Active Vibration Reduction of Rigid Rotor By

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J. ONDROUCH, P. FERFECKI, Z. PORUBA

ACTIVE VIBRATION REDUCTION OF RIGID ROTOR BYKINEMATIC EXCITATION OF BUSHES OF JOURNAL BEARINGS

Received Prispjelo: 2008-12-24Accepted Prihva}eno: 2009-08-14

Original Scientific Paper Izvorni znanstveni rad

INTRODUCTION

Centrifugal forces caused by the unbalance are themain source of vibration in rotating machines. The per-

fect balancing is very expensive, or it is not possible atall. Besides the unbalance distribution can vary in time,caused by abrasion or by support modifications. To sup-

press excessive vibration, higher damping is used firstby passive tools such as absorbers and dampers, or byactive tools are mostly used.

Nowadays, the use of magnetic bearings is proposedfor the active control. In other cases, the use of differenttypes of actuators such as pneumatic, hydraulic, electro-magnetic [1, 2] was proposed for the active control.However, no one of them can show significantly better

properties.Recently piezoelectric actuators are used for active

control frequently. Very promising results wereachieved in the case of active control of a rotor sup-ported by rolling bearings [3].

The aim of this work is to obtain the basic knowledgefor the adjustment of the test stand designed to investi-gate the possibilities how the rotor vibration can be re-duced by kinematic excitation of bearing bushes usingthe piezoelectric actuators. The actuators are located inhorizontal and vertical directions where the direct kine-matic excitation of the bushes is supposed (Figure 1).

ANALYTICAL STUDY AND INSTABILITY

LIMIT DETERMINATION

The investigated rotor system has the followingproperties: (i) the rotor is symmetric and rigid, (ii) therotor is supported by two identical journal bearings andthe oil-film forces are determined using the solution ofReynolds equation for cylindrical, short and by cavita-tion influenced bearing, (iii) a static unbalanced shaft isassumed, (iv) the centrifugal force caused by unbalanceacts in the symmetry plane and (v) the rotor rotates at aconstant angular velocity.

Under the given assumptions the motion equationsof the rotor system lateral vibration in a stationary coor-dinate system x, y, z(Figure 2) can be written as:

METALURGIJA 49 (2010) 2, 107-110 107

Possibilities of active lateral vibration reduction of a symmetric, rigid rotor supported by journal bearings are gi-

ven. They were obtained by computational modelling. Efficiency of the feedback P and PD controllers in the sta-

ble revolution interval was examined. The linearized rotor system model was used. The results of the theoretical

analysis are assigned for a testing stand where the bearing bush motions are deactivated by piezoelectric ac-

tuators connected to the controllers.

Key words: bearing, controller, rotor system, vibration reduction

Aktivno prigu{ivanje vibracija krutog rotora kinematskom pobudom blaznica kliznih leàjeva.

Ovaj rad prikazuje mogu}nosti aktivnog prigu{ivanja lateralnih vibracija simetri~nog, krutog rotora sa kliznim

leàjevima kori{tenjem ra~unalnog modeliranja. Istraèna je u~inkovitost povratne sprege P I PD kontrolnih

sklopova u stabilnom okretnom interval. Kori{ten je linearizirani model rotorskog sustava. Rezultati teoretske

analize bit }e kori{teni za rad na ispitnom stolu, gdje se pokreti leàjnih blazinica kontroliraju piezoelektri~nim

pokreta~ima spojenim sa upravlja~kim sklopovima.

Klju~ne rije~i: leàj, upravlja~ki sklop, sustav rotora, prigu{ivanje vibracije

ISSN 0543-5846

METABK 49(2) 107-110 (2010)UDC UDK 669.14-418:539.37:620.17=111

J. Ondrouch, *P. Ferfecki, Z. Poruba, Department of Mechanics, *Struc-tural Integrity & Materials Design, VSB Technical University ofOstrava, Czech Republic

Figure 1. Rotor system control layout

DSpace V B-TUO http://hdl.handle.net/10084/77574 18/06/2012


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my F y y z z y y z z me t y J&& ( & , & , & , & , , , , ) cos( )J J B J B J B J B= + w w2 ,

mz F y y z z y y z z

me t

z

J

&& ( & , & , & , & , , , , )

sin(

J J B J B J B J B= +

+ w w2 )- mg, (1)

where yJ, zJ, &yJ , &zJ and &&yJ , &&zJ are displacements, speedsand accelerations of shaft journal centre in the horizon-tal and vertical vibration plane, respectively (Figure 2),yB , zB and &yB , &zB are displacements and speeds of bear-ing bushes centre in the horizontal and vertical vibration

plane, Fy is the horizontal and Fz vertical component ofthe oil film force, m is the shaft mass, eJ is the shaft un-

balance, is the angular velocity of the shaft rotation, tis time andgis the gravitational acceleration. In the caseof an uncontrolled rotor system yB , zB , &yB , &zB are equalto zero.

