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The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS14.004
Dynamic responses and vibration characteristics for an inclined rotor
with unbalanced magnetic excitation
Xueping Xu1, Qinkai Han, Fulei Chu
Department of Mechanical Engineering, Tsinghua University
Beijing, China
Abstract: The eccentricity is one of the most common
trouble sources in electrical machines. The angular
eccentricity in which case the unequal air-gap length is
related to the axial coordinate was rarely studied. This
paper aims to investigate the vibration characteristics of an
eccentric rotor with both the radial and angular
eccentricity in the three-dimensional space. The air-gap
length of eccentric rotor is derived and the electromagnetic
excitation which consists of the electromagnetic force and
torque is obtained. The gyroscopic effect is taken into
consideration and the dynamic equations of the rotor
system with four degrees of freedom are established. The
electromagnetic excitation, static angular misalignment
are investigated for their effects on the dynamic response in
both the time domain and frequency domain, respectively.
Simulation results illustrate that the electromagnetic
excitation cannot be ignored in the dynamic model. The
axially inclined angle determines the vibration amplitude,
while the mean of steady response depends on the
orientation angle. Keywords: Rotor, Angular eccentricity, Air-gap length,
Electromagnetic excitation, Deflection angle
1. Introduction
Due to an increasing concern about the quality of motor,
the requirements as regards the noise and vibration levels
of the motor are increasingly stringent and wider in scope,
which has been one of the important issues in the design of
three phase synchronous motor [1]. The eccentric rotor
motion of an electrical machine distorts the air-gap field
and produces a net radial force on the rotor. The effect is
referred to as the unbalanced magnetic pull (UMP) [2-3].
The phenomenon of UMP was observed already in the
early twentieth century [4]. The coupling interaction of
UMP and structure may cause unwanted vibrations [5],
give rise to stability problems [6], affect the wear of
bearings [7] and even produce a rub between the rotor and
the stator. The potential hazards is remarkable. Moreover,
if the air-gap magnetic field distortion induces undesirable
unbalanced torque, the matter becomes worse. Therefore,
the electromagnetic excitation generated by the interactions
of the stator and the rotor must be chiefly considered during
the electromagnetic design stage.
The calculation of UMP is essential for the analysis of
vibrations and the optimal design of electrical rotating
machinery. Many methods have been presented in
literatures for calculating the UMP. The two common
approaches are analytical method and the finite element
method (FEM). Although the FEM has been widely applied
to study the UMP [8-10], the analytical method still
[email protected]
receives much attention for the reason that it can provide
an observation into the origins and key factors in the
production of UMP. Earlier publications [11-16] focused
primarily on the theoretical formulation of UMP and the
studies about analytical method in the early stage were
mainly linear equations. Behrend [16] calculated UMP
based on the assumption that UMP was in proportion to
eccentricities. Later on, Covo [17] took the effects of
magnetic saturation into consideration and improved the
linear equations. Calleecharan et al. [18] studied an
industrial hydropower generator and the UMP was
characterized to be a linear spring with negative
electromagnetic stiffness coefficient. Werner [19]
established a dynamic model for an induction motor with
eccentric excitation by taking a radial electromagnetic
stiffness into account. Although linear expressions are
convenient to use, the preciseness is reliable only when the
eccentricity is small enough. Funke et al. [20] drew
attention to the fact that there existed a nonlinear
relationship between UMP and eccentricity. Many
researchers have introduced nonlinear methods to
determine UMP in the last two decades. Smith et al. [21]
studied the UMP by winding analysis and investigated the
effects of the principal harmonic on UMP. Li et al. [22]
adopted the conformal mapping method to calculate UMP
in a slotted permanent magnet motor with rotor eccentricity.
Lundström et al. [23] obtained the UMP which was due to
deviations in generator shape through the law of energy
conservation. Im et al. [24] applied the Ampere’s circuital
law to investigate the magnetic field which was distorted
by the non-uniform air-gap. The most commonly adopted
analytical method for calculating the magnetic flux density
to obtain UMP is the air-gap permeance approach [25-30].
Pennacchi [25-26] presented a model based on the actual
position of the rotor inside the stator to calculate UMP, and
the author broke through the limitation of circular orbits.
