Vibration mitigation of a flexible bladed rotor dynamic system with passive dynamic absorbers

Accepted Manuscript

Vibration mitigation of a flexible bladed rotor dynamic system with passive dynamic absorbers Ghasem Ghannad Tehrani , Morteza Dardel PII: DOI: Reference:

S1007-5704(18)30268-5 https://doi.org/10.1016/j.cnsns.2018.08.007 CNSNS 4621

To appear in:

Communications in Nonlinear Science and Numerical Simulation

Received date: Revised date: Accepted date:

14 December 2017 12 July 2018 19 August 2018

Please cite this article as: Ghasem Ghannad Tehrani , Morteza Dardel , Vibration mitigation of a flexible bladed rotor dynamic system with passive dynamic absorbers, Communications in Nonlinear Science and Numerical Simulation (2018), doi: https://doi.org/10.1016/j.cnsns.2018.08.007

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Highlights 

Here the application of tuned mass damper (TMD) and nonlinear energy sink (NES) absorbers on a flexible bladed rotor system is accounted.



Absorbers are placed on each of disk and blade.



The bifurcation diagrams of the system without and with absorbers couplings



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are investigated.

The results show that passive vibration absorbers can reduce the vibration of flexible multi- blade rotor at higher clearance values but they have limitation in

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PT

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lower values of clearances.

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Vibration mitigation of a flexible bladed rotor dynamic system with passive dynamic

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absorbers Ghasem Ghannad Tehrania and Morteza Dardel1b

(a)- Ghasem Ghannad Tehrani, MSC student of Mechanical Engineering, Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran, [email protected].

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(b)- Morteza Dardel, Associate Professor of Mechanical Engineering, Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran, [email protected].

Abstract

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The main objective of this paper is to study the application of tuned mass damper (TMD) and nonlinear energy sink (NES) on vibration mitigation of a flexible bladed rotor system.

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TMD tunes to a specific frequency of the main system and reduces its amplitude only in a vicinity of that frequency. NES is another well-known passive dynamic absorber comprises a mass, nonlinear spring, and linear or nonlinear damper. The nonlinearity of this absorber

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allows it to reduce the vibrations of the main system in a wider frequency range. In this study to reduce the contact of flexible bladed rotor system, the mentioned passive

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absorbers are attached both to the disk and to the tips of the blades, such that to deactivate the effect of disk’s eccentricity force and stabilizing the behavior of the system. The

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optimum parameters of the absorbers are obtained through optimization. The bifurcation diagrams, and Lyapunov exponent of the system are obtained with respect to different parameter and rotational speed are obtained and consequently, the efficiency of the absorbers are investigated. The obtained results show how these passive absorbers can decrease the vibration of the rotor system and remove the contact at higher clearance 1Corresponding

author: Department of Mechanical Engineering, Babol Noshirvani University of Technology, P.O. Box 484, Postal Code: 47148-71167, Shariati Street, Babol, Mazandaran, Iran. [email protected].

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values between the blades and the stator; although, there are limitations in their performances at lower values of clearance. Key words: Flexible blade, bladed rotor, contact, multi- bladed rotor optimization, complex

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averaging, TMD, NES.

Nomenclatures: Eccentricity of the disk

, , ,

,

,

Stiffness of the contact

Rotational speed of the disk

Friction coefficient of contact

Disk, stator & shaft radiuses Displacement field of the disk’s geometric center Displacement field of the disk’s mass center position of an arbitrary point of the blade

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,

Mass, moment of inertia & damping of the disk

Swing & Lateral displacement of the blade

,

Length of the blade & bearing Density of the blade

( )

mode shape of the blade

( )

generalized coordinate of the blade

,

radial clearance of the bearing & eccentricity of the journal

PT

,

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,

centrifugal & normal rubbing forces

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,

,

Attitude angle

angular position of the th blade

Viscosity of the oil film

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Bending deflection of the blade

Displacement field of the journal’s center

,

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number of the blades

,

cross section area & moment of inertia of the blade’s

,

cross section area & moment of inertia at the blade’s root

,

breadth at the root & the tip of blade

Stiffness of the Shaft ,

, Mass, Damping, Linear & Nonlinear stiffness of the bearing system

, ,

3

,

Masses, Damping, stiffness of TMD

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,

thickness at the root & the tip of blade

,

Young modulus of the blade

,

,

Masses, Damping, stiffness of NES Displacement of TMD & NES

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1- Introduction Rotordynamic systems are one of the most applicable operational machines in numerous industrial centers such as marine, aerospace, powerhouses etc. Hence, so much care about their design and dynamical behavior are required. One of the most probable consequences of the operation of such systems is the contact phenomena occurrence.

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This problem is mostly as the result of the eccentricity forced produced by the unbalanced disk and could result in other undesirable issues such as blade loosening and buckling, permanent bow in the shaft etc. In recent years, so many studies have been conducted involving with analyzing and solving such problems; however, few of them concentrated on providing a solution to overcome such defects.

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Cveticanin [1] analyzed a Jeffcott rotor considering the strong nonlinear elastic property of the shaft. The governing equations of motion are solved applying Krylov – Bogolubov

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method. Then the influences of the factors such as damping, hydrodynamics etc. on the system’s vibrations are studied. Wan and Jian [2] studied the vibration characteristics of

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two rotors that supported with journal bearings at both ends. The long bearing assumption is assigned to the oil – film supports in order to derive the lubricant

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dynamical equations. Adopting some of the parameters as the control parameter such as rotating speed, Poincare’ section, bifurcation diagrams, and Lyapunov exponents are obtained in order to analyze the behavior of the system. Chasalevris et. al. [3] studied a

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rotor – bearing system with the purpose of the journal bearing wear influence on the system’s response. Using the continuous wavelet transform, sub and super – harmonics resonances are observed in the system’s response. Saeed and EI-Gohary [4] analyzed the vibrations of a horizontally supported Jeffcott rotor system considering rub-impact force. The multiple-scales method is utilized to obtain a second-order approximate solution. The results are extracted using bifurcation analysis and Lyapunov first method.

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Popprath and Ecker [5] studied the rotor stator contact in a suspended Jeffcott rotor. Their results show that the damping of the stator suspension has a great influence on the type of the system’s motion. Shen et. al. [6] considered a rotor bearing system supported by oil film and with initial bow, and studied the rub impact. The influences of rotating speed, initial bow length, phase angles between mass eccentricity direction and bow

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direction on rub impact phenomena are studied through bifurcation diagrams, Lyapunov exponents, Poincare’ sections and orbit maps. Inayat-Hussain [7] studied bifurcations in the response of a rigid rotor supported by load sharing between magnetic and auxiliary bearings, which occurs during contact between the rotor and the auxiliary bearings. Numerical results revealed the existence of period-doubling bifurcation with periods 2,

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4, 8, etc. as well as quasi-periodic and chaotic vibrations. Parent et. al. [8] studied the blade – casing contact phenomena of aircraft engine model focusing on the dynamic stability of the system. In this research, first of all the linear behavior of the systems is analyzed, where dynamic coupling between different components of the system is investigated. Then, the nonlinear model of the system because of contact phenomena is

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investigated. Chen [9] concentrated on the blade- casing rubbing interaction. In the first step of this research, a model of blade – casing rubbing of an aero – engine system is

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proposed and the effects of number of the blades and the variations of the blade – casing clearance on the rubbing force are investigated for several rubbing conditions such as

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single-point, multi-point, local-part, and complete-cycle. Then, the rubbing model is applied to a complete model of rotor dynamic system and the casing acceleration under the rubbing force is determined. Finally, an experimental verification of the whole study

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is conducted where the results are fully verified. Thiery et. al. [10] analyzed an initially misaligned multiple blade Kaplan turbine system with blade – stator impacts, and in

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order to determine the safe operation of the system, different dynamics motion in the form of Poincare’ sections and bifurcation diagrams are studied. Ma et. al. [11] revised the rubbing force expression between blade and stator in a bladed rotor system, and the influences of parameters such as penetration depth, casing stiffness etc. on the normal rubbing force are analytically and experimentally determined.

