The phasing effect of the slot/magnet combination on rigid-elastic vibration is addressed by incorporating the cyclic symmetry of permanent magnet (PM) motors. Expanding research is also carried out to achieve more general findings in rotary power-transmission systems widely available in practical engineering. To these aims, model-free analysis is used to deal with the effect via superposition treatment. The results imply that the vibration induced by temporal-spatial excitation can be classified into rotational, translational, and balanced modes, all of which have rigid and elastic vibrations having specific base and/or contaminated deflections, and the elastic vibration can be of the standing, forward traveling, and backward traveling waves. These modes can be suppressed or excited depending on whether particular algebraic relationships are satisfied by slot/magnet combination, excitation order, and base and contaminated wave numbers. Since the analysis is independent of any models, specified magnetic force, and rigid-elastic vibration, analytical results regarding the expected relationships can be naturally created due to the structural and force symmetries of the PM motors. Because of this, similar results can be found for other rotary systems basically consisting of a rotary rotor and a stationary stator both having equally-spaced features, apart from the PM motors, typically including the turbine machines having fluid field and planetary gears with a mechanical contact. As an engineering application, the proposed method can serve as a fundamental tool when predicting or even suppressing the possible excitations associated with particular vibration modes in the mechanical and electrical designs of the symmetric systems. The superposition effect and analytical predictions are verified by the finite element method and strict comparisons against those from disk-shaped structures in an existing study.

References

References
1.
Huang
,
B.
, and
Hartman
,
A.
,
1997
, “
High Speed Ten Pole/Twelve Slot DC Brushless Motor With Minimized Net Radial Force and Low Cogging Torque
,” U.S. Patent No. 5,675,196.
2.
Tang
,
R. Y.
,
1997
,
Modern Permanent Magnet Machines: Theory and Design
,
Mechanical Industry Press
,
Beijing
.
3.
Hwang
,
S. M.
,
Eom
,
J. B.
,
Jung
,
Y. H.
,
Lee
,
D. W.
, and
Kang
,
B. S.
,
2001
, “
Various Design Techniques to Reduce Cogging Torque by Controlling Energy Variation in Permanent Magnet Motors
,”
IEEE Trans. Magn.
,
37
(
4
), pp.
2806
2809
.10.1109/20.951313
4.
Williams
,
M. M.
,
Zariphopoulos
,
G.
, and
Macleod
,
D. J.
,
1994
, “
Performance Characteristics of Brushless Motor Slot/Pole Configurations
,”
Incremental Motion Control Systems and Devices Symposium (IMCSD 94)
,
San Jose, CA
, June 14–16, pp.
145
153
.
5.
Zhu
,
Z. Q.
, and
Howe
,
D.
,
2000
, “
Influence of Design Parameters on Cogging Torque in Permanent Magnet Machines
,”
IEEE Trans. Energy Convers.
,
15
(
4
), pp.
407
412
.10.1109/60.900501
6.
Hanselman
,
D. C.
,
1997
, “
Effect of Skew, Pole Count and Slot Count on Brushless Motor Radial Force, Cogging Torque and Back EMF
,”
IEE Proc.: Electr. Power Appl.
,
144
(
5
), pp.
325
330
.10.1049/ip-epa:19971205
7.
Hwang
,
C. C.
,
Wu
,
M. H.
, and
Cheng
,
S. P.
,
2006
, “
Influence of Pole and Slot Combinations on Cogging Torque in Fractional Slot PM Motors
,”
J. Magn. Magn. Mater.
,
304
(
1
), pp.
e430
e432
.10.1016/j.jmmm.2006.01.207
8.
Chen
,
S. X.
,
Low
,
T. S.
,
Lin
,
H.
, and
Liu
,
Z. J.
,
1996
, “
Design Trends of Spindle Motors for High Performance Hard Disk Drives
,”
IEEE Trans. Magn.
,
32
(
5
), pp.
3848
3850
.10.1109/20.539193
9.
Bi
,
C.
,
Jiang
,
Q.
, and
Lin
,
S.
