Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 3, pp. 751-763, Warsaw 2017
DOI: 10.15632/jtam-pl.55.3.751

STABILITY ANALYSIS AND DYNAMIC BEHAVIOUR OF A FLEXIBLE

ASYMMETRIC ROTOR SUPPORTED BY ACTIVE MAGNETIC BEARINGS

Molka Attia Hili, Slim Bouaziz, Mohamed Haddar

Laboratory of Mechanical Modeling and Production (LA2MP), National School of Engineers of Sfax (ENIS),

University of Sfax, Tunisia; e-mail: Molka.hili79@gmail.com

The objective of this work is to show the influence of dynamic characteristics of Active
Magnetic Bearings (AMBs) on the stability and dynamic response of an asymmetric and
unbalanced rotor. Indeed, AMBs have been successfully applied in several industrial machi-
nery facilities. Their main advantages are the contactless working principle, frictionless su-
spension and operation in very high speeds. Firstly, the AMBs dynamic support parameters
havebeen obtained through electromagnetic theory.Then, a generalized systemequations of
motion have been derived using the finite element method. The motion of a rotor the shaft
cross-section of which is asymmetric is generally governed by ordinary differential equations
with periodic coefficients. Floquet’s theory is used to investigate the stability of this system
of equations. Finally, numerical simulation results are presented and discussed.

Keywords: asymmetric rotor, finite element, Floquet’s theory, dynamic coefficients, stability

1. Introduction

A spinning system serves as amodel formany rotatingmachinery elements. It is generally com-
posed of a flexible shaft on which a flexible or rigid disk is mounted and supported by bearings.
Bearings have a considerable effect on the dynamic behaviour of such systems. Recently, AMBs
are increasingly used, especially in machines operating at very high rotational speeds, because
of their many advantages (no lubrication, very long life, supporting hard environments, precise
control, low power use and high-speed operating) compared to rolling elements, hydrodynamic
or elasto-hydrodynamic bearings.

On the other hand, the presence of defects is a major concern in rotating machinery; they
generate some important loads and vibrations and also stability problems. Asymmetric cross
section of the shaft is among commonly encountered defects, it is usually due to machining
defects.

The study of rotating systems supported by AMBs bearings and analysis of machine faults
has resulted in an extensive body of publications.

Lei and Palazzolo (2008) presented an approach for the analysis and design of magnetic
suspension systems with a large flexible rotor dynamic model including dynamics, control and
simulation. Inayat-Hussain (2007) presented a numerical study to investigate the response of
an unbalanced rigid rotor supported by AMBs. The mathematical model of the rotor-bearing
system used in that study incorporated non-linearity arising from the electromagnetic force-coil
and current-air gap relationship, and the effects of geometrical cross-coupling. The response of
the rotor was observed to exhibit a rich variety of dynamical behaviour including synchronous,
sub-synchronous, quasi-periodic and chaotic vibrations. Inagaki et al. (1980) studied a multi-
-disk fully asymmetric rotor with longitudinal variation of the shaft cross section. The temporal
equations of motion were obtained using the transfer matrix method. The unbalance response
was deduced by the harmonic balance method.



752 M. Attia Hili et al.

Oncescu et al. (2001) proposed modifications into a classical finite element procedure deve-
loped for rotors with symmetry to incorporate the effect of shaft asymmetry and used Floquet’s
theory to investigate the stability of a general system of differential equations with periodic
coefficients.
Recently, Inayat-Hussain (2010) studied the dynamics of a rigid rotor supported by load-

-sharing between magnetic and auxiliary bearings for a range of realistic design and operating
parameters. Numerical results of that work show that the unbalance parameter is the main
factor that influences the dynamics of the rotor-bearing system. It was also shown that the
non-synchronous vibration response amplitude of the rotor with a relatively small unbalance
magnitude can be reduced by decreasing the magnitude of the friction coefficient. Tsai et al.
(2011) developed a wavelet transform algorithm to identify magnetic damping and stiffness
coefficients of the driving rod with a set of 4-pole AMBs. This work further revealed that the
identified second-order damping coefficient is negative for a specific rod displacement and speed.
The dynamics of the rotor-AMBs system in the axial direction is unstable. Bouaziz et al. (2011)
investigated the dynamic response of a rigid misaligned rotor mounted in two identical AMBs.
Three simplifiedmodels of current biased radialAMBswere presented,where four, six and eight
electromagnetswerepoweredbyabias current and the respective control current.Results of that
work show that angular misalignment is such that the 2× and 4× running speed components
are predominant in spectra of vibration. Their magnitudes vary with the number of magnets in
the bearing. Bouaziz et al. (2016), proposed a dynamical analysis of a high speed AMB spindle
in the peripheral milling process. The time history of the response, orbit, FFT diagram at the
tool-tip center and the bearings dynamic coefficients were plotted to analyze dynamic behavior
of the spindle.
Most of the papers found in the literature concerningmagnetic bearings are interested in the

