Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 2, pp. 393-406, Warsaw 2017
DOI: 10.15632/jtam-pl.55.2.393

FAULTS DETECTION IN GAS TURBINE ROTOR USING VIBRATION

ANALYSIS UNDER VARYING CONDITIONS

Benrabeh Djaidir, Ahmed Hafaifa, Abdallah Kouzou

AppliedAutomation and Industrial Diagnostics Laboratory, Faculty of Science andTechnology,University ofDjelfa,Algeria

e-mail: b.djaidir@univ.djelfa.dz; hafaifa.ahmed.dz@ieee.org; kouzouabdellah@ieee.org

Monitoring of rotating machines is a very important task in most industrial sectors, which
requires a chosen number of performance indicators during the exploitation of such kind
of equipments. Indeed, for understanding the undesirable phenomena complexity of the
industrial systems under operation, a reliable and an accurate mathematical modeling is
required to ensure the diagnosis and the control of these phenomena. This work proposes
development of a fault monitoring system of a gas turbine type GE MS 3002 based on
vibration analysis technique using spectral analysis tools. The obtained results prove the
effectiveness of the presentedmonitoring tool approachwhich is applied on the gas turbine,
for avoiding the operation under vibration mode and for generating optimal performance
during the exploitation of the gas turbine.

Keywords: default bearings, dynamic behavior, faults detection, gas turbine, monitoring
system

1. Introduction

Vibration analysis of rotating machines in industrial plants is widely used to ensure premature
faults diagnosis before these machines may break down. Gas turbines are the most important
pivotal rotating machines belonging to the family of internal combustion engines that are used
intensively in oil and gas industrial plants. Indeed, it is used to achieve the main function of
generating mechanical energy in form of shaft rotation from kinetic energy of gases produced
in the combustion chamber (Djaidir et al., 2015, 2016; Djeddi et al., 2015, 2016; Eshati et al.,
2013; Jurado andCarpio, 2006; Kim et al., 2011). However, thesemachines are subject to some
phenomena inducing important vibrations that can be very critical to their mechanical state.

The work presented in this paper proposes a supervision approach based on vibration ana-
lysis, to ensure a permanent fault diagnosis of the studied gas turbine, where the main aim is
to prevent failures that may cause malfunctions and to fulfill an optimal operation availability
of such a machine. Indeed, there are several methods that have been used for characterization
andmodeling of the gas turbine parameters. However, thesemethods are less exploitable for the
control of such systems due to its complexity. The proposed approach in this work is based on
the analysis of the magnitude and the harmonics spectrum of signals resulting from measured
vibrations on the gas turbine under different failures thatmay affect this kind ofmachine during
its normal operation.

Gas turbines are widely used in many industrial applications such as oil industry where
they are mainly used for wells re-injection and gas transportation along the pipelines. These
machines are subject to various instability factors that are influencing directly their operating
performances. Therefore, in order to improve the performances of these systems, many resear-
ches previously built mathematical models that represent the real time operation state more
accurately and instantaneously. Madhavan et al. (2014) developed a vibration model based on



394 B. Djaidir et al.

damage detection of rotor blades in a gas turbine engine. They showed that the developedmodel
can be used to carry out the analysis of mechanical components failures detection of the system
under study. Liu et al. (2015) studied the influence of large scale wind turbine blade vibration
based on the influence of the aerodynamic effects. Usman et al. (2015) presented the impact of
operating conditions and exhaust gas recirculation on the performance of rotors dynamics of
a gas turbine system. However, the current development of new technologies has allowed rese-
archers and industrial practices to improve the performance of gas turbines and to contribute
effectively to developing new methods and algorithms for controlling of such systems. In this
context, many industrial monitoring systems are being developed by several researchers to en-
sure fault detection, as well as for the diagnosis of defects (Ali et al., 2015; Djaidir et al., 2015,
2016; Djeddi et al., 2015, 2016; Günyaz, 2013; Krzyzynski et al., 2000; Liu et al., 2015; Sanaye
and Tahani, 2010; Simani and Patton, 2008).
The present paper deals with themodeling of gas turbine defects based on vibration analysis

usingmathematical techniques, where a number of assumptions representing different operating
parameters of the studied gas turbines have been taken into account. In this paper, the analysis
of dynamic behavior of rotor vibration of the gas turbine type GE MS 3002 is proposed. This
gas turbine rotates at high speed up to 7100tr/min and it is supported by bearings thatmay be
subject to faults during normal operation, whichmay contribute directly to the deterioration of
the gas turbine in a rapid manner.
The obtained in this paper results show clearly the effectiveness of the presented modeling

of the defects in a gas turbine. The presented approach allows one to build a diagnostic strategy
based on the vibration behavior analysis. On the other side, the experimental procedure for
tracking the anomalies in the gas turbine is presented. It is based on the use of the obtained
vibration signal spectrum and the associated analysis.

