ROTOR-BEARING STABILITY

ROTOR-BEARING STABILITY by

Edgar J. Gunter, Jr. Department of Mechanical .Engineering

University of Virginia

Edgar!. Gunter Jr. is a n associate professor of Mechan ical Engineering at the Univers ity of Virgin ia. He was with Clark Brothers Company, Olean , New York during the period from 1956 through 1959, and was with Franklin Institute, Philadelphia, Pennsylvania during the period from 1962 through 1965.

He received a B.S. (Mechan ical Engineering) Degree in 1956 from

Duke University, received a M.S. (Engineering Me· chan ics) Degree in 1961, and a Ph.D. (Engineering Mechanics) Degree in 1965 from University of Penn sylvania.

He is a m e mber of several professional societies. He has several publications, technical n otes, and technical reports in the area of bearings.

ABSTRACT

This paper p resents a survey o f the var ious mecha · n isms tha t c a n cause instabil i ty i n ro tor bear ing systems. The occu r rence of self-exc ited instabi l i t y is o f par t icu lar importance t o the manufacturers and u sers o f modern turbomachinery part icular ly w i th the p resen t trends to· w ards h i gh speed and l oadin g condi t ions.

The paper p resent s s tabi l i ty data on pla in and mul · t i lobed jou rnal bearings and shows the effect o f unbal · ance and externa l loading on the n onl inear ro tor w h i rl orb i ts.

I n addit ion to the stabi l i ty characterist ics of h ydro· dyn amic bear ings, i nst ab i l i ty due to i n ternal fri c t i on , aerody n amic c ross coupl ing and seals i s also discussed . The paper discusses how many of these ins tabi l i t i es may be avoided b y proper bear ing and suppor t design .

I n addi t ion to the theoretical predic t ion o f ro tor wh i rl orb i ts, ac tua l case h i stor ies of ro tor i nst ab i l i t y are presented.

I NTRODUCTION

The object o f th is paper i s to review some o f the mechan isms tha t can crea te nonsynchronous whi r l mo· t i on in a ro tor-bearing system. This mot ion may be man i fested by a sub-synchronous o r super harmonic mot ion o r combina t ion of var ious mot ions and i ts direc· t ion may be forward o r backward precession . There are many m echanisms which can cause whir l ing o r n o n · synchronous p recess ion i n a rotor system b u t t he most ser ious ones o f consequence i n t urbomachinery i s t he occurrence o f self-exci ted wh i rl ing such as caused by hydrodynamic bear ing instabi l i ty , i n terna l fr ict i on , o r

1 19

aerodynamic cross coupl ing . When these condi t ions a r e encountered the v ibrat ion may b e s o severe as t o cause destruc t ion o f the rotor . I t i s therefore imperative for the plant eng ineer to be ab le to recognize the occurrence of potent ia l ly dangerous wh i r l mot ion so a s to change the ro tor operat ing cond i t ions o r speed. I t i s preferable tha t whir l mot ion be taken i n to considera t ion in turbo· machinery design so as to m i n imize the possib le occu r· renee o f se lf-exci ted wh i r l mo t ion i n operat io n .

As the operat ing req u irements o f speed a n d power becomes i n creasingly more demand ing o n turbo rna· ch inery , condi t ions are encountered i n w hich the tu r · h ines and compressors are runn ing a t many t imes the ro tor f irst c r i t ica l speed. When th i s occur s the system may be susceptible to whir l o f many forms. In the past decade w h i rl mot ion has become o f i n c reasi n gl y more concern to the engineer and cons iderable l itera tu re has been publ ished on var ious aspects o f i t . N o w t h a t elec· t ron ic instrumentat ion has developed t o measur e ro tor mot ion and analyze the frequency componen ts i t i s pos· sihle to accurately moni t o r the wh i r l mo t ion exper ien ced in rotat ing machinery u n der ac tua l f ield condi t ions.

A-: -�---+----

I I �-�- -�- _ _!L_ -�-

1

Figure 1. Schematic Diagram of Flexible Ro to r.

120 PROCEEDINGS OF THE FIRST TURBOMACHINERY SYMPOSIUM

I t i s impossible i n th i s p resen ta t ion to adequa te ly exam i n e a l l the var ious aspects of whir l ing i n h i gh speed ro ta t ing machinery . I t is the purpose o f this paper, h o wever t o educate and acquaint the engineer w i th some o f the various modes o f whir l mot ion experienced w i t h t urbomachinery .

Description of the System The usua l h i gh -speed ro tor m a y be considered a s

a con t i nuous elast ic body w i th var iable m ass and ine r t i a properties a l ong i ts length as shown in Figure l A . The shaf t u sual ly h as a t tached t o i t such components as tur· bine wheels , impel ler disks, spacers , e tc . I n the calcu la · t i on o f ro tor c r i t i ca l speeds, s t ab i l i t y , and u nbalance response , mos t computer codes t rea t the ro tor as a l umped m ass system i n which the ro tor i s represented by an elast ic shaft t o wh i ch i s a ttached N- mass s ta t ions . I f the moments o f iner t ia o f each sec t ion are ignored, then the s ta t ions may he considered as concentrated masses as shown i n Figure lc . This assumpt ion leads to the n eglect o f the gyroscopic moments ac ti n g on the ro tor which i s u nacceptable for cer ta in h i gh speed ro tors o r overhu n g configurat ions .

Figure 2 represent s a typical cross sec t ion o f an i deal ized ro tor t aken a t the Nth sta t ion . The to ta l angu·

Figure 2 . Rotor Cross Section .

l a r veloci t y o f the system i s given by the t ime rate o f change o f a l i ne f ixed i n the disk a n d i s represent ed b y w. B y examina t ion o f the configura t ion o f Figure 2 ,

TABLE 1 . FORCES ACTING O N ROTOR-BEARING SYSTEMS (Ref. Rieger)

Source of Force

1. Forces transmitted to foundations, casing, or bearing pedestals.

2. Forces generated by rotor motion.

3. Applied to rotor

Description

Constant, unidirectional force Constant force, rotational Variable, unidirectional Impulsive forces Random forces

Rotating unbalance: Residual, or bent shaft. Coriolis forces

Elastic hysteresis of rotor

Coulomb friction

Fluid friction

Hydrodynamic forces, static.

Hydrodynamic forces, dynamic. Dissimilar elastic beam Stiffness reaction forces

Gyroscopic moments

Drive torque Cyclic forces

Oscillating torques

Transient torques Heavy applied rotor force

Gravity Magnetic field, stationary or

rotating.

Axial forces

Application

Constant linear acceleration. Rotation in gravitational or magnetic field. Impressed cyclic ground-or foundation-motion. Air blast, explosion or earthquake. Nearby un-

balanced machinery. Blows, impact.

Present in all rotating machinery.

Motion around curve of varying radius. Space applications. Rotary-coordinated analyses.

Property of rotor material which appears when rotor is cyclically deformed in bending, torsionally or axially.

Construction damping arising from relative motion between shrunk fitted assemblies.

Dry-friction bearing whirl. Viscous shear of bearings. Fluid entrainment in turbomachinery. Windage. Bearing load capacity. Volute pressure forces. Bearing stiffness and damping properties. Rotors with differing rotor lateral stiffnesses.

Slotted rotors, electrical machinery, Keyway. Abrupt speed change conditions Significant in high-speed flexible rotors with disks.

Accelerating or constant-speed operation Internal combustion engine torque and force

components.

Misaligned couplings. Propellers. Fans. Internal combustion engine drive. Gears with indexing or positioning errors Drive gear forces Misaligned 3-or-more rotor-bearing assembly. Non-vertical machines. Non-spatial applications. Rotating electrical machinery

Turbomachine balance piston. Cyclic forces from propeller, or fan. Self-excited bearing forces. Pneumatic hammer.

PROCEEDINGS OF THE FIRST TURBOMACHINERY SYMPOSIUM 121

the following defi n i ti ons o f whi r l ra t io may be s tated:

cf>/w = shaf t wh i rl = !-Synchronous precission or "whip ratio" =I= 1-Nonsynchronous Precess ion

8/ w j ournal whir l ra t io

a/ w system whirl o r precession ra t io

For the case o f s teady-s t a te unbalance synch ronous precess ion w i th isentropic bear ings ( 17 1 , the confi gura t ion formed by 0, 011, OJ> C, N, i s constant and precesses w i th an angular veloc i ty equal to the ro tor angular veloc i ty w.