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Guo et. al. [12] used passive absorbers (TMD & NES) in order to mitigate the vibration amplitude of a rotor system. In this paper, two kinds of passive dynamic absorbers (NES & TMD) are taken into account and are optimally designed and applied. Bab et. al. [13] used some smooth nonlinear energy sinks in a rotor-bearing system supported by journal bearings, for the purpose of vibration attenuation. In order to obtain the most

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effective performances of the NESs the genetic algorithm is applied to provide the optimized parameters of NESs. Taghipour and Dardel [14] investigated the effect of single and double DOFs NESs on the behavior of a two-DOF dynamical system in steady state condition. It is concluded that dynamical system with single DOF of NES shows less robustness to change in parameters and amplitude of external forces in comparison to

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double absorbers. Abbasi et. al [15] applied a high – static low – dynamic stiffness (HLDS) suspension to a rotating machine for the sake of vibration control. To minimize rotor and bearing vibration, a multi-objective optimization formulation is considered, and the optimum parameters of the HLDS are obtained using genetic algorithm. It is concluded that low dynamic stiffness can eliminate non – periodic behavior of the

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system. Parseh et. al. [16] studied the influence of NES on a nonlinear vibration of beam, modeled according to Euler-Bernoulli and Timoshenko theories. Using complex

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averaging and continuation method, the steady state response of the system based on resonance condition is investigated. Taleshi, et. al. [17] studied the behavior of a simply

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supported plate using NES. Changing the parameters and the location of the NES, and using complex averaging and arclength continuation method, an optimized passive targeted energy transfer of the plate was obtained. Furthermore, the performance of

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NES and TMD was compared with each other. Saeed and Kamel [18] worked on controlling the vibrations of a Jeffcott rotor implementing an active control technique, by

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a proportional- derivative (PD) controller using two pairs of electromagnetic poles. Zapoměl et. al. [19], used magnetorheological squeeze film damping devices for vibration suppression of rigid rotors. They used bilinear material modeling for magnetorheological fluids. They showed that changing magnetic induction in the lubricating film makes it possible to achieve optimum performance of the damping device in a wide range of the rotor operating speeds. Bab et. al. [20] investigated the effects of a number of smooth nonlinear energy sinks (NESs) located on the disk and 6

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bearings on the vibration attenuation of a rotor blade journal bearing system under excitation of a mass eccentricity force. The nonlinear energy sinks on the bearing have a linear damping and an essentially nonlinear stiffness. It is shown that the optimum NESs can remove the instability region from the system response. In this paper, passive dynamic absorbers are suggested to eliminate the undesirable

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vibrations produced by both rigid disk and the flexible blades in a rotor dynamic system. To this end, the absorbers are placed on both disk and the blades. For the disk, the absorber is mounted in the radial direction to resist the eccentricity force and for the blades, the absorber is attached to the tips of the blades aiming at their bending vibrations suppression. In addition, an optimization procedure is proposed to obtain the

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parameters of absorbers.

The outline of the current study is as follows: In section 2, a schematic of the rotor dynamic systems is given and the governing equations of motion are derived. Then, the application of passive dynamic absorbers on the disk and the blades are suggested and the related equations of motion are derived. Furthermore, in order to optimal design of

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the absorbers, the semi-analytical response of the system in presence of dynamic absorbers using complex averaging method are derived and the resulted responses are

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solved through an optimization process where the optimal values of the absorbers’ parameters are achieved. The obtained results are presented and discussed in section 3.

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In this section, the behavior of the rotor dynamic system without and with the presence of passive dynamic absorber are investigated and compared. The conclusion is given in

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the last section where the influence and efficiency of passive dynamic absorbers

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application and the proposed optimization process are debated.

2- Modeling of the problem

In Fig. 1,

and

are the geometrical and mass center of the disk respectively and its

eccentricity is demonstrated by . The mass moment of inertia of the disk is specified by The rotational speed of the disk is denoted by

.

which is assumed constant and counter

clockwise. The length of blades is presented by and the radiuses of the disk and stator are 7

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shown by

and

respectively. The derivation of the equations of motion and the

analysis of the whole system are based on the following assumptions: The shaft is assumed to be massless and has a uniform stiffness

.

ii.

The stator is rigid.

iii.

Axial and torsional vibrations are negligible.

iv.

The longitudinal displacement of the blade is neglected.

v.

The motion in the swing direction of the blade and the axial deformation of the shaft

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i.

are neglected

In the following sections, using the Lagrange method the equations of motions of the

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system with and without absorbers are derived.

Figure 1: Rotor dynamic system.

2.1 Modeling and equations of motion of the rotor dynamics system

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without absorber

2.1.1 Modeling of flexible blade motion

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The flexible blade is modeled according to a clamped - free Euler–Bernoulli beam, in which the shear deformation of cross sections is neglected. Considering Fig. 2, the position vector

[

]

[

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⃗

PT

of an arbitrary point such as of the blade is in following form.

where (

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(1)

) are the displacements of the geometric center of the disk in the global

coordinate system and ( (

]

) are the position of an arbitrary point of the blade. In addition,

) are the swing and lateral motion of the flexible blade respectively. For convenience,

the slope of transverse deformation is denoted by

which represents an infinitesimal

rotation angle of the blade in the local coordinate system.

is a rotational transformation

matrix from local coordinate system to the global (inertial) coordinate where:

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[

] (

Here

) . /, with

is the number of the blades and the relation

) . / denotes the angular position of the th blade.

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(

(2)

Figure 2: Blade, disk and contact model.

According to the fifth assumption:

and

⃗

0

1

[

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arbitrary element of the blade is:

, therefore the position vector of an

][

]

(3)

The first derivation of the position vector in global coordinate and its kinetic energy are

M

given respectively as follows: ⃗̇ ( ( ̇

))

))

∫ |⃗̇ |

( (

CE

(

) ̇

AC

(

∫

(

∫

(

) )

̇)

] ̂[ ̇

̇)

(5)

̇ (

̇ ( ̇

∫ ( )

(4)

]̂

)

̇ ̇ ̇ ̇

̇)

|⃗̇ |

∬

6∫ (

( (

PT

(

( ̇

ED

[ ̇

̇ ) )

̇ ) ̇ )

̇

where

9

7

̇

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4

. /5 4

̅

. /5

4

. /5 4

. /5

̅

(6)

The first and second area moments of inertia for the blade are: ∬ and

and (

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and

∬

are the taper ratios in breadth and thickness directions. Furthermore, (

)

) are the breadth and thickness at the root and the tip of the blade respectively.

In addition,

and

represent the cross section area and moment of inertia at the blade’s

root.

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The potential energy of the blade comprises the effects of bending, centrifugal force and normal rubbing force, which is

∫ ∬ ( )4

)

∫ ∬4

(

In the latter expression

) 5

,

(

) 5

(7)

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(

( ) and

ED

∫

demonstrate the Young modulus of the blade,

centrifugal and normal rubbing forces respectively. Applying the condition of ∬

)

∫

(

)

∫

( ) is defined as follows:

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where

(

CE

∫

the potential energy of blade reduces to the following expression.