2. Dynamic vibration behavior in gas turbine

Gas turbines are heavily utilized in industry, where they are used mainly for producing and
ensuring important mechanical energy (Ford et al., 2013; Guasch, 2013; Lee et al., 2013; Rah-
moune et al., 2015). They are used for driving of fixed appliances where important output power
is required such as in electrical power generator, in powerful compressors, in powerful pumps,
etc. Therefore, the analysis of dynamic behavior of the gas turbine is attracting much attention
from the researcher and industrialists. It is based on the study of vibration behavior of the
main element of the gas turbine which is the high pressure rotor (HP) (Madhavan et al., 2014;
Rahmoune et al., 2015).
The vibration signal is presented by the following expression

x(t)=Asin(wt+φ) (2.1)

where w is the frequency of vibration, φ is the phase and A the amplitude of vibration in
micrometers [µm].
The vibration speed ν(t) and the vibration acceleration a(t) are obtained by differentialing

vibration expression (2.1) as follows

ν(t)=
dx(t)

dt
=Awcos(wt+φ) a(t)=

dν(t)

dt
=−Aw2 sin(wt+φ) (2.2)

Practically, all units are based onmillimeters and seconds.
Based on equations (2.1) and (2.2), the following expressions can be deduced

|x|= |ν|
w
=
|a|
w2

|ν|= |x|w= |a|
w

|a|= |ν|w= |x|w2 (2.3)



Faults detection in gas turbine rotor using vibration analysis... 395

where x is the displacement, ν is the vibration speed, a is the vibration acceleration and w is
the frequency of vibration.
Equation (2.3) highlights the importance of the choice of physical variables for datameasu-

rement andmonitoring of the gas turbine.Compared to themodel based on speedmeasurement,
the displacementmeasurements are used to reduce the high and themedium frequencyvibration
effects, which can amplify the low frequencyvibration components. In this case, radial vibrations
in the gas turbine are studied in the simple case of a circular cross-section of the rotor in the
transverse position.
Thedominant torsional displacement is the rotation of the cross sections givenby theangular

displacement α. The simplified displacement is used as follows

u1(x1,x2,x3, t)≈ 0 u2(x1,x2,x3, t)≈−x3α(x1, t)
u3(x1,x2,x3, t)≈x2α(x1, t)

(2.4)

For the deformation calculations, the Hamilton functional is used

ε11 = ε22 = ε33 = ε23 ε12 =−
1

2
x3
∂α

∂x1
ε13 =

1

2
x2
∂α

∂x1
(2.5)

The Hamilton functional construction is given by the following equation

H(α) =

t1
∫

t0

l
∫

0

[1

2
ρI1
(∂α

∂t

)2
− 1
2
GI1
( ∂α

∂x2

)2
+M1α

]

dx1 dt

I1 =

∫

S

(x22+x
2
3) dx2 dx3

(2.6)

The rotor motion is given by

ρI1
∂2α

∂2t2
−
∂

∂x1

(

GI1
∂α

∂x1

)

=M1 ∀x1 ∈ [0, l], ∀t∈ℜ (2.7)

whereG=E/[2(1+ν)] is the shear modulus of the rotor in the limitat conditions x1 =0 and
x1 = l

α(0, t) = 0 GI1
∂α

∂x1
(0, t)= 0

α(l,t) = 0 GI1
∂α

∂x1
(l,t)= 0

(2.8)

The conditions in the initial state are given by α = 0 and the condition GI1 = ∂α/∂x1 = 0
when no torsion at the free end is provided. The stress on the rotor surface is determined by
using the following equation



















σ11
σ22
σ33
σ23
σ13
σ12



















=





















E νE νE 0 0 0
νE E νE 0 0 0
νE νE E 0 0 0

0 0 E
2(1+ν)