,
2005
, “
Unbalanced-Magnetic-Pull Induced by the EM Structure of PM Spindle Motor
,”
8th International Conference on Electrical Machines and Systems
(
ICEMS 2005
), Nanjing, China, September 27–29, pp.
183
187
.10.1109/ICEMS.2005.202509
10.
Song
,
Z. H.
,
Han
,
X. Y.
,
Chen
,
L. X.
, and
Tan
,
R. Y.
,
2007
, “
Different Slot/Pole Combination Vibro-Acoustics of Permanent Magnet Synchronous Motor
,”
Micro Motor
,
40
(
12
), pp.
11
14
.10.3969/j.issn.1001-6848.2007.12.004
11.
Huo
,
M. N.
,
Wang
,
S. Y.
,
Xiu
,
J.
, and
Cao
,
S. Q.
,
2013
, “
Effect of Magnet/Slot Combination on Triple-Frequency Magnetic Force and Vibration of Permanent Magnet Motors
,”
J. Sound Vib.
,
332
(
22
), pp.
5965
5980
.10.1016/j.jsv.2013.05.022
12.
Rahman
,
B. S.
, and
Lieu
,
D. K.
,
1991
, “
The Origin of Permanent Magnet Induced Vibration in Electric Machines
,”
ASME J. Vibr. Acoust.
,
113
(
4
), pp.
476
481
.10.1115/1.2930211
13.
Chen
,
Y. S.
,
Zhu
,
Z. Q.
, and
Howe
,
D.
,
2006
, “
Vibration of PM Brushless Machines Having a Fractional Number of Slots Per Pole
,”
IEEE Trans. Magn.
,
42
(
10
), pp.
3395
3397
.10.1109/TMAG.2006.879072
14.
Wang
,
S. Y.
,
Xu
,
J. Y.
,
Xiu
,
J.
,
Liu
,
J. P.
,
Zhang
,
C.
, and
Yang
,
Y. H.
,
2011
, “
Elastic Wave Suppression of Permanent Magnet Motors by Pole/Slot Combination
,”
ASME J. Vibr. Acoust.
,
133
(
2
), p.
024501
.10.1115/1.4002954
15.
Jang
,
G. H.
, and
Lieu
,
D. K.
,
1991
, “
The Effect of Magnet Geometry on Electric Motor Vibration
,”
IEEE Trans. Magn.
,
27
(
6
), pp.
5202
5204
.10.1109/20.278787
16.
Chang
,
S. C.
, and
Yacamini
,
R.
,
1996
, “
Experimental Study of the Vibrational Behaviour of Machine Stators
,”
IEE Proc.: Electr. Power Appl.
,
143
(
3
), pp.
242
250
.10.1049/ip-epa:19960184
17.
Long
,
S. A.
,
Zhu
,
Z. Q.
, and
Howe
,
D.
,
2001
, “
Vibration Behavior of Stators of Switched Reluctance Motors
,”
IEE Proc.: Electr. Power Appl.
,
148
(
3
), pp.
257
264
.10.1049/ip-epa:20010255
18.
Tseng
,
J. G.
, and
Wickert
,
J. A.
,
1994
, “
On the Vibration of Bolted Plate and Flange Assemblies
,”
ASME J. Vibr. Acoust.
,
116
(
4
), pp.
469
473
.10.1115/1.2930450
19.
Kim
,
M.
,
Moon
,
J.
, and
Wickert
,
J. A.
,
2000
, “
Spatial Modulation of Repeated Vibration Modes in Rotationally Periodic Structures
,”
ASME J. Vibr. Acoust.
,
122
(
1
), pp.
62
68
.10.1115/1.568443
20.
Chang
,
J. Y.
, and
Wickert
,
J. A.
,
2000
, “
Response of Modulated Doublet Modes to Travelling Wave Excitation
,”
J. Sound Vib.
,
242
(
1
), pp.
69
83
.10.1006/jsvi.2000.3363
21.
Chang
,
J. Y.
, and
Wickert
,
J. A.
,
2002
, “
Measurement and Analysis of Modulated Doublet Mode Response in Mock Bladed Disks
,”
J. Sound Vib.
,
250
(
3
), pp.