PT

and ∬

10

( )4 (

) 5

∫ (

)

(8)

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∫

( )

*

(

∫

(

)

.

(

)

/

)

(

)

.

(

(9)

/ )

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( )

+

4

.

/

√

√

|| ⃗ |

In above equation

.

/

M happens

⃗⃗⃗⃗

(

⃗⃗⃗⃗

(

AC

⃗ |

at

non-conservative

the

place

)̂ }⃗ )̂

CE

section):

by

PT

contact

done

ED

(

work

5

(10)

| designates the penetration depth. Other quantities are as

follows:

The

is given by [11]:

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Furthermore, the normal contact force between blade and casing

⃗⃗⃗⃗

of

the

)

(11)

forces

is

neutral

(it

is

assumed

point

of

the

⃗⃗⃗⃗

that

blade’s

the

cross-

(12) (13)

,

(

)

(

)

-̂

,

(

)

(

)

-̂

⃗ ⃗

(14)

(

),

(

)

(

)

-

(

),

(

)

(

)

-

11

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where

the

generalized

force

in

the

Furthermore,

the

assumed

mode

deformation.

The

following

solution

Lagrange

method is

is

method

used

assumed

as

to

is

describe

the

⃗

. the

lateral

⃗

/.

blade’s

response

of

the blade: )

∑ ( ) ( ) ( )

where

is

the

(15)

time-dependent

generalized

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(

coordinate

( )

and

is

an

admissible function for a clamed – free blade and is given by: (

)

(16)

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( )

2.1.2 Modeling of bearing motion and hydrodynamic lubrication

6

4

5

4 (

ED

where

57

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Reynold’s equation in two-dimensional form for hydrodynamic lubrication is [2]: (17)

) is the oil film thickness and

is the angle of

minimum thickness/attitude angle of the oil film. In this expression

and

are the

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eccentricity of the journal and radial clearance of the bearing respectively. In addition is defined as the dimensionless eccentricity of the journal. It should be mentioned

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that due to the symmetry of the system, just one of the bearing are taken into account [13].

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Figure 3: Journal – bearing configuration.

Using the Reynold’s equation and short bearing assumptions (variation of pressure along the circumferential direction is negligible in comparison to the variation of pressure along the axial direction), the pressure distribution of the oil film is given as below: ( )

(

)

.(

̇) ̇

12

/4

5

(18)

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Hydrodynamic forces acting on the journal’s surface in radial and tangential directions are:

̇)

*(

∫∫ ( )

(

)

(

( It is worthy to note that for

)

(19)

) ̇

̇)

*(

+

+

(20)

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∫∫ ( )

/ ̇

.

(

)

due to evaporation of oil film, and having axial

airflow, the pressure is zero according to Gumbel boundary condition [13]. Considering

(

)

(

)

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equilibrium for journal center:

(21) (22)

Substituting Eq. (19) and Eq. (20) into Eq. (21) and Eq. (22), the equations for variations of and

would be derived which will be shown in the section 2.2. The kinetic, potential

( ̇

̇ )

)

(

(23)

)

PT

(

̇ )

ED

( ̇

M

and dissipation energies of the bearing are given as following:

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2.1.3 Modeling of Disk motion The mass center displacement of the disk using the relative motion concept is:

AC

(24) (25)

The kinetic energy of the disk is given by: ( ̇

̇ ) (26)

,( ̇

)

( ̇

) -

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The potential and dissipation energy of the disk is: ((

)

2.2

(27)

) )

(28)

̇ )

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( ̇

(

Equations of motion

( ) as independent variables and applying the Lagrange method, the

Taking

equations of motion are derived. For expressing the equations of motion in dimensionless form,

√

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the following dimensionless parameters, are introduced. ̅ ̅ ̅ ̅

̅

̅

̅

̅

CE

̅

̅

̅

̅

̅

ED

̅

̅

̅

PT

̅

̅

̅

M

̅

̅

̅

(29) ̅

̅

̅ ̅

Accordingly, the equations of motion are as follows: ̅

̅ ∫

AC

4

̅

̅

̅

̅

̅5

̅ ∑

̅

̅ ̅ ̅

∫

̅

̅

( ̅)

̅ ∑

̅

̅

̅

̅

14

̅

∫

̅

( ̅) ̅

̅ ∑ ̅ ( )∫

̅

̅

(30) ( ̅) ̅

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̅

) ̅

̅̅ 4

̅ ∫ ̅

̅

̅

̅

̅ ̅

̅

̅5

̅ ̅ ̅

̅ ̅

̅

̅

̅ ∑

̅ (̅

∫

∫

̅

̅) ̅

̅ ∑

( ̅) ̅

̅ ̅

̅

∫

̅ ̅

( ̅)

̅

̅

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( ̅

̅

̅ ∑ ̅ ( )∫

̅

̅

( ̅)

̅

(31)

̅ ̅

̅

̅ ∫

4

̅

̅

∫

̅

̅

( ̅)

̅5 ̅

̅

̅ ̅ ∑ ̅ ( )∫ ̅

AC

̅ ̅

̅

̅

̅ ̅

̅ ∫

( ̅)

̅

̂

̅

̅

̅

̅

̅ ∑

̅ ̅

( ̅) ( ̅)

( ̅)

( ̅) ( ̅)

̅

̅)

4

PT

̅

̅

CE

̅ ∑ ̅ ( )∫

̅ (̅

∫

( ̅) ( ̅)

̅ (∑ ̅ ( ) ∫ ̅

̅

̅

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̅ ̅ (∑

̅

) ̅

̅

∫ ̅

( ̅)

( ̅) ( ̅)

̅

̅5

M

̅

ED

(̅

̅

̅

)

̅

̅ ∑

̅ ( )∫

̅

̅

( ̅) ( ̅) ̅

)

(32)

̅

̅

̅

̅ ∑ ̅ ( ) ∫ ̅ ( ̅) ( ̅) ( ̅) ̅

̅

( ̅)

̅ ̅

̅ ̅

( ̅ ̅

̅ ̅

̅ ̅

(̅

̅

15

̅ ̅

)

(33)

)

(34)

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. ( . [(

) ̅

{

)

(

)

/

(

{ (

) ̅

[

̅ ̅̅

(

)

̅

)

̅ ̅ ] ̅ ̅ { }

(

)

(

)

(

)

(

)

(

)

}

]

(35)

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)

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(

/

}

In Eq. (32), ̂ is the proportional damping and is defined in Appendix A.

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2.3 Equations of motion of the rotor system with passive absorbers The unwanted vibrations due to the disk’s eccentricity and rotating blades cause

ED

perturbations in the system’s operation which result in serious problems. In order to reduce the influence of these undesirable vibration sources, the application dynamics

PT

passive absorbers (TMD & NES) are suggested. In the following sections, the equations of motion of the system with passive absorbers on the disk and blades are derived and

CE

through an optimization method, the optimal values of the absorbers components such as mass, stiffness, and damping are obtained. It should be mentioned that the equations of

AC

motion of the system art derived for two cases: case 1: TMD on disk – NES on the blade, and case 2: NES on disk – NES on the blade.

2.3.1 TMD absorber on disk According to Fig. 4, the TMD is attached to a rod which passes the center of the disk and is placed in the space between the disk and the shaft. Hence, the absorber is restrained to a

16

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radial movement along the rod.

in addition,

of the absorbers, respectively. Parameter center of the disk, which can vary from

are the damping and stiffness coefficients

is the angle between the rod and the mass

to . Here

is angular displacement of the rod,

represents shaft’s radius, ⃗ is the displacement vector of the TMD’s mass, are the disk’s mass center displacements and coordinates (

and

) are the displacement

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components of the absorber’s mass.

