Unbalance behaviour of a multi-disk rotor with mass eccentricity, residual shaftbow and disk skew

Unbalance behaviour of a multi-disk rotor with mass

eccentricity, residual shaftbow and disk skew

Citation for published version (APA):

Crooijmans, M. T. M. (1983). Unbalance behaviour of a multi-disk rotor with mass eccentricity, residual shaftbow and disk skew. (DCT rapporten; Vol. 1983.020). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1983 Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

UNBALANCE BEHAVIOUR OF A MULTI-DICK ROTOR WITH MASS ECCENTRICITY, RESIDUAL SHAFTBOW AND DISK SKEW

by Marc Crooijmans

Eindhoven

21 november 1983 WE 83.20

These investigations were supported (En part) by the Netherlands Foundation

IOE

Technical Research (STW), Future Technical Science BranchlDivision o f the Netherlands Organization for the Advancement o f Pure Research (ZWO).

-2-

Contents

Introduction

...

page 3 The rotor model

...

p age 5

Discrete modelling

...

p age 9

Equations of motion

...

p age 7 6 Residual shaft bow

...

p age 20 Solution of the equations of motion

. . .

p age 21 Computer program and examples

...

p age 22

A trend which can be observed in the design of rotating machinery is the need for larger capacity and higher speeds. Therefore, in order to save the costs and increase the reliability, ik is necessary -to have a good un- derstanding of the dynamic behaviour o € the rotor. An important part o€ the dynamic behavour is due to unbalances. Several authors [I],

[a],

[ 3 ] , [4],

[5] have reported on this subject. They all build simple o r more soghis- ticated rotor models ia order to describe the unbalance behaviour numerically and verify their rotor models with experimental results. One main feattire of their research is the improvement of balancing procedures. Another feature is the description o f several unbalances which car, not be avoided such as thermal deformations, magnetic force influences ,material losses, manufacturing tolerances etc. All o f these unbalances may produce large displacements which can result in damage OE the machinery.

Basically, unbalance behaviour occurs when the rotor is rotational non-symmetric. This casses forces and moments that are perpendicular to the central axis o f the rotor. The unbalance forces and moments rotate with the same speed as the rotor and tkeir magnitudes are proportional to the square of the rotox speed.

WE will discuss some unbalances and their deviations with respect to

the ideal rotational symmetric rotor.

Firstly, the rotor mass can be distributed unequally with respect t o the shaft centre line. Examples are rotors with eccentrically assembled rotor- parts, shafts with unequally distributed material density and rotors that are significantly damaged. This effect can be modelled by applying an ec- centric mass in the F d e d r^oGel.

Secondly, a rotor part, for example a blade-vane can be mounted without making right angles with the shaft centre 1ine.We will call this kind of unbalance disk skew. Due to this unbalance so called gyroscopic effects will influence the rotor behaviour. We will show that these gyroscopic effects can be modelled by the disk skew angle and the difference between polar and equatorial moments of inertia.

-4-

The last example o f an unbalance is the residual shaft bow. It can be tem- porari.ly caused by thermal or magnetic effects or permanently due to manufactoring inaccuracy. The model parameters o f this unbalance are the displacements o f the shaft centre line with respect to the ideal straight line through the bearing centres.

Large displacements al" the shaft perpendicular to the rotor centre line can occur both in the neighbourhood of the critical speed and at very high speeds. When the rotor speed comes near the critical speed, damping and several non-linearities will restrict the shaft displacements and determine rotor behaviour. When very high speeds are reached all kinds of instability mechanisms will determine rotor behaviour. In this case, the influences a f

the unbalances will be small.

In this report we will not investigate the balancing process itself but we will concentrate on the behaviour o f the unbalanced rotor. We do not consider non-linearities in the system. We will assume that the rotor behaviour will be stationary and synchronously which will syrnplify D E S

method of solutlon. Special attention will be paid to the formulation o f the equations of motion and to the verification o f the results obtained in recent literature. This was done in order t o study the assumptions that are usually and to obeserve t h e complications when the problem have t o be d w s J L c u û ü t .

made

The rotor model

The rotor model consists of a number of shaft elements and disks and is mounted in bearings. The shaft element with element number

i

will be located between the structural nodes i and (i+l). A node can be the starting- or ending point of the rotor shaft or it can be the connection n n i n f between two adjacent shaft elements. This: i s indicated in figure

1 .

fig.1: Example of a rotor with 4 disks, 9 nodes and 8 shaft elements.