0 0 0

0 0 0 E
2(1+ν)

0 0

0 0 0 0 0 E
2(1+ν)







































0
0
0

−x3αx1
−x2αx1
0



















(2.9)

In practice, gas turbines systems are complex and are generally non-stationary. Consequently,
their non-linear behavior makes the modeling step of the vibrations very difficult. Hence, im-
plementation of a reliable predictive tool is required to ensure the control and the diagnosis of



396 B. Djaidir et al.

such systems. For this purpose, vibration analysis techniques are used, tested and validated in
this paper.
The vibration analysis used is based on the real vibration data collected on the studied gas

turbine at different phases of operation. For the present application, the points ofmeasurements
are shown in Fig. 1, where a moving accelerometer is used for these measurements for both
levels. Indeed, there are three sensors that are used with accelerometers on different positions
(horizontal, vertical and axial) at each level, as shown in Fig. 1.

Fig. 1. Experimental installation of the gas turbine system

The recorded vibrations of the studied gas turbine can be quantified by three fundamental
quantities. The displacement, the speed and the acceleration. The model considered comprises
two entrances, the room temperature and the fuel mass flow rate which is expressed as follows

ṁc =
√
∆P
1

2
ρµ2 =

√
∆P

k
⇒ ṁc =

1

k

√
∆P (2.10)

The air mass flow ṁa is given by

ṁa = k
′(P2−P1)

ṁa = k
′(PCD−Patm)

(2.11)

It is necessary to control the combustion of the gas turbine by regulating the ratio of the
efficiency F given by

F =
T3−T2
ηbCV +T3

(2.12)

The sensors are installed on the gas turbine to provide dynamic behavior measurements of the
high pressure (HP) turbine shaft speed, the low pressure (BP) turbine shaft speed and the
temperature of the blades.

3. Gas turbine parameters modelling and analysis

All the rotating machines used in different industries applications are subject to more or less
vibrations. These vibrations are due basically to poor distribution of mobile masses, especially
in the rotor, which is essentially caused by imperfect machining during their fabrication or a
lack of homogeneity of themetal constituting the rotor. Indeed, the origin of vibrations observed
in a gas turbine can be caused by different phenomena such as (Galindo et al., 2013; Günyaz,
2013; Krzyzynski et al., 2000):



Faults detection in gas turbine rotor using vibration analysis... 397

• Unbalance of the rotor,
• Rapid braking of the shaft,
• Amechanical failure; such as a broken blade,
• Permanent deformation of the rotor.

The rotating part of the gas turbine consists of two shafts on which several wheels holding the
blades are mounted, where the line connection between the bodies of the two shafts is ensured
by bearings. Therefore, vibration behavior of suchmachines extremely depends on these compo-
nents. It is obvious that each such rotatingmachine is exposed to dynamic excitations extremely
diverse and specific to its mode of operation. At the same time, the fixed andmoving parts are
vibrating structures, depending on their modal characteristics and their vibration damping ca-
pacities. Each part of the system (fixed ormobile) generates dynamic vibration constraints that
cause fatigue of the rotatingmachine parts. It is impossible for a rotatingmachinewhich is desi-
gned, evenwithmore andmore advancedmaterials, not to vibrate. However, themanufacturers
of such machines try to improve the quality of materials and design to built rotating machines
that can supportmore severe vibration constraints.
The researchwork presented in this paper allowsmodeling andanalysis of dynamic vibration

based on the spectral analysis by considering various defects that may occur in this type of
machines.

3.1. Spectral analysis

TheFourier Transform (FT) can convert a periodical signal into a series sum (whichmay be
infinite) of sine and cosine functions. Suppose a real signal x(t) is defined with a finite energy.
The Fourier Transform of the signal x(t) is defined by the following equation

FTf{x(t)}=X(f)= 〈x(t),e2πift〉=
+∞
∫

−∞

x(t)e−2πift dt (3.1)

A discrete signal x(k) of length N presenting the number of values during a fixed interval, can
be presented as follows

X(m) =
N−1
∑

k=0

x(k)e−
2πimk

N (3.2)

On the other side, the spectrum of an aperiodic signal is given by its FT

X(ω) =FT [x(t)] (3.3)

where FT [x(t)] is spectrum of x(t).
The spectral energy density (SED) in the case of a discrete signal is defined through the

discrete Fourier transform as follows

Px(m)= |x(m)|2 Sx(m)= |X(m)| (3.4)