Whirli ng i n tu rbomachinery can be genera ted b y many i nfluences act ing o n rotor. Table 1 represents a summary o f the var ious forces ac t ing on ro tor -bear ing systems as compiled b y Rieger ( 48) .

Causes of Rotor Whirling in Turbomachinery

There are many factors which can lead to ro tor whirling i n turbomachinery . Some of the mechanisms tha t can i nduce ro tor wh i rl mot ion are given as follows:

1 . Hysteretic o r I n ternal Frict ion Damping: a . Shrink fi ts ; b . Fric t ion i n gear type coupl ings ; c . Shaft internal damping.

2 . Hydrodyn amic f lu i d f ilm bearings and seals.

3. Aerody namic cross coupl i ng forces.

4. Dry fr ic t ion whir l (ro tor rub) .

5. Pulsat ing Ax ial Loads and Torques.

6. Asymmetr ic shaft in g p roperties .

7. Entrained f luid in rotor (part ia l ly filled centr ifuge.)

8. Gyroscopic induced wh i rl ing .

9. Subharmonic and superharmonic whir l ing in d uced by sys tem n onl inear i t ies.

10. Electromagnetic forces.

Table 2 represents a summary o f the var ious causes of rotor wh i rl mot ion and some of i ts characterist ics as to whirl frequency, the speed at which i t occu rs , and comments on i ts general behav ior . There are several o ther mechanisms which h ave n o t been inc luded such as electromagnet ic effects b u t this table should represen t the maj o r sources of i nterest i n t urbomachinery . I n general , the wh i rl behav ior and frequency response tha t the var ious mechanisms man i fes t themselves i s extremely complex and Table 2 is n o t to be taken as i nd ica t ive of the complex behavior .

I t i s no t the i n ten t of th i s p resenta t ion to discuss all of the whirl mechanisms o r to p resent the mathematical foundat ions o f each one o f these as th is would be prohibi ta t ive and beyond the scope o f this paper. I t i s however the pu rpose of this paper t o m ake the reader aware tha t wh i r l mot ion o ther than tha t d i rect ly at tr ibuted t o u nbalance exists and tha t this motion can have di re consequences i n h i gh speed turbomachinery i f not properly accounted for .

The most ser ious form o f whirl in a turboro tor i s self-excited mot ion . This self-excited o r nonsynchronous precessive mot ion i s defined as a phenomeno n i n w h ic h t h e excita t ion forces induc ing t h e v ibrat ion are con trolled by the mot ion . Th i s i s i n con tras t t o a forced v ibra t ion such a s ro tor unbalance i n which the external exc i ta t ion i s a funct ion o f t ime only . I n a forced v ibra t i on due t o an u nbalanced o r bowed ro tor , t he ro to r response i s synchronous o r equal t o running speed. I n a self-exci ted whi rl , which i s often referred t o a s sus ta ined transien t mot ion , the whir l mot ion i s super imposed upon the runn ing frequency . The superposit i on o f the var ious frequencies l e ad to the u nusual r o t o r orb i ts o f ten observed o n h i gh speed rotors . Note t ha t mon i tor ing ro to r mot ion through a synchronous tracki n g f i l ter such as used in b alanc ing w i ll lose the whir l mot i on .

A n ac tual turbomach ine may experience whir l from one o r the i nteract ion o f several o f the mechan i sms l isted in Table 2 . If a spec t rum analys is i s taken o f a ro tor under var ious condi t ions o f speed and loadin g, i t wi l l u sually be observed tha t some form of whi rl mot ion (nonsynchronous p recession) wi l l be present w h ic h i s superimposed upon the synchronous o r unbalanced response mot ion . I t i s therefore o f cons iderable importance t o the manufacturer and plan t operator t o be able t o i dent i fy the occurrence o f a po tent i al ly dangerous wh i rl mot ion tha t may requ i re shu t t ing down the u n i t o r cu rta i l i ng i t s operat ing condi t ions o f speed o r l oading. The u lt imate obj ec ti ve however o f the fundamental u n ders tand ing o f wh i rl i s t o design s table turbomachinery t ha t wi l l no t encoun ter self-exci ted whir l mot ion throughout i t s opera t ing speed over a range of loading condit ions from n o load to surge.

S ince the obj ect i ve of th i s paper is to survey the problem o f ro tor-bear ing ins tabi l i ty , on ly the maj o r sources o f self-excited whir l ing wi ll b e considered.

Hysteretic or Internal Friction Induced Whirling

At the turn o f the cen tu ry , the maj o ri t y of the pumps and c ompressors were rec iproca t ing and the ro ta t ing u n i ts were massivly designed t o operate well below the ro tor f i rs t bend ing cr i t ical speed . The 1 920's saw a trend reversin g the rotor-design concepts o f a prev ious decade and turb ine and part icular ly compressor and pump manufacturers were beginn ing to cons t ruc t lighter w eigh t , h igher speed rotors to opera te wel l above the f i rs t c r i t ical speed.

As more manufacturers went to the flexib le ro tor design , several encoun tered severe operat ing difficulties when operat ing well above the first cr i t ical speed. These problems were at first a t t r ibuted to the lack o f proper balanc e. I n the Uni ted S tates a t this t ime, Genera l Electric encountered a series of fai lures o f blast furnace compressors desi gned t o operate above the first cr i t ical speed. These machines were subj ec t t o occasiona l f i t s o f more o r less v iolent v ibrat ion o f unknown orig in . Dur ing t h ese dis turbances the shaf t wou ld vibra te a t a low frequency which i n some c ases could be v i sually ob served. The phenomenon was therefore called by shop men and engin eers "shaf t wh ipp ing ." D r. B . L. ·Newkirk ( 40) o f t he Gener al Elect r ic Research Labora tory was


called i n to i n vest igate the n a ture o f the fai lures. He set u p a series o f experimen ts wi th several u n i ts t o observe the rotor dynamic behavior . I t was observed tha t a t speeds above the f i r s t c r it i cal speed, these un i ts would enter in to a v iolent whir l ing i n which the rotor center l ine p recessed at a ra te equal t o the f irs t cr i t ical speed. I f the u n i t ro ta t iona l speed were increased above i ts i n i ti a l whir l speed, the w hir l ampli tude would i ncrease, leadin� to eventua l rotor failure . To further i nvesti�ate a l l aspects and contr ibut ing factors to th i s problem, an experimenta l tes t ro tor w as constructed to s imulate a typical compressor u n i t. Upon extensive tes t ing o f th i s uni t , the fol low ing impor tan t fac ts were uncovered concern i ng th i s phenomenon :

I. The onse t speed o f whir l ing o r wh i rl ampl i tude w as u naffected by refinement i n ro tor b alance.

2 . Whir l in� always occu r red above the fi rs t cr i t icaf speed, never be low i t .

3 . W h i r l threshold speed could v a r y w idely between machines o f s imi lar construct ion .

4. The p recess ion {or whirl l speed was cons tant regardless of the u n i t rota t ional speed .

5. Whir l ing was encountered only w i th b u il t -up ro tors .

6. I ncreasing the foundat ion flexib i l i ty wou ld in crease the threshold speed o f ins tabi l i ty .

7. D istor t ion o r misalignment o f the beari n g hous i ng wou ld i ncrease s tabi l i ty .

8 . I ntroducing damping in to the foundat ion w ou l d i ncrease the wh i rl threshold speed .

9. lncreasin� the axial thrust bearing load wou ld i n crease the wh i r l threshold speed .

1 0. A small d is turbance was sometimes req u i red t o i n i ti a te the whir l mot ion in a w el l balanced ro tor.

TABLE 2. CHARACTERISTICS OF ROTOR WHIRL MOTION.

Cause of Whirl

Hysteretic or Internal Friction Whirl

Hydrodynamic fluid film bearings

Aerodynamic Excited Whirl

Dry Friction Whirl

Pulsating Axial Load and Torque Induced Whirl

Asymmetric Shaft Whirl

Gyroscopic Induced Whirl

Entrained fluid in rotor

Frequency

N > N, Nr = N,

N Nr ;:;2 -2- when

(half-frequency whirl)

g Nr = N, when -e-N > 2N, (resonant whip)

N > N, N, = N,

Nr -n N

N > N,

a c < 1

> 1

Nr = ( � , 1 , � , 2 ) N

N,r <N< NIX

N, N 3

Nr = ± n N

N, < N < 2N, N, = N,

N 2 , N

Subharmonic and Super- N r harmonic Whirl Induced N by Nonlinearity

N, ,2N,, N., % N,, % N,, 2Nl, N.

