thermal bending of the rotor to rotor-to-stator...

International Journal of Rotating Machinery2000, Vol. 6, No. 2, pp. 91-100Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2000 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach SciencePublishers imprint.

Printed in Malaysia.

Thermal Bending of the Rotor Due toRotor-to-Stator Rub

PAUL GOLDMAN*, AGNES MUSZYNSKA and DONALD E. BENTLY

Bently Rotor Dynamics Research Corporation, 1711 Orbit Way, Bldg. 1, Minden, NV 89423, USA

(Received 5 September 1997;In finalform 21 July 1998)

The rotor thermal bending due to the rotor-to-stator rubbing can lead to one of three types ofobserved rotor lateral motion: (1) spiral with increasing amplitude, (2) oscillating betweenrub]no-rub conditions, and (3) asymptotical approach to the rotor limit cycle. Based on themachinery observations, it is assumed in the analytical part of the paper that the speed scaleof transient thermal effects is considerably lower than that of rotor vibrations, and that thethermal effect reflects only on the rotor steady-state vibrational response. This responsewould change due to thermally induced bow of the rotor, which can be considered to slowlyvary in timefor the purpose of rotor vibration calculations. Thus uncoupled from the thermalproblem, the rotor vibration is analyzed. The major consideration is given to the rotor whichexperiences intermittent contact with the stator, due to predetermined thermal bow,unbalance force, and radial constant load force. In the case of inelastic impact, it causesan on]off, step-change in the stiffness of the system. Using a specially developed variabletransformation for the system with discontinuities, and averaging technique the resonanceregimes of motion are obtained. These regimes are used to calculate the heat generatedduring contact stage, as a function of thermal bow modal parameters, which is used as aboundary condition for the rotor heat transfer problem. The latter is treated as quasi-static,which reduces the problem to an ordinary differential equation for the thermal bow vector.It is investigated from the stability standpoint.

Keywords: Rotor, Vibrations, Thermal bow, Rub, Averaging technique

1. INTRODUCTION

Rotor-to-stator rub, an unwelcome contact be-tween rotating and nonrotating elements of amachine, can be one of the most damagingmalfunctions of rotating machinery. Generated bysome perturbation of normal operating conditions

that causes an increase of rotor vibration level, and/or an increase of the rotor centerline eccentricity,the rub can maintain itself, and gradually becomemore severe. The self-generating feature of thisphenomenon originates from the interactionbetween rub-related thermal effects and lateralvibrational response of the rotor. Starting from

Corresponding author. Tel." (702) 782 3611. Fax: (702) 782 9236. E-mail: psg@bncl 17psg.bently.com.

91

92 P. GOLDMAN et al.

pioneering works of Taylor (1924) and Newkirk(1926), the unwinding spiral vibrations of rotorsare documented in several papers (Black, 1968;Kroon and Williams, 1939; Dimaragonas, 1973;Kellenberger, 1979; Natho and Crenwelge, 1983;Hashemi, 1984; Smalley, 1987; Muszynska, 1993).In addition to the spiral response, Dimaragonas,(1974) described an oscillating mode of shaftvibration, occurring during the transition fromthe spiraling to a steady-state mode. A similarresult from an improved rotor dynamic model wasobtained by Muszynska (1993).The problem of rub-related heat distribution was

discussed by several authors, for example, Sweets(1966), Kellenberger (1979) and Smalley (1987).The most complete analysis of the heat transferproblem associated with rub is given in the book byDimaragonas and Paipetis (1983). In all referencedliterature the analysis of the shaft bow, resultingfrom the uneven temperature distribution due torub, is based on an approximation on the meanflexural rotation of one end of the shaft in relationto the other (Goodier, 1958).

The idea of the discontinuous variable transfor-mation applied in this paper for the rub dynamicsanalytical description appeared first in the paperby Zhuravlev (1978), and was expanded later byPetchenev and Fiddling (1992) and Goldman andMuszynska (1994a,b; 1995a,b).

2. MATHEMATICAL MODEL OF THERUBBING ROTOR

An isotropic rotor in its lateral mode motion is con-sidered (Fig. 1). The rub (or more generally any localnonlinearity which creates a thermal effect) occursat the shaft axial location 12. If the rotary inertiaand shear stresses are neglected, the equations ofmotion of the rotor can be expressed as follows:

02 IEjO2(7-r),lm+- 0l2

mgfZeJ + O, v/7-1 (1)

where x(1, t), y(1, t) are horizontal and verticaldisplacements of the rotor at the axial location

M2(/’/) BendingMonet to Moment ofInertia Ratio:I(1,) | hermai BowMmle

Temlratut differenceT’

(1, t) x +jy (x, +jy,e "

L

FIGURE Physical model of the rubbing rotor.