It can be shown that the energy of the signal, which is defined as the integral energy transferred
at each instant x(t), which can also be expressed as the integral of energy contributions carried
by each frequency given by

E =

+∞
∫

−∞

|x(t)|2 dt=
+∞
∫

−∞

|x(ω)|2
dω

2π
=

+∞
∫

−∞

|x(f)|2 df (3.5)



398 B. Djaidir et al.

The energy spectrum of the signal is replaced by the square module of the FT of the squ-
are module of cn by the spectrum of energy |x(ω)|2 = |x(f)|2 of x(t) with limT→∞Tcn =
∣

∣

∣

∫+∞
−∞
x(t)e−iωt dt

∣

∣

∣.

In this paper, the estimation method of the spectral density is proposed for the vibration
faults detection and isolation in gas turbine. Thismethod presents a good spectral analysis tool
from the point of view of calculation and allows obtaining good results for a large class of signals
in the studied gas turbine.

3.2. Vibration analysis

Asmentioned previously, the gas turbines are subject to unstable phenomena, because the shaft
cannot be perfectly balanced. The presence of residual unbalance on the shaft line is counted
by adding point masses. The addition of a large unbalance to the shaft is explained by the loss
of a blade on one of the disks. The imbalance is characterized by its kinetic energy when the
mass mb is located at the point B in the plane of the disk at a distance d from its geometric
centreC, Fig. 2.

Fig. 2. Centrifugal forces due to unbalance

The coordinates of the unbalance in the fixed coordinate system are given by

x(t)=uB +dcosΩt ⇒
x(t)

dt
=ub−dΩ sinΩt

z(t)=wB +dsinΩt ⇒
z(t)

dt
=wb+dΩcosΩt

(3.6)

Its kinetic energy is given by

TBal =
1

2
mb[(u̇

2
B + ẇ

2
B)+Ω

2d2+2Ωdu̇B sinΩt−2ΩdẇB cosΩt] (3.7)

The term Ω2d2/2 is constant and will not intervene in the equations. The mass of the imba-
lance is negligible in comparison to the rotor mass; the expression of the kinetic energy can be
approximated by

TBal ≈mbdΩ(u̇B sinΩt− ẇB cosΩt) (3.8)

By introducing the generalized coordinates, it can be rewritten as follows

TBal ≈mbdΩ(ȦuẎ 3B+ḂuY 2B+ĊuY 2B+Ḋu)sinΩt−(ȦwẎ 3B+ḂwY 2B+ĊwY 2B+Ḋw)cosΩt (3.9)

Theunbalance generates strongvibrations at the operational rotation speed of the turbine.They
appearwhen the gravity axis of the rotor (equilibriummass axis) does not correspond to its axis



Faults detection in gas turbine rotor using vibration analysis... 399

of rotation. This unbalance is caused by an inhomogeneous distribution ofmass around the axis
of rotation. In the case of turbocharger stationary motion, where the bearing part is used to
guide the rotating shaft and the radial load is small and comprised primarily of the unbalance,
which may be reduced if necessary. On the other side, the ball bearing damage causes several
major factors; such as surface material fatigue as a result of constraints that produce flaking
and cracking. Furthermore, the superficial fatigue can be aggravated by some effects such as
insufficient lubrication, surface conditions, shock. Also for the elements in contact, the wear can
aggravate the unstable phenomena during the operation of the machine.

For a ball bearing, Fig. 3, it comprises nb number of balls, and rotates at the frequency fr
(shaft rotation frequency).Thepresenceof avibrationdefect creates shocksbetween the rotating
elements of the channel whose frequencies are defined by the expressions in equations (3.9) and
(3.14).