Guo et al. [27] obtained an analytical expression of UMP
for different number of pole-pair by expressing the air-gap
permeance as a Fourier series. Many researchers applied
Guo’s results to determine UMP afterward. For instance,
Gustavsson et al. [28] studied the effects of UMP on the
stability of a 70MVA hydro-generator by simplifying the
rotor to be an Euler-Bernoulli beam. Wu et al. [29] analyzed
the stability of a synchronous generator model under UMP
and mass eccentric force. Zhang et al. [30] studied the
nonlinear dynamic characteristics of a rotor-bearing system
with rub-impact for hydraulic generating set under the
UMP.
Although the analytic calculation for UMP has been
extensively investigated in literatures mentioned above,
only the translational eccentricity was taken into
consideration. Almost all the researches dealt with the case
that the rotor is parallel with respect to the stator. In short,
it is the plane problem that has been widely investigated
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and only two translational DOFs (degree of freedom) were
analyzed consequently. However, the rotor angular
misalignment always exists to some extent due to
manufacturing tolerances, wear of bearings, poor
maintenance and shaft bending. This means an unequal air-
gap along the axial direction occurs when the rotor’s
geometric axis is not parallel to that of the stator. Thus, the
degree of eccentricity is not constant in different axial
position and rotational DOFs besides translational DOFs
are required. The rotor system has to be modeled in the
three-dimensional space consequently. This kind of
eccentricity can be defined as inclined eccentricity and
should be treated as a variable circumferential eccentricity.
Moreover, the simultaneous existence of radial static
eccentricity and inclined eccentricity is more probable in
reality. As a result, it is of great importance and significance
to investigate this problem.
The air-gap length is determined not only by the cross
section itself but also by the axial coordinate when the
inclined eccentricity is taken into consideration. The
calculation process of the electromagnetic excitation
becomes complicated consequently. Some scholars
investigated this issue recently. Yu et al. [31] studied the
incline UMP in a permanent magnetic synchronous motor
by numerical simulation method. Ghoggal et al. [32]
proposed an improved method for the modeling of axial
and radial eccentricities in induction motors. Li et al. [33]
conducted the analysis of a three-phase induction machine
with inclined static eccentricity according to the simulation
and experiment results. However, they adopted the
simplified Fourier series method similar to Guo et al. [27]
to calculate the electromagnetic force, which has a great
limit in the accuracy and range of the computing process.
Dorrell [34] put forward a model for assessing UMP due to
rotor eccentricity in cage induction motors which takes
axial variation into consideration. Kelk et al. [35] and Faiz
et al. [36] studied the trapezoidal flux tube between each
stator and rotor teeth and brought forward an expression of
permeance function. However, the electromagnetic torque
was neglected and the investigation on dynamic
characteristics of the rotor system were not covered in the
literature above. Tenhunen et al. [37] dealt with the
combination of radial eccentricity and symmetric inclined
eccentricity based on the hypothesis that the force
distribution has the spatial linearity property. But the actual
electromagnetic force and torque is nonlinear as is known
widely. Therefore, a proper model which is investigated in
the three-dimensional space scope is necessary and the
calculation of nonlinear electromagnetic excitation
including the electromagnetic force and torque with
accuracy is meaningful indeed.
The air-gap length of the eccentric rotor with both the
radial static eccentricity and inclined eccentricity is derived.
And the electromagnetic excitation including the
electromagnetic force and torque is obtained based on the
permeance approach. The dynamic equations of the rotor
system in the three-dimensional space are presented. The
numerical method is adopted to solve the equations and
make the summations. The effects of electromagnetic
excitation are investigated. The static angular misalignment
are analyzed for their effects on the dynamic response in
both the time domain and frequency domain, respectively.
Finally, some conclusions are presented.
2. Dynamic Model of the Rotor System
For the analysis of the electromagnetic excitation acting on
the rotor, the following assumptions are made in this
investigation: (a) The rotor and stator are both perfect
cylinders, which means their axes are straight. (b) The
stator is assumed to be the rigid body in comparison with
the rotor and only the vibration of the rotor is analyzed. (c)
The permeability of the rotor iron and the stator is infinite
and the motor has smooth poles. (d) The effects of leakage
flux, magnetic saturation are neglected. (e) Axial motion is
neglected and only the transverse vibration is investigated.
