4, 5]. The following observations were noted: the stator resonates at all characteristic frequencies when using one-point excitation but does not resonate at the 3rd characteristic vibrational frequency when using two-point excitation.

In keeping with the observations of the above men-tioned previous study, this paper describes resonance characteristics of two types of commonly-used EPS motors: the 8-pole-12-slot and the 10-pole-12-slot

motors. The relationship between the excitation modes and characteristic vibrational modes is clarified by conducting a coupled magnetic field - structural analysis using 3-D finite element method (3D- FEM) and carry-ing out a modal analysis on a prototype. Finally, the 8-pole-12-slot and 10-pole-12-slot motors are compared in terms of their vibration.

2. Analysis Model and Method

2.1 Analysis Model

Analysis models of the 8-pole-12-slot and 10-pole-12-slot motors in this study are shown in Fig. 2. The same stator is used for both motors, and the major

Motor

Steering wheel

ECU

Tire

Tire

Fig. 1. Electric power steering system.

Stator

Rotor

Permanentmagnet

Stator

(a) 8-pole-12-slot motor (b) 10-pole-12-slot motor Fig. 2. Analysis models.

_______________________ Correspondence: N. NIGUCHI, Department of Adaptive

Machine Systems, Graduate School of Engineering, OsakaUniversity, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan Email: [email protected]

*1 Osaka University

Regular Paper

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日本 AEM 学会誌　Vol. 21, No.3 (2013)

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Table 1 Major dimensionsOuter diameter of the rotor 38.8Outer diameter of the stator 80.0

Axial length of the rotor 42.5Axial length of the stator 42.5

Unit: mm

Electromagnetic force

Magnetic flux density

Magnetic field analysis

Nodal force method

Frequency domain analysis

Characteristicfrequency

Modal analysis

Fig. 3. Flowchart of the coupled analysis.

dimensions are shown in Table 1. The coils are not wound because the vibrational characteristics only under driven condition is considered.

The stator and rotor yoke are formed by laminated silicon steel sheets 50JN470, and permanent magnets (Br = 1.24 T) are used. In this study, the motors are formed only by the rotor yoke, permanent magnets, and stator. Therefore, the case, bracket, shaft, etc. are removed.

2.2 Analysis Method

In order to calculate the resonance characteristics, a coupled analysis shown in Fig. 3 is employed. In the magnetic field analysis, 3-D FEM analysis using Eq. (1) is conducted.

M� rot)rot(rot 0�� (1) where � and �0 are the reluctivity, A is the magnetic vector potential, and M is the magnetization. One electromagnetic force period generated in the stator is calculated by using the nodal force method.

On the other hand, in the modal analysis, Eq. (2) is employed.

0)( 2 �� ukm� (2) where m is the mass matrix, k is the stiffness matrix, uis the displacement, and � is the frequency.

Finally, the frequency domain analysis using one electromagnetic force period is conducted:

Fukcm �� )( 2 �� j (3) where c is the damping matrix, and F is the electromag-netic force vector.

3. Fundamental Excitation Mode

The rated rotation speed of the EPS motor in this study is 1200 rpm. In order to calculate the excitation mode of the stator due to the rotor, the rotor is driven at a speed of 1200 rpm. The frequencies of the radial electromagnetic force of the 8-pole-12-slot and 10-pole-12-slot motors are shown in Figs. 4 and 5, respectively,

and the electromagnetic force distributions at the center of the teeth facing the air gap are shown in Figs.6 and 7, respectively.

From Figs. 4, 5, 6, and 7, the fundamental excitation modes of the 8-pole-12-slot and 10-pole-12-slot motors are square (4-point excitation mode) and ellipsoidal (2-point excitation mode), respectively, which are the maximum common divisor of the pole and slot number,

05

101520253035404550

1 2 3 4 5El

ectro

mag

netic

forc

e (N

)Frequency of the electromagnetic force (Hz)

160 320 480 640 800

Fig. 4. Frequency of the electromagnetic force (8-pole-12-slot motor).

0123456789

1 2 3 4 5

Elec

trom

agne

tic fo

rce

(N)

Frequency of the electromagnetic force

200 400 600 800 1000

Fig. 5. Frequency of the electromagnetic force (10-pole-12-slot motor).

0100200300400500

1

2

3

4

5

6

7

8

9

10

11

12 Tooth number

Radial electromagnetic force

(N)

Fig. 6. Excitation mode of the 8-pole-12-slot motor.


403(71)

0

50

100

150

2001

2

3

4

5

6

7

8

9

10

11

12Tooth number

Radial electromagnetic force

(N)

Fig. 7. Excitation mode of the 10-pole-12-slot motor.

and their frequencies are 160 and 200 Hz, which are equal to twice the frequency of the driving current.

