Estimating Influence of Inertial Resistance of Throttle for Hydraulic Balancing Device on Rotor Axial Vibration 2012 Procedia Engineering

7/30/2019 Estimating Influence of Inertial Resistance of Throttle for Hydraulic Balancing Device on Rotor Axial Vibration 2012

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Procedia Engineering 39 ( 2012 ) 261 274

1877-7058 2012 Published by Elsevier Ltd.doi: 10.1016/j.proeng.2012.07.033

XIIIth International Scientific and Engineering Conference HERVICON-2011

Estimating Influence of Inertial Resistance of Throttle forHydraulic Balancing Device On Rotor Axial Vibration

A. Korczak a, V. Martsynkovskyy b, S. Gudkov c, c*aSilesian University of Technology, Faculty of Energy And Environmental Engineering, St. Konarski 18 Street, 44-100 Gliwice,

Poland b,cSumy State University, Department of General Mechanics and Dynamics of Machines, 2, Rimsky-Korsakov Street,

Sumy 40007, Ukraine

Abstract

There had been considered axial vibration of the rotor equipped with an automatic balancing system for axial forces.There were constructed amplitude and phase frequency characteristics, as well as evaluated influence of inertialresistance into throttling channels of auto discharge system. There were identified damping properties of resistancedevices.

2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Sumy State University

Keywords : Rotor; dynamics; hydraulic balancing device; throttles; inertial resistance; vibration amplitude; critical frequency.

1. Introduction

While operating, the centrifugal pumps with automatic balancing systems sometimes expose higheraxial vibration of the rotor [1], which can be explained either by resonance in the system of rotor autocharge, or self-exciting oscillations due to loss of the system dynamic stability. The rotor axial vibrationgives rise to the significant stress pulsations in the relief disk and in cross-section of the shaft, and may

also cause the elevated transverse vibration of the rotor. In this regard, calculating the amplitude andphase frequency characteristics of the balancing system and testing its dynamic stability are essential toensure the reliability of high-speed and high-pressure pumps.

* Corresponding author. Tel.:+38-0542-333594; fax:+38-0542-333594. E-mail address : [email protected].

Available online at www.sciencedirect.com


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262 A. Korczak et al. / Procedia Engineering 39 ( 2012 ) 261 274

At calculating the dynamic charalumped-parameter system making axwhich the corresponding steady-statethe static calculation [1, 2].

2. Equation of rotor axial vibration

Based on Newton's 2nd Law, the eq

,+=++ k T F kz zc zm z

where: k is an adjusted (reduced) sttheir previous compression; T is

( ) 22 5,0 p A AF c z += is a balancing p

Fig. 1. Design scheme for balancing system

There is performed dividing of all

20* pk n= and go over to dimen

+=++ uuT uT 2

21 ,

0

22

0

221 ,, =

=

= pk

pc

T pm

T

n

z

p AF 2

20

,+

===

teristics, the rotor with the balancing device is coial oscillations relative to the position of static eqalues of pressure, face clearance and flow rate are

uation of axial vibration can be written as follows (Fi

iffness of the pressing-out device elastic elements; an axial force being balanced while acting oressure force acting on the balancing disk.

the terms of the equation by the conditional hydrossionless variables:

00

2 ,

=

pk

,20

, H z

u p AT

n

== ,

2

2,5,

= .

sidered as ailibrium, foretermined by

. 1):

(1)

is a value of n the rotor;

tatic stiffness

(2)

(3)


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where: m - mass of the rotor; c - coefficient for damping the rotor axial vibration; , thedimensionless axial forces, the dimensionless previous stress of the pressing-out device; 0 - thearea of the impeller inlet funnel;

- nominal pump discharge pressure.

On introducing the operator of differentiation with respect to time, dt d p = , there is written equation

of axial vibrations in the operator form:

,)(1 +=u p D

(4)

,)( 222

11 pT pT p D ++=

is an own operator of the rotor axial vibrations.