For the shaft elenierits we use Bernouilli-Euler beams: the mass o f the shaft elements will be included in the model. We introduce a space-fixed reference

( x , y , z ) with vector basis

(z

,e

,e

1

with

e

colinear and coincident with the undeformed rotor centre line. Each node i has four degrees of freedom: two small displacements ui and u and two small rotations ui and ui with respect to the space-fixed reference [figure 2).

X Y Z z

i

x Y ipx ipY

The upper index i refers $9 t h e nGdal-pûint-nümbes in question. The

influence of the displacements in the z-direction will be AegleCted. Each disk centre has to coincide with one of the nodal-points i, j u s t as each bearing location. These four degrees o f freedom for a nodab point and the rotor angular speed about the z-axis determine the translation and rotation of a disk or the position of the shaft at a bearing location.

I

-5-

Fig.2: A ) Displacements and 8 ) rotational degrees of freedom for nodal point i of the rotor model.

n t

E i g . 3 : Bearing model.

The disk parameters are the macs (m) and the central body fixed axial and

equatorial moments of inertia (I and I

1 .

These moments of inertia enable

theory).

The centre of the bearing coincides with one of the nodes (i). We will assume that the bearings are isotropic and the situation in a plane is shown by fig. 3 . In this figure we use the spring with stiffness k the spring and the damper are assumed to be linear.

with the stiffness k and the viskous damper d. The stiffnesses cpf

In the rotor model we take into account three kinds of unbalance:

1) an eccentric mass centre of the disk with respect to the ~

shaft centre line,

2 ) a skewed disk and

3 ) a shaft with residual bow.

The eccentricity of the disk is represented by the eccentricity vector

8

with amplitude E . This vector rotates in the x-y-plane of the fixed

reference with angular velocity w . At t=O the angle between this

vector and the x-axis will be a (fig 4). constant

Y

The disk skew is

rotating in the fixed reference with angular velocity w . The amplitude o f

this rotation vector is T and at t=û the angle between the rotation vector

and the x-axis will be fi (fig.5).

- 8 -

Y

fig.5: Rotation vector €or disk skew at t = O .

At time t the matrix representation E and T of the vectors

E ( t )

and

P ( t )

with respect to the fixed reference can be found easily:

v1 VI

The residual bow of the shaft will be represented by the displacements uir

1 1 ir ,,ir ,,ir _{f û ï} each nodai _point i. x f

Discrete modelling (of the disk).

In this section we will derive a set of equations of motion f o r a single disk. As mentioned earlier, we assume that the whole rotor-system will rotate with a constant angular velocity w with respect to the fixed z-axis. Firstly, we will concentrate on the rotations of the disk and secondly on the translations, both with respect to the fixed reference (x,y,z). Finally

those will be combined into two equations of motion with complex

variables describing the dynamic behaviour of the disk.

For the derivation of the equations of motion for the rotation of a

1 1 1

we introduce a second reference basis (x r y ,z I . This basis will have equations

disk the

plane of the disk. The rotation of the disk with respect to the reference 1

basis (x ,y , z I can be described by a single rotation about the z

same origin as the fixed basis. The x 1 and y 1 axes will remain in the

1 1 1 _axis.

ykly

reference ( x

1 1 1

, y , z 1 w i t h -? 1 X X

-?O-

With the assumption thût the angülar rotations u and u but also the

i n i t i a l disk zikzw given by the vector T are small, enough, we can use a

linearized theory and describe the rotation of the d i s k wi-izh zespect t o the reference basic (x,y,z) with the small Bryant angles 6 and 6 It can easily be seen that 6

3 IPx IPY

1 2 '

and 6 2 will be:

1

The rotation of the disk with exception of the rotation with constant an- gular velocity w with respect t o the z-axis will be called "base-

rotation" .This base rotation can be described by -the rotation vector ;. The matrix-representation of this vector i t ~ l with respect to the f i x e d reference will be :

This rotation is associated with a rotation matrix Ii:

where - I = 1

i,

is a skew-symmetric ~~atrixrepresentation of a tensor which just as the vector 6 completely describes the base rotations of the diz.k. For an

arbitrary with respect to the

fixed and the rotating reference we have: -9

1

5

= a.x

vf

= g .x VI 1 t VI +

We will now write the total angular velocity vector w as the sum of the base rotation vector wb and its relative rotation vector wr. For the matrix representation of these vectors with respect to the fixed reference

za,

yfb

and yfr we have:

i

with :

For the equation o f motion we start with the balance of angular momentum with respect to the fixed reference:

In this equation the symbol J is used €or the matrix representation o€ the moments o f inertia “censor 3 with respect to the fixed reference (x,yiz). The problem is now that the matrix J is a function of time due to the rotation 1 of the disk. It should be easier to deal with the matrix representation

2

of this tensor with respect t o the (x , y Iz ) reference because this matrix is not a function of time (The constant rotation w does not influence the moment of inertia).