Fig. 3. Geometric characteristics of the bearing

The frequencies relative to the defects on each rotating elements are functions of the passage
frequency of the rotating element in the outer ring. It is given by the following equation

fbe =
frnb
2

(

1−
d

D
cosφ

)

(3.10)

Thepassage frequencyof the rotating element in thedefected inner ring, is givenby the following
equation

fbi =
frnb
2

(

1+
d

D
cosφ

)

(3.11)

The defects frequency on the ball are given by

fB =
frnb
2

[

1−
( d

D
cosφ

)2]

(3.12)

The passage frequency of the defected ball in the outer ring or in the inner ring is presented as
follows

fc =
fr
2

[

1−
( d

D
cosφ

)]

(3.13)

Taking into account the modulation of the signal by the frequency fc, the above equations are
modified, so the defects in the inner ring become

fbi =
frnb
2

(

1+
d

D
cosφ

)

±fc (3.14)



400 B. Djaidir et al.

And the defects on the ball are expressed as

fB =
frnb
2

[

1−
( d

D
cosφ

)2]

±fc (3.15)

This signal modulation leads to functional modeling imperfections of the examined gas turbine.
In the following Section, the proposed approach based on spectral analysis at a low range fre-
quency will be applied to a practical case which is considered in a real machine in the presence
of an unbalance anomaly.

4. Industrial investigations

In this studyof the vibration behavior based on the analysis ofmeasured signals on a gas turbine
installed in the gas center of TIMZHERT, at Hassi R-Mel, south of Algeria, has been examined.
Figure 4 shows the studied gas turbine with the specifications given in Table 1. This machine
undergoes vibration effects observed during its operation. The measurements were carried out
on this turbine via amobile accelerometer for ensuringdiagnosis of various vibration phenomena
occuring at bearingNo. 1 of the studied gas turbine.While guaranteeing themaintenance policy,
vibrations aremeasured, processed and used for diagnosis to assessmechanical conditions of the
examined turbine.

Fig. 4. Studied GE 3002 gas turbine

Table 1. Specifications of the examined turbine

Parameter Value or symbol

Designer: General Electric GE

Type MS 3002

Serial No. 226293

HP speed axial compressor 7100tr/min

No. floor wheel (s) HP 01

No. axial compressor stage 15

BP speed 6500tr/min

No. BPwheel 01

Turbine power 80% 9400 CV

Fuel consumption rate (100%H 27◦C) 3.84m3/h

Pressure of exhaust gas 1009.3 bars

global peas 63000k



Faults detection in gas turbine rotor using vibration analysis... 401

4.1. Applications results

In this Section, the obtained experimental results using the measurements performed on
the studied gas turbine are presented. The results were performed on each bearing (bearing 1
and 4) in the vertical direction and with a rotation speed of the shaft ω = 7100tr/min. The
characteristic frequencies of the bearing defects are listed in Table 2. The values given in Table
were drawn on the basis of dimensions of the bearing and on the basis of rotation frequency
of the shaft, using the frequency fr = V (tr/min)/60s = 7100/60 = 118.33Hz (where V is the
shaft rotation speed and nb =15 is the number of beads).

Table 2.Characteristic frequencies of bearing defects

fr [Hz] fbe [Hz] fbi [Hz]

118.33 748 1029

According to the obtained results of measurements, it is found that the measured values on
the turbine at bearing 1 indicate the presence of vibrations in the vertical direction over a range
of frequencies (0-10000Hz). In order to diagnose this vibration failure, a spectral analysis has
been carried out. The obtained experimental results using the experimental conditions show the
amplitude of the signal versus the frequency, see Figs. 5 and 5b. It is important to clarify that
the bearing life depends on the load acting on the shaft, on the rotational speed and on the
point of application of the force.

Fig. 5. Vibration signal: (a) of a defected bearing, (b) of a bearing with a fault located in the inner ring

The gas turbine is operated by a set of commands that support themachine during starting
andaccelerating till the loadingphase.This automatic control helps tomaintain the systemtobe
stable. Themeasurements of variables such as pressure, velocity and acceleration are performed
using an instrumentation device, as shown in Fig. 6.



402 B. Djaidir et al.

Fig. 6. Position of installed sensors in the studied gas turbine

Analysis of various elements of the studied gas turbine was carried out on this installation,
starting from the study of rotation speed and its frequency, which must define the parameters
useful for the proposed monitoring system. This is based on the knowledge of variations of
the expected unstable phenomena, given by characteristic frequencies of possible defects such
as unbalance, the alignment and the attachment. Also, the resulting vibrations due to forces
developed or transmitted by the bearings caused by the unbalance have been analyzed. To
analyze these phenomena,Movilog 2 with FFTmono channel and the accelerometer with speed
of 100mV/g type (HS-AM004) are used, see Figs. 7a and 7b.