Comments

Occurs in flexible rotors due primarily to shrink fits and built up parts. Often requires unbalance or initial impulse to start whirl. Violent whirl may occur at first critical speed, disappear and reappear at a higher speed. Realignment of coupling may improve system.

Often referred to as oil whip, or half frequency whirl. Characteristics change with a flexible shaft where 8/C > 1 and is referred to as resonant whip. Large unbalance may suppress whirl motion.

Whirl motion change with change in power output. May be caused by seals, balance pistons or turbine or compressor tip clearance effects. May not be eliminated by change of unbalance.

Usually initiated due to rubbing caused by large unbalance leading to backward whirl motion. Can lead to catastropic failure on overhung rotor when interacting with disc gyroscopics.

Occurs in long flexible rotors under large power levels. Pulsating torque may induce large lateral shaft whirling usually equal to N,.

Caused by difference in shaft stiffness in two directions. Most violent near first critical speed. Keyways and slots should be avoided. Requires unbalance to initiate motion. May cause difficulty in balancing rotor.

External excitation or friction rub may excite rotor higher order forward and backward resonance modes. Ball bearing imperfections may initiate backward and forward whirl. Large transient whirl may develop with high acceleration rates with bowed shaft or skewed discs.

Instability encountered in high speed centrifuges. May occur in any high speed turbine or compressor if liquid, oil or steam condensate becomes inadvertently trapped in the internal cavity of a hollow rotor. Also produces large unbalance vibrations and is impossible to balance.

Requires unbalance or external excitation to initiate and appears frequently in under damped rotors. May combine with gravity in heavy horizontal rotor to produce a secondary critical speed. Half frequency whirl motion produced often confused with oil film whirl. Effect may be often aleviated by rotor balance, alignment of bearings or coupling.

ROTOR BEARING STABILITY 123

I t became clear to Newkirk tha t the ro tor dynamic behav ior cou ld not b e a t t r ibuted t o a cr i t ica l speed resona nce, s ince the h i gh v ibrat ions encoun tered always occurred above the first c r i ti ca l speed and refinement of ba lance had n o effect upon diminishing the whi r l ampl i tudes. There was no th ing i n the l i tera ture a t t ha t t ime to i nd icate tha t any mode o f mot ion, other than synchronous wh irl, was poss ible . Dur ing the course of the i nvestiga t ion, a theory of the cause of the v ibrat ion was postulated by A . L . Kimball (26) . K imbal l suggested tha t forces n o rmal to the plane o f the deflected ro tor could be produced by the hysteresis o f the meta l under �roing a l terna te s tress reversal cycles. Newki rk concluded tha t th ese ou t -o f-phase forces· could also be developed by a d i sk shrunk on a shaft . Newk i rk was unable at the time to explain why i n creased bearing o r foundat ion flexib i l i ty would improve s tability.

Newkirk concluded tha t the i n ternal fr iction created by shrin k fits o f the impellers and spacers was the predominate cause o f the observed whirl ins tabi l i ty. He h ad observed tha t when al l shr ink fits were removed from the experimen ta l ro tor , no wh i rl ins tabil i ty could develop.

Kimbal l (26) , at Newk irk's sugges t ion , cons tructed a special test rotor w i th rings on hubs shrunk on the shaft . He d id indeed confi rm Newki rk's conclusion that the fric t iona l effect of shr ink f i ts is a more act ive cause o f shaf t whir l ing than the internal fr ic t ion within the shaf t i tself . Measuremen t s showed tha t, even with the ra ther l igh t shr inkages used i n the tests, the effec t ive i n ternal fr iction may be increased from two to five t imes its or igi n al va lue. In fac t, Kimball found that long clampin g f i t s always lead to trouble w i th h i gh speed rotors .

For the case of a hub or a s leeve which i s fastened to a shaf t which i s afterward deflected, e i ther the surface fibers o f the shaf t must s l ip inside the sleeve as they alterna tely elongate or con trac t, o r the sleeve i tself mus t bend a long w i th the shaf t . Usu ally both ac t ions occu r s imul taneously to an exten t which depends upon the t igh tn es s o f t he shr ink f i t , and the rel a t i ve s tiffness o f t h e two par ts . H . D . Taylo r ( 41), after conduct ing n umerous tes ts wi th var ious hub con figura t ions, con cluded tha t the axia l con tac t length o f shr i nk f i ts should be as sho r t as permissible and as t igh t as possible without exceed ing the yield s t rength o f the m aterial.

Robertson ( 491 repor ts tha t even shor t, h ighly s tressed shr ink f i ts are not ent i rely devoid o f the problem. He s tates tha t even small , t i gh t shr ink fi ts may develop whir l i ns tabi l i ty, provided the ro tor i s given a su ffic ient ly large i n i t ia l d i s tu rbance or d isplacemen t to in i t ia te rela t ive in terna l s l ippage i n the f i t . I f long shr ink f i t s such as compressor wheels and impeller spacers are employed, i t i s importan t that these components be undercu t along the central region of the inner bore so tha t the contact area i s res t r ic ted to the ends o f the shr ink fi t . Robertson ( 49 I shows severa l designs of hubs and bosses which h ave been found to be benefic ia l i n reduc ing i n ternal fr ic t ion effects.

Robertson also concludes tha t a s imi lar effect can be produced by any friction wh ich opposes a change of the deflect ion of the shaft, such as fric t ion which exists

at the connect ions o f f lexible coupl ings, and gea r type coupl ings . He refers to this group o f fr ict i on forces as hys teret ic forces .

Followin g the analys is o f Smith (52 I, the equat ions o f mot ion o f a s ingle mass ro tor on d amped elas t ic supports are g iven as fol lows :

mx + ( u + vI x + w v' uv y + K x = mew�cos wt { 1 )

my+ (u +vi y- w v'uv y + K y = mew�si n wt {2) where

( K )2 v = C"

K" ;K.

IS the s ta t ionary damping c oeffi c ien t represen t i ng the effec t o f damping in the bear ing supports

_ C ( Kh ) u - I Kh+K. is the s ta t ionary damping coeffi c ien t represen t i n g the e ffect o f dampi n g i n the bear ing supports

Note that the ro tor equat ions of motion are c ross coupled by the inf luence o f i n terna l fri c t ion . When there i s s ta t ionary damping but no in ternal fri c t ion dampi ng, the equa t ions o f mot ion are uncoupled and the free o r trans ien t motion i s a d amped osci l la tory behav ior and i s always s table .

I t can be shown tha t the th reshold o f s tab i l i ty speed for this s imple system is given by the followin g relat ionship ( 1 7) .

I+ GJ (�:') J (3)

Hence i t i s concluded tha t a system wi th i n terna l damping bu t n o externa l damping i s uns table a t a l l speeds above the cr i t i ca l speed w,.. An ana lys i s o f the precess ive m o tion shows tha t the whir l rate i s approxi mately equal to the ro tor system cr i t i ca l speed w,.. An i nvest igat ion o f the unbalance response of the system indica tes that external damping restr ic ts the u nbalance response ampli tude but the i n ternal d amping does no t .

Figure 3 represents the d imens ion less r o t o r s tabil i ty threshold vs . support flexib i l i ty coeffic ien t R . Figure 3 shows tha t i n general the ro tor s tabil i t y t hreshold i s a lways equa l o r greater than the sys tem cr i t i ca l speed and i s a func t ion o f the i n terna l damping, suppor t dampi ng, and s t i ffness coeff ic ients . No te tha t i f no damping i s i n troduced in to the foundat ion the s tabi l i ty c r i te r i a reduces to

w. = We (4)

The above relat i onsh i p impl ies tha t i f bea r i ng flex i b i l i ty i s incorpora ted i n t o the system wi th o u t bearin g damping, then the sys tem cr i t ica l speed and the whi r l t hreshold speed w i l l be reduced. Thi s c r i te r ia therefore is i nadequate to explain the observat ions o f Newk i rk tha t greater s tabi l i ty can be achi eved b y founda t ion flexi b i l i t y on ly . The s tabi l i ty cr i ter ia for a symmetr i c founda t ion s ta te s tha t b o th foun dat ion flexib i l i t y and dampi n g mus t be inc luded to i nc rease ro tor s tabi l i ty . Smith


2

10.----.-----.--�.-,--.-,----.---. .. -,

8.0�---+-----r--+-,_��L-r-T--1--��-1

6.0 �---+-----r--+--P'-+-f----tf----+ or-r--H

e 4.o 1-------l-----+-,/ al:f � 2.0 �--1---:A---:A---+"7"1----+�--+--:o :::; iii i! (I) 1.0 ��:1�:!�� :!5 o.8 f-

sTABILITY CONDITION! i 0.6

13 0.4 a:: :1: .... a:: � 0.2

W<WcR. � (1+�2)

C2 C2 ROTARY DAMPING COEFFICIENT O • c; ; C1 STATIONARY DAMPING COEFFICIENT

O.I L--�--�-��--�---L-�� 0.1 0.2 0.4 0.6 0.8 1.0 2.0 4.0 6.0 8.0 10

STIFFNESS COEFFICIENT, R (DIM)

Figure 3. Stability Threshold of a Flexible Rotor With In ternal Friction on a Symm etric Elastic Bearing Support.

