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Abstract--In this paper, the unbalanced magnetic pull
and eccentric mass force which caused vibration of the rotor
are introduced. The virtual displacement method is
employed to solve the two kinds of force. The inherent
characteristics of rotor are analyzed through finite element
method and experiment. The dynamic response and axis
locus of rotor under the action of unbalanced magnetic pull
and eccentric mass force are studied in detail. The
relationship between the vibration and rotating speed of
rotor is concluded. The theoretical basis for the rotor safe
and reliable operation of the high speed permanent magnet
synchronous machine is provided.
Index Terms-- Unbalanced magnetic pull, eccentric mass
force, dynamic response, high speed permanent magnet
synchronous machine.
I. INTRODUCTION
The high speed permanent magnet synchronous
machine (HS-PMSM) becomes one of the great concerns
in electrical engineering since it has small size, high
efficiency and power density [1-4]. Due to the high
rotating speed, the vibration of rotor becomes a
significant problem. So it is very necessary to study the
vibration sources and the dynamic response of rotor for
its safe and reliable operation.
The unbalanced magnetic pull (UMP) and eccentric
mass force (EMF) will act on the rotor when the rotor of
electrical machine is eccentric. If the frequency of the
two kind forces is equal or close to the rotor natural
frequency, the vibration of rotor will arise. So the two
kind forces become the main source of the rotor
vibration. But now, the study on rotor vibration
characteristics under the UMP and EMF are still not too
much. Such as, Belmans R has studied the UMP in the
two pole induction machines [5], Zhu Z. Q. has studied
the effect of rotor eccentricity and magnetic circuit
saturation on acoustic noise and vibration of single-phase
induction motors [6], and Fruchtenicht has studied the
self-excited transverse vibration of the rotor [7]. In this
paper, the virtual displacement method is employed to
solve the UMP of the eccentric rotor, the dynamic
response and axis locus of rotor under the action of the
UMP and EMF are studied in detail, and the relationship
between the vibration and rotating speed of rotor is
concluded.
This work was supported by National Natural Science Foundation of
China, Project No. (50907064).
II. UNBALANCED MAGNETIC PULL AND ECCENTRIC
MASS FORCE ANALYSIS OF HS-PMSM
A. Unbalanced Magnetic Pull Analysis
There are always two kind eccentricities of rotor,
shown in figure 1. One is static eccentricity which the
stator and rotor center axes are parallel but not overlap.
The other is dynamic eccentricity which the stator and
rotor centers are overlap at static state, but when the rotor
is rotary, the stator and rotor centers are not concentric.
Fig. 1. Eccentricity diagram of rotor
When the rotor is dynamic eccentricity, it will be acted
by UMP and EMF at the same time. But the UMP which
is generated by the dynamic eccentricity is much smaller,
so it can be omitted. In this paper, the UMP that is
generated by the static eccentricity will be calculated
when the rotor’s eccentric distance of Y direction is
0.5mm and a 12 slots, two poles HS-PMSM will be
studied.
Through the virtual displacement method, the UMP of
rotor is calculated and shown in figure 2 and 3.
Fig. 2. X-UMP and spectrum of rotor
Fig. 3. Y-UMP and spectrum of rotor
Research on Vibration Characteristics of Rotor
of High Speed Permanent Magnet Synchronous
Machine
Cheng Fang, Haiping Xu, Shaoshen Xue, Shan Xue
Key Laboratory of Power Electronics and Electric Drive, Institute of Electrical Engineering, Chinese Academy of Sciences,
Haidian District, Beijing 100190, China
It can be seen from the figure, the UMP contains DC
component, 2nd harmonic and many higher harmonic.
These higher harmonic are caused by stator slots and the
influence of higher harmonic on rotor vibration will be
emphatically researched. If the stator is slotless, the UMP
will only contain DC component and 2nd harmonic, and
can be formulated as in (1).
⎩
⎨
⎧
−=
=
tF
tF
y
x
ω
ω
2cos5.00892.48
2sin405.0 (1)
B. Eccentric Mass Force Analysis
If the rotor has eccentric mass, there will be much
higher centrifugal force acting on the rotor of HS-PMSM.