After the linearization of the nonlinear oil-filmforces in the surroundings of the instant revolutions [4]the obtained equations of motion around the static equi-librium position will be the following:

mz B z B y K z K y B z

B y K

zz zy zz zy zz

zy yy

&& & & &

&

J J J J J B

B

+ + + + = +

+ + z K y me t zyB B J+ + w w2 sin( )

,

my B y B z K y K y B y

B z K

yy yz yy yz yy

yz yy

&& & & &

&

J J J J J B

B

+ + + + = +

+ + y K z me t yzB B J+ + w w2 cos( )

(2)

where

BF

yB

F

z

BF

yB

F

z

yy

y

yz

y

zy

z

yy

z

= =

= =

&,

&,

&,

&

, (3)

KF

yK

F

z

K

F

z K

F

y

yy

y

yz

z

zz

y

zy

z

= =

= =

, ,

,

(4)

are the damping and stiffness coefficients.The linearized motion equations can be used for the

determination of the stability limit of the rotor system,static equilibrium position and they are suitable for thesolution of small vibration in case of rotor revolutionsunder the limit of the whirl-type instability.

To determine the stability limit the Hurwitz stabilitycriterion can be used. From the linearized motion equa-tions (2) with zero right hand side, the characteristicmultinomial of the s argument can be obtained in theform:

N s a s a s a s a s a

a m a m B B

a B

p

yy zz

( )

, ( ),

= + + + +

= = +

=

44

33

22

1

42

3

2 yy zz yz zy zz yy

zz yy yz zy yy zz z

B B B m K K

a K B K B K B K

- + +

= - + +

( ),

1 y yz

yy zz yz zy

B

a K K K K

,

,0 = -

(5)

and the Hurwitz determinant

H

a a

a a a a

a a a

a

=

1 0

3 2 1 0

4 3 2

4

0 0

0

0 0 0

(6)

NUMERICAL SIMULATIONS

Data corresponding to the test stand of the company

TECHLAB, Ltd. were used [5]. The journal bearingswere approximated as short bearings with following pa-rameters: bearing length 15 mm, bearing radius 15,01mm, shaft journal radius 14,97 mm, dynamic oil viscos-ity 0,004 Pas and ambient pressure 0 Pas.

According to the Hurwitz stability criterion (6) therevolutions at the stability limit 12980 rpm were deter-mined.

After the calculation of bearing forces and after theirlinearization the damping and stiffness coefficients forrevolutions under stability limit were obtained.

To examine the possibilities of the active reductionof rotor lateral vibration the uncontrolled rotor systemmodel and models of rotor systems with feedback P, andPD controllers (Figure 3) were created in the softwareMATLAB-Simulink. The input data for the revolutionsof 2000 rpm, 4000 rpm and 6000 rpm and for unbalancemass of 0,0001 kgm can be found in Table 1.

Two time dependences of obtained vibration resultsin vertical direction at 4000 rpm for the rotor system,one without control and the other with a P controller (kp= 50) are compared in Figure 4. (mind the big difference

108 METALURGIJA 49 (2010) 2, 107-110

J. ONDROUCH et al.: ACTIVE VIBRATION REDUCTION OF RIGID ROTOR BY KINEMATIC EXCITATION OF BUSHES...

Figure 2. Rigid rotor system with journal bearings

Table 1. Parameters of the investigated rotor system

Parameters

Revolutions

2000 4000 6000

/rpm /rpm /rpm

Byy /k g/s 4,984103 4,844103 4,815103

Byz /kg/s -1, 129103 -5,716102 -3,820102

Bzy /kg/s -1, 129103 -5,716102 -3,820102

Bzz /kg/s 5,535103 4,989103 4,880103

Kyy /kg/s 2,361105 2,393105 2,400105

Kyz /kg/s 4,993105 1,002106 1,505106

Kzy /kg/s -6,023105 -1,057106 -1,541106

Kzz /kg/s 1,365105 1,247105 1,223105

mew2 /N 4,40 17,50 39,40

m /kg 0,3875



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in displacement scale). The vibration results in horizon-

tal direction, at 4000 rpm for the rotor system withoutcontrol and with PD controllers (kp = 50, kd = 5 s) arecompared in Figure 5. The orbits fora rotor systemwith-out control and the one with P controller are compared,(in different scales), in Figure 6.

