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