Figure 4: Schematic diagram of rotor with single TMD.

( ̇

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The kinetic, potential and dissipation energies of the TMD are given as below: ̇ )

(36)

̇

The displacements and velocities of the disk and TMD in horizontal and vertical directions

(

)

̇

)

̇

( ( ̇

̇

̇

)

(

( ̇

(

) ) )

̇

̇

) (

(37)

)

ED

(

M

in the global coordinate system are as follows:

̇

̇

PT

Considering following dimensionless parameters: ̅

CE

̅

(38)

The equation of motion of the TMD is: ̅

AC

4

̅

̅

5

̅ ̅

̅

̅

̅

(39)

2.3.2 NES absorber on disk According to Fig. 5 the configuration of the NES attachment is just like TMD and the only difference is in the potential energy term. The difference is due to the potential energy

17

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produced by nonlinear spring of the NES. To this end, an oblique attachment of the springs is considered as is shown in Fig. 5. For NES with a radial displacement of energy is

, potential

(see Appendix B).

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Figure 5: Schematic diagram of rotor with single NES.

Dimensionless parameters NES are: ̅

(40)

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̅

Hence, the governed equation of motion of the NES is given as following: ̅

̅

̅

2.3.3 NES on the blade

5

̅

̅

̅

̅

̅

(41)

M

4

ED

In addition to placing absorbers on thedisk, an NES is placed at tips of the blades for its bending vibration mitigation. For this purpose, a block with curved shape corners is

PT

considered where a rectangular casing is placed within the encasement for the placement and displacement of the NES. According to Fig. 6,

,

and

are the mass, nonlinear

CE

stiffness and linear damping of NES respectively. Furthermore, it is assumed that this

AC

absorber has a displacement along the length of the rectangular casing.

Figure 6: Schematic of on blade absorber (NES).

The position vector of the absorber’s is: ̂

⃗ |

[

(

|

) ̂

(

18

|

) ̂]

(42)

ACCEPTED MANUSCRIPT

Expanding the sine and cosine in the latter expression and taking into account the small deformation of the blade .

|

|

/ following expression is

obtained for the position vector of the absorber: ⃗ (

)

( |

)

] ̂

[

(

)

( |

)

] ̂

(43)

CR IP T

[

Computing the first derivative of ⃗ , the kinetic energy of the on –blade NES is expressed as

̇

6

̇

̇

)

̇

̇

̇

̇

̇ ̇ 5

̇

̇

PT

4 ̇

̇ ̇ 5

CE

̇

(

̇

ED

̇

̇

5

̇ ( ̇

̇

̇

)

̇ ̇

M

4

4

AN US

follows:

̇

(44) ̇

̇

̇ ̇ ̇

̇

̇

̇

̇ ̇

̇ ̇ ̇

⌉

AC

The potential and dissipation energies of this absorber are: ̇

(45)

Defining dimensionless parameters: ̅

̅

(46)

The dimensionless equations of motion of the on blade NES is:

19

ACCEPTED MANUSCRIPT

̅

̅

(

̅∑

∑

( )

̅

̅

( ̅) ̅ ( )

( ̅) ̅ ( )

∑

∑ ( )

)

∑

( ̅) ̅ ( )

∑

( )̅ ( )

(̅

̅) ∑

)

( )̅ ( )

̅

∑ ̅

∑ ̅∑

̅

( ̅)

( )

( ̅) ̅

∑ ̅

( ̅) ̅ ( )

(47)

∑ ( )̅ ( )

AN US

̅

(

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̅

Hence, the equations of motions of the rotor dynamic system in presence of on the disk and on the blade absorbers are given by Eqs. (A10) - (A.12) of Appendix C. It should be noted that the governing equations of bearings and lubrication remain unchanged since the influences of the passive dynamic absorbers are just on the disk and

ED

M

blades.

2.4 Optimal values of TMD and NES parameters

PT

As was discussed before, unwanted vibrations result in emerging of contact and therefore some residual stresses and deflections in blades, disk, and bearing and in some cases crack

CE

on the disk. Hence, eliminating such critical situations and improving the working condition of the system is vital. Since estimation and changing the parameters of the

AC

absorbers in order to achieve a desirable dynamical behavior is inappropriate and inaccurate, optimization with regard to a predefined cost function is proposed for determination of the absorbers’ parameters. To this end, it is assumed that there is no contact between blades and stator. This would help to eliminate the nonlinear terms in the equations of motion. The considered rotor dynamic system is a self-excited system because of the effects of Coriolis, sliding, tangential and other accelerations. Therefore, dynamics of such systems is in the form of parametric system in which stiffness, inertia and damping

20

ACCEPTED MANUSCRIPT

matrices are dependent on time, which makes the optimization process so difficult. Therefore, it would be better to rewrite the equations independent of time, for which the complex averaging method is utilized.

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2.4.1 Complex form of the equations of motion In the complex averaging method, the equations of motion are written in terms of amplitude and phase of the response. This method is a semi-analytical method and gives nearly an exact solution for periodic solution. Here 1:1 response is considered, which means that the condition in which the frequency of response of system is equal to variables of , ̅

, and are expressed as follows. ̅

̅

̅

̅

PT

̅

CE

̅

̅

̅

,̅

(48)

̅

̅

in which ̅

̅

̅

̅

AC

̅

̅

ED

̅

̅

̅

M

̅

̅

̅

̅

̅

̅

̅

AN US

excitation frequency. The displacements and their derivatives in terms of complex

̅

are the static part of the displacements, and ̅ , ̅ , ̅ and ̅ are the

, and ̅

complex conjugates of

,

,

and

respectively. These variables are written in the

following complex form:

21

ACCEPTED MANUSCRIPT

(49) In the latter expression

and

are arbitrary parameters used for the definition of

complex mentioned variables where: | |

√

| |

√

√

| |

| |

√ (50)

(

̅

)

( ̅

) ̅

(

)

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( ) ̅

Then, by replacing latter complex variables in Eq. (48) and substituting the resulted displacements and their derivatives into governed equations of motion of the system in Eqs. (39, 47, A10, A11 and A12) and averaging of the obtained equation with respect to the , the equations of motion in terms of phase and amplitude are

AN US

exciting frequency

obtained. Then, the equations are set into a following standard form: ,

̇ - 2 ̇3

,

-2 3

* +

(51)

where Eq. (54) represents dynamics and static parts of the equations of motion altogether.

M

In addition, * + is the excitation vector. The nonlinearity due to the nonlinear stiffness of the NES makes the dynamic and static parts of the equations dependent to each other.

ED

Separating the real and imaginary parts of Eq. (51), a system of nine unknowns and nine

PT

equations is obtained which consists of dynamic and static parts.

CE

2.4.2 Optimization

There are varieties of methods for optimum design of the absorbers like

AC

method, the integral of the amplitude in terms of frequency is minimized. In

,

etc. In method on

the other hand, the maximum amplitude in terms of frequency is minimized. In this research, another procedure for optimization is proposed and developed. In the beginning, the structure without the absorbers is considered. A design point is selected from the simulations or based on the experience. For example, this point can be selected at conditions where resonance happens or in cases that the system has large

22

ACCEPTED MANUSCRIPT

amplitudes of vibrations. The goal is to determine the system’s parameters under following conditions: 

The amplitude of the primary system reaches a desirable value: Objective function The equation of motion must be satisfied: Constraint function



The stability of the system should be guaranteed: Constraint function



The absorber’s parameters should be within the allowed interval (upper and lower bound): Constraint function

∑(

AN US

The objective function is determined as follows:

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

)

It should be mentioned that for the steady-state response,

and

(52) must be set to zero. The

amplitudes | | | | that represents the amplitude of the disk’s center in the vertical and horizontal directions are considered as the cost function of the optimization process. Then,

M

using fmincon function in Matlab, the values of each absorber’ parameters can be

ED

computed.