(52 ) , G un ter (20 ) , and Kellenberger (25 ) h ave demons t ra ted tha t i t w as the i n fluence o f an i so tropic bear in g s t i ffness t h a t w a s responsible for t h e improved s tab i l i ty observed b y Newkirk even in the absence o f damping in the bear i ngs .

When the s tab i l i ty threshold i s exceeded as shown i n Figure 4a for the l inear system as represented b y Equat ions l and 2 , t he r o t o r mot ion becomes unbounded and grows exponent ia l ly w i th t ime. In an actual system the ro tor orb i t forms a l im i t cycle due to the presence o f non l inear effect s e i ther in the shaft or the bear in gs as shown i n Figures 4 b and c. Var ious authors such as Tondl (59) and Dimentberg (l3J have chosen cons iderably more complex m odels in which the i n terna l damping i s i ndependent o f frequency, n o nl inear o r amp l i t ude dependen t .

Figure 5 represents t he s tabi l i ty cr i ter ia o f Tond l and represents the r o t o r externa l dampin g D and t he s tabi l i ty parameter H ( Z) versus frequency r a t i o v . The funct ion H ( Z I i s a func t ion of the i n ternal rotor fric t ion , external damping and ro t o r unbalance response . The ro tor i s u n stable when the D2 and H (Z I c urves i ntersect . This f igure demons tra tes the impor tan t conc lus ion tha t i f a ro tor becomes uns table j us t above the cr i t ica l speed that i t w ill not n ecessar i ly be un stable a t al l h i gher speeds. For l i gh t damping the ro tor is u n stable above t h e cr i t i ca l speed . As t h e damping i s i n creased t he r o t o r i s uns t able on l y f o r a v e ry n arrow range above the cr i t i ca l speed from the speed ra t io of v � t o v:1• The rotor restabi l izes above the cr i t ica l speed and is stable un t i l the ro tor speed i s i nc reased to a much h i gher speed v 4 where i n s tab i l i ty i s again encountered.

The s tabil i t y c r i ter i a o f Tondl may wel l explain the unusual rotor wh i r l mo t i on reported by S todola (56) over 40 years ago in wh ich cer ta in gear t ype coupl ings would cause large n onsynchronous whir l mot ion over a shor t operatin g speed range .

CONDITIONS:

Wcr • 7 06 RAD /SEC W1 • 1412 RAD/ SEC W • 1 500 RAD/SEC

(b) FI NITE ORBIT (c) FINITE ORBIT NONLINEAR SYSTEM -8 •0.01 NONLINEAR SYSTEM-8•0.04

(a) UNSTABLE ORBIT LINEAR SYSTEM -8 • 0

w > w,

Figure 4. Effect of Non linearity o n Rotor Motion A bove the Threshold of Stability.

D2 H(v)

::!: 9 210"4

.. _ uiJ -"

� <5 � STABLE Q. uo·• :::E � :::E 11.1 .... Ul > Ul

0 �'I I

11.1 ..J m � Ul 2 :::l

STAB L E

... 2 2 3 4

wlwc •., SPEED (DIM)


U NSTA B L E

...

Figure 6. Photographs of Ro tor Motion With Internal Friction (Ref. Kushul') .

Figure 5. Rotor Stability With Internal Damping (Ref. Tondl) .

u p s truc ture of a long wooden spind le inserted ove r a thin steel shaft . I t is easy t o v i s ua l iz e h o w such a l ong shr ink f i t cou ld l ead to s tab i li ty problems. Robertson reported i n 1935 ( 49) that long con tin u o u s shr ink fits invariab ly lead to d i fficu l t i es when opera t ing above the ro tor f i r s t cr i t ica l speed.

An extensive i nvestigation of the self excited whir l m otion o f h i gh speed text i le spindles was conduc ted b y Kushu l' ( 17) . The sp indles were composed o f a b u i l t

Figure 6 represents t ypica l r o t o r o rbits obtained b y Kushu l' on the text i le sp indles o pera t ing above t h e s ta b i l i ty t hreshold. A t the t ime Kushu l' recorded this da ta ,

10 8 6

10 8

(/) 6 ..J � 4 u..i 2

3H

3V FRACTIONAL FREQUENCY WHIRL

PROBE 4 PROBE 3 PROBE 2 PROBE I

i I r-- 3.75 l l ' � � ' +T+-tr-·Er-·-B--w---l.5 �67� -.12.25 � -col 2.25 f.1.oo5� i

§ 01�------------------------------------��-¥�-------------------------------------+--� 3V ::::i 10 � 8 <( a::

� a::

I�UNDOWN 01<( zu Oi= frllo: rnu

3V 2V

4,000 6,000 8,000 0 2,000 4,000 6,000 8,000 ROTOR SPEED, RPM

Figure 7. Unbalance Response and Self-Excited Instability With a Three-Mass R o to r System.

15 -olE "-0::

,..,(.)


precis ion electronic proximi ty probes to mon i tor the rotor motion were not ava i lable so h e had t o resort to an opt ica l system. H e a ttached a f in e n eedle to the spi ndle end and obta ined the fol low ing p ic tures by photograph ing the resu l t ing mot ion under a microscope. Figure 6a represent s the spindle wh i rl mot ion at the s tabil i t y threshold . As the ro ta t i ona l speed was i nc reased, Kushul' observed tha t the mo tion could abruptl y change to the orbit as shown in Figure 6b in wh ich the whir l ra te i s % of ro ta t ional speed . This behav ior cannot be obtained i n a l i nea r system.

Figure 7 represents the unbalance response and self-exci t ed wh i r l ins tabi l i ty encountered w i th a three· mass ro tor system due to i n ternal ro tor fr ic t ion . The ro tor was r u n from 0 to 1 0,000 RPM and ro tor t races o f the mot ion were obta ined at the d i fferent probe loca· t ions along the shaft in the h or izonta l and ver t ica l d irection . The ro tor has th ree c r i t ica l speeds i n the opera t ing range at approxima tely 2,000, 6,000 and 10 ,000 RPM . Note t h a t below t h e f i rs t cr i t ical speed super h armon i c osc i l la t ions a r e encoun tered due to n on-linear i ty i n the ro tor and bear ing supports . Superharmoni c osci l la t ions a re often experienced in machinery and are u sual ly n ot o f serious consequences.

The c r i t i ca l speeds of this par t icu lar model are similar to those shown i n Figure 8 . The first i s a symmetric bend ing o f the shaft , the second i n a con ica l m ode and the thi rd mode i s a symmetr ic bend ing i n which the ampl i tude a t the beari ng locat ions i s out o f phase to the ampli tude a t the center shaf t and the maximum ampl i tude

I D I .; I I § I . I �

1-

I a I ; I

I I I i I

,._,Ill' OUTleR. ftms

CAS!

II I • 31�1 "'"

II Z • 6"159 "'"

.. I

!fl .: I

I I

i s experienced a t the bearings . I t i s o f i n terest to n ote tha t a four th cr i t i ca l speed i s encountered a t a much h i gher speed wi th the three mass model . Thi s par t icu lar mode i s c aused b y the gyroscopic effects o f the d isks. Figu re 7 shows tha t as the rotor approaches the th i rd c r i t ical speed, the maximum ampli tude i s experienced at probe locat ions 4 and 1 which correspond to the bearin g ends . The mot ion inc reases rapidly due to rotor unbalance and i s synchronous. Probe 2V near the ro tor center shows l it t le i nc rease i n ampl i tude as i t approaches the third c r i ti ca l speed . Abrupt ly the center o f the shaf t a t s ta t ion 3 j umps in to a v io len t self-exci ted whir l i n s ta· b i l i t y . Des truc t ion o f the ro tor would have resul ted i f t h e u n i t were not i mmediately shu t down. T h e mecha· n i sm tha t caused th i s ins tab i l i ty is obvious ly more com· p l ica ted than the s impl i fied sys tems described b y equat i on 1 and 2 and i s more c losely s imulated by the non -l inear sys tem of Tondl ( 60 I . Although u nbalance i s n ormal ly n o t a maj or i n fluence i n ro tor ins tab i l i ty , i t was seen i n th is case that l arge unbalance was n eces· sary t o i n i ti a t e the wh i rl i n s tabi l i ty . This observa t ion i s s imi lar t o the ear l i e r conc lus ions of Newkirk ( -t.O I on the wh i r l mot ion encountered with cent r i fu gal compres· sors due to the i n ternal fr ict i on of shr ink f its .

