Conventional Model

Impact of Degree of Spatial Harmonics

by Calculating Natural Frequency

Prediction of Natural Frequency

(Mathematical Methods)

Calculation of Impact of Degree Factor

of Spatial Harmonics

Exception Spatial Harmonics

with Small Impact from the Analysis

Analysis of Main Vibration Factors

by Separating Vibration Source

Time / Spatial Harmonic Separation

of Radial Force (FEA)

Extraction of Major Vibration Factor Orders

Selection of Design Variable

for Vibration Reduction

Improved Model

• Process of the vibration analysis using spatial harmonics and time harmonics of radial force density

I. Introduction

In this paper, we propose a process of the vibration analysis using spatial harmonics and time harmonics of radial

force density as shown in Fig.. For this purpose, this study suggests the impact of degree factor to limit the order of

spatial harmonics and deduct the component that has a significant influence on vibration through time, spatial

harmonics separation, and calculation of the vibration. Finally, the vibrations of the conventional model and improved

model were analyzed, and the validity of the result was verified through the manufactured prototype and vibration test.

II. Impact of Degree of Spatial Harmonics by Calculating Natural Frequency

Description Symbol Value Unit

Peak value of the phase current Imax 14.5 ApkRotation speed - 5000 rpm

Stator outer diameter Dso 96 mm

Rotor outer diameter Dro 60 mm

Airgap length g 0.5 mm

Stack length Lstk 26.95 mm

A. Specification of Conventional Model

B. Prediction of Natural Frequency and Calculation of Impact of Degree Factor of Spatial Harmonics

( )21Ω

2

11

cc

c

c

r

c

rmr

vρ

E

RπM

K

πf

−=

=

• Circumferential Mode Shape

• Vibration Velocity for r-th mode

• Natural Frequency in r-th mode

( ) ( ) 222 /2/11

mrrnmmrrn

m

ffζff

h

+−

=

2

1

=

mr

mv

f

fk

=

=1n

rnr VV

mrn

mrc

stksirnrn hP

ωM

LDπfπV

22=• Vibration Velocity for r-th space harmonics

and n-th time harmonics

Mode 3Mode 0 Mode 2Mode 1 Mode 5Mode 4

• Magnification Factor

• Vibration Influence Factor

0

10,000

20,000

30,000

40,000

50,000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

Na

tura

l F

req

uen

cy [

Hz]

Circumferential Mode

0

1

2

3

4

5

6

7

8

9

10

0 0.5 1 1.5 2 2.5 3

Ma

gn

ific

ati

on

fa

ctor

fr/fm

ζm = 0.05

ζm = 0.1

ζm = 0.2

ζm = 0.3

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

Vib

rati

on

In

flu

ence

Fa

cto

r

Circumferential Mode

• Cross-section of conventional model

III. Analysis of Main Vibration Factors by Seperating the Vibration Source

A. Time / Spatial Harmonics Separation of Radial Force Density

( ) ( ) ( ) ( )tθBμ

tθBtθBμ

tθp rtr ,2

1,,

2

1, 2

0

22

0

−= ( ) ( )

=

=

=0 0

cos,r n

e

rn θrtωnPtθp

• Radial Force Density

ωt = π/2 ωt = 3π/4ωt = 0 ωt = π/4

θ = 0deg

0

200

400

600

800

0 45 90 135 180 225 270 315 360

Ra

dia

l F

orc

e D

ensi

ty

[kN

/m2]

ωt

0

20

40

60

80

100

120

140

160

180

200

0 4 8 12 16 20 24 28 32 36 40

Ra

dia

l F

orc

e D

ensi

ty [

kN

/m2]

Spatial Harmonics

10fe

8fe

6fe

4fe

2fe

0fe

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0 4 8 12 16 20 24 28 32 36 40

Vib

rati

on

Vel

oci

ty [

m/s

]

Spatial Harmonics

10fe

8fe

6fe

4fe

2fe

0fe

Time Harmonics

2fe 4fe 6fe 8fe 10fe

Spatial

Harmonics4, 8, 16, 20 4, 16, 28 0, 12, 24 4, 8, 20, 32 4, 8, 16

B. Extraction of Major Vibration Factor Orders

C. Improved Design for Vibration Reduction

RroRtap = Rro – dtap

dtap

• Objective Function

1. Minimizing the cogging torque

2. 20% reduction in the electromagnetic vibration

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Total 0fe 2fe 4fe 6fe 8fe 10fe 12fe 14fe 16fe 18fe 20fe 22fe

Vib

rati

on

Vel

oci

ty [

m/s

]

Time Harmonics

• Time / Spatial harmonic separation of the vibration velocity • Vibration velocity according to the vibration frequency

0

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0 1 2 3 4 5 6 7 8 9

Co

gg

ing

To

rqu

e [N

m]

Vib

rati

on

Vel

oci

ty [

m/s

]

dtap [mm]

Vibration @ 5,000rpm

Cogging Torque

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Total 0fe 2fe 4fe 6fe 8fe 10fe 12fe 14fe 16fe 18fe 20fe 22fe

Vib

rati

on

Vel

oci

ty [

m/s

]

Time Harmonics

Conventional Model

Proposed Model

• Cross-section of conventional model

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

4 8 16 20 4 8 16 28 0 12 24 4 8 20 32 4 8 16

Vib

rati

on

Vel

oci

ty [

m/s

]

Time Harmonics

Conventional Model

Proposed Model

2fe 4fe 6fe 8fe 10fe

• Vibration and cogging torque

according to the center point of rotor.

• Vibration of the proposed model and conventional model • Selected Time / Spatial harmonics

of the vibration velocity

• Selected Time / Spatial Harmonics

• Test Condition : 4Nm @ 5,000rpm

In this study, the electromagnetic vibration review process was proposed. Components with a large influence on

vibration were derived through influence factors for limited spatial harmonic dimensions and time-space harmonics

separation and vibration. To verify the proposed process, the design was evaluated to reduce the electromagnetic

vibration by reducing the vibration other than the 2fe component, which constitutes the main frequency. It was verified

through experiments.

The proposed process is not an electromagnetic-structure interlocking analysis but a method to examine the results of

electromagnetic analysis. There is an advantage that vibration can be examined simultaneously in the electromagnetic

optimum design process.

IV. Test and Verification

V. Conclusion

Acceleration sensor

Recorder

Conventional Model Proposed Model Rotor Stator & Housing

Motor

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Vib

rati

on

Vel

oci

ty [

m/s

]

Frequency [Hz]

Conventional Model

Proposed Model

6fe 8fe 10fe 12fe2fe 4fe

• Test results of the proposed model and conventional model

• Manufactured prototypes and test equipment settings

• Test Results

• Comparison of the test results with the proposed method

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

2fe 4fe 6fe 8fe 10fe 12fe 2fe 4fe 6fe 8fe 10fe 12fe

Vib

rati

on

Vel

oci

ty [

m/s

]

Time Harmonics

Conventional Model

Proposed Model

Proposed Method Test Results

Iksang Jang , Won-ho Kim

Design of 8p12s IPMSM for Minimization of Electromagnetic Noise and Vibration