dynamic response of unbalanced or misaligned rotors. The shaft is generally considered rigid or
massless.
On the other hand, the papers dealing with asymmetric shafts, consider that the shaft is

supported by two identical elastic bearings. The coefficients of stiffness and damping are given
arbitrarily. Stability study is very limited.
In this paper, dynamic characteristics of ActiveMagnetic Bearings (AMBs) is first be deter-

mined. Then a model of an asymmetric rotor supported by two magnetic bearings is presented
using the finite element procedure. A stability analysis will be conducted while showing the
influence of various parameters of the bearings on stability areas. In the sameway, the dynamic
response of the asymmetric shaft will be calculated and analyzed.

2. Bearing modelling

The electromagnetic bearing studied is formed by four electromagnets (n = 4) placed in the
bearing around the rotor and producing an attractive force (Fig. 1).
Using the electromagnetic theory, the electromagnetic resultant forces produced by every

pair of the electromagnets in x and y directions are expressed as (Inayat-Hussain, 2010)

Fx =λ
[( I0− ix
C0−ux

)2
−
( I0+ ix
C0+ux

)2]

Fy =λ
[(I0− i0− iy
C0−uy

)2
−
(I0+ i0+ iy
C0+uy

)2]

(2.1)

where C0 is the nominal air gap, i0 is the bias current (to produce neutralizing force due to
weight of the rotor), I0 is the steady state current in the coil, ux and uy are respectively the
shaft displacements in the x and y directions, λ is the global magnetic permeability expressed
as

λ=
µ0AN

2

4
cosθ (2.2)



Stability analysis and dynamic behaviour of a flexible asymmetric rotor... 753

Fig. 1. Electromagnetic bearing

whereA,µ0, θ andN represent, respectively, cross section area of an electromagnet, permeability
of vacuum, half angle between the poles of the electromagnet and the number of windings in the
coil, and ij (j = x,y) represents the control current expressed using a proportional-differential
(PD) controller as

ij = kpuj +kdu̇j j=x,y (2.3)

where u̇j is shaft velocity in the j direction, kp is the proportional gain and kd is the differential
gain.
By replacing (2.3) respectively in (2.1), we obtain

Fx = a





(

1−
kpux
I0
− kdu̇x
I0

1− ux
C0

)2

−

(

1+
kpux
I0
+ kdu̇x
I0

1+ ux
C0

)2




Fy = a





(

1− i0
I0
−
kpuy
I0
−
kdu̇y
I0

1−
uy
C0

)2

−

(

1+ i0
I0
+
kpuy
I0
+
kdu̇y
I0

1+
uy
C0

)2




(2.4)

where a=λI20C
2
0.

Electromagnetic forces that depend on the shaft centre displacement and velocity are line-
arized (first order) around the equilibrium position (Bouaziz et al., 2016). This will provide the
classic model of a bearing with four stiffness and damping coefficients (Fig. 2)

{

fx
fy

}

=−KB

{

ux
uy

}

−CB

{

u̇x
u̇y

}

(2.5)

whereKB is the bearing stiffness matrix expressed as

KB =

[

Kxx Kxy
Kyx Kyy

]

=−









(∂Fx
∂ux

)

0
0

0
(∂Fy
∂uy

)

0









(2.6)

andCB is the bearing stiffness matrix expressed as

CB =

[

cxx cxy
cyx cyy

]

=−









(∂Fx
∂u̇x

)

0
0

0
(∂Fy
∂u̇y

)

0









(2.7)



754 M. Attia Hili et al.

In this model, the stiffness and damping cross-coefficients of the bearings are neglected. Nume-
rical differentiationmethod is selected for determination of the dynamic coefficients. The partial
derivatives are evaluated by the finite difference central method.