For matrix representation D of the angular moment of momentum vector

8

with respect to the fixed reference we may write according to ( 5 ) :

1 1 1 the v1 1 B =

R.2

VI

-12-

where D1 is the matrix representation o f

6

with respect t o the reference (x ,y

1 1 -1

, z ) . Substituting ( 9 ) in ( 8 ) gives:

- 1

d 1

5

=

R.:

-k

R . Z ( O

) (10)

1

Now

0

can be written as:

1 1

Dl = J .W

v1 -a

Because the matrix 2 1 is not a function of time we get:

* 1

1 '1 -a

M = B.D -i-

R.3

. w

v) v1

We now return to the fixed reference by using:

( 1 2 )

which results in:

- i i . t *

= R.R .D _VI -i- R.2 .E .w _-.a í 1 4 )

- t

For R.& we find the skew symmetric matrix ( w ) with associated vector tub:

-b

The transformation for the matrix J is given by:

1 t

-

Substitution o f ( 1 5 ) and ( 1 6 ) in ( 1 4 ) leads to:

fn

= (w

).Z.w

4- 2.:

-b -a -a

The unknown factor in this equation is the matrix representation

2

of

the moment of inertia tensor with respect t o the fixed reference. For the only representation J 1 we have: 3.

-

3 = O O J P Using ( 1 6 ) this gives:

1

J Z J

for small values of 6 1 and ö 2 . So finally we g e t :

i4 = (g

j.2.w t2.w

YI b -a -a where:

( q )

=

1

O O ti2 I

o

o

-6

[-a,

i l

o ,

ö = -w.~.sin(wttfl)iu 1 ipx O O J (20) 62 = w.r.cos(wttf3)+u @Y

-14-

Substitution of (21) in (20) finally leads t o the equations of motion for the rotation of the disk:

2 M = J U tJ W.U -W .T.(J -J ) . c ~ s ( w ~ + @ ) x t cpx P cpY t P M = T " -J .w.U -w2.~.(J -J ).sin(wt+@) Y ot'ucpY P cpx t P Mz =

o

(22b)

We can disregard the equation for the rotation with respect t o the z-axis which does not yield relevant information. After introducing the definition of the complex quantities:

u = u +j.u

tB cpx cpY M = M + j . M

X Y

j = J(-l)

we can write the two equations of motion (22a) and (22bj as one complex equation of motion:

J

.u

-1.J .w.u -M = w2.r.(J -J ).e j@.,jwt

t i p P cp t P (24)

The equations of motions for the translation of the mass centre o f the disk follow directly by applying Newton's second law with respect t o the fixed reference:

where

F = x11.u

yr ma

The result is: 2 Fx = m.u - m . w .e.cos(wti-a) X F = m.U -m.w2.E.sin(wt+a) Y Y F = O Z

We can disregard the last equation. Using the complex quantities:

u = u +j.u

X Y

F = F -4-j.F

X Y

b

the equations (27a) and (27

1

can again be combined into one complex equation :

2 ja jwt

m.ri-F = m . w .&.e .e ₁₂₉₁

Finally the equations of motions (29) and ( 2 4 ) for a single disk (translations iiïìd ïûtâtiûns) can be cûmkhed in one ::,at~ix eyuatim:

or

i "i i 'i i i

M u i-

-

f = f

-d '-d -d mex

-16-

Equations of motion

The complete set of equations o f motion of our rotor model is composed o f

the discrete models for the disks, the bearings and the shaft e1ement.s. In this section we will derive the relations for the bearings and the shaft

elements. we will compose the equations of motion of our compleke

~ rotor model.