The four DOFs (two translational and two rotational DOFs)
are discussed in this paper. Let the midpoint of rotor in the
axial direction be the origin of coordinates, and the O-xyz
orthogonal coordinate system is established. The radial
static eccentricity can be decomposed in the x-direction and
y-direction. Fig.1 demonstrates an inclined rotor with four
DOFs in the three-dimensional space. As Fig.2 shows, the
cross section of z=0 is selected to analyze the radial
eccentricity of an inclined rotor in detail.
Rotor Stator
Fig.1 Schematic diagram of an inclined rotor with four DOFs
Fig.2 The radial eccentricity of an inclined rotor
in the Oxy plane
In Figs.1~2, x and y is the deflection angle around
the x-axis and y-axis, respectively. sO is the geometric
center of the stator and O is the initial geometric center of
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the rotor. The linear distance between sO and O is 0r
which is radial static eccentricity usually caused by
installation of rotor. is the direction angle of static
eccentricity. rO is the geometric center of rotor, which is
decided by r and . The linear distance between O and
rO is r which stands for dynamic eccentricity mainly
brought by unbalanced mass distribution of the rotor. It is
assumed that the coordinate of rO is ( , )x y which
represents the location of rotor in the coordinate system and
2 2r x y . is the position angle of rotor with
reference to x-axis and cos x r , sin y r . is
the air-gap angle with respect to x-axis.
As shown in Fig.3, the static angular misalignment is
analyzed in a tapered surface and characterized by two
angles ( and ). is the axially inclined angle around
the z-axis and is the orientation angle in the Oxy plane.
Assuming that the coordinate of point A in the z-axis is (z,
0, 0), and it moves to point A due to static angular
misalignment. We can obtain
( sin cos , sin sin , cos )z z z OA (1)
Fig.3 Static angular misalignment of an inclined rotor
The transformation matrices of rotation in x-axis and y-
axis are
1 0 0
0 cos sin
0 sin cos
ox x x
x x
R
(2)
cos 0 sin
0 1 0
sin 0 cos
y y
oy
y y
R
(3)
The comprehensive transformation matrix is
cos sin sin cos sin
0 cos sin
sin sin cos cos cos
y x y x y
oy ox x x
y x y x y
R R R
(4)
As Fig.4 shows, point A is converted to point A with
the rotor rotating around two axes (x-axis and y-axis). The
new cross section containing point A is parallel to Oxy
plane.
Fig.4 The rotation around two axes and coordinate transformation
The point along the axis of stator corresponding to A is
sO in the cross section after rotation. The coordinate of A
can be obtained as follow:
cos sin sin cos sin sin cos
0 cos sin sin sin
sin sin cos cos cos cos
sin cos cos sin sin sin sin cos cos sin
sin sin cos cos sin
sin sin sin cos
y x y x y
x x
y x y x y
y x y x y
x x
x
z
z
z
z z z
z z
z
cos cos cos sin cos siny x y yz z
(5)
Let A be the coordinate origin and the A x y
coordinate system forms. The displacement of transverse
vibration in the new cross section is the same with the cross
section of z=0, which is based on the assumption that the
rotor is a rigid body. The geometric relationship in vector
form is
0 0( cos , sin ,0)r r sO O (6)
sin cos cos sin sin sin sin cos cos sin
sin sin cos cos sin
sin sin sin cos cos cos cos sin cos sin
y x y x y
x x
x y x y y
z z z
z z
z z z
OA
(7)
0
0
cos sin cos cos sin sin sin sin cos cos sin
sin sin sin cos cos sin
sin sin sin cos cos cos cos sin cos sin
y x y x y
x x
x y x y y
r z z z
r z z
z z z
sO A
(8)
The following equations are introduced here:
0 cos sin cos cos sin sin sin sin
cos cos sin
y x y
x y
X r z z
z
(9)
0 sin sin sin cos cos sinx xY r z z (10)
It is supposed that the vibration of the stator is far smaller
than the rotor and the axis of the stator is stationary. The
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radial static eccentricity is only related to the radial distance
in a certain cross section of z-direction. As shown in Fig.4,
the radial static eccentricity and direction angle will be
refreshed in different cross section.