3.1

PT

3- Results and discussions

Numerical simulation based on different rotational speed values

CE

In the following sections, the behavior of the rotor dynamic system for different rotational speed is investigated. At the beginning of the analysis, the system’s response with four–

AC

bladed rotor having no passive absorbers is studied, then the system is analyzed in presence of on-blades and on-disk absorbers and the results are compared with the primary system. It should be mentioned that, following analysis is presented for two values of the clearances between the blades tips and the stator, where clearance is determined by the following formula: (53)

23

ACCEPTED MANUSCRIPT

In this equation

,

and

represent the radius of the stator, the radius of the disk and

the length of the blades respectively. The values of for the length of the blades and accordingly, two

and

are considered

are obtained. The values of system

parameters are mentioned in Table 1 [6].

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Table 1: Parameters of rotor dynamic system [6].

3.1.1 Rotor dynamic system response without passive absorbers and

AN US

clearance of

In this section, the behavior of the system under different rotational speed and for the clearance value of

is studied. In order to have a better understanding of the

system’s response, the time responses of the four – bladed rotor for four different rotational speeds are depicted in Fig. 7. It should be noted that in the these figures, line ,

,

,

) represent the tip displacement of first, second, third

M

figures of (

and fourth blade respectively. According to the time responses in Fig. 7, the system has a . Further

ED

complete periodic motion while working under the rotational speed of

augmentation of rotational speed generates the undesirable vibrations of the system and contact phenomena occurrences. These complex behaviors can be implied from multi

PT

periodic and irregular responses of the blades tips for higher rotational values of . It should be noted that the red structural line designates the location of

CE

the stator and the intersection of the blades trajectories with this line shows the contact

AC

occurrence.

Figure 7: Time response of a four – bladed rotor for

Then, the behavior of the rotor dynamic system for rotational speed variations is taken into account According to Fig. 8, the system has a regular periodic behavior for small values of rotational speed. Then, by the increase of rotational speed, the multiperiodic responses

24

ACCEPTED MANUSCRIPT

emerge. These claims have been justified in Fig. 7. In addition, there are critical and unstable response regions around

and

amplitude increase in accordance with time. After

where the vibrations the vibrations decrease and the

system is so called in the region of dominant inertia. In these diagrams, undesirable

CR IP T

responses, repetitive bifurcations and sequential contact phenomena are observable.

Figure 8: Blade’s tip displacement of four – bladed rotor with .

and

AN US

Due to the improper behavior of the system for different rotational speed, the application of passive dynamic absorbers is suggested and their influence on the system’s response is investigated in the following section. The optimum parameters of the absorbers are obtained through the proposed optimization process and the obtained results are presented in Table 2. Since all of the blades have rather similar responses, just the first

ED

M

blade is taken into account and the results can be extended to the rest of them.

PT

3.1.2 Rotor dynamic system response with passive absorbers at

Fig. 9– A shows the system’s response in presence of TMD (disk) – NES (blade). What can

CE

be drawn is that utilizing two passive absorbers simultaneously is quite efficient since the safe operating condition is obtained until

. In this safe region, the system has a

AC

periodic behavior and the contact is eliminated. However, unstable responses for does still exist.

On the next step, the performance of NES attachment on both the disk and the blades is investigated. What can be perceived from Fig. 9 – B is that although the application of NES on the disk produces a periodic response and removes the contact, it is not as efficient as TMD since the safe working condition provided is around the previous configurations.

25

which is smaller than

ACCEPTED MANUSCRIPT

Figure 9: Blade’s tip displacement of the four – bladed rotor with ̅

and

.

For clarification of the system’s behavior during its operation, the Lyapunov exponent of the first blade is shown in Fig. 10. According to this diagram, there is no chaotic response

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within the system’s response due to the negative values of the Lyapunov exponent in absence and presence of the absorbers. In addition, the FFT diagram for the displacement of the first blade in its local coordinate and the FFT diagram of the disk are depicted for . According to these diagrams, since FFT spectrum has some dominate frequencies, this means that the frequency content of response is limited, and periodic or quasi-periodic

AN US

are occurred.

Figure 10: Lyapunov Exponent of the first blade, FFT of the first blade and of the disk for

M

in absence and presence of the absorber for

ED

3.1.3 Rotor dynamic system response without passive absorbers at

PT

In order to determine the efficiency of passive absorbers, the previous analysis is repeated for another clearance value. Considering Fig. 11, the time responses of the blades tips of the ; except

CE

four – bladed rotor system is rather similar to what was obtained for

for the fact that due to smaller clearance, the numbers of contacts during time passage have

AC

increased.

Figure 11: Time response of a four – bladed rotor for

.

Similar to the previous procedure, in order to observe the vibration response of the system thoroughly, the bifurcation diagrams of the blades displacement are depicted considering

26

ACCEPTED MANUSCRIPT

rotational speed as the control parameter. In Fig. 12 the response of a four – bladed rotor system is shown. The displacement of the tip of each blade demonstrates that around the system starts experiencing contact phenomena as well as multi – periodic responses. Besides there are some unstable situations near

and

where the

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vibration amplitudes are increased by time.

Figure 12: Displacement of the blade’s tip for four – bladed rotor, with 2.

and

AN US

In the following section, the same procedure is repeated, and the application of passive absorbers is proposed and their performances are evaluated. The optimum values are presented in Table 2.

M

Table 2: Dimensionless values of the absorbers.

ED

3.1.4 Rotor dynamic system response with passive absorbers for

PT

Fig. 13A shows the displacement of the blade’s tip with absorbers of TMD on disk, and NES on blade. It can be understood from the diagrams that application of the absorbers makes

CE

the system have a periodic response without any contact until the rotational speed of However after this speed, the contact is occurred and near the

.

the multi – periodic

AC

responses are emerged in the system’s behavior. It should be noted that the increase in the rotational speed, increases vibration amplitude. In addition, around

resonance like

behavior occurs. With increase in rotational speed, the systems’ amplitude response decreases; however, multi – periodic response and the contact are still existed. In Fig. 13B, with NESs placed to the disk and the blades, the system is behaving safely until . After that, the contact is occurred and multi – periodic is emerged. Here, a

27

ACCEPTED MANUSCRIPT

resonant response similar to the previous configuration can be seen where near the systems amplitude intensifies. Quite after this critical situation, attenuation in the course of response is observable; however, multi – periodic responses as well as the contact phenomenon do still existed.

̅

.

and

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Figure 13: Blade’s tip displacement of the four – bladed rotor with

Lyapunov exponent and FFT analysis similar to the previous section are presented in Fig. . With negative value of the Lyapunov exponent for the

AN US

14 for the system with

system without and with absorbers, the response is non- chaotic, and from FFT analysis it is revealed that the response has multi-frequency or quasi- periodic nature. Again, this is

M

due to low frequency content of the FFT spectrum.

Figure14 : Lyapunov Exponent of the first blade, FFT of the first blade and of the disk for

ED

in absence and presence of the absorber for

PT

3.1.5 Rotor dynamic system response without passive absorbers at

CE

Fig. 15 demonstrates the vibration response of a rotor dynamic system, where the value of the clearance between the tip of the blades and the stator is set to

AC

systems’ response for

. Unlike the

mm the contact starts emerging from the beginning.

