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

1xxp13@mails.tsinghua.edu.cn

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

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

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

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)

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

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

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

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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