The dynamic balance must be done. The criterion is GB-
T9239 and precision grade is G0.4 of HS-PMSM. So the
EMF of rotor can be calculated as in (2) ~ (4). In this
paper, the rotating speed of rotor is 60000rpm and the
mass of rotor is 9kg.
mkgmmg
n
Geper
83 1037.6/0637.010
2
60 −
×=⋅=
⎟
⎠
⎞
⎜
⎝
⎛×
×
=
π
(2)
mmgmeU perper ⋅== 57.0 (3)
NUF per 5.22
2==
ω
(4)
The formula (2) is the residual unbalance per unit mass
of rotor after dynamic balance. The formula (3) is the
residual unbalance of the whole rotor. And the formula
(4) is the EMF of rotor at 60000rpm. The EMF is a
rotating force wave with uniform amplitude, and it can be
divided into two parts of x direction and y direction,
shown in equation (5).
⎩
⎨
⎧
=
=
tF
tF
y
x
ω
ω
sin5.22
cos5.22
1
1 (5)
III. INHERENT CHARACTERISTICS ANALYSIS
The dynamic analysis of rotor includes two parts, the
inherent characteristics and the dynamic response. The
inherent characteristics contain the natural frequency and
mode shapes of rotor. In this paper, the inherent
characteristics are analyzed through finite element
method and experiment.
Fig.4. Finite element model of rotor
There are two kinds of materials along the radial of
rotor of HS-PMSM that are NdFeB and GH4169. In order
to simulate the actual rotor, three-dimensional models
were established for the rotor modal analysis. It needs to
consider the overall structure and size of the rotor for the
rotor inherent characteristics analysis. In this paper, the
rotor of HS-PMSM includes a solid PM, an enclosure and
both ends of the shaft. The finite element model is shown
in Figure 4. The length of PM is 124mm and total length
of rotor is 410mm. By the finite element method, the
first-order and second-order bending mode and natural
frequencies are calculated, shown in Figure 5.
Fig.5 Simulation result of first and second rotor bending mode
Other natural frequencies of higher modes are shown
in table I.
TABLE I
NATURE FREQUENCY OF EAC H MODE
Mode order Natural frequency (Hz)
First-order oscillating mode 105.03
First-order bending mode 1680.2
Second-order bending mode 2673.1
First-order axial transition mode 4013.2
Third-order bending mode 4537.1
Second-order axial transition mode 6123.7
Fourth-order bending mode 6621
The hammer impact method is used for the mode
experiment. The experiment device schematic diagrams
are shown in the figure 6 and 7. Through the test, the
first, second bending mode and damping ratio are
obtained, and shown in figure 8. Comparing the
experiment and the simulation result, they are very close.
The mode analysis results are also verified correct.
Fig.6. Schematic of rotor Modal experiment
Fig.7. Rotor modal test equipment of HSPM machine
Fig.8 Experiment result of first and second rotor bending mode
IV. DYNAMIC RESPONSE ANALYSIS
The dynamic response is the core part of this paper. It
reflects the vibration of rotor and provides theoretical
basis for the rotor safe operation. In this paper, the
dynamic response of rotor under the action of the UMP
and EMF is studied in detail.
A. Analysis method of the rotor dynamic response
The motion equations of rotor system can be expressed
as formula (6).
fKCM =++
δδδ
(6)
If the external loads are UMP and EMF, the motion
equations of rotor system can be expressed as formula (7).
⎩
⎨
⎧
+=++
+=++
yy
xx
FFyKyCyM
FFxKxCxM
1
1
(7)
Where, M is mass matrix of the whole system, C is
damping matrix and K is stiffness matrix. Fx1 and Fy1 are
the x direction and y direction parts of EMF. Fx and Fy
are the x direction and y direction parts of UMP.
The rotor dynamic response can be calculated by
ANSYS. But factors these in the formula (7) must be
calculated firstly. The mass matrix M and stiffness matrix
K can be calculated through the finite element
geometrical shape and material property. The damping
matrix C is usually calculated by the method of mass and
stiffness factor and the formula is shown as (8), the factor
α and β is a function of damping ratio ζn, shown as (9).
KMC
β
α
+= (8)
22
n
n
n
βω
ω
α
ζ
+= (9)
The damping ratios of first and second bending mode
have already obtained through the mode experiment.
They can be used to calculate the α and β. Considering
the frequency components of UMP and EMF, and decide
to select the frequencies of first-order oscillating mode
and first-order bending mode in the calculation of
damping, the damping ratios is ζ=ζ1=ζ2=0.356%. Final,
the factors are calculated by formula (9), namely,
α=4.422545429 and β=0.000000634.