Fig. 2. TwoDOF bearingmodel

3. Equation of motion

Themathematical model (Fig. 3) consists of a flexible asymmetric shaft, one rigid disk and two
active magnetic bearings.

Fig. 3. Rotor bearing systemwith AMBs

Thefinite element procedure for rotorswith the symmetric shaft is considered.Modifications
are made to accommodate the effect of shaft asymmetry (Oncescu et al., 2001).

3.1. Equation of motion of the shaft

The shaft is considered to be flexible. It is characterized by its kinetic and deformation
energies. Its motion results from transverse displacement (ux,uy) and bending deformations
(θx,θy) in the x- and y-planes (Fig. 4).

Because of the shaft asymmetry, the sectional moments of inertia Ix and Iy are not identical,
consequently, the kinetic energy of the shaft can be represented by Oncescu et al. (2001)

Ts =
ρS

2

L
∫

0

(u̇2x+ u̇
2
y + u̇

2
z) dz+

ρIm
2

L
∫

0

(θ̇2x+ θ̇
2
y) dz+2ρImΩ

L
∫

0

θ̇xθy) dz

+
ρId
2

L
∫

0

(θ̇2x+ θ̇
2
y) dzcos(2Ωt)+ρId

L
∫

0

(θ̇xθy) dz sin(2Ωt)

(3.1)



Stability analysis and dynamic behaviour of a flexible asymmetric rotor... 755

Fig. 4. Shaft modelling and corresponding DOF

where ρ is material density, Im = (Ix + Iy)/2, Id = (Ix − Iy)/2 are respectively deviatory and
mean area moments of inertia of the shaft cross-section (Ix and Iy are the second moments of
area about the principal axes x and y of the shaft).
Rayleigh’s dissipation function of the disk is (Gosiewski, 2008)

Ed =
1

2

L
∫

0

cs(u̇
2
x+ u̇

2
y) dz+

1

2

L
∫

0

ci[(u̇x+Ωuy)
2+(u̇y −Ωux)

2] dz (3.2)

where cs and ci are respectively coefficients of external and internal damping.
If shear deformations are neglected, the strain energy of the shaft is (Oncescu et al., 2001)

U =
1

2

L
∫

0

EId
{[(∂2uy
∂z2

)2
−
(∂2ux
∂z2

)2]

cos(2Ωt)−2uxuy sin(2Ωt)
}

dz

+
1

2

L
∫

0

EIm
[(∂2ux
∂z2

)2
+
(∂2uy
∂z2

)2]

dz

(3.3)

whereE is Young’s modulus.
The finite element used to discretize the shaft consists of two node beam elements where

each node has four degrees of freedom: two lateral displacements and two bending rotation
angles. Applying Lagrange’s formalism to this system permits the development of the equations
of motion of asymmetric shaft (Oncescu et al., 2001)

Ms(t)δ̈s+(ΩGs+Cs)δ̇s+Ks(t)δs =0 (3.4)

with

Ms(t)=Mss+Md,ccos(2Ωt)+Md,s sin(2Ωt)

Ks(t)=Kss+Kd,ccos(2Ωt)+Kd,s sin(2Ωt)

whereMss is the mass matrix of the symmetric shaft (Batoz and Gouri, 1990), Md,c andMd,s
are mass matrices induced by the asymmetry of the shaft, Gs is the gyroscopic matrix of the
shaft,Cs is the dampingmatrix, Kss is the stiffness matrix of the symmetric shaft (Batoz and
Gouri, 1990), Kd,c andKd,s are stiffness matrices induced by the asymmetry of the shaft, δs is
the vector of shaft DOFs.

3.2. Equation of motion of the disk

The center of mass of the rigid disk coincides with the elastic center of the shaft
cross-section. The nodal displacements vector of the disk in fixed co-ordinates is given by:
δD = {ud,x,ud,y,θd,x,θd,y}.