Figure 9 represents t h e wh i rl orbi ts o f an u nbalanced rotor w i th i n ternal fr ict ion on anisotropic sup· por ts. The ro tor i s s imi lar t o the configura t ion repre· sen ted in Figure 6 for the l i nea r case except tha t the suppor t s t i ffness in the X d i rec t ion has been reduced to % i ts or ig inal value. This has caused the ins tab i l i t y speed to i ncrease from 1412 r ad/sec to 3230 r ad/sec.

II! • Tl2ll "'"

1-�

a J "I

I I I d

I I I ; I I § I f I

N. • 1'1610 "'"

Figure 8. Critical Speeds and Mode Shapes for a Three-Mass Rotor Includin g Gyroscopics.

PlWCE.EDINGS OF THE FIRST TURBOMACHINERY SYMPOSIUM 127

STEADY STATE WHIRL ORBITS OF AN UNBALANCED ROTOR CONDITIONS'

C) W=400

W=3200

/

W=SOO

w .. 101

M "'0.25LB SEC2/!N. Ky" K2"" 250,000 U3/!N. Kx = 125,000 LB/IN. Oa "' l)z "' 200 RAD/SEC Wcx "' 517 RAD/SEC Wcy,..707 RAO/SEC Ws "' '3230 RAD/SEC

WQ WQ O W=2000

0 r� 0

W"l500

U , Q W=IOOO

W<>900

Figure 9. Whirl Orbits of an Un6alanced Rotor Tllith In ternal FrictZ:on on Anisotropic Supports.

Thus i t i s seen w it h certain cases o f i nstabil i t y the threshold speed m ay be improved by supporting the rotor on an an iso t;opic supports . \\1hen the bearin g characterist ics are unsymmetr ical the SYHchronous o r runn ing speed o rh ib due to unbalance an�

. :wt c ircu lar hut are ell i pt ical . The �ystem has two ent1cal speed�; one correspond ing to the X d irec t ion and one for the Y d i rect ion . When the rotor i s opera t ing a t the speed corresponding to the horizonta l critica l

;speed the . el l ipse

i s oriented predominantly along the honzonlal aXI6. A_s

the speed is i ncreased the el l ipse r_hange;; from a

_ho.n

zontal to a vert ical orien ta t ion . When the speed 1s lll· creased above the cri t ical speeds, the orbi t changes to circular orb i t wi th a rad ius equal lo the u nbalance eccent r ic i t y . [pon approach ing the s tabi l i ty threshold a t ;)200 rad /see the transient mot ion no longer damps out but grows rapidly . Even when the ro tor i s . below the s tab i l i ty threshold a t .3 ,000 rad see there 1s a small eompo;1en t of non-synehronous >�h i rl mot ion present in the orbi t . I t i s o ften observed m turbomachmery tha t a smal l componen t of wh i r l mo t i on exis ts before the threshold o f s tabi l i ty i s cncoun tered.

Fluid Film Bearing Whirl

On e of the m aj o r �ou rces of wh i rl i n s tab i l i t y i n lurhomachinery i s tha t caused by flu i d film bear ings and sea l s . Figure 10 representB a n ideal ized hydrodynamic hear ing

-configuration . Cnder no rma l operat ing

condi t ions the shaf t and the bear ing sur face a re separa ted bv a f i lm of o i l . The rotat i(-,n of the sha f t i n the conv�rgi n g f i lm area bui lds u p a pressu re prof i le w h ich supports the shaft as given h} the follow ing wel lknown I\eyn olds equat ion .

(! iJO

(� )� L

;\ ( [ l

+ 2 () [ uz

[ i)p fPp (10 i) [ T/l IPp

2(€1)) 1 () - iJO

pH ])

] Op 7ll

[

+

] pH ]

(5)


BEARING GEOMETRY Figure 10. Hydrodynam ic Journal Bearing Configuration .

I n addi t ion t o t h e bearin gs , there a r e other components in a typ ica l t urbomach ine such as the sea ls i n which the sha f t and s ta t ionary sea l o r cas ing i s separated by a f i lm of f lu i d . I n t h i s c a s e t h e seal can generate a hydrody n am i c pressu re d i s t r ibu t ion a round the shaft and a c t a s a bear ing. The forces induced on a shaf t may c rea te a cond i t ion kno w n a s o i l fi lm or ha l f-frequency ro to r w h i rl . For example Figure l l represen ts a t yp i ca l p res· sur e d is t r ibut i on d eveloped in a hydrodynamic j ou r n a l bear ing for a p a rt icu lar eccentr ic i ty . I n t h i s c a s e the

f i lm i s considered to cav i tate when the pressure d i s t r i b ut ion i s be low the o i l vapor pressure . If t he f i lm does no t cav i ta te in a p la in j ou rnal bear ing then the bearing force-d isp lacement a t t i t ude angle i s 90° and the system w i l l be u nstable at a l l speeds.

Figure 12 represents the whir l mo t i on of a vert ica l ba lanced ro tor runn ing at 6,500 RPM. The rotor is given a smal l i n i ti a l d isplacemen t and i t o rb i ts ou tward a t a p recession ra te which i s approxim a tely lh the ro ta· t i ona l speed. I f the ro tor were permi t ted t o r u n i n th i s


condi t ion the heari ng would eventual ly fai l s imi lar to the hear ing shown in Figure 13. Figure 1 3 i s a photograph of a j ourna l hear ing that w as operated above the s tabi l i ty threshold and shows fai lure due to massive fat igue p i t t ing. The large orbi t o f the shaft caused a ro ta t ing pressure prof i le s imi lar to tha t shown in Figure l l which eventua l ly caused massive fat igue p i t t i ng o f the j ournal s urface. I t i s obvious tha t the l arge orbi t ing represented in Figure 12 w o uld he u nacceptable for con t inuous opera t ion .

Figure 1 4 represents the wh i r l mot ion o f the ver t ica l ro tor w i th a n unbalance added to the shaft. The u n balance eccentr ic i ty r a t i o E M U = 201/c of the hear ing rad ia l clearance. No te that when the rotat in g unbalance force has been added to the ro tor the orbi t i s smal ler . The orbit i s now hounded and forms a l im i t cyc le due t o the nonl inear i ty o f the f i lm. Hence i t i s seen tha t for the case o f the vert ical ro tor, the addit i on o f unba l ance ac tua l ly reduces the ro tor whir l orbi t. There h ave been n umerous inc iden ts where manufacturers o f wa ter lubr icated vert ica l p umps h av e used p la in j ourna l hear· i ngs sat isfactori ly. This may he expla ined by the e ffect of u nbalance on the ro tor mot ion and also the effect

y

of hear ing misa l ignment . Var ious i nvest igators have seen tha t s l i gh t m isalignment o f the hearin g hous ing w il l promote s tabi l i ty. The p la in j ou rna l hear ing how · ever i s a poor choice f o r use i n a ver t ica l ro tor -hear i ng system.

I t i s o f i nterest to no te tha t i nves t iga t ions w i th gas hear ings for gyroscopes have shown that qui t e often the m an u facturer would produce a hear ing w i th lower s ta· hi l i ty character is t ics as the qual i ty con tro l would i n · crease. As more care was made i n produc ing a wel l a l igned rotor-hear ing and a c i rcu lar geometry, the s ta · h i l i t y characteris t ics of the system w ou ld decrease. Such effect s as hear ing misal ignment, e l l ip t i c ity , and surface i r regular i t ies h ave been known t o change the hear in g s tabi l i ty characteris t ics .

The orbit as shown i n Figure 1 4 represents a com· b ina t ion o f synchronous motion due to unbalance and ha l f frequency whirl motion due t o the hear ing flu i d fi lm. By examinat ion o f t h e s ize o f t h e i nner lobe w i th respect t o the orb i t, the ra ti o o f the synchronous t o the nonsynchronous whir l componen t can h e est imated. For example Figure 1 5 represents var ious combinat ions o f

X= 0.05 Y=-G.02

•

X

--- ---------------

- ---- --- - ------ ---- ---

---·----------·---------

·---------------·--------

--------- - -�-�-, _____________ _

--·--- ---·----------------- - ---

.. , ............. . - --- - ------- -----

-- - ---- --- --- ----------- -- --- ---- --- -

- - ---------- --- -- --------- - -- - -------- -

-----·--·------------�--------------------

x = o.ooo Y = o.ooo

P=O·O

r 0-257

2'ii

Figure 11. Journal Bearing Pressure Distribution Considering Cavitation.