THERMAL BENDING OF THE ROTOR 93

in the stationary coordinates, ’-x/jy; EJ isbending stiffness, m is mass linear density,/- is avector ofthermal bending in stationary coordinates,/(l, t) v(1, t)eJ, /, is a vector of thermalbending in rotating coordinates, is rotative speed,-t+, Y is a distributed unbalance vector inthe coordinate system rotating with the rotor atthe particular axial location l, Q- Qx +jQy isa complex vector describing linear density of theexternal and nonconservative forces. Assumingthat the contact/no-contact situation at the axiallocation 12 generates a radial reaction force

F2U- -OKu([T2] c)

{1 if 172] > c(2)

o- -,IFzl x+0 ifl&l < c

where KT is the local stiffness of the stationaryobstacle, Y2 7(/2, t), c is the radial clearancebetween the rotor and the obstacle, the vector Qcan be presented in the following form"

Q -c(1)+ q(l)h() + (1 +

()

Here.C(/) is a damping linear density, q(l), ?(1) areradial side-load forces linear density and theirangular orientation, respectively, fis a dry frictioncoefficient, L is the length of the rotor and (... isthe Dirac function. The thermal bow appears due toan uneven temperature distribution along the rotorcaused by the friction force-generated heat. Thelatter can be characterized by the heat rate densityg(l, 9, t) per unit area ofthe rotor cross section (is anangular coordinate, Fig. 2). Considering the parti-cular area element R2dld (R2 is the rotor externalradius), on the rotor surface around the point withaxial coordinate l, the friction force jfN is appliedto the rotor ir CN-- deN CN+ deN and

& -dl & +dl (it is assumed that the rotor-to-stator contact occurs at a single location only),where 9N- 2--+ arg(&) is the angular posi-tion of the friction force (Fig. 2). The frictionforce, therefore, can be expressed in a form of a

distribution over the rotor surface as follows:

Of,, offL/2 2ra -L/2

/4/

(...) is defined for angles as a periodic withperiod of 2. Since the rotor velocity at the contactlocation can be approximated as R2, the heat ratedensity g(l, , t) per unit area equals to the frictionforce power per unit area. Taking into accountEq. (4), it can be presented as follows:

L 6 (9+-arg(7)). (5)

The thermal conductivity equation with initial andboundary conditions

OZT 10T 10T O2T 10TOr

/ +r 7 012 at

t=0 + fiT =0,I=0,L

+r _x(,e,t),

()

describes heating of the shaft due to rub. HereT= T(r,,l,t) is a difference of temperatures

FIGURE 2 Cross-section of the rotor at the rub axiallocation 12.


between the environment and the material point ofthe shaft with coordinates 1, r, at an instant t,and , ,/ are thermodynamic constants, R1 is therotor internal radius. To complete the problemformulation, a relationship between the temperaturedistribution and thermal bow has to be derived. Inorder to accomplish this, the assumption is madethat the rotor can be considered within the limitsof Euler’s beam theory. In this case, the thermalstress-related bending moments in the rotatingcoordinate axes Xr, Yr are as follows:

fRR2

My-- -)T db ETrZ cosbdr My(t, 1),

Mx d ETr2 sin dr Mx(t, ),

where XT is the thermal expansion coefficient, Eis Young’s modulus of elasticity.

According to the Castigliano’s theorem andEqs. (7), the vector fir(l, t) of the thermal bow inthe rotating coordinates can be expressed in theform of convolution:

where M(1, la) is a bending moment at the axiallocation la, resulting from the unit load applied atthe axial location 1. For simplicity it is assumed thatthe rotor has constant modulus of elasticity E forall cross-sections along its length. Due to the localcharacter of the rotor heating (Fig. 1) the convolu-tion can be approximated as shown in the secondof Eqs. (8). An example of a simply supported rotorwith overhung impeller, shown in Fig. 1, has the

following expression for M(I, 12):

n m-Z2l- 12 (Lrn_l,,7

Obviously the expression -(M(1,12)/I(12)) deter-mines the mo+de shape of the thermal bow (seeFig. 1) while i?(t) plays the role of the modal timevector-function. The equations which describe thelatter can be derived from Eqs. (6) by integraltransformations (see Taylor (1924) for details), withthe assumption that all thermal processes arelimited to the rotor area with the same cross-sectiondefined by internal radius R1(12) and externalradius R2(12), and averaged over the yet unknownperiod of mechanical oscillations (e is a smallparameter)"

dre \. i

where

DT(1 2)[k (1 q- 32) q- /(1 2)]