756 M. Attia Hili et al.

The kinetic energy of the disk, by considering the effect of the unbalance, is expressed as
(Oncescu et al., 2001)

Td =
1

2
Md(u̇d,x+u̇

2
d,y)+

1

2
J(θ̇2d,x+θ̇

2
d,y)+2Jθ̇d,xθd,y+mudΩ[u̇d,x cos(Ωt)+u̇d,y sin(Ωt)] (3.5)

where Md and J are mass and moment of inertia of the disk, Ω is angular speed of the rotor,
mu is unbalance mass (assumed to be small if compared with Md), d is the radius defining
location of the unbalance.
Rayleigh’s function of the disk energy dissipation is (Gosiewski, 2008)

Ed =
1

2
cs(u̇

2
d,x+ u̇

2
d,y)+

1

2
ci
[

(u̇d,x+Ωud,y)
2+(u̇d,y −Ωud,x)

2
]

(3.6)

where cs and ci are respectively the coefficients of external and internal damping.
The application of Lagrange’s equations for the disk only gives

MDH+ δ̈(ΩGD+CD)δ̇D+KDδD =Fu(t) (3.7)

whereMD,GD,CD andKD are respectively mass, gyroscopic, damping and stiffness matrices
of the disk,Fu(t) is the unbalance vector.

3.3. General equation of motion of the rotor

By assembling the elementarymatrices of shaft elements, disks andbearings (as expressed in
Section 2), we obtain a systemofn second order differential equations andnunknown functions,
where n is the number of DOFs of the rotor. The global equations of motion are

M(t)δ̈+(C+ΩG)δ̇+K(t)δ=Fu(t) (3.8)

whereM(t) andK(t) are periodicmatrices of period T1 =π/Ω, for which the time dependency
is due to shaft asymmetry, C is a constant matrix including damping effects of AM bearings,
G is the gyroscopic matrix, Fu(t) – unbalance vector of period T2 =2π/Ω and δ is the vector
of global DOFs.
The equations of motion are therefore parametric in nature, which usually causes a stability

problem. Floquet’s theory will be used to determine the zones of instability.

4. Floquet’s theory

Floquet’s method is a mathematical tool for solving parametric differential equations, such as
(3.8). It involves computation of a transfermatrix over oneperiodofmotion (Dufour andBerlioz,
1998).
The study of stability of the steady state solution of system (3.8) can be reduced to study

of stability of the trivial solution of the associated homogeneous system.
A state-space model for system (3.8) (with Fu(t)=0) has the form

Ẋ=A(t)X (4.1)

where A(t) is a m×m (m= 2n) periodic matrix called the dynamic matrix of period T , and
X= {δ, δ̇}−1 is the state variable vector.
The transfer matrix Φ(T,t0) (Bauchau and Nikishkov, 2001) is by definition a matrix that

relates the initial solution X(t0) to the solution X(T) obtained at t=T , so

X(T)=Φ(T,t0)X(t0) (4.2)



Stability analysis and dynamic behaviour of a flexible asymmetric rotor... 757

The period T of the matrix A(t) is divided into n intervals of equal length h=T/n (t0 < t1 <
... < tn−1 <tn). Between the two solutions of (Eq. (3.7))X(ti+1) andX(ti), there is a relation

X(ti+1)=Φ(ti+1, ti)X(ti) (4.3)

whereΦ(ti+1, ti) is the elementary transfer matrix.We can easily notice that

Φ(T,ti)=Φ(T,ti+1)Φ(ti+1, ti) (4.4)

The matrix Φ(T,t0) can be obtained by iterative calculation based on relation (4.4). We start
withΦ(T,T)= Im and gradually gets Φ(T,tn−1),Φ(T,tn−2), . . . ,Φ(T,t0).
Todetermine the elementary transfermatrixΦ(ti+1, ti), severalmethodshavebeenproposed.

In this work, Newmark’s method is used.
Furthermore, it is shown that the stability of the trivial solution of equation (3.7) is fully

defined by the eigenvalues of the transfermatrix over one periodΦ(T,t0), known as the charac-
teristic multipliers of system. The trivial solution is asymptotically stable if the modulus of all
m eigenvalues is less than one, and is unstable if the modulus of at least one of the eigenvalues
is greater than one.

5. Numerical results

The spinning system investigated in this paper is shown in Fig. 4. The disk is mounted on the
shaft at 3L/4. The system parameters are given in Table 1.