VERTICAL BALANCED ROTOR

N • 6500 fi'H fl • 1.00 IN. L • 1.00 IN. C • 5.00 MILS TflSHAX • 1. 39 5 • O.Qit8 55 • 0.012

X(T=Ol =0·1

Ill. IIIIlS

WT • 0.00 W • 1800 LB. llJ.S • 1. 000 flEYNS FHAX • 2'i93.0 LB. Atl)

OOCUAS AT li.96 CYCLE WS • 2.1i5 E5 • 0.811i

Figure 12. Journal Orbit of a Balanced Vertical Rotor With Small Initial Velocity for 5 Cycles.

synchronous and ha l f frequency whir l . I f a smal l component o f half frequency whir l motion is s uperimposed upon the synchronous motion then the c i rcu lar o rb i t begin s to appear as two c i rcles, one super imposed upon the other . A s the h al f-frequency wh i r l componen t i n creases, the s i ze o f the i nner lobe decreases . A comparison o f the figures i n Figure 15 t o the orb i t i n Figure 1 4 ind i ca tes that the ha l f-frequency w h i r l motion i s

Figure 13. Photograph o f Failed Journal Bearing Opera ted A bove the Stability Threshold.

VERTICAL UNBALANCED ROTOR

N • 6500 fl"" fl • 1.00 JN. l • 1.00 IN. C • 5.00 "JLS Tf\StllX • 1.31 s. 1.733 ss c 0.1133 El4U • 0.20 su • 1.'1'16 TAOttlX ., 1. 10

... .,..

NT • 0.00 W • SO LB. tl.liiS • 1. 000 fiEYNS FltAX • 65. 7 LB. AND

OCCURS AT 0. 58 CYClE WS • Z.'IS ES • 0.211 FU • 59.95 LB. FIJAATIO • 1. 20 ESU • 0.2'tll

Figure 14. Journal Orbit of an Unbalanced Vertical Rotor for 5 Cycles (N = 6500, Emu = 0.2) With Damped Half-Frequency Whirl.

approximate ly equa l to the synchronous unba lance com· ponent [A=B=.5].

Figure 1 6 represents the s tab i l i t y threshold based on the shor t j ou r n al approximation cons ider ing var ious values o f ex te rna l l oad ing . For example the l i n e W t = 1 represent s t he hor izon ta l ro tor . Values above the threshold l i n e wi l l be u n stable and below the l ine w il l b e s table . For example the stabil i t y con tour for the h or izon ta l ro to r shows that if the bear ing eccentr ic i ty i s above . 75 the sy s tem w i l l be s tab le for a l l speeds. Conversely if the s tab i l i ty p arameter w . is less than 2 .5 the bear ing w i l l be s tab le for a l l va lues of eccentr ic i ty . A s the va lue o f the external loading parameter W t de· c reases, the s tab i l i t y boundar ies of the sys tem decreases. The value o f Wt = 0 represents the vert ica l ro to r and the system wi l l be u n stable for a l l speeds and eccentr ic i · t i es .

F igure 17 represents t he orbit o f a ba lanced hor i · zonta l ro tor a t the s tab i l i ty threshold . The ro tor i s released f r o m the or ig in and precesses a t approx imately 1;2 frequency wh i r l in a d imin ish ing orbi t un t i l i t reaches the s teady s tate eccen tr ic i ty o f 0.2. The beari n g steady s ta te load-deflect ion a t t i tude angle i s abo u t 70°. I f the ro tor speed i s now i ncreased from 6,500 to 1 0, 000 RPM,

ROTOR BEARING STABILITY 1 3 1

the s tabi l i ty parameters w. i n creases from 2.-!.5 to -t..O. According to the s tabi l i ty p lo t of Figure 16 the rotor i s we l l above the s tab i l i t y threshold and hence w i l l be un stable. F igure 1 8 shows tha t the ro tor i s h ighly unstable and precesses ou tward w i th approximately % frequency whi r l mo t ion. The orb i t wi l l con t inue to grow un t i l it reaches an eccent r i c i t y o f approximately .7 in which i t then forms a l im i t cycle. Al though the bearing does not immedi a te ly fai l , the large whirl motion i s undesi rable because o f the poss ibil i ty o f fatigue p i t t ing as i l lus t ra ted in the bear ing o f Figure 1 3 .

I f, h owever, u nbalance equiva lent to 20/{ of the bearing clearance is i n t roduced the whi r l orbit i s con·

siderably reduced from tha t experienced w i th t h e ba l anced rotor as sho w n i n Figure 1 9 . This however i s an undes i rable prac t i ce because o f the h igh t r an s m i t ted forces . For example the ro ta t ing u n balance l oad i s 1 50 lbs . wh i le the max imum force t ransm itted i s 213 lbs . Thu s the dynamic transmiss ib i l i t y i s greater tha n 1. I n a properly designed rotor-bear ing supp o r t s y stem, the dynamic transmiss ib i l i ty should be less t han l to achieve good attenua t ion of t h e forces t r ansmi t ted through the s t ruc ture .

The s tab i l i ty character is t ics o f the c i rcular j ou r n al bear ing are rather l i m i ted and therefore are n o t app l ica b l e t o modern h igh speed turbo-mach inery because of the

ANALOG COMPUTER TRACES OF VARIOUS COMBINATIONS OF

SYNCHRONOUS AND HALF-FREQUENCY WHIRL

A•0. 9 8•0.1

A•0. 4 8•0.6

A• MAGNITUDE OF SYNCHRONOUS

WHIRL COMPONENT

A•0.8 8•0.2

A•0.3 8•0.7

8 • MAGNITUDE OF HALF -FREQUENCY

WHIRL COMP ONENT

A•0.7 8•0.3

A•0.2 8•0.8

y

A•0.6 8•0.4

A•O.I 8•0.9

Figure 15. A n alog Computer Traces of Various Componen ts of Synchrono us and Half-Frequency Whirl.


30

20

10

,...., 9 :::!!: 8 0 7 '-'

� 6

5 -..;, a� "

Ill a

4

3

2

I 0

16...;1

8

6

4"""" 3

2 � ,I

UNSTABLE WT=I.q_... v � ST�BLE

I / 0.5 _..,.

I I l/ WT=� /(mg) I I X I y I 0.25/

0.2 0.4 0.6 0.8 ECCENTRICITY, e0 (DIM]

I

1.0

Figure 16. Stability Map for the Short Journal Bearing Considering Consta n t Loading.

possib i l i ty of self-exci ted whir l ing. One of the ways o f improving t h e s tab i l i ty o f t h e plain j ourna l hear ing i s by means o f t h e addi t ion o f a p ressure d a m t o preload the hear ing. I t can he seen from the s tab i l i ty char t o f Figure 1 6 t h a t i f t h e load parameter W t i s greater than 1 then the s tab i l i ty character ist ics o f the heari n g w i l l improve. For example w i th a W t va l ue o f 6 the heari n g w i l l he s table f o r a l l eccentr ic i ty above .45 a n d for speeds below W8 = 6. Example 1

Consider the followi n g sample calcu la t ion o f t he s tabi l i ty character is t ics o f a two stage c en t r i fu ga l compressor o n w a te r lubricated p ressure dam hear ings . The hear ing character is t ics are as follows:

D = 2 . 5 i n . Cr = 0.00 1 5 i n . M = 1 .0 X 1 0-7 Pocket Depth = 0.03 1 i n .

Wg = g c 507,

L = 0.75 i n . W /hrg = 50 lb . N = 1 9,100 RPM.

w = 2000 Preload = 1 00 lb .

3 .94, W t = 3 .00

From Figu r e 16 assuming E < 0.5 for Wt = 3 .00, w" = 4.3 Threshold Value s ince w, > w/wg system i s m a rgina l ly s table .

There are o ther w ays to improve hea r i ng s tabi l i ty o ther than the pressure dam hearing such a s t h e exter· n al ly pressur i zed hear ing, t i l t i ng pad, o r the m ul t ilohe hear ing.