4 (1-N2)(kl-+-th) q- h lq-t2q-8N3-+-N(1--N2)-

fR2(12)2

Since, according to Eqs. (8),/3r is proportional tothe temperature ;?- f i?i2 di, it follows from thefirst Eq. (10) that d/3r/dr O(e/3r) has a higherorder of smallness than fir itself. As a consequence,the thermal bow in Eq. (1) can be considered as a

parameter, and the system dynamics become essen-

tially decoupled from the thermal part ofproblem.


The rotor lateral mode modal coordinatesffq Uq + jVq (q 1,2,..., oo) togetherwithortho-gonal modal functions % (q-1,2,..., oo) can beintroduced in a way that the following equationsare satisfied:-- Z 7]q(l),q(t) + ej(p Z Z]q(l)(q -4-

k=l k=lL M1 (l,/2)

dl rCq. (12)q(l)1(/2)

In this case, Eq. (1) can be written in modalcoordinates as follows:

Here Mq, Dq and Kq are modal mass, damping andstiffness ofthe qth mode, q f mTq dl is a modalunbalance, PqeJTq f qeJ/lq dl is a modal radialside-load force. The analysis of Eqs. (13) whichrepresent the mathematical model of the system,is performed below under the following physicalassumptions:

(1) The system of applied radial side-load forcesmaintains the static position of the rotor veryclose to the stationary obstacle at the axiallocation 12. The rotor-to-stator contact isintermittent due to the dynamic action of theunbalance.

P/ejzk --jc Kk (1 A/),Ek

-j /Xq (qNq

(14)

as follows:

2,1- c ZAq (1- o-A),q=l

for statically "loose" case (no contact),cr-

-1 for statically "tight" case (contact),

where cA is an absolute value of a gapbetween the rotor static position and thestator. According to physical assumptions,A << 1. The coefficient A is used below as ameasure of smallness.

(2) One mode in particular, the kth mode, forexample, governs the contact at the axiallocation 12. It means that this mode delivers amuch higher contribution into [F(12, t)] than allother modes:

o(1),]q(/2) //q O(x/-) (q - k).

In almost all practical situations this is true.

(3) The vertical (imaginary) direction for eachmode is chosen opposite to the direction ofthe corresponding radial side-load force: ")/q--3rc/2, q- 1,2,

New variables are introduced as follows"

(q#k),

(16)

In this case, the absolute value of the shaftstatic displacement at the rub axial location is

where (u+jv) and rq (qk) are dynamiccomponents of Eqs. (13) solutions, h is the


nondimensional distance between the rotor and anobstacle at the axial location 12,

s- l(12, t)]/(c/) 0(1), or- arg((12, t)).

With the new set of variables h, u, Iq (q- k)using Eqs. (16) and (13) can be rewritten as follows:

of smallness, but the conditions (19) of the switchare precise.

3. MECHANICAL RESONANCES ANDHEAT GENERATION

h" + n(1 -t- @2)h crn AH + O(A2),+ + o( x:)

tt 2 2Fu + nZqFq -jOcrnpqh+ x/q[aqej(+cq) -+-7rq,sTeJ(+cv)]+ zx +

where

(17)

The rotor resonance responses to the unbalancewhich are considered below are associated withthe leading mode, or in mathematical terms, aredescribed by the first two equations (17). It meansthat the rest of the Eqs. (17) determine only"forced" solutions, which are defined at eachsequential approximation by the previous approx-imation to the solution of the first two equations.Therefore, it is important to consider the latter in

H -2nkh’n2ku2sin(g) + c) + sr(er/, + n 1) sin(g) + cr) + -- u’2 + Z(n n2q)Im rq],

qv/:k

U- -2nu + eca cos(g) + a) + /r,srcos(g) + at) + Opngh u +

Wq -2gqnqE + Opqnqah u +Dk,q

’q 2Mk,qUk,q

U,q Kk,q 2 U,q Jk e,qmk,q nk,q 2 akq O(1)c2

f O(1) t, "’" d

d

p,q gfk’q p2 gfKk,q -k l’2k-l-/kZ

I(/2)7rq" cA27rq,STejecT, 7rq, --TrqM1 (/2,12).