Table 1. System parameters

Parameter Symbol Value

Permeability of vacuum µ0 4π ·10
−7Wb/Am

Number of windings around core N 300

Half angle between poles of electromagnet θ 22.5deg

Bias current i0 0.5A

Differential gain kd 42.4A·s/m

Proportional gain kp 14869A/m

Rotor length L 300mm

Shaft diameter dS 10mm

Disk diameter dD 100mm

Young’s modulus E 2.1 ·1011Pa

Density ρ 7.85gm/cm3

Poisson’s ratio ν 0.28

Unbalance mass Mu 5gm

Moment of inertia J 0.005Kg·m2

Rotor running speed Ω –

Figures 5 and 6 show respectively the dependency of stiffness and damping coefficients of
AMBs on the air gapC0 between the stator and the rotor for different values of the steady state
current in the coil I0. Figures 7 and 8 show respectively the dependency of stiffness anddamping
coefficients of AMBs on the air gap C0 between the stator and the rotor for different values of
cross-sectional area of one electromagnet A.
Kxx,Kyy,Cxx andCyy are strongly influenced byC0, I0 andA. Indeed, their values decrease

considerably with C0, but regularly increase respectively with I0 and A. This observation was
proven by Bouaziz et al. (2011) when they determine the dynamic coefficients of AMB with



758 M. Attia Hili et al.

four, six and eightmagnets. It is clear that the four coefficients become constant for large values
ofC0 (fromC0 =2mmand irrespective of I0 andA). It should also be noted that the values of
damping coefficients are significant compared to those frequently encountered in literature for
modeling of elastic bearings.

Fig. 5. Stiffness coefficients (A=200mm2)

Fig. 6. Damping coefficients (A=200mm2)

Fig. 7. Stiffness coefficients (I0 =3A)



Stability analysis and dynamic behaviour of a flexible asymmetric rotor... 759

Fig. 8. Damping coefficients (I0 =3A)

Since the coefficients of stiffness and damping are variable according to the parameters of
the bearings, the stability of the studied asymmetric rotor varies as well. So try from the layout
of different regions of stability to find the optimal parameters providing better system behavior
(a minimum of regions of instability). The variable parameter is the factor of shaft asymmetry
defined as the rate between the deviatory (Id) and themean areamoments of inertia of the shaft
cross-section (Im): fsa= Id/Im.

We are interested in the range of higher speeds because electromagnetic bearings are used
in applications at high speeds.

Figure 9a shows regions of instability obtained for the studied system for factors of shaft
asymmetry varying from 0 to 0.3 with increments by 0.05. The results have been obtained by
varying the rotational speed between 10000 and 80000rpm,with increments by 100rpm.The air
gap C0 is 0.5mm, the steady state current in the coil I0 is 3A, and the effective cross-sectional
area of one electromagnet A is 200mm2.

Fig. 9. Instability regions: (a)C0 =0.5mm, I0 =3,A=200mm, (b) [30000-40000rpm],C0 =0.5mm,
I0 =3,A=200mm

For the symmetric shaft, no instability interval has been identified around the four shaft
critical speeds: 17685, 36350, 63840 and 80100rpm (according to the Campbell diagram). For
the asymmetric shaft, four regions of instability appear for a factor of shaft asymmetry of 0.1
and for widths increasing with the shaft asymmetry. For a shaft asymmetry of 0.25, there are
5 critical speeds delimiting the three regions of instability: 15492, 19639, 31600, 40475, and
55765rpm.



760 M. Attia Hili et al.

Figure 9b shows the influence of internal and external damping on the instability region
[30000-40000rpm]. The width of the stability zone decreases substantially by introducing the
internal and external damping. This behavior is also observed for the other two instability
regions. These coefficients are generally negligible compared to those provided by AMBs and
cannot in this case considerably influence the stability of the system studied.

Figure 10 depicts the regions of instability respectively obtained for three different values
of C0: 0.5mm, 1.5mm and 3mm.

Fig. 10. Instability regions, I0 =3,A=200mm

It is evident that the instability regions are highly dependent on the air gap C0. Indeed,
for a small value of C0 (C0 = 0.5mm), the stiffness coefficients of AMBs are very important
(Fig.5). They increase the critical speed of the system and, therefore, the stability regions are
also affected.

We note the presence of three regions of instability for C0 = 3mm against two for
C0 =1.5mm.

The second region of instability begins from a factor of shaft asymmetry of 0.2 for
C0 =1.5mm,while it starts from a factor of shaft asymmetry of 0.1 forC0 =3mm. In fact, for
greater values of C0, the stiffness coefficients of AMBs undergo a slight decrease. The positions
of instability regions remain therefore almost unchanged, while their size and numbers increase
withC0. This last behavior is explained by a decrease in damping coefficients of the AMBs.