Figure 20 represents the mot ion o f a n external ly pressurized gas hear ing grinder spindle operat ing a t 52,500 RPM. I n Figure 20a the spindle shows on ly a smal l synchronous orbit o f 1 80 JL i n . Wi th on l y a smal l change in the opera t ing speed, the spindle d eveloped the l arge whir l orbit as shown in Figure 20h. F urther i n crease in speed beyond th i s point would have resul ted in destruc t ion o f the grinder sp indle . Extreme care mus t h e exercised i n opera t ing gas hearin g machinery above the s tab i l i ty threshold . Appl icat ions o f gas hear ings to smal l h i gh speed turhomachinery are now b ei n g devel· oped us ing external ly pressurized, t i l t i ng pad, and fo i l hear ings.

The m u l ti lohed hearin g conf igu ra t ion a s s hown i n Figure 2 1 h as superior s tab i l i ty characteri s t i c s a s com· pared to the plain j ourna l hear ing. Figure 22 represen ts t he s tabi l i ty o f the three l obed hear ing configura t ion for var ious values of preload . A zero preload represent s an ax ia l groove hear ing. I t i s o f i n terest t o n ot e tha t

BALANCED ROTOR N • 6505 APH

A • 1.00 IN.

L • 1.00 IN.

C • 5.00 HILS

TASHRX • 1. 27

s • 1.81!5

55 • 1.81!5

ND.Lo&l

W • 1!7 LB.

HU•5 "' 1. 000 AEYNS

FHAX • 59. 9 LB. AND

OCCURS AT 0.53 CYCLE

ws • 2.1!5

ES • 0.200

Figure 17. Journal Orbit of a Balanced Horizontal Rotor at the Stab ility Threshold (N = 6500, W = 50, C = 0.005, LID = %).


BALANCED ROTOR N • 10621 APH

A • 1.00 IN.

L • 1.00 IN.

C • 5.00 HILS

TASHAX • 3. 32

s • 3.013

ss. 0.753

Nll.l02llll'l

W • 'l7 LB.

HU•S • 1.000 AEYNS

FHAX • 156.2 LB. AND

OCCUAS AT 5. 00 CYCLE

WS • 'l.OO

ES • 0.130

Figure 18. Journal Orbit of a Bala nced Horizon ta l Rotor A bove the Stability Threshold (N :::::; 10,621, w :::::; 47, c :::::; 0.005).

the ax ia l groove bear ing does n o t appear to have super ior s tabi l i ty characteris t ics to the p la in j ourna l bear ing . Al though improvements i n s tab i l i ty have been experi· enced w i th the axial groove bear ing in comparison to the plain j ourna l bear ing, i t i s fe l t that this effect may be due t o the misa l ign men t . Note tha t the s tabi l i ty threshold of the mul t i lobe beari ng with a pre load factor o f 5 = 0.6 i s Ws = 9 which i s over twice the stabi l i ty value o f the pressure dam bear ing i n the prev ious exam· pie. In the vert ical pos i t ion , the axia l groove bearing ( 8 = 0) i s completely uns table. The s tabi l i ty threshold o f the opt imum three-lobe bear ing o f L/D = l .O, with an offset factor 5 = l.O (completely convergin g fi lm) and preload of 0 .5 is gi ven by

30MO�-tL (

CR )3 NIU')I =

Improved s tabi l i ty i s obtained by means o f a flex i b l e suppor t o r a f loat in g bush damper as shown i n Figure 23 . Figure 23 represents the s tabi l i ty threshold vs . the bush clearance for a par t i cu lar rotor conf igu ra t ion . Note tha t a maximum stabi l i t y threshold for the p la in j ourna l bearin g i s 2.5. If, however, the bush clearance i s made

too large o r t oo smal l then there i s l i t tle improveme n t i n s tab i l i ty.

The effec t of ro tor flex ib i l i ty can have a pronounced effect on the threshold speed and the oi l f i lm whi rl he· hav ior in a ro tor as shown by var ious i nves t igators such as Hor i (24), Lund (35), Ruhl (50) and o thers . For example Figure 2-t represents t he s tabi l i t y character is t ics and ampl i tude and frequency measuremen ts ob ta ined by Hor i w i th a f lexible rotor in a j ou rn a l bear ing. In Hori 's f igure the s tabi l i t y threshold i s over twice the ro tor f i rs t c r i t ical speed as shown i n Figure 2-t a,b and c . Note a l so tha t the ro tor ampl i tude does no t become unbounded above the threshold speed but forms a l im i t cycle .

A erodynamic Induced Whirling

I n addit i on t o s tab i l i ty p roblems created by hydro· dynamic bear ings in turbomachinery another importan t cause o f ins tab i l i ty is aerodynamic cross coup l ing forces and ins tabi l i ty created by labyr in th seals and balance p is tons . Stodola , in h i s ear l ier work on s team turb ines, reported on the i n fluence of leakage in crea t ing i nsta· bi l i ty problems in steam turb ines . Alford has reported

UNBALANCED ROTOR N • 10621 APH

A = 1.00 IN.

L = 1.00 IN.

C • 5.00 HILS

TASHRX • 'l.SS

s • 3.013

ss. 0.753

EHU .. 0.20

su • 0.9'l1

TADHAX • 1.'l2

Nil. I 0211115

W = 'l7 LB.

HU•S • 1.000 AEYNS

FHRX • 213.8 LB. AND

OCCUAS AT 0.93 CYCLE

WS • 'l.OO

ES = 0.130

FU • 150. 'l6 LB.

FUAATIO • 3.20

ESU • 0.331

Figure 19. Journal Orbit of an Unbalanced Horizontal Rotor A bove the Stability Threshold (N :::::; 10,621 RPM, W :::::; 47, C :::::; 0.005, Em u :::::; 0.20).

134 PROCEEDINGS OF T HE FIRST TC RBOMACHINERY SYMPOSIVM

( 1) on aerodynamic cross coupl ing: effec ts in caus ing s tab i l i ty prob lems in j e t a i rcraf t engines. A l though a cent r i fugal compressor may be designed to have a s table bearing configurat ion. such as a t i l t i ng pad arrangemen t i n a mu l t i s tage cen tri fugal compressor, i t may be sus· cept ible t o aerodynamic ins tab i l i ty under h i gh power levels when run apprec iab ly above the f i r s t cr i t i ca l . Recently there h ave been a n umber o f cases o f aerody· n amic exci ta t ion in cen tr i fugal compressors caus ing severe s tabi l i t y problems.

Figure 25 represents the ampl i tude vs. speed of the NASA Brayton cycle experimenta l cen t r i fugal compres· sors . The compressor i s a s i ngl e s tage cent r i fuga l u n i t w i th b a l l bear ings moun ted on f lexible supports . When several inc idents o f fai lure were encoun tered in this sys· tern non-con tac t ing probes were placed on the u n i t to m o n i tor the ro tor mot ion . Figure 25 shows tha t as the rotor speed approached the f irs t c r i t ical speed around 1 7,500 RPM super-harmoni c mot ion was obta ined i n addi t i on t o the synchronous mot ion . The superharmonic mot ion i s approximately equa l t o the second c r i ti ca l speed. As the speed was further i n creased the ampl i tude a t the secon d cr i t ica l speed d id not increase dur ing accelera t ion . I t w as on l y upon a reduc t ion o f speed tha t the second cr i t ica l was exc i ted . There have been several inc idents reported in w h ich this phenomen a i s so severe t h a t t h e ampl i t ude upon reduc t ion o f speed i n creases at the second cr i t i ca l and does n o t reduce un t i l be low the f i r s t cr i t i ca l speed. A s the speed was fur ther i n creased t o 52,000 RPM severe w h i rl ins tab i l i t y was encoun tered a t the compressor wheel as shown b y the enclosed orb i t . Figure 26 represent s the NASA Brayton cycle tes t ro tor a t var ious power levels a t 52,000 RPM . Figure 1 represent s a superimposed pic ture o f t h e ro tor mot ion a t no load and par t load . Under no load the orb i t i s a smal l synchronous orb i t . As the power level

i s increased on the compressor wheel the orb i t hegin s to increase as s h o w n b y F igures 1 , 2 , 3 , a n d - L For example i n Figure 6 the ampl i tude wou ld ha,·e caused a des truc t ion of the ro tor i f the un i t were not shut down . The w h i r l ra t io observed i n th i s orb i t i s approximately l '5 of runn ing speed, or around 10,000 RPM.