3. Equations of unsteady flow in throttles

The equation of unsteady turbulent flow in the i-th throttle can be written as:

ir iaiiiii p Q J Q R p +=+=2 ,

where 1+= iii p p - full pressure drop on the th throttle,2iiia Q R p = , iiir Q J p = - pressure

expenditures to overcome effective resistance and reactance, iQ - instantaneous flow rate, i R and i J - the

coefficients of effective friction resistance and reactance (inertial resistance) of the fluid ( )2,1=i .Effective resistance at each instant of time is determined under the formulas for steady flow,

neglecting its weak dependence on the frequency of the pressure pulsations [3]. For the turbulent flow,

the effective resistance coefficients make 2 / 1 ii g R = , where the conductivity of the ring and facethrottles is computed under formulae [2]:

( )[ ] 5,0111111 2 / 5,02

+= l Rg ,

(5)

( ) ( )[ ] 5,0222322 25,0

++= zl z R Rg ,

,04,0 06,02

- friction coefficients for self-field of the turbulent flow for the annular and faceclearances, respectively; 21 =1,01,15 coefficients of local (input) resistances; 232 R Rl = .

Reactance is determined in the accordance with the Law concerning rate of change of momentum iK :

ir iiiiiiiiii p f F Qlv f lK F

dt dK

==== ,, ,


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,iiir i Ql p f = iiii

iir Q J Q f

l ==

.

On comparing the left and right sides, we here obtain

, / iii f l J = (6)

where; - momentum coefficient taking into account the uneven distribution of the fluid velocity over thechannel cross section; ii l f , - cross-sectional area and length of the i-th channel; for laminar flow in a circular

channel ,2,1= and for the turbulent flow 1 ; ;

fvQ =

v - average speed over the section f .With the account of the flow rate expression iaii pgQ = the pressure losses to overcome the

inertia of the fluid take the form:

)( iaiiir pgdt d J p = = 1** , += iiiaii p p p p p ,

where

).3,2,1(),( 1` === + i p pgdt d

J p p p p iiiiiir ii

Provided that inertia is not taken into account, iiir p p == *,0 . The last expressions are nonlineardifferential equations of the first order concerning pressure .i p

Henceforth we herein restrict ourselves to consideration of small deviations of variables in comparisonwith their steady-state values, which fact allows passing to the linearized equations (to the equations in

variations):

.)(2

)(01

1001

+=

+

++

ii

iiiiiiiii

p p

p pg p pg

dt d

J p p

Subscript "0" indicates the values of the variables in the steady (equilibrium) state defined by a static

calculation. In the steady state 0=Q , 0= ir p 00 ii p p = , therefore:

),(5,0 10

0

0

0+

=ii

iii

iiii p p

p

Q J g

g

Q J p p

000000)1(00 , pg pgQ p p p eiiiii === + . (7)


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The conductivity of the cylindrical throttle at a constant eccentricity (the rotor radial vibrations are not takeninto account) is independent of the axial displacement of the rotor and keeps the constant value, so .01 =gThe conductivity of the face throttle is determined by the second formula (5) and taking into account that underthe steady condition .01 =g , the variation of its time derivative is reduced to the form:

uugg n5,0

22 5,1= , .5,15,10

205,0022 uu

guugg n ==

Further, to shorten the description, the signs of variations are omitted. After passing to dimensionlesspressures

/ p p ii = and clearances , / 2 zu = the equations (6) are reduced to the form:

uT T T T 1222221221111111 , =++=+ ,

or in the operator form:

( ) ( ) pT D pT D 12222122111111 , =+= ; 1)( 11 += pT p D ii , (8)

where

0

01 2 i

ii p

Q J T

= , i

ii

i

p p

p p

==

0

0 , ,ko

io

k ik i

i

p p

u

===

=

10

00

0i ,

33. (9)

Provided that inertial resistance is not taken into account, iiiii p p DT J ==== *11 ,1,0 .

4. Flow rate balance equation

Equations (8) contain unknown pressure i and pressure 2 into the hydraulic balancing devicechamber. To calculate these pressures, there is used the flow rate balance equation, which in contrast tothe static equations 02010 QQQ == must take into account the flow rate values for displacement

zQv

=2 and for compression E V pQ p / 222 = , where 2V is volume of the chamber, and E is an

adiabatic bulk modulus of fluid. Thus, the flow rate balance equation takes the form

2221 pv QQQQ ++= or E V p z pg p pg / 22223

2211++=

. (10)

The last equation is nonlinear (the flow rates 321 ,, QQQ are nonlinear), so it is necessary to linearizethem, transiting to the equations in variations. Taking into account that in equilibrium position

200*21001 , p p == , we herein omit the signs of variations:

( ) .22

3

2 22

*220

0

0

02*1

10

0 p E

V u z

p

Qu

u

Q p p

p

Qn

++

+=


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Having multiplied this equation by 0102 Q and after transition to dimensionless pressures takinginto account designations (9), we herein obtain:

u pQ

p zu

EQ pV

n

n

0

1012122

0

1022*1

22 +++

=

.