(18)

The conditions (3) of the contact/no-contact switchnow become very simple:

0- f0 ifcrh>0,ifcrh <_ 0. (19)

Note, that the right-side terms of Eqs. (17) arecalculated with an accuracy up to the second order

generating approximation, which is defined by neg-lecting all right-side terms. The second of Eqs. (17)in generating approximation has a simple solution:u= pcos0, p’ =0, 0’= nk, The first of Eqs. (17) ingenerating approximation, together with the switchconditions (19), is more complex. Its solution can bebuilt using piecewise integration, and connecting


conditions at the ends of continuity intervals:

l+x()p2l+x()p2 1-5 sin

+ cos (20)

where S is constant referred as an amplitudeparameter,

where Q, F, U, W are functions of new variables,determined by the system of Eqs. (17) and . isan integration variable. Equations (22) have therequired format, with three fast rotating phases ,0, and two slow variables S and p. This allowsapplication of the Averaging Method [7]. Note thatat this point, the equations are limited to the termsof the first order of smallness, and only small

-1

(21)

The relation between h and S (amplitude param-eter) and (phase) is shown graphically in Fig. 3.It also shows the relation between the amplitudeparameter S and overall amplitude A. Equations(20) and (21) constitute a variable transformation.This transformation allows to introduce the rotorvertical response amplitude parameter S and"phase" , and is used to convert the originalsystem (17) into the form with three rotating phases, , 0 and two slow variables S, p:

S’ AQ(S, p, , O, p, ’q(S, )) + O(A2);

’ co coAF(S, p, , O, ,’q(S, )) + O(A2)

p,_ A U(S, p, ,, O, , rq(S, ,)) sin O+ O(A2);

nk

zxu(s, p, o, r (s, ))cos00t- IIk -- O(A2)pnl

’--1 (22)

and a number of noncritical variables

rq _jcrp2qnq fo0

sin + O(zX) (23)

right-sided terms of Eqs. (22) have discontinuities,as they change with the fast rotating phase .As it results from the expression (21), the ratio

co/nk is contained within the following limits:

co 2<--< if or- +1,-n:-1 + 1/X,/’I _+_p2

2 cop2<--<v/l+ ifcr--1.

+ l/V/1 +p2- nk(24)

These inequalities, together with Fourier analysis ofthe right-side terms of Eqs. (22), show that possibleresonances occur when:

1-n-0, 1-ico-0 (i- 1,2,3,...),2n co 0 (for cr -1). (25)

The analysis of Eqs. (22) from the standpoint ofbalance between the supplied and dissipated ener-

gies, allows the stationary resonance solutions forthe case of nk 0, ic 0 (i-- 1,2, 3,... ), or

for the case ofa combinational resonance: n 0and 2n-co=0. The first resonance occurs whenthe rotative speed of the shaft f2 is close to the rotor


A) Normal-loose caseh or-1

S-I- no contact contact

B) Normal-tight case

hno contact i contact

FIGURE 3 Variable nondimensional distance h (c- 721)/c/X as a function of phase

leading mode natural frequency vk. It could eitherbe accompanied by the vertical resonance

2nk-a--0 or not. This regime, referred to as a

horizontal mode resonance, is described in detailin Taylor (1924). Important thing to know abouthorizontal mode resonance is that, due to thesymmetric heating, the thermal bow does not occur.

The sequence of resonances 1-ico(Sa)=O (i=1,2, 3,... ), referred to as vertical mode resonances,is of interest because it creates uneven heatingwhich results in a thermal bow. The resonance

frequency equation according to Eqs. (21) deter-mines the zeroth approximation Sa to the slowvariable S as a function of the ratio f/iu in thecorresponding resonance zone. A simple analysisof Eqs. (21), together with inequalities (24), showsthat for each value of 1,2,... it can be satisfiedonly within the range of rotative speeds determinedby the following inequality:

This means that the main, 1 (synchronous)regime of rotor vibrations (i- 1) occurs at rotativespeeds higher then unaltered resonance frequency

u of the leading kth mode, the subsynchronous1/2 regime (i 2) occurs at rotative speeds higherthen 2u, and so on. In the case of normal-loosesituation, (or-+ 1) the maximum rotative fre-quency of the corresponding resonance regime islower then 2iu, while in the normal-tight situation