4. Modal Analysis

A modal analysis of the stator was conducted. The computed characteristic frequencies and mode shapes are shown in Fig. 8. If we extrapolate Figs. 4 and 5, we can see that the ellipsoidal (1062 Hz) characteristic vibrational mode exists in between the 24th (960 Hz) and 28th (1120 Hz) excitation modes for the 8-pole-12-slot motor, and 10th (1000 Hz) and 12th (1200 Hz) for the 10-pole-12-slot motor, respectively. Therefore, it is clear that the 1062-Hz electromagnetic force does not exist in the electromagnetic force of both motors. This becomes a problem in our study since it becomes clear that the stator cannot be excited at its ellipsoidal charac-teristic frequency with the data form the previous section.

However, in FEM analysis, for study purpose, it is possible to artificially generate the 1062-Hz electro-magnetic force. The amplitude is obtained by linearly interpolating the amplitudes of the adjacent frequencies, which are 960 and 1120 Hz for the 8-pole-12-slot motor and 1000 and 1200 Hz for the 8-pole-12-slot motor.

Frequency domain analyses of the 8-pole-12-slot and 10-pole-12-slot motors were conducted at the character-istic frequency of 1062 Hz, and also at arbitrarily chosen, non-characteristic frequency of 962 and 1162 Hz.

The deformation modes of the 8-pole-12-slot motor (equivalent to 4-point excitation) are shown in Fig. 9. At the non-characteristic vibrational frequencies, almost no deformation can be observed, but a large ellipsoidal deformation was observed at the characteristic frequen-cy.

Results obtained from the 10-pole-12-slot motor (equivalent to 2-point excitation) were qualitatively the same. Almost no deformation could be observed at non-characteristic frequencies, but at 1062 Hz, the defor-mation was also ellipsoidal.

1062 Hz 2678 Hz

2805 Hz 4409 HzFig. 8. Characteristic vibrational mode of the stator.

962 Hz

1062 Hz

1162 Hz

Displacement(m)

5E�9

�5E�9

Fig. 9. Deformation modes of the 8-pole-12-slot motor.

Incidentally, the frequency domain analyses were also conducted at the other characteristic frequencies, and the deformation modes were equal to the character-istic vibrational mode. Therefore, we can deduce that the deformation modes at characteristic frequencies are dominated by the characteristic vibrational mode, and it does not depend on the excitation mode.

5. Experimental Modal Analysis

In order to verify the characteristic frequencies and characteristic vibrational modes, an experimental modal analysis was carried out.

First, the modal characteristic frequencies were measured by a hammering test. The frequency response function is shown in Fig. 10 along with the computed

404


(72)

characteristic frequencies. The corresponding computed characteristic frequencies of 1062, 2678, and 4409 Hz are observed. However, the 2805-Hz characteristic frequency is not observed. This is because the deforma-tion is axial and the analysis model does not consider the laminations and YAG welding which the prototype stator employs.

Next, the characteristic vibrational modes will be verified through an experimental modal analysis. The setup is shown in Fig. 11. A 2-, 3-, and 4-point excita-tions from the outside of the stator can be given simul-

-20

-10

0

10

20

30

40

50

60

0 1000 2000 3000 4000 5000

Acc

eler

ance

(dB

)

Frequency (Hz)

1062 Hz 2678 Hz 4409 Hz

Measured

Computed

2805 Hz

Fig. 10. Measured frequency response function.

Shaker

Shaker

Stator

Accelerationpickup

Fig. 11. Experimental setup.

-180

-90

0

90

180

Base phase Measured phase(deg)

Fig. 12. Ellipsoidal mode (1062 Hz).

-180

-90

0

90

180

Base phase Measured phase

Fig. 13. Triangle mode (2678 Hz).

-180

-90

0

90

180

Base phase Measured phase(deg)

Fig. 14. Square mode (4409 Hz).

taneously from even angular intervals. In this measure-ment, only the 2-point excitation is used. Two accelera-tion pick-ups are employed: one is fixed to the stator and the other measures a point on the stator every 30 deg. The measured characteristic vibrational modes can be obtained by calculating the phase difference between the two acceleration pick-ups and are shown in Figs. 12, 13, and 14. From these graphs, the computed character-istic vibrational modes are verified.

6. Excitation Mode and Characteristic Vibrational Mode

In order to clarify the relationship between the exci-tation mode and characteristic vibrational mode, an experiment using the setup shown in Fig. 11 was con-ducted. The 2-, 3-, and 4-point excitations with the same phase were given to the stator at a frequency of 1062, 2678, and 4409 Hz. The corresponding accelerations are shown in Figs. 15, 16, and 17, respectively. From these graphs, resonance could be observed when the excita-tion mode is equal to the characteristic vibrational mode.