379
400
.10.1006/jsvi.2001.3942
22.
Wu
,
X. H.
, and
Parker
,
R. G.
,
2006
, “
Vibration of Rings on a General Elastic Foundation
,”
J. Sound Vib.
,
295
(
1–2
), pp.
194
213
.10.1016/j.jsv.2006.01.007
23.
Kim
,
H.
, and
Shen
,
I. Y.
,
2009
, “
Ground-Based Vibration Response of a Spinning, Cyclic, Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects
,”
ASME J. Vibr. Acoust.
,
131
(
2
), p.
021007
.10.1115/1.3025847
24.
Kim
,
H.
,
Khalid
,
N. T.
, and
Shen
,
I. Y.
,
2009
, “
Mode Evolution of Cyclic Symmetric Rotors Assembled to Flexible Bearings and Housing
,”
ASME J. Vibr. Acoust.
,
131
(
5
), p.
051008
.10.1115/1.3147167
25.
Wang
,
S. Y.
,
Xiu
,
J.
,
Xu
,
J. Y.
,
Liu
,
J. P.
, and
Shen
,
Z. G.
,
2010
, “
Prediction and Suppression of Inconsistent Natural Frequency and Mode Coupling of a Cylindrical Ultrasonic Stator
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
,
224
(
9
), pp.
1853
1861
.10.1243/09544062JMES1993
26.
Kumar
,
A.
, and
Krousgrill
,
C. M.
,
2012
, “
Mode-Splitting and Quasi-Degeneracies in Circular Plate Vibration Problems: The Example of Free Vibrations of the Stator of a Traveling Wave Ultrasonic Motor
,”
J. Sound Vib.
,
331
(
26
), pp.
5788
5802
.10.1016/j.jsv.2012.07.032
27.
Parker
,
R. G.
,
2000
, “
A Physical Explanation for the Effectiveness of Planet Phasing to Suppress Planetary Gear Vibration
,”
J. Sound Vib.
,
236
(
4
), pp.
561
573
.10.1006/jsvi.1999.2859
28.
Wang
,
S. Y.
,
Huo
,
M. N.
,
Zhang
,
C.
,
Liu
,
J. P.
,
Song
,
Y. M.
,
Cao
,
S. Q.
, and
Yang
,
Y. H.
,
2011
, “
Effect of Mesh Phase on Wave Vibration of Spur Planetary Ring Gear
,”
Eur. J. Mech. A/Solids
,
30
(
6
), pp.
820
827
.10.1016/j.euromechsol.2011.06.004
29.
Ouyang
,
H. J.
,
2011
, “
Moving-Load Dynamic Problems: A Tutorial (With a Brief Overview)
,”
Mech. Syst. Signal Process.
,
25
(
6
), pp.
2039
2060
.10.1016/j.ymssp.2010.12.010
30.
Chen
,
D. L.
,
Wang
,
S. Y.
,
Xiu
,
J.
, and
Liu
,
J. P.
,
2011
, “
Physical Explanation on Rotational Vibration Via Distorted Force Field of Multi-Cyclic Symmetric Systems
,”
13th World Congress in Mechanism and Machine Science (IFToMM’11
),
Guanajuato, Mexico
, June 19–25.
31.
Kahraman
,
A.
,
1994
, “
Natural Modes of Planetary Gear Trains
,”
J. Sound Vib.
,
173
(
1
), pp.
125
130
.10.1006/jsvi.1994.1222
32.
Nicolet
,
C.
,
Ruchonnet
,
N.
, and
Avellan
,
F.
,
2006
, “
One-Dimensional Modeling of Rotor Stator Interaction in Francis Pump-Turbine
,”
23rd IAHR Symposium on Hydraulic Machinery and Systems
,
Yokohama, Japan
, October 17–21.
33.
Dai
,
J. C.
,
Hu
,
Y. P.
,
Liu
,
D. S.
, and
Long
,
X.
,
2011
, “
Modelling and Characteristics Analysis of the Pitch System of Large Scale Wind Turbines
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
,
225
(
3
), pp.
558
567
.10.1243/09544062JMES2257
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