2iuk is a minimal rotative frequency. This agreeswith practical observations of rubbing rotor beha-vior (Choi and Noah, 1987). The parameter p,which affects the width of the frequency band foreach regime, characterizes the stiffening effect ofthe rotor-to-stator contact. Equations (22), afteraveraging in proximities of vertical resonances

(see details in Zhuravlev (1978)), allow for the follow-ing stable stationary solutions:

+ sign(G) arccos7rk(1 n)[(1 +p )nk 1]qkG

bp2 @/1 1/Sa COSO + ncr/SasinO)’i- 1,2,3,...

:r arccos

G(Sa,p)

{ P)Cr/1Sa + 7r(1 + p2 in:

(27)


where Sa is a function of the rotative speed,determined by the solution of resonance equation

ico(Sa) 0 (i 1,2, 3,...), /- bej kakej"k + ST(kTrk, + n 1)ej"T is an equivalentvector of unbalance which includes a componentdue to the thermal bow. It defines the series ofx, 1/2x, 1/3 x,... regimes in the rotative speed

bands roughly described by the inequality (26).The relationship of the vertical response phase

’ ( -t- O/i) and overall amplitude A with rotativespeed ratio for the vertical modes, calculated fora particular set of parameters or, p, and b, ispresented in Fig. 4. Based on the resonance solu-tions (27) and expressions (11), the heat generatingvector () can be calculated:

<) jffcAKf crV/Sza + Pz(SZa 1)

27ri(1 / p2)+jcos

jcos(1 e-2jo) +ngl? +p2)

[1--e-2J9(cos26--jnkV/l+p2sin2(5)]}ej7,

1000

[ 00

Rotative speed versus k-th mode natural frequency ratio

3 5

Rotatlve speed versus k-th mode natural frequency ratio

FIGURE 4 Phase and nondimensional amplitude of therotor-to-stator distance versus rotative speed to natural fre-quency ratio for the series of Ix, 1/2x, 1/2x, 1/4x resonances. Cal-culations are made for the parameters cr-- 1, p 2, b/ 4.

where Sa is determined by the second of Eqs. (25)and Eqs. (21) as a function of rotative speed f.Since ()) is essentially the forcing function inEq. (10) for the thermal bow, its behavior deter-mines the behavior of the thermal bow. Figure 5depicts ()) in a polar plot format for differentresonance regimes i= 1,2, 3, 4.

4. SUMMARY AND CLOSING REMARKS

This paper outlines the modeling of thermal/mechanical effects of one of the most destructivemalfunctions in rotating machinery: the rotor-to-stator rub. The thermal/mechanical problem ispartially uncoupled by the assumption that thethermal process is relatively slow. As a result, therotor thermal bow remains in the mechanicalequations as a parameter which can be considereda constant. The combination of the AveragingMethod and the assumption that the thermalprocesses are quasi-static allows the heat transferproblem to reduce to a vectorial ordinary differ-ential equation (Eq. (11)), with the heat generat-ing equivalent vector as a forcing function. Thisequation allows a realistic estimate of the thermal

.2120

21

240

270

60

3O

FIGURE 5 Heat generating vector (} versus rotativespeed to the natural frequency ratio in the polar plot format.Heavy pot indicates the position of equivalent unbalancevector b. cr-- 1, p--2, b/ 4.


bending mode (Fig. and Eqs. (9)), and can beapplied not only to rub-related heating. The otherpossible application is an effect of journal bearingdifferential heating, described in the paper byKeogh and Morton (1993).The heat generating equivalent vector for the rub

is calculated by application of the discontinuesvariable transformation and resonance version ofthe Averaging Method to the mechanical part oftheproblem. Based on the heat generating equivalentvector behavior, predictions can be made on theone of three possible thermal bow behaviors:

Asymptotic approach to the equilibrium stateof the thermal bow.Increasing spiraling motion ofthe thermal bow inthe direction opposite to rotation.Slow oscillations of the thermal bow.

The analytical algorithm described in this paper(Eqs. (20), (21), (23) and (27)) has a high potentialas a valuable research and prediction tool forinvestigating rub and thermal effects in rotatingmachinery.

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Childs, D.W. (1982), Fractional frequency rotor motion due tononsymmetric clearance effects, Journal of Engineering forPower, 533-541.

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