The radial static eccentricity in any cross section can be
obtained by the projection of A in the Oxy plane and the
expression is
2 20r X Y (11)
And the new direction angle of the corresponding cross
section is
0
0
arccos 0
2 arccos 0
XY
r
XY
r
(12)
The axial position of the cross section is
sin sin sin cos cos cos cos
sin cos sin
x y x y
y
z z z
z
(13)
The air-gap length is a function of the air-gap angle, time
and axial position. The unified air-gap length can be
approximately expressed by the equation as follow
0 0( , , ) cos( ) cos( )t z r r (14)
where 0 is the average air-gap length when the rotor is
centered. Assuming a special case 0x y , and then
the z coordinate is unnecessary. Eq. (14) is reduced to
0 0( , ) cos( ) cos( )t r r consequently. It
is exactly the plane problem that has been widely
investigated [38-40].
The air-gap permeance can be calculated as
0( , , )( , , )
t zt z
(15)
where 0 is the vacuum magnetic permeability.
As shown in Fig.5 ,the resultant fundamental
magnetomotive force (MMF) of air-gap for a synchronous
generator under symmetric load can be expressed as
( , ) cos( )cF t F t p (16)
where 2 2 2 sinc r s r sF F F F F is the amplitude of
the resultant MMFs, arctan[ cos / ( sin )]s r sF F F
is the initial angle of the resultant MMFs, rF and sF are
the amplitudes of the fundamental MMFs of the excitation
current of the rotor and the armature reaction current of the
stator, respectively, is the supply electrical frequency,
p is the number of pole-pair, and is the inner power
factor angle.
Fig.5 The MMF of a synchronous generator
The magnetic flux density distribution of the air-gap is
( , , ) ( , , ) ( , )B t z t z F t (17)
The Maxwell stress on the rotor surface is approximately
expressed as
2
0
( , , )( , , )
2
B t zt z
(18)
The UMP of an infinitesimal element in the x-direction
and y-direction are obtained by integrating the horizontal
and vertical components of the Maxwell stress over the
surface of the rotor.
2
0(z ) ( , , )cosump
xF R dz t z d (19)
2
0(z ) ( , , )sinump
yF R dz t z d (20)
where R is the radius of the rotor.
The resultant UMP can be computed by integrating the
infinitesimal element along the z direction. Substituting Eq.
(13) into Eq. (19) and Eq. (20), we can obtain
2 2
2 0( , , )cos
Lumpx L
F R dz t z d
(21)
2 2
2 0( , , )sin
Lumpy L
F R dz t z d
(22)
where L is the axial length of the air-gap.
The right-hand rule is applied here. The electromagnetic
torque on an infinitesimal element of the cross section
around the x-axis and y-axis positive direction are
respectively as follow:
2
0( ) ( , , )cosump
yM z R z dz t z d (23)
2
0( ) ( , , )sinump
xM z R z dz t z d (24)
Substituting Eq. (13) into Eq. (23) and Eq. (24), and the
resultant electromagnetic torque is obtained by integrating
the infinitesimal element along the z direction.
2 2
2 0( , , )cos
Lumpy L
M R z dz t z d
(25)
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2 2
2 0( , , )sin
Lumpx L
M R z dz t z d
(26)
The unbalanced mass excitation of rotor in the x-
direction and y-direction can be expressed by following
equations:
2 cosexF ma t (27)
2 sineyF ma t (28)
where m is the mass of the rotor, a is the mass
eccentricity of the rotor and is the rotating speed of the
rotor.
A Jeffcott rotor model with four DOFs is adopted in
this paper. The whole rotor system is shown in Fig. 6. The
general case is taken into consideration, which means the
disk is not fixed in the middle of the axis. In the figure, 0L
is the distance between the bearings. L is the distance
between the rotor and the left bearing.