Finally,

~

As mentioned already, we use Sernouilli-Euler beans for the shaft elements. We assume that the beam is axisymmetric with respect to the z- axis. means that the rotation o f the beam may be neglected an6 we can describe the displacements and rotations with respect t o the fixed

reference. o f

shaft-element number i. The matrixrepresentation u i of this vector with respect to the fixed reference is:

This

We will introduce the vector o f nodal degrees O E freedom, ;i _{e r} -e

The

mass matrix Pi i and a stiffness matrix K i .

equations of motion o f the shaft element can b

-e -e

d by means of a descrik

ui

is the second derivative o f u i with respect to time. The column f i is

the vector with the forces and moments: -e ..e

f i = me Y i Y i+l Y it1

+

j.M Mi Fi”+ 1.F X L Mitlt 1.M Y

The mass matrix o € the shaft element reads:

.

M = (m /420). -e e ‘6 3.1, 3.1e 2.1, 2 22.1, 54

where m is the mass

matrix reads: e K = (2EIe/le 3 -e 22.1, 4.1,

*

13.1, -3.1, 2 13.1, 156 -22. le 1-13.1e -3.12 e -22.1 e 4.1 e (34)

and 1 is the length of the element. The stiffness e -6 3.1, 2 le -3. le

1

-6 -3.1e 6 -3. le 13.1 e l2 e -3.1 2.1 2 e e (35)

where EIe is the bending-stiffness.

As mentioned earlier, the bearings are represented by linear springs and a viskous damper. We will introduce a vector o€ nodal displacements at

(09) :LE) anayn X

+

c n' n T

_n

T -8

t-Ti i

where 4 i5 the transposed of matrix h

.

Our equations, now, must follow the condition :

n n n

= o

( 4 2 )

where n is the number 3f shaft elements and n is the number of nodal points. Using ( 3 0 a ) , (321, ( 3 7 ) and ( 4 0 ) it follows from ( 4 2 ) that:

e d

d T i i i n e T i i *

n

[.E L .M .L

+

. E 4 .M .Lil.u i-

i=l-d -d -d i=l e -e --e w t n

-20-

Residual s h a f t bow

Having the equations of motion (441, we can easily incorporate the effect of residual shaft bow. The displacement vector with respect to the bowed position of the system a t rest will be called the dynamic bow

td

The equation :

t -

~

3

must be satisfied. The vector ut will be affected by forces due to masses or dampers. However, forces due to stiffnesses will only affect t h e dynamic bow

Ud

From (44) and (45) it follows that t' Y or Y L M.u t 0.u

+

x.zt

= F t

-

\.t -t w- e% (47)

The term

E.?:

accounts for forces due to the residual shaft bow and can be seen as a static, initial stress situation.

Solution of the equations of motion

In the introduction we have mentioned that we will restrict ourselves t o stationary and synchronous motions. This means that the displacement vector rotates with a constant speed w . The displacement vector can now be written as :

j w t =

where u is a time-independent column-matrix. The same can be said o f the residual bow and the excitation vector: They all rotate with a constant speed w .

- t a

Using (481, ( 4 9 ) and (50) Pt follows from ( 4 7 ) that:

2 r

[-w .E

-+

j.u.0 i-

=

Fexa

i-

This is a complex linear s e t o f equations and the solution can be found with existing numerical routines.

-22-

Computer program and examples

A u s e r f r i e n c t l y coinputer program i s de;ieloped, based o n t h e theory disctisued in t h i s r e p o r t . The procjraii, i s callet= ASI. The i n p u t d a t 8 i s entered i n t o

t h e program by means o f i n t e r a c t i v e ccmmunizûlion wikh t h e u s e x . The proyrsm s u p p l i e s i n f o r m a t i o n a b o u t t h e r o t o r i ~ i ~ d e l , the u n b a l a n c e s , p r i n t e d o u t p u t

aid plotted ~ ~ - L p . t t .

Two exâmples o f c a l c u l a t i o n s w i t h AS4 will be g i v e n . Tke r e s u l t s of b o t h c a l c a l a t L o ~ s will be v e x i f l e d with d a t a p r e s e n t & i n l i t e r a t u r e .

-ûverhuns r o t o r w i t h d i s k skew

The f i r s t examplci. i s an overhung mCor wikIn d i s k skew 2reseiited by Salornone

and Cunter [ I ] . i n figlire 7 the r e l e v a n t r c t ~ z päx&itlei;ErS axe i n d i c a t . e d . The

v a l u e s o c .- 2 I ’- t parameters 2x2 sriisei:tcd i n table ? .