Fig. 11. Instability regions: (a)C0 =1.5mm,A=200mm, (b)C0 =1.5mm, I0 =3A

Figure 11a shows regions of instability obtained for three values of I0. It is observed that
regions of instability decreas with the increasing I0. By increasing I0, we can also remove a



Stability analysis and dynamic behaviour of a flexible asymmetric rotor... 761

region of instability, which proves the importance of control. Figure 11b presents the regions of
instability obtained for three values ofA. It is also observed that regions of instability decrease
with the increasingA.

We can conclude that the stability is strongly linked with the dynamic coefficients of the
bearings. A significant change of stiffness coefficients results in a change of the critical speed
of the system and, consequently, a change in the positions of instability areas. At the same
time, an increase in the damping coefficients, automatically leads to a decrease in the size of
instability regions. C0 and I0 are the most influential parameters on the rotor stability. Based
on this analysis, the parameters of AMBs that offer greater stability to the asymmetric rotor
are:C0 =1.5mm,A=200mm and I0 =5A.

To better understand the instability phenomenon of asymmetric rotors, it is important to
calculate and analyze the dynamic response. The dynamic response is observed at the disk-to-
shaft attachment. A spectralmethod is used to estimate the dynamic response (Attia Hili et al.,
2006).

The frequency response along the x direction (power spectral density) of the rotor in the
frequency range of 0 to 250Hz is shown in Fig. 12a. The factor of asymmetry is 0.25 when the
shaft running speed is 420rpm.

Fig. 12. (a) Frequency response,C0 =1.5mm, I0 =3A,A=200mm
2. (b) Experimental frequency

response, Lazarus et al. (2010)

The frequency response is essentially characterized by:

• A dominant peak at the rotational frequency (Ω = 7Hz) indicating the presence of an
unbalance.

• Several peaks located at frequencies of odd multiples of the rotational frequency
(3Ω,5Ω,7Ω,.. .) indicating the simultaneous presence of an unbalance and shaft asym-
metry as well as representing themodulation phenomenon. Indeed, themass and stiffness
terms are time variables having the frequency equal to two times of that of rotation, whe-
reas the excitation frequency (unbalance force) being equal to the rotational frequency.

• Two peaks presenting the first and second natural frequencies of the rotor (in xz-plane):
Ωn1 = 59Hz and Ωn2 = 203Hz. Supplementary harmonics emerge in the frequency re-
sponse and are located following the relation: Ωj = Ωni ± 2jΩ (j > 1, i = 1,2). The
fundamental and first secondary harmonics related to the natural frequenciesFn1 andFn2
are respectively marked with spots and squaremarks. These peaks correspond to parame-
tric quasi-modes characterizing every linear time-varying system like an asymmetric shaft.
These results are in good agreement with the experimental results given in Fig. 12b and
found by Lazarus et al. (2010). Indeed, the authors measured the frequency response of



762 M. Attia Hili et al.

an asymmetrical shaft. These experimental results are filtered to remove the spin speed
subharmonics. The fundamental and first secondary harmonics related to the natural fre-
quencies Ωn1 = 15Hz and Ωn2 = 23Hz are, respectively, marked with red square marks
and with blues pots.

We deliberately chose a low speed rotation (although magnetic bearings operate at high
speeds) tovalidate thenumerical simulations.Finally,wenote that the samebehavior is observed
in the yz-plane. The parametric quasi-modes also appear regardless of the running speed used.

6. Conclusion

In this paper, a finite element procedure for rotor-AMBs systems is generalized to include the
effects of shaft asymmetry.

Firstly a model describing electromagnetic bearings (with four electromagnets) has been
developed allowing to calculation of the dynamic coefficients which are mainly influenced by
the air gap C0 between the stator and the shaft, the effective cross-sectional area A and the
bias current I0. Then, analysis of the stability by Floquet’s theory shows that the stability is
stronglydependenton theparameters of thebearings (whichallowdeterminationof theoptimum
parameters providing better behavior). The stability of the system is improved by the choice of
AMBs parameters leading to an increase in the damping coefficients.

Thedynamic response identifies theasymmetry of the shaft by thepresence of oddharmonics
of the rotational frequency and parametric quasi-modes in frequency spectra. These features
could be useful in the detection of shaft faults in diagnosis of rotating machines.

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Manuscript received August 24, 2016; accepted for print January 5, 2017