Fig. 7. (a) Used accelerometer; (b) Movilog 2 with FFTmono channel

This is donewith the input of balancing data for the selected rotation speed (V =118rpm),
rotor mass (M =3400kg) and rotor radius (R=75mm). At the rotational speed of 7100rpm,
where fr = 118.33Hz in the frequency range of 0-10000Hz, the large amplitude peaks are
observed, where fr is the rotation frequency of the shaft as shown in Fig. 8. This figure is the
continuous display of the spectral density of the measured vibration signal within frequency
regions corresponding to proper modes of the inner ring.

The appearance of such components as the shaft of the turbine (HP rotor) runs with the
defectedbearing is explainedby thepresenceof oscillations in the torque load.Thesefluctuations
have the feature to occur at the same characteristic frequency of the defect. In the present
diagnosticmethod, the appearance of these components is used as a reference for diagnosing the
presence of the anomalies in the studied gas turbine.

The levels shown in Tables 3 present the overall levels from the accelerometers within the
frequency ranges (1000Hz-10000Hz).



Faults detection in gas turbine rotor using vibration analysis... 403

Fig. 8. Vibration signal of the bearing with a fault located in the outer ring

Table 3.Overall levels from accelerometers of the studied gas turbine

SC1 Kurtosis
Crete Acc. [mg] Speed [mm/s]
factor 10000Hz 1000Hz Pic

3.0 4.6 208.8 2.9
RV 6.9 7.0 89.8 1.3

2.9 4.5 61.8 2.6

3.0 5.0 32.0 0.2
HP RH 7.0 7.0 13.3 0.1

2.9 5.2 7.0 0.2

2.7 4.5 160.6 0.7
AX 6.3 7.1 69.9 0.2

2.6 4.4 45.6 0.7

Fig. 9. Amplitude of the unbalance response: (a) test 1, (b) test 2

For unbalance defects, the operator can compensate for this misalignment by adding or
removing known masses at specific locations on the rotor. These imbalance defects are mainly
due to themanufacturing errors. They can explained by the presence of poor homogeneity of the
materials used in the rotor structure. In order to verify the capacity of the system to balancing
strong unbalances, additional balancing tests were carried out. Figures 9a,b and 10a,b show
results of tests with two balancing speeds, themain goal is to reduce the overall vibration levels
at all speeds and to kept the amplitude of vibration to be stable.



404 B. Djaidir et al.

Fig. 10. Amplitude of the unbalance response: (a) test 3, (b) test 4

A misalignment is occurred when the peak amplitude is captured, generally up to 2 or
3 times the rotation frequency. The vibration measurements are in function of choice of the
measurement points. In the studied case, the bearings are chosen as the points of measurement
and data collecting on the examined gas turbine. Figures 11a and 11b successively present the
evolution of the spectral structure (time signal) when a fault alignment is occurred.

Fig. 11. (a) Timemeasurement signal of the acceleration in the presence of alignment defects;
(b) temporal signal without defects

Ata speed of 7100rpm, for a frequency range of 0-5000Hz, apeak is observed (the shaft rota-
tion frequency) which characterizes the misalignment fault. In Fig. 11b, the fault is directional,
which is the case of the detected and identified misalignment on the gas turbine rotor. The re-
sults obtained were used to analyze the signature of an axial vibration fault. Unlike unbalanced,
this defect is often seen in the axial direction.

5. Conclusion

The understanding of the dynamic vibration behavior in gas turbines allows identification and
localization of the source of the vibration anomaly as well as quantification of the defect. In this
paper, the rotor dynamics is studied to ensure the improvement of security and performance
of the studied system. The developed model allows one to analyze the dynamic behavior of a
HP turbine rotor type MS 3002 rotating at a high speed and supported by bearings having
defects that are presenting instabilities areas varying with the rotation frequency and creating
furthermore new critical frequencies. On the other side, it is shown that some important effects



Faults detection in gas turbine rotor using vibration analysis... 405

can occur. Firstly, the wear of components regarding the rotor elements or mass distribution
which can generate new critical speeds. Secondly, the defect in the inner ring of a bearing can
generate the passing frequency which varies with the speed of rotation. Thirdly, the alignment
effect manifests itself as a large amplitude component of the rotation frequency of the rotor in
the radial direction, sometimes in the axial direction in the case of a rotor cantilever.

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Manuscript received September 28, 2015; accepted for print April 28, 2016