1 C

R

f i g 7 : Some parameters o f an overhung r o t o r

k, = k c , = c, T-. 5 . 1 2 % N.s/mm = 1 3 f 7 . 1 1 1 N / m 2 1 L: = 8 5 0 . 2 mm. d = 201.6 m m . I 2 E = 0 . 2 5 mm. i = 0 . 0 6 8 4 degrees. 7 2 I = 4.07284~10 kg.mm. 7 2 I, = 2 . 0 3 6 4 2 ~ 1 0 kg.mnì. P

Salomone and Gunter have calculated the rotor responses of five unbstlaricr

cases :

-

1

-

2 a positive disk skew angle (+TI

-

3 a negative disk skew angle ( - T )

-

4 both 1 and 2 {+Z-!-T) - 5 b o t h 1 and 3

(+€-TI

a radiai snbstlznce mass eccentricity ( + E ]

The amplitudes and phase angles o f the translation o f the nodal pclints at the near bearing, t h e far bearing and the disk location were calculated. The results o b t a i n e d with AS1 nat& w i t h tkoss obtaine6 in [ 1

1.

Figures 2 an2 9

show t h 2 amplitudes and phase angles o f khe nodal point translations et the

disk : G t d t i o i ” . * o r variakle speeds.

-Jeffcott-rotor with residual shaft bow

The second example is a Jeffcott-rotor on rigid supports with E ~ S ~ ~ U G : shafi

bow presented 3Jy Nicholas, Cunter and Alfaire [;i]. In figure ?O the rator parameters are indicated and the parameter ITI~IIPS ure =resente3 is t a b l e 2.

We will introduce the critical damping

eer:

where k is the beading stiffness of the shalt and m is the disk mass. e

~ ;sU0Q k g .

1

= 109U mm.

d = 59 mm.

We approxmateá che ïigïd suppfjxts by taking high values foï the lincaï bearing stiffness. in zi?eir article [ 2 ] , Nicholas et al. have considered several values o f e,, the angle between the residual shaft bow u and the

mass E (fig.101, and several damping values d . The results o f

some o f the calculations with AS-1 are presented in fig. 11, 1 2 , 7 . 3 and 1 4 .

These

(apart from the scaiicgj.

r

eccentricity

results match completely with tiie resuits obtained by Nïchalas et al.

T ' k = ,ix.ic'ij/mm ' ui = z x ï i x z . 8 1 raais CE' cr E = 'i mm. c = 68,2354 N s j i n m u = . 5 m m . r

- -

d (Ns/mm)'d/ccr' 13.64708 6.82354 20.47062 34.11770

'

amplitude (mm) 0.1 0.2 0.3 0.5 line no ~~ 5 1 ' 68.2354 1

.o

I

phase (degrees) ,?

.I*

!

i

fig, *I 4 : Phase a n g l e a g a i n s t rotor speed for variable v a l u e s

Salamone I E. J. i G u n t e x E. 3 .

,

Cynchranous Unbalance Response af ai2

Overhung Rotor w i t h Z i s k Skew, Transactions of the ASPIE, Journal of

Engineering for Power, Vol.102, No.4, October I?8L1, pp.749-755.

~

N i c h o l - s s , 3 . C . , G u n t ~ r , E . J . , ~ ~ ~ ~ ~ ~ r ~ , P . E . , Effect of Residual Shaft Eic..;

.

,l

on Unbalmce R e s ~ o n s r and Eaiancincj of a Single Mass Flexib2-e Rotoz,

Part 1: Unbalance Response, Transactions of the ASME, Journal of

Engineering for Power, Vol.98, No.2, April 1 9 7 6 , pp.17i-181.

Kikuchi,K., A n a l y s i s o f Unbalance Vibration o f Rotating shaft Systems w i t h zany Bearings and Disks, ~ u l l e - t i n of “Le JUME, vc:.13? No.6:, J u l y 1970, pp.864-372.

Flack,R.D.,Rooke,J.H.,Bielk,J.R.,Gunter,~.~., Comparison of the Unbalance Response o f Jeffcott RotorLi with S h a f t Zow and S h a f t Ri;nout, Journal o f Mechanical Design, ASME Paper 81-DET-46, hos: H.eetlïlcj

SeFtEember 20-23, 1931, 1 1 p ,

C a F k o r C . r ~ ~ ~ n a , T . , Bcr?ancing o f Flexible iilotors by the Camplez; ~ o d a :

Method, ASMIE, Journal a4 Mechanical Design, ASME Paper 8?-OET-46; :si