Fig.6 The supporting structure of the rotor system
When the gyroscopic effect is taken into consideration,
the coupling effect between the displacement and
deflection angle is more apparent. The differential
equations of the rotor system are
11 11 14
22 22 23
32 33
41 44
e umpy x x
e umpx y y
umpd x y x x
umpd y x y y
mx c x k x k F F
my c y k y k F F
J H k y k M
J H k x k M
(29)
where ( 1,2)iic i is damping coefficient of the system,
( 1,2,3,4)iik i are the independent stiffness coefficients,
14 41 23 32, , ,k k k k are the coupling stiffness coefficients,
21
4dJ mR is the rotational inertia of rotor, pH J is
the moment of momentum and here 2p dJ J is the polar
rotational inertia. The shaft of the system in this paper is a
cylinder, so the stiffness coefficients satisfy the following
relationships: k11= k22, k33= k44 and k14= k41= k23= k32.
Stiffness coefficients can be calculated by parameters from
the shaft itself and rotor locations.
The dynamic equations can be simplified as
0 0 0
0
M q M q
C M q K q F
(30)
where the equivalent mass matrix, stiffness matrix and
damping matrix are
0 0 0
0 0 0
0 0 0
0 0 0
d
d
m
mM
J
J
,
11 14
22 23
32 33
41 44
0 0
0 0
0 0
0 0
k k
k kK
k k
k k
,
11
22
0 0 0
0 0 0
0 0 0
0 0 0
c
cC
H
H
The equivalent external force and the motion parameters
are respectively as follow:
ump ex x
ump ey y
umpx
umpy
F F
F FF
M
M
,x
y
x
yq
The initial motion parameters are assumed to be zero if
not specially mentioned. By solving Eq. (30), the
simulation results are obtained.
3. Simulation Results and Discussion
The parameters are as follows: m=18.15kg, a=0.5mm,
k11=k22=1.7692×106N/m, c11=c22=81.9Ns/m, ω=50Hz,
k33=k44=1.474×105N/m, Ω=10Hz, k14=k41=k23=k32=-
2.949×105N/m, L0=75cm, Fc=684A, 𝜂�=0, δ0=2.2mm,
R=59mm, μ0=4π×10-7, L=0.1551m. The other parameters
will be provided in specific discussions. The numerical
simulation are conducted by Matlab
3.1 Effects of electromagnetic excitation
The vibration characteristics of the rotor system with and
without electromagnetic excitation are analyzed when other
initial conditions are identical. The vibration behaviors in
x-direction are investigated for simplicity, which is based
on the similarity of motion parameters in the x-direction
and y-direction. The integration step and total simulation
time is set 0.0001s and 4s, respectively. When the static
angular misalignment and radial static eccentricity are
equal zero, the steady-state responses of the displacement
and deflection angle are displayed in Fig.7.
Fig.7 Time history of the displacement and deflection angle
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4-2
-1
0
1
2x 10
-4
t (s)
X (
m)
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4-4
-2
0
2
4x 10
-4
t (s)
x (
rad)
electromagnetic excitation ignored
electromagnetic excitation considered
electromagnetic excitation ignored
electromagnetic excitation considered
Page 6
The displacement and deflection angle illustrate the
feature of simple harmonic motion, and their period for
both of the cases are the same. When the electromagnetic
excitation is taken into account, the vibration amplitude
(displacement and deflection angle) is much bigger than
when the electromagnetic excitation is ignored. It indicates
that electromagnetic excitation will significantly increase
the vibration of the rotor system, which will easily result in
so large amplitude that the rub-impact between rotor and
stator occurs. The effects of electromagnetic excitation
cannot be omitted and should be considered in the process
of dynamic modeling.
The spectra of displacement and deflection angle are as
Fig.8 shows, and the frequency components of them are
basically the same. The spectra of displacement is
discussed if not specially mentioned. For the case that
electromagnetic excitation is ignored, there exist 40.5Hz
and the rotating frequency ( ) brought by unbalanced
mass excitation. One of the natural frequencies of the rotor
system is 40.48Hz by calculating. This frequency is close
to the four times of the rotating frequency. It may be excited
by this reason. The natural frequency is reflected in the
steady response in this case. When electromagnetic
excitation is taken into consideration, the natural
frequencies disappear. 100 3 , 100 and 200 3
besides the rotating frequency ( ) are discovered. They
are results of joint action between the supply electrical
frequency (50Hz) and the rotating frequency ( ).