In addition, the multi – periodic response and contact phenomena is observable in all over the systems’ response. Furthermore in

the system experiences resonance

response, which disappears as the rotational speed, increases. Moreover, in the interval of the dynamical response is attenuated, and the repetitive bifurcation windows in presence of contact is occurred.

28

ACCEPTED MANUSCRIPT

Figure 15: Displacement of the blade’s tip for four – bladed rotor, with

and

2.

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For better clarification of obtained results following description are given. For very low rotational speed, the behavior of the system in the absence of absorber has single periodic response as shown in Figs. 7 (a) and 11 (a). However, as the rotational speed increases, the system shows a multi-periodic response, as shown in Figs. 7 (b), (c) and 11 (b), (c). This claim could be verified by Figs. 8 and 12 as well. In these figures, the Poincare sections for

AN US

blade’s tip displacements at different frequencies are shown. The number of points in this section show the period of behavior. When there are limited numbers of points in each rotational speed, a multi-periodic with rational or irrational frequencies are presented. Hence since there are limited numbers of points in Figs. 8 and 12, the multi-frequency or quasi-periodic responses are presented in this system. After attaching the absorbers, the

M

frequency content of response is reduced, and behavior with lower frequency content is obtained. In Figs. 8, 9, 12, 13 and 15 around

and

, sudden jump in the amplitude

ED

of the system is occurred. From nonlinear dynamics, it is known that when sharp decrease in amplitude of nonlinear oscillation is presented, there is a nonlinear response between

PT

these two stable responses. For rotor without absorber this jump is more evident.

CE

3.1.6 Rotor dynamic system response with passive absorbers for

AC

Here, the performance of both passive absorbers configuration is evaluated for a smaller value of clearance. According to Fig. 16, none of the absorbers attachments are able to prevent the contact occurrence. Hence, the disability of the dynamic absorbers in vibration attenuation in such low clearance values can be understood. This is due that, in low clearance values, absorbers cannot sufficiently be displaced to counteract with mass eccentricity.

29

ACCEPTED MANUSCRIPT

Figure 16: Blade’s tip displacement of the four – bladed rotor with ̅

.

Numerical simulation based on different eccentricity values

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3.2

and

The eccentricity value alongside the clearance and rotational speed is another important parameter, which plays a significant role in analyzing a rotor dynamic system. In this section, the eccentricity is selected as a control parameter and the efficiency of passive absorbers are taken into account. The analysis is done for two clearance values. In addition, and it is because according to Fig. 8 and Fig. 12 the system

AN US

the rotational speed is set to

shows a rather critical behavior in this value. Furthermore, for

the passive

dynamic absorbers were able to improve the system’s behavior considering Fig. 9 and Fig. 13.

M

In the following sections, the optimization process for designing absorbers is done in terms , and ̅

.

ED

of two different eccentricity values: ̅

3.2.1 Rotor dynamic system response without passive absorbers for

PT

and

CE

The displacements of the blades tips are shown in Fig. 17. According to these results, for small values of eccentricity, there is no sign of contact. On the other hand, growth in the eccentricity causes the system to behave multi – periodically and to experience repetitive

AC

contact phenomena.

Figure 17: Blade’s tip displacement of four – bladed rotor with ̅

30

.

,

and

ACCEPTED MANUSCRIPT

3.2.2 Rotor dynamic system response with passive absorbers for and Here the efficiency of passive absorbers application is investigated. In Fig. 18 the effect of TMD on disk – NES on blade (A) and NES on disk – NES on blade (B) attachments are as the design point and their parameters are the

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shown with considering ̅

same as presented in Table 2. The results demonstrate that the absorbers at smaller clearance values act as an unbalance mass and cause the instability of the system, while, near ̅

their best performances are observable due to the improvement of the

AN US

system’s behavior.

Figure 18: Blade’s tip displacement of four – bladed rotor with ̅

M

Here the designs of the absorbers are based on

,

and

.

̅

and their parameters are

presented in Table 3. According to Fig. 19, it is apparent that the region of efficient

ED

performance of the absorbers is where the eccentricity is set as the design point. It can be perceived that the system’s behavior is refined around

̅

even though the

PT

eccentricity has its maximum value.

CE

Table 3: Dimensionless values of the absorbers.

AC

Figure 19: Blade’s tip displacement of four – bladed rotor in presence of a) TMD (disk) –

NES (blade), b) NES (disk) – NES (blade) with

,

and

̅

.

3.2.3 Rotor dynamic system response without passive absorbers for and 31

ACCEPTED MANUSCRIPT

In Fig. 20, the system’s response is shown for another clearance. Here, the behavior is quite like the previous section and the repetitive contact occurrences as well as multi – periodic responses are observable during system’s operation.

Figure 20: Blade’s tip displacement of four – bladed rotor with .

and

CR IP T

̅

,

3.2.4 Rotor dynamic system response without passive absorbers for

AN US

and

Like the previous section, here the analysis is done adopting two eccentricities as the design point for obtaining the absorbers components. In Fig. 21, the system’s behavior is shown based on ̅

and the parameters are presented in Table 2. Previous

results can be concluded here as well since the best performances of the absorbers are near .

M

̅

ED

Figure 21: Blade’s tip displacement of four – bladed rotor in presence of a) TMD(disk) – NES(blade), b) NES(disk) – NES(blade) with

,

and

CE

PT

.

The parameters of the absorbers provided based on ̅

and are shown in Table 4. As

is expected, the efficient regions of the absorbers performance are in a span of considering Fig. 22.

AC

̅

Table 4: Dimensionless values of the absorbers.

32

̅

ACCEPTED MANUSCRIPT

Figure 22: Blade’s tip displacement of four – bladed rotor in presence of a) TMD(disk) – NES(blade), b) NES(disk) – NES(blade) with

,

and

̅

.

With regards to the obtained results, it can be said that inccording to Fig. 8 the system experience no contact phenomena for

CR IP T

without the absorbers for

.

Howeve, in Fig. 9, the system in presence of the absorber does operate desirably until where its safe functional limit is doubled. The same story could be observed for where the system with the absorbers is able to work until the

according to Fig. 13A. Therefore, the proper performances of the absorbers could be

AN US

justified for such values of the absorbers. On the other hand, for

due to the

very small gap between the blades and the stator, not only does the system experience the contact phenomena from the very beginning, but also the has multi-periodic response. While, after the absorbers implementation, the system has a one-period behavior for

M

. Furthermore, the same acceptable performance of the system in the presence of the absorbers could be observed for different values of eccentricity. For instance, according , the contact phenomena is occurring for ̅

. In this case,

ED

to Fig. 17 for

the absorbers are designed in way which the system could operate safely for higher values

PT

eccentricity as shown in Fig. 18.

CE

4- Conclusion

This paper mainly focuses on applying passive dynamic absorbers for vibration mitigation

AC

of a flexible bladed rotor system. Through a suggested optimization process, the optimal parameters of the absorbers are determined. In order to evaluate the efficiency of the absorbers two bifurcations analyzes are done based on the variations of rotational speed and eccentricity. These analyses are performed for different clearances and the following results are acquired:

33

ACCEPTED MANUSCRIPT

-

The TMD (on disk)– NES (on blade) configuration is an efficient structure for

-

clearance values of

and

rotational speed

.

since the system is able to operate in higher values of

The configuration of NES (on disk) – NES (on the blade) is less efficient than the previous one; however, the contact is discarded for

. ).

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The TMD – NES configuration can eliminate the first unstable region (

-

On the other, the NES – NES configuration was not effective in comparison to the previous configuration. Furthermore, the secondary unstable region ( ) is presented in both absorbers configurations.