B. The Results Analysis of Rotor Dynamic Response
Sometime, the stator and rotor centers are not
concentric because of the processing and assembly, and
the eccentric mass of rotor is usually existent. The static
and dynamic eccentricities always exist simultaneously.
So the UMP and EMF will act on the rotor at the same
time when the rotor is rotary with high speed. The figure
9-12 are the vibration displacement and spectrum of x
and y direction when the speeds of rotor are 10000rpm,
20000rpm, 50406rpm and 60000rpm.
(a) Vibration displacement and spectrum of x-direction
(b) Vibration displacement and spectrum of y-direction
Fig. 9 Vibration displacement of rotor under the effect of UMP and
EMF at n=10000rpm
(a) Vibration displacement and spectrum of x-direction
(b) Vibration displacement and spectrum of y-direction
Fig. 10 Vibration displacement of rotor under the effect of UMP and
EMF at n=20000rpm
(a) Vibration displacement and spectrum of x-direction
b) Vibration displacement and spectrum of y-direction
Fig. 11 Vibration displacement of rotor under the effect of UMP and
EMF at n=50406rpm
(a) Vibration displacement and spectrum of x-direction
(b) Vibration displacement and spectrum of y-direction
Fig. 12 Vibration displacement of rotor under the effect of UMP and
EMF at n=60000rpm
The rotor is raised along the y direction by the DC
component of UMP. The main source of rotor vibration is
the fundamental frequency vibration caused by the EMF.
When the speeds of rotor are 10000rpm and 50406rpm,
the 10th and 2nd harmonic of UMP cause the rotor first
bending resonance respectively, and the vibration
response of the corresponding harmonic is more apparent.
In this paper, the frequencies of first-order oscillating
mode and first-order bending mode of rotor are 105.03Hz
and 1680.2Hz. The UMP contains 2nd and many higher
harmonic. It is obvious that only the speed of rotor is
lower, these higher harmonic of UMP may be closer to
the frequency of first-order oscillating mode and first-
order bending mode, and will make the rotor resonance.
Along with the increase of rotor speed, the higher
harmonics of UMP will not cause the first-order
oscillating and first-order bending resonance. But much
higher order bending resonances may be caused. The
frequencies of these higher order bending resonances are
usually very high, and amplitudes are small. So they are
often ignored.
The loci of rotor center under the action of UMP and
EMF at different rotate speed are shown in figure 13-16.
Fig. 13 Locus of rotor center Fig. 14 Locus of rotor center
at n=10000r/min at n=20000r/min
Fig. 15 Locus of rotor center Fig. 16 Locus of rotor center
at n=50406r/min at n=60000r/min
With the rotor speed increases, the amplitude of EMF
is also on the increase, but the rotor response caused by
the EMF is to reduce. The amplitude of UMP remains
unchanged, and the frequency increases when the rotor
speed increases. So the response caused by the UMP
reduces faster than the response caused by the EMF, and
the latter will be the major component at high speed. The
locus of rotor center will be from the circle with many
petals to the ellipse, and circle in the end. When the
speeds of rotor are 10000rpm and 50406rpm, the locus of
rotor center was irregular curve because of the resonance.
The vibration trend is to reduce along with the increase of
speed.
V. CONCLUSIONS
The UMP and EMF which cause vibration of the rotor
are calculated. The UMP contains DC component, 2nd
harmonic and many higher harmonic. The EMF is a
rotating force wave with uniform amplitude, and only
contains fundamental wave.
The inherent characteristics of rotor are analyzed
through finite element method and experiment. Through
the experiment, the first, second bending modes and
damping ratios are obtained. They can be used to
calculate the damping matrix C for the dynamic response
analysis.
The dynamic response and axis locus of rotor under the
action of UMP and EMF are studied in detail. The main
source of rotor vibration is the fundamental frequency
vibration caused by the EMF. When the speed of rotor is
lower, the vibration response caused by UMP will be
more apparent. With the rotor speed increases, the locus
of rotor center will be from the circle with many petals to
the ellipse, and circle in the end. The vibration trend is to
reduce along with the increase of speed.
ACKNOWLEDGMENT
This paper’s work has been supported by National
Natural Science Foundation of China, Project No.
(50907064). Please allow the authors to express our
cordial acknowledgment here.
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