Electromagnetic excitation is the external excitation source
acting on the rotor system. It will strengthen the feature of
forced vibration, and meanwhile weaken the free vibration.
In addition, the nonlinear factor of the electromagnetic
excitation makes the frequency components vary
complicatedly. The combined frequencies of the supply
electrical frequency and the rotating frequency appear.
These results coincide with the 2-DOF investigations
which has been performed by Guo et al. [27]. Furthermore,
the frequency components of displacement and deflection
angle are almost the same, which means the coupling
effects of bending vibration and rotational vibration really
exist. Therefore, the investigations of a 4-DOF rotor system
are significant and necessary.
Fig.8 The spectra of the displacement and deflection angle
As shown in Fig.9, the rotor shaft orbit is a standard
circle when the electromagnetic excitation is neglected.
However, the orbit expands and finally forms a circle with
petals round its circumference for the other case. The effect
of electromagnetic excitation may certainly aggravate the
vibration of the rotor, which put a danger to the stability and
safety of the rotor system.
Fig.9 Rotor shaft orbit of the system with and without electromagnetic excitation
3.2 Effects of static angular misalignment
The axially inclined angle ( ) and orientation angle ( )
which characterizes static angular misalignment can be
investigated respectively. When the orientation angle is
zero, the time history of displacement and deflection angle
for different axially inclined angles are as Fig.10 shows.
The mean of displacement within a period deviates from
zero in the case of 0 . Moreover, with the increase of
axially inclined angle, the amplitude of displacement
increases nonlinearly. The troughs change little while the
crests vary greatly. The asymmetrical increase of crest and
trough indicate that the air-gap in the x-axis positive
direction is shorter, which needs great attention. However,
the deflection angle increases slightly for different axially
inclined angles. The change of crest keeps pace with the
trough. It can be concluded that the axially inclined angle
mainly have an influence on the displacement. And the
deflection angle alters due to the coupling effects of
displacement and deflection angle.
Fig.10 Time history of displacement and deflection angle for different axially inclined angle
0 50 100 150 200 250 300 350 40010
-15
10-10
10-5
100
Frequency (Hz)
Dis
pla
cem
ent
am
plit
ude (
m)
0 50 100 150 200 250 300 350 40010
-15
10-10
10-5
100
Frequency (Hz)
Angle
am
plit
ude (
rad)
electromagnetic exciation considered
electromagnetic exciation ignored
electromagnetic exciation considered
electromagnetic exciation ignored
200-3
200-3
100-
100-3
100-
100-3
-1.5 -1 -0.5 0 0.5 1 1.5
x 10-4
-1.5
-1
-0.5
0
0.5
1
1.5x 10
-4
x (m)
y (
m)
electromagntic exciation considered
electromagntic exciation ignored
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4-2
0
2
4x 10
-4
t (s)
x (
m)
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
-2
0
2
x 10-4
t (s)
x (
rad)
=0.002 rad, =0
=0.005 rad, =0
=0.008 rad, =0
=0.002 rad, =0
=0.005 rad, =0
=0.008 rad, =0
Page 7
As shown in Fig.11, the shape of rotor shaft orbit for
different axially inclined angle is distinct. If the axially
inclined angle equals zero, the orbit is a center symmetrical
circle with petals around its circumstance. The orbit is
merely a axisymmetric elliptic with petals for
0.002 rad and 0.005 rad . Furthermore, the orbit
for 0.008 rad owns no symmetry. The displacement
increases nonlinearly and the amplitude in the x-direction
increases faster than in the y-direction. The case of
0.008 rad is apparently different from the other two
cases. The amplitude varies severely and the orbit is
geometrically irregular. After several attempts to increase
the axially inclined angle continuously, the unstable state of
the rotor system occurs. It reminds us that the axially
inclined angle cannot exceed a certain range and should be
as small as possible from the perspective of security.