Further, we introduce designations of time constants and differential operators:

0

022

20

0

n323

10

0

n222 3

2,

2,

2Q

z p p

EQ pV

T p p

EQ pV

T nn

=

=

= ,

(11)

1)(,1)( 22222222 +=+= p p M pT p D ,

and we reduce the flow rate balance equation to the following form:

u p M p D )()( 2212121222 = . (12)

From equations (8), it follows:

).(1

),(1

122212

2211111

1 puT D pT

D =+=

Having Substituted these expressions into (12), we herein obtain an equation concerning 2 :

( ) ( ) u M puT D

pT D

D 221122212

12211111

22211 += .

Then we bring it to a common denominator and group the members

( ) ( )u pT M D D D D pT D D D D 122212111112211121112221211 =+ . (13)

Based on (3) 2 = , therefore, having lettered the operators of a controller and a device detectinginfluence by mistake

( ) ( ) ( )u pT M D D p M D pT D D D D p D

12221211111211122212112=+=

2; , (14)

we herein write the equation of a controller in the following form:

( ) ( ) ( )u p M p D p D 211122 = . (15)

Further we perform the multiplication in expression (14) and group the addends in powers of p :


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( ) 32213023221302 ;)( b pb pb pb p D +++=+++= ,.

where

( ) ;1,,, 12322121112212112212212110 +=++=+== T T T T T T T

(16)

1,),(, 32211212112212212110 =+=+== bT bT T bT T b .

5. Controller transfer function

From equation (15), it is possible to determine the transfer function with the error, that is, with thedeviation of the clearance (of the controlled variable) from its steady-state value. With due account of theexpressions of the operators (15), the controller transfer function with the error takes the form

( ) ( )( ) 32213032

21

30

12

21

b pb pb pb

p D

pW u ++++++=== . (17)

In the case of the harmonic effects, it represents the frequency transfer function or dynamic stiffness of the controller, which allows at the first approximation to estimate the natural frequency of axial vibrationsof the rotor and the stability of the system as well.

For the steady-state condition, 0== dt d p and the transfer function of the controller become thecoefficient of the hydraulic balancing device static stiffness:

( )12

13

31 1

10

+===

ab

W su . (18)

The dynamic stiffness possesses important informative features: it allows setting the range of variations for the natural frequencies of the rotor axial vibrations and at the first approximation, toevaluate the system stability.

As for the harmonic signals, which are considered in the frequency transfer functions, thedifferentiation operator i p = . On introducing such a change, we herein obtain the frequency transferfunction of the controller or a complex dynamic stiffness of the system:

( )( )

( )( )022123

02

212

31

2

21)( aaiaa

bbibbi Di M

iW u

+

+=== . (19)

At 0= the last expression represents the controller dimensionless static stiffness: .)0( suW = At

22

221

0

01)( T a

biW u

=== . (20)


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We herein obtain the maximum modulus value corresponding to the stiffness of conventionally non-flowing part, the stiffness of which is only caused by the compressibility of the fluid in the chambers of the hydraulic balancing device.

Further, we herein disjoint real and imaginary parts in (19).

[ ]V iU iW uuu )()()( 1 +== , (21)where

( )( ) ( )( )( ) ( )

( )( ) ( )( )( ) ( )

.

,

20

22

221

23

12

302

202

212

3

20

22

221

23

02

202

22

12

312

3

aaaa

bbaabbaaV

aaaa

bbaabbaaU

u

u

+

=

+

+=

(22)

If the real and imaginary parts are positive 0,0 >> uu V U , they respectively represent the positiveadditions to the dimensionless stiffness

of the pressing out device and to the relative external

damping 2 . The negative values uU and uV destabilize the system, and while 2T V u = , the system is

at the oscillatory boundary and while nuU = , the system is at the aperiodic stability boundary. In theabsence of the external damping ( )02 the rotor stabilization is provided by damping the controller.The condition 0>uV comes to the inequality

( )( ) ( )( )0123022022123 > bbaabbaa ,

which can be used for preliminary (with some allowance) assessing the stability of the system. Inparticular, for the pivot without additional external throttle and not taking into consideration the fluidinertia, the condition 0>uV is reduced to form [1]

21

22222

+

> T or 2010

0

0

2

3 p p p E

zV

e

Documents

Estimating Influence of Inertial Resistance of Throttle for Hydraulic Balancing Device on Rotor Axial Vibration 2012 Procedia Engineering