Passive absorbers application is not efficient enough for small clearance values (

-

) and other strategies such as active control should be utilized.

AN US

-

Taking the eccentricity as the control parameter, the absorbers are just able to provide a safe working condition near the designed eccentricity point.

-

The proposed optimization process is trustworthy and can be generalized to any

M

dynamic system.

ED

Appendices Appendix A

-* ̈ +

,

-* ̇ +

,

-* +

(A.1)

* +

CE

,

PT

The linear equations of motion of the beam solely are of the form below:

Applying eigenvalue problem and modal coordinates, the proportional damping of the

AC

beam is defined as follows: ,

-

(

)

(A.2)

Following change of coordinate is introduced. * +

, -* +

(A.3)

Hence with introducing this transformation and multiplying the Eq. (A.1) in , - , we have:

34

ACCEPTED MANUSCRIPT

, - ,

-, -* ̈ +

, - ,

-, -* ̇ +

, - ,

-, -* +

* +

(A.3)

Then the equations of motion are transferred to the modal coordinates, and are written to the following form. -* ̈ +

̂ * ̈+

,

-* ̇ +

̂ * ̇+

,

-* +

̂ * +

* +

(A.4)

* +

CR IP T

,

From Eq. (A.4) the proportional damping is defined as follows. ̂

, - ,

-, -

,

, -

-

̂

[

̂

[

]

(A.5)

], -

AN US

̂

In the latter expression is damping ratio and is usually small. In this paper, this parameter is equal to

.

M

Appendix B

Consider the arrangement of NES shown in Fig. A.1. In this arrangement, spring has initial , and with displacement of

ED

length of

direction, the spring has length of √

. Hence the elongation of spring is written to the

AC

CE

PT

following form.

in the direction perpendicular to its initial

35

ACCEPTED MANUSCRIPT

Figure A.1: The arrangement of NES.

√

(A.6)

4 For

CR IP T

The force produced by the displacement of the spring is: 5

√

(A.7)

and Taylor expansion of Eq. (A.7) is written to the following form.

AN US

(A.8)

The potential energy of NES is the integral of force of

, and has the

(A.9)

M

following form.

with respect to

ED

Appendix C

The equations of motion of the rotor dynamic system in presence of on-disk and on- blade

PT

absorbers are given as follows. ̅ ̅ ∫

̅

CE

4 (

AC

̅

∑ (̅)

̅

( ̅

̅

̅

̅5

∑

̅

̅ ̅

( ̅ ) ̅ ( ))

̅

̅∑

∑

( ̅)

( ̅) ̅

∑ ̅

( ̅)

∫ ̅

∑

36

̅

̅

̅ (A.10)

̅

( ̅ ) ̅ ( ))

ACCEPTED MANUSCRIPT

̅

( ̅)

̅(

∑

∑ ̅ ( )∫

( ̅ ) ̅ ( ))

(

( ̅)

)

( ̅)

(̅

̅

̅ )

∑ ( ̅) ̅ ( ) ̅

̅

)

̅ ̅

̅

∫

̅

AN US

̅

̅

̅

̅

̅ ̅

̅

∑

̅

∑ ( ̅)

( ̅

̅ ̅

CR IP T

̅∑

̅ ̅

̅

AC

CE

PT

ED

M

̅

∫

37

(̅

̅) ̅

̅

ACCEPTED MANUSCRIPT

(

∑

(

̅

∑

̅∑

( ̅)

̅

∑ ̅

̅ ̅ ̅

̅

∫

( ̅) ̅

̅

( ) ̅ ( ))

̅

̅

̅

∫

( ̅)

̅

(A.11)

(̅

̅ )

∑ ( ̅) ̅ ( )

̅

̅

̅

PT

)

̅

̅ ̅

̅

∫

̅ (̅

̅) ̅

̅

CE

̅

( ̅ ) ̅ ( ))

( ̅)

)

̅ (

̅

̅

̅

ED

∑ ̅ ( )∫

̅

∑

∑

̅

̅

∑

̅

( )

̅ ̅

( ̅)

̅(

(̅

̅ ̅

̅

( ̅)

̅

( ̅ ) ̅ ( ))

∑ ̅∑

̅

̅

∑ ( ̅) ̅

̅

̅5

M

̅

̅

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4

̅

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̅ ̅ ∫

̅

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̅ ̅ (∑

̅

4∫ ̅

( ̅) ( ̅) ̅

̅

∫̅

4

̅

(̅

( ̅) ̅ 5

̅

∑ (̅)

̅

̅

∫ ̅

( ̅) ( ̅) ̅5

̅

4

∫̅

( ̅) ̅5

∑ ( ̅ ))

38

̅

)

(A.12)

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(̅ ∑

̅ ∑

(̅) ( ) ∑ ̅

( )

( ̅ ∑ (̅) ̅

∑ ( ̅ ))

(̅ ∑

(̅)

( ̅ ∑

̅ ∑ ( ̅ ))

∑ ̅

̅∑

(̅)

(̅)

∑

(̅) ( ) ∑

(̅)

̅

( ̅)

∑ (̅)

̅

̅

̅

∑ ( ̅ ))

(̅

̅) ∑

∑ (̅) ( ) ∑ (̅)

̅

( ̅) ( ̅)

̅

̅

̅

(̅)

̅

∫ ̅

( ̅) ( ̅)

̅5

̅

∑ ̅ ( ) ∫ ̅ ( ̅) ( ̅) ( ̅)

̅

̅ )̅∑

(̅

( ̅ ))

̅

( )

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( ̅) ( ̅)

Appendix C:

̅

∑ (̅)

(̅) ( ) ∑ (̅)

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̅

( ̅)

(̅)

∑

̅ ̅ ̅ ∑ ̅ ( ) 4∫

̅ ̅ ∑ ̅ ( )∫ ̅

̅ ∑ ̅ ( )∫

∑ ( ̅ ))

(̅)

∑ ̅

(̅)

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̅

(̅)

∑

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(̅

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̅

In this section a review of the concepts of dominate stiffness, damping and inertia regions

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in frequency response function of linear single degree of freedom vibration system is considered. The frequency response of a one – degree freedom mass – spring – damper under an external harmonic force has the following, which is shown in the following Fig. A2.

√(

)

(

)

√(

)

(

)

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√

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Figure A.2: The concepts of dominate stiffness, damping and inertia regions in frequency response function of linear single degree of freedom vibration system In low dimensionless frequencies, for example in the domain of

,

, hence the force of the spring is the dominant force. Accordingly, in this region

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and

,

increasing or decreasing the stiffness of spring value, the amount of displacement can be

ED

decreased or increased as well. In damping dominant region, for example in the domain of , the force of damper is the dominant force. Adjusting damping coefficient in

PT

this region, the amount of displacement especially in the resonance frequency can be controlled. In this region vibration amplitude is approximately

. For

the

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inertia force is dominated where by increasing the actuating frequency, the amplitude of .