Fig.11 Rotor shaft orbit of the system for different
axially inclined angles
The spectral characteristics of displacement and
deflection angle are displayed in Fig.12. When the axially
inclined angle is 0.002 rad, frequency components consist
of 0, , 2 , 3 , 100 3 , 100 2 , 100 , 100,
100 , 100 2 , 200 3 , 200 2 , 200 and
200. While if the axially inclined angle is 0.005 rad,
frequency component of 4 appears. Moreover, when ,
more frequency components including 5 , 100 4 ,
100 3 and 200 2 are discovered. It can be inferred
that the axially inclined angle will produce some constant
frequencies (0 Hz, 100 Hz and 200 Hz). And with the
increase of the axially inclined angle, the higher multiples
of the rotating frequency are induced. The larger the axially
inclined angle, the more complicated in the spectra.
Fig.12 The spectra of displacement and deflection angle for different axially inclined angles
Not only the axially inclined angle has a great effect on
the vibration characteristics of the rotor system, but also the
orientation angle in the cross section may influence the
vibration behaviors. The effects of the orientation angle on
the time-domain waveform of displacement and deflection
angle are as Fig.13 shows. The tendency of displacement
and deflection angle in the same conditions are apparently
different. The displacement are similar when the orientation
angle is 4 and 7 4 . While the deflection angles are
totally different in this situation. In addition, when the
orientation angle is 4 and 3 4 , the deflection angles is
almost the same. However, the displacement changes a lot.
The other cases can be analyzed similarly. The orientation
angle plays an important role in the response of the rotor
system and should be analyzed specifically.
Fig.13 Time history of displacement and deflection angle for different orientation angles
The rotor shaft orbit for four different orientation angles
( 4,3 4,5 4,7 4 ) are displayed in Fig.14.
When the axially inclined angle remains the same, the
shape and size of the orbit for different orientation angles
-1 -0.5 0 0.5 1 1.5 2 2.5
x 10-4
-1
-0.5
0
0.5
1
1.5x 10
-4
x (m)
y (
m)
=0.002 rad, =0
=0.005 rad, =0
=0.008 rad, =0
0 50 100 150 200 25010
-10
10-8
10-6
10-4
10-2
Frequency (Hz)
Dis
pla
cem
ent
am
plit
ude (
m)
0 50 100 150 200 25010
-10
10-8
10-6
10-4
10-2
Frequency (Hz)
Angle
am
plit
ude (
rad)
=0.002 rad, -=0
=0.005 rad, =0
=0.008 rad, =0
=0.002 rad, =0
=0.005 rad, =0
=0.008 rad, =0
0 2
4
5
100-
100+2
100+3
0
2
4
100
100-
100-3
100-2
100+2
100+3
200-2
200-2
200+2
100-4
3 100-3
100-4
100-2 100
200-3
200200-
100+
3
5200+2
100+
200
200-
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4-2
-1
0
1
2x 10
-4
t (s)
x (
m)
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4-4
-2
0
2
4x 10
-4
t (s)
x (
rad)
=0.005 rad, =/4
=0.005 rad, =3/4
=0.005 rad, =5/4
=0.005 rad, =7/4
=0.005 rad, =/4
=0.005 rad, =3/4
=0.005 rad, =5/4
=0.005 rad, =7/4
Page 8
are identical. However, the location of the orbits are
different. The outer contour of the orbit is an ellipse and the
major axis is in the direction of orientation angle, which
means the rotor is easier to contact the stator in this
direction. It may be induced that the axially inclined angle
determines the vibration amplitude, while the mean of
steady response depends on the orientation angle.
Fig.14 Rotor shaft orbit of the system for different orientation angles
4. Conclusions
The dynamic equation of the rotor in the three-dimensional
Cartesian coordinate system was established. The effects of
electromagnetic excitation were investigated. The static
angular misalignment were analyzed for their effects on the
dynamic response in both the time domain and frequency
domain, respectively. Main conclusions can be summarized
as follows:
1) The electromagnetic excitation can increase the
vibration amplitude of the rotor system and should be taken
into consideration in the dynamic model.
2) The frequency components of the displacement and
deflection angle is the combination of the electrical supply
frequency and the rotating frequency. The electromagnetic
excitation will weaken the free vibration and strengthen the
forced vibration.
3) The axially inclined angle determines the vibration
amplitude, while the mean of steady response depends on
the orientation angle.
Acknowledgments
The research work described in this paper was supported by
the Natural Science Foundation of China (Grant no.
11272170).
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