The summary of the obtained results is in Table 2.

CONCLUSIONS

The performed analytical study and numerical simu-lations have shown, that by active kinematic excitationof the bushes of the bearings, it is possible to reduce sig-nificantly the rotor lateral vibration, in the case of a rotor

METALURGIJA 49 (2010) 2, 107-110 109


Table 2. The amplitudes of steady-state, forced vibra-tion and the vibration centre position

ControllerRevolu-

tions

/rpm

Direction

/-

Magnitu-de

/m

Equili-brium po-

sition/m

Withoutcontroller

2000y 8,210-6 6,910-6

z 8,210-6 -1,610-6

4000y 1,710-5 3,710-6

z 1,710-5 -4,410-7

6000y 2,610-5 2,510-6

z 2,610-5 -2,010-7

P

controller

kp = 50

2000y 1,610-7 2,410-7

z 1,610-7 -8,410-7

4000y 3,410-7 2,310-7

z 3,410-7 -7,410-7

6000 y 5,110

-7

2,610

-7

z 5,110-7 -7,010-7

PD con-troller

kp = 50

kd = 5 s

2000y 2,810-8 2,410-7

z 2,810-8 -8,410-7

4000y 5,610-8 2,310-7

z 5,610-8 -7,410-7

6000y 8,510-8 2,610-7

z 8,510-8 -7,010-7

Figure 3. Rotor system with two dimensional control cir-cuit

Figure 4. Time history of the displacement in the verticalvibration direction of the uncontrolled rotor sys-tem and thecontrolled onewith P feedbackcon-troller

Figure 5. Time history of the displacement in the horizon-tal vibration direction of the uncontrolled rotorsystemand thecontrolled onewith PD feedbackcontroller



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supported by journal bearings excited by the unbalancein the range of working revolutions, i.e. under the insta-

bility limit.In the case of a two-dimensional control circuit with

two identical P controllers there was a reduction ofsteady-state vibration amplitude in both directions. For2000 rpm, 4000 rpm and 6000 rpm, and for the chosen

controller constant of kp = 50 the vibration amplitudewas smaller approximately 51 times. In the case of twoidentical PD controllers the reduction rate was approxi-

mately 300, for the steady-state forced vibration ampli-tude at 2000 rpm, 4000 rpm and 6000 rpm, with the usedcontroller constants kp=50 and kd=5s.

REFERENCES

[1] A. El-Shafei, A. S. Dimitri: Controlling Journal BearingInstability UsingActive Magnetic Bearings, In Proceedingsof ASMETurbo Expo2007, PaperGT2007-28059, Montre-al, Canada, 2007.

[2] O. Bonneau, E. Lecoutre, J. Frne: Dynamic behaviour of arigid shaft mounted in an active magnetic bearing, In Proce-edings of ISROMAC-7, Honolulu, Hawaii, USA, 1998,30-37.

[3] A. Muszynska, W. D. Franklin, D. E. Bently: Rotor ActiveAnti-Swirl Control, Journal of Vibration,Acoustic,Stressand Reliability in Design, 110 (1988) 2, 143-150.

[4] E. Krmer; Dynamics of Rotors and Foundations, Sprin-ger-Verlag, Berlin, 1963,

[5] J. [imek: Device for bearing active control with the aim tosuppress the rotor instability, Research report nr. 07-405,TECHLAB Ltd., Praha, 2007, in Czech.

Acknowledgements:This research work has been supported by the grant

No.101/07/1345 of the Grant Agency of the Czech Re-public, by the research project MSM6198910027 of theCzech Ministry of Education and by the institutional re-search plan No.AVOZ20760514 of the Institute ofThermomechanics of the Academy of Sciences of theCzech Republic. The help is gratefully acknowledged.

Note: The responsible translator for English language is Ing. LadislavKovac, IMR SAS Kosice, Slovakia

110 METALURGIJA 49 (2010) 2, 107-110


Figure 6. The orbits for the same dates for rotor systemwithout control and with P controller


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Active Vibration Reduction of Rigid Rotor By