AC

vibration is approximately given by

Reference

[1] L. Cveticanin, Free vibration of a Jeffcott rotor with pure cubic non-linear elastic property of the shaft, Mechanism and Machine Theory (2005) 40: 1330–1344 [2] C. Wan, C. Jian, Non-linear dynamic analysis of dual flexible rotors supported by long journal bearings, Mechanism and Machine Theory (2010) 45: 844–866 40

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[3] Athanasios C. Chasalevris, Pantelis G. Nikolakopoulos, Chris A. Papadopoulos, Dynamic Effect of Bearing Wear on Rotor-Bearing System Response, Journal of Vibration and Acoustics (2013) 135: 011008-1 – 011008-12, doi:10.1115/1.4007264 [4]N. A. Saeed, H. A. El-Gohary, On the nonlinear oscillations of a horizontally supported

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Jeffcott rotor with a nonlinear restoring force, Nonlinear Dynamics (2017) 88: 293-314. [5] S. Popprath, H. Ecker, Nonlinear dynamics of a rotor contacting an elastically suspended stator, Journal of Sound and Vibration (2007) 308: 767–784

[6] X. Shen, J. Jia, M. Zhao, Nonlinear analysis of a rub-impact rotor-bearing systemwith initial permanent rotor bow, Journal of Applied Mechanics, (2008) 78: 225–240

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[7] Jawaid I. Inayat-Hussain, Bifurcations in the response of a rigid rotor supportedby load sharing between magnetic and auxiliary bearings, Meccanica (2011) 46:1341–1351. [8] Marie-Oceane Parent, Fabrice Thouverez, Fabrice Chevillot, Whole Engine Interaction in a Bladed Rotor-to-Stator Contact, ASME Turbo Expo 2014: Turbine Technical Conference

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and Exposition, Doi: 10.1115/GT2014-25253.

[9] G. Chen, Simulation of casing vibration resulting from blade–casing rubbing and its

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verifications, Journal of Sound and Vibration (2015) 361: 190 – 209. [10] F. Thiery, R. Gustavsson, J.O. Aidanpää, Dynamics of a misaligned Kaplan turbine with

PT

blade-to-stator contacts, International Journal of Mechanical Sciences (2015)99: 251–261

CE

[11] Hui Ma, Xingyu Tai, Qingkai Han, Zhiyuan Wu, Di Wang, Bangchun Wen, A revised model for rubbing between rotating blade and elastic casing, Journal of Sound and

AC

Vibration (2015) 337: 301–320. [12] C. Guo, M. A. AL-Shudeifat, A. F. Vakakis, L. A. Bergman, D. M. Mcfaral, J. Yan, Vibration reduction in unbalanced hollow rotor systems with nonlinear energy sinks, Nonlinear Dynamics (2015) 79:527–538. [13] S. Bab, S. Khadem, M. Shahgholi, Vibration attenuation of a rotor supported by journal bearings with nonlinear suspension under mass eccentricity force using nonlinear energy sink, Meccanica (2015) 50:2441-2460. 41

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[14] Javad Taghipour, Morteza Dardel, Steady state dynamics and robustness of a harmonically excited essentially nonlinear oscillator coupled with a two-DOF nonlinear energy sink, Mechanical Systems and Signal Processing (2015) 62-63:164–182. [15] Amirhassan Abbasi, S. E. Khadem, Saeed Bab, M. I. Friswell, Vibration control of a rotor suspension, Nonlinear Dynamics (2016) 85: 525–545.

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supported by journal bearings and an asymmetric high-static low-dynamic stiffness

[16] Masoumeh Parseh, Morteza Dardel, Mohammad Hasan Ghasemi, Steady state dynamics of a non-linear beam coupled to a non-linear energy sink, International Journal of Non-Linear Mechanics (2016) 79:48–65.

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[17] Maryam Taleshi, Morteza Dardel, Mohammad Hadi Pashaei, Passive targeted energy transfer in the steady state dynamics of a nonlinear plate with nonlinear absorber, Chaos, Solitons and Fractals (2016) 92: 56–72.

[18] N. A. Saeed, M. Kamel, Nonlinear PD-controller to suppress the nonlinear oscillations Mechanics (2016) 87: 109-124.

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of horizontally supported Jeffcott-rotor system, International Journal of Nonlinear

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[19] Jaroslav Zapoměl, Petr Ferfecki, and Jan Kozánek, Modelling of magnetorheological squeeze film dampers for vibration suppression of rigid rotors, InternationalJournal of

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Mechanical Sciences, (2017), 127, 191-197. [20] Saeed Bab, S. E. Khadem, Majid Shahgholi, Amirhassan Abbasi, Vibration attenuation of

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a continuous rotor – disk journal bearing system employing smooth nonlinear energy

AC

sinks, Mechanical Systems and Signal Processing, (2017)84: 128–157.

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Figures:

AC

CE

PT

ED

Figure 1: Rotor dynamic system.

43

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CE

PT

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Figure 2: Blade, disk and contact model.

Figure 3: Journal – bearing configuration. 44

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CE

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Figure 4: Schematic diagram of rotor with single TMD.

Figure5: Schematic diagram of rotor with single NES.

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(a)

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ED

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Figure 6: Schematic of on blade absorber (NES).

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(c)

AC

CE

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(b)

(d)

Figure 7: Time response of a four – bladed rotor (

47

,

,

,

) for

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Figure 8: Blade’s tip displacement of four – bladed rotor with

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.

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and

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b) NES(disk) – NES(blade)

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a) TMD(disk) – NES(blade)

Figure 9: Blade’s tip displacement of the four – bladed rotor with

AC

CE

PT

ED

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̅

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.

and

AC

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Figure 10: Lyapunov exponent of the first blade, FFT of the first blade and of the disk for in absence and presence of the absorber for

50

(b)

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(a)

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(c)

51

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(d) Figure 11: Time response of a four – bladed rotor (

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,

AC

CE

PT

ED

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.

52

,

,

) for

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), with

and

,

,

2.

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PT

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,

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Figure 12: Displacement of the blade’s tip for four – bladed rotor(

AC

a) TMD(disk) – NES(blade)

b) NES(disk) – NES(blade)

Figure 13: Blade’s tip displacement of the four – bladed rotor with ̅

53

.

and

AC

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Figure14 : Lyapunov exponent of the first blade, FFT of the first blade and of the disk for in absence and presence of the absorber for

54

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Figure 15: Displacement of the blade’s tip for four – bladed rotor, with

and

CE

PT

ED

M

2.

AC

a) TMD(disk) – NES(blade).

b) NES(disk) – NES(blade).

Figure 16: Blade’s tip displacement of the four – bladed rotor with ̅

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.

and

PT

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Figure 17: Blade’s tip displacement of four – bladed rotor with

AC

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̅

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.

,

and

b) NES(disk) – NES(blade)

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a) TMD(disk) – NES(blade)

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Figure 18: Blade’s tip displacement of four – bladed rotor with ̅

and

.

B)

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CE

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A)

,

Figure 19: Blade’s tip displacement of four – bladed rotor in presence of a) TMD(disk) – NES(blade), b) NES(disk) – NES(blade) with

, .

57

and

̅

PT

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AC

a)

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Figure 20: Blade’s tip displacement of four – bladed rotor with ̅

b)

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.

,

and

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Figure 21: Blade’s tip displacement of four – bladed rotor in presence of a) TMD(disk) – NES(blade), b) NES(disk) – NES(blade) with

,

and

̅

.

b)

AC

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a)

Figure 22: Blade’s tip displacement of four – bladed rotor in presence of a) TMD(disk) – NES(blade), b) NES(disk) – NES(blade) with

, .

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and

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Tables:

Table 1: Parameters of rotor dynamic system. )

( )

( )

(

)

( )

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(

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√

Table 2: Dimensionless values of the absorbers TMD – NES ̅

ED

M

(

̅

NES – NES (

)

̅

CE

PT

̅

)

AC

Table 3: Dimensionless values of the absorbers

̅

TMD – NES (

)

̅

NES – NES ̅

(

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)

̅

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Table 4: Dimensionless values of the absorbers TMD – NES ̅

(

̅

)

NES – NES )

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CE

PT

ED

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(

61

̅

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̅