However, the accelerations due to the 2- and 4-point excitations are higher than that due to the 3-point excitation at 4409 Hz. This is because the deformation direction corresponds to the excitation direction as


405(73)

shown in Fig. 18. On the other hand, the acceleration due to the 4-point acceleration is lower at 1062 Hz. This is because the deformation is suppressed by the excita-tion as shown in Fig. 19.

From these graphs, resonance can be observed when the characteristic vibrational mode is an integer multiple of the excitation mode. In mathematical terms, reso-nance can be observed when the characteristic vibra-tional mode equals m×N, where m is a positive integer and N is the excitation mode.

The fundamental excitation mode is decided by the greatest common divisor between the number of poles and slots. Therefore, the excitation mode should be selected so that it does not correspond to the characteris-tic vibrational mode and its divisor. For example, 6-pole-9-slot and 8-pole-12-slot motors do not have a 2-point excitation mode. Therefore, the vibration is small at ellipsoidal characteristic vibrational mode.

0

50

100

150

200

250

300

0 60 120 180 240 300 360

Acc

eler

atio

n (m

/s2 )

Measured position (deg)

Two-point excitationThree-point excitationFour-point excitation

Fig. 15. Measured acceleration at 1062 Hz.

0

5

10

15

20

25

30

35

0 60 120 180 240 300 360

Acc

eler

atio

n (m

/s2 )




0

5

10

15

20

25

0 60 120 180 240 300 360

Acc

eler

atio

n (m

/s2 )




Direction of excitation Direction of deformation

Fig. 18. 2-point and 4-point excitation modes for a square characteristic vibrational mode.

Direction of excitation Direction of deformation

Fig. 19. 2-point and 4-point excitation modes for an ellipsoidal characteristic vibrational mode.

7. Comparison of 8-Pole-12-Slot and 10-Pole-12-Slot Motor

In order to verify the measured relationship between the excitation mode and characteristic vibrational mode, magnetic field analyses where the rotation speeds of 7965 and 6372 rpm were given to the rotors of the 8-pole-12-slot and 10-pole-12-slot motors, respectively, were conducted. When the rotors of the 8-pole-12-slot and 10-pole-12-slot motor are rotated at 7965 and 6372 rpm, the fundamental excitation mode frequency of both

406


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3E�5

�3E�5

8-pole-12-slot motor 10-pole-12-slot motor

Enlarged

Displacement(m)

3E�3

�3E�3

Displacement(m)

Fig. 20. Deformation at 1062 Hz.

the motors are 1062 Hz, which corresponds to the ellipsoidal characteristic frequency.

It is observed that the deformation of the 10-pole-12-slot motor is much larger than the deformation of the 8-pole-12-slot motor. This verified the above mentioned relationship between the excitation mode and character-istic vibrational mode.

8. Conclusion

This paper described the resonance characteristics due to the electromagnetic force by conducting a cou-pled magnetic field - structural analysis and carrying out measurements on a prototype. The deformation mode at characteristic frequencies was dominated by the charac-teristic vibrational mode.

The relationship between the excitation mode and characteristic vibrational mode was clarified. Resonance could be observed when the characteristic vibrational mode was an integer multiple of the excitation mode.

Finally, the 8-pole-12-slot and 10-pole-12-slot mo-tors were compared in terms of the deformation during the ellipsoidal vibrational mode. The deformation of the 10-pole-12-slot motor was much larger than the 8-pole-12-slot motor.

In future works, the characteristic vibrational mode of the motor, which includes the rotor, coil, case and so on, will be investigated.

References

[1] T. Sun, J. Kim, G. Lee, J. Hong and M. Choi, “Effect of Pole and Slot Combination on Noise and Vibration in Permanent Magnet Synchronous Motor,” IEEE. Trans.Magn., Vol. 47, No. 5, pp. 1038-1041, 2011.

[2] M. Islam, R. Islam and T. Sebastian, “Noise and Vibra-tion Characteristics of Permanent Magnet Synchronous Motors Using Electromagnetic and Structural Analyses,”Proc. Energy Conversion Congress and Exposition 2011,pp. 3399-3405, 2011.

[3] R. Islam and I. Husain, “Analytical Model for Predicting Noise and Vibration in Permanent-Magnet Synchronous Motors,” IEEE. Trans. Indust. Appl., Vol. 46, No. 6, pp. 2346-2354, 2010.

[4] S. Noda, F. Ishibashi and K. Ide, “Vibration Response Analysis of Induction Motor Stator Core (Vibration Re-sponse of Distributed Excitation and Multipoint Excita-tion),” J. Jpn. Soc. Mechanical Eng., Vol. 59, No. 562, pp. 1650-1656, 1993, (in Japanese).

[5] S. Noda, L. C. Siong and S. Mizuno, “Natural Frequency Analysis of Motor Laminated Core,” Proc. Dynamics & Design Conf. 2008, No. 615, 2008, (in Japanese).

resonance characteristics of an electric power steering motor

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