A n u mber of i nc idents of self-exci ted aerodmamic ins tabi l i ty h ave been repor ted i n mu lt i -s tage cen tr i fu ga l compressors . I n a n umber o f these cases i t h a s been found that a change in bear ing character is t ics alone was rela t ive ly ineffec t ive in s tab i l i z ing these ro tor sys tems. However , one successfu l approach tha t has been used to s tabi l ize ro tors subj ec t t o aerodynamic ins tabi l i t y has been the use o f a squeeze f i lm damper support system incorporated wi th the bear ing. Figure 27 represents the rotor o rb i t o f a seven s tage t urbo-compressor before and a fter s tabi l i za t ion w i th a squeeze f i lm bear ing. The u pper lef t figure represen t s the u nstable ro tor orbi t a t a d i scharge pressu re o f 1 75 ps ig. I f the d i scharge pres· sure were i ncreased above this value the orbit wou ld h ave caused des t ruc t ion o f t he ro tor . The u pper r igh t f igure represen ts the s tab le ro tor orb i t a t the fu l l d i s · charge pressure o f 650 ps ig a f t e r i n t roduc ing the squeeze film damper . Upon f i l t ra t ion o f the orbi t on ly a smal l componen t o f frac t ional frequency whi r l remained in the system. There have been several unsuccessful a t tempts t o s tabi l ize a turborotor sufferin g from aerodynamic i ns tabi l i ty by means o f a squeeze f i lm damper suppor t . A recent ana ly s i s o f the s tab i l i t y o f a f lexible ro to r w i th aerodynamic c ro s s coup l i ng h as shown tha t the suppo r t system mus t b e careful ly t u n e d t o t h e r o t o r i n order t o promote s tabi l i t y . For example Figure 28 represents the s tabil i ty character is t i cs for var ious va lues o f suppor t s t iffness K. vs. suppor t damping. Va lues above the r eference l in e 0 represent a rea l posi t ive roo t and ind i ca t e an unstable system, wh i l e those be low the l ine i nd i ·

MOT I O N OF A G A S B E A R I N G R OTOR AT T HE TH R E S HO LD O F STA B I L I TY

A. STABLE SY NC HRONOUS PRECESSION

W < Ws

B . UNSTABLE NONSY NC H RONOUS PRECESSI O N

W >W s

CONDITIONS :

Ws • 5 5 0 0 RAD SEC , A • 3. 5

SCALE : I MAJOR DIVISON • IOOp. in

Figure 20. Gas Bearing Rotor at the Threshold of Stab ility (N 52,500 RPM) .


Figure 21. Multilobed Bearing Configuration .

cate s tabi l i t y . Note that the opt imum s tab i l i ty i s ob ta ined i n thi s case w i th a support damping o f 1 ,000 lh -sec/ in and a support s t i ffness between 50,000 and 1 00,000 lh/i n . If the damping i s e i ther too smal l or too large the system wi l l he un stable . Conversely if the support s t i ffness increases to 250,000 lh/ in the system wil l also he u n stable. In general, squeeze fi lm d ampers are h igh ly n o n l inear sys tems and i f the damper clearance i s not properly selected and the damper correc t ly a l igned, the non -l inear i ty can eas i ly cause the damping characteristics to he excess ive thus defeat i ng t h e p urpose o f the support system.

SUMMARY

It is seen that there are many factors which can cause wh i r l i ng i n turhomachinery . Pr inc ipa l mechan isms o f concern to the compressor des ign engineer are

h ydrodynamic hearin g i n s tab i l i ty , i n ternal f r i c t ion effect and aerodynamic cross coupl ing . These effects can cause ser ious self-exc i ted whir l ins tab i l i ty i n a turbo -ro tor which can lead t o des truc t ion o f the sys tem. If turbom achinery i s to he bui lt that is going to run several t imes greate r than the f irs t c r i ti c a l speed then i t is important that the system h e c hecked for s tabi l i ty and that proper elec t ron ic mon i to r ing equipment h e p laced on the mach ine during opera t ion so a s t o detect the possib l e occurrence o f a dangerous wh i r l mot ion which may cause des truct ion of the machine .

ABBREVIATED B IB L IO G RAPHY ON ROTOR-BEARIN G STA B IL ITY

l. Alford, J. R., "Protect in g Turbomachinery from Self-Exc ited Rotor Whirl," Journal o f Engi neerin g for Power, Trans. ASME, Ser ies A, No. 4, pp. 333-344, October 1 965.

PROCEEDINGS OF THE FIRST 1TRBOMACHINERY SYMPOSIL'M

!.0 v I C\1

I 0 0 0 II II I vo C(l 00

I I I I I I

' I I

w --- 3 - LO B E B E A R I N G - · - PL AI N JOUR N A L B E A R I N G - - - --- 180 ° LO N G B E A R I N G - - - AX I A L GROOV E B E A R I N G

0 . 2 0 .4 0.6 0 .8 E C C E N T R I C I T Y R AT I O , E0 ( D I M )

Figure 22. Stab ility of a Three-Lobed Bearing for Various Values of Preload.

2 � 1\llen� E. E ., "A Coupl ing Desjgn to lVIin hnize l)y .. n amic lns tabi l i t ies o f H igh Speed Sys tems" Eng i m�erin g Proceedi n gs o n H igh Speed Flexib le Cou p l i n gs, p . 4 1, Pennsy lvan ia S ta t e Universi t y, Jan . l %3, p. 1 50 -l 5H.

3 . Badgley, R . H ., ' 'Turborotor I n s tab il i ty : Dynamic L1nhalance, Gyroscopic and Var iable Speed Effect s w i th Fin i te-Length Cav i tated, Flu id -Fi lm Bearings," Ph. D. Thesis, Cornel l Lniversi t y, I th aca, Ne11: York, 1 %7.

4. Badgley, R . H. and J . F . Booker, ''Turbo In stabi l i t y : Effect o f I n i t i a l Trans ien t s on Plane Mot ion," Jour n a l o f Lubrica t i on Tech ., Trans . ASME., p p . 625-633, October 1 ()69.

5 . B el lenot C.. "The Effect o f Fr ic t ion o n the S tab i l i t v o f a R� t a ting Shaf t, " Brown Bover i Rev iew, Voi. 49, No . %, 1 962, p . '-1<8-55.

6. Bo1nnan, R . M., L. C . Col l ingwood and J . \V. i\iidgley, "Some Factors ,<\ffeding the W'h i rl Ins tab i l i t y o f a Journa l Bear ing, Par t l ," Paper No. 2 , Lubr i ca t ion and Wear Group Convent ion, 1 ')6:3 .

7. Bowman, R. L. C . Col l ingwood and .J . W . Midg-ley . "Some Faetors Affec t i ng the Whir l Inslah i l i tv o f .

a Journa l Bear ing, Par t 2.'' Paper No. 5, Lubrf. ca t ion and Wear Group Second Conven t i on, East · hourne, May 1 964.

8 . Bi l le t t , H. A., "Shaf t \Vhir l Induced by Drv Fr ic t ion," The Engineer, Vol . 220, pp . 1 1 3 -7 1 -1, Oc tober 20, 1 965.

9. Cheng, H. S . and P. R. Trumpler, "Stabi l i t y of the H igh S peed Journa l Bear ing under S teady Load," J . o f Eng . for Indust ry, Trans . AS:VIE, Ser ies B, Vol . HS, p . 2 7�,, l%3.

LO . Choudhury, P. de and E . .l . Gunter , " lhnamic S ta b i l i t v o f ];' lexible Rotor-Bear ing S1s tem�." !{(,search Lah�ra tor ies for the Engineer in g

· Sc iences, t n i ver·

si ty o f Virgin i a Repor t N o . M E- 1 010- 1 0l-70U, Dept. of Mechan ica l Engineer ing for NASA Lew i s Hesearch Center, Decem!Jer l !}/0. 2 J.3P.

l l . Cot)per, S., " Prel iminary I nves t iga t ion o f () i l Fi lms for Contro l of Vibra t ion," Proceedings o f the Lubr ica t ion and Wear Conven t ion, I . Mech . E., London, England, 1 96:1.

1 2 . Crandal l, S . H ., and P . J. Brosens, "On the S t ab i l i t y o f Rota t ion of a Holor \\ i t h H.ota t iona l t'mymmetr i · ca l Iner t i a and S t iffness Propert ies ," Trans . ASME, Journa l o f Appl ied Mechan ics, Vo l . 28, Series E , No . '� , p . 567-510, Dec . 1 % 1 .

1 3 . D imentherg, F. M ., " Flexural Vibrat ions o f Rota t · i ng Shaft s," B u t terwo r th and Co ., L td ., London, 1 961 .

1 4. Ehric h, F ., "The I nfluence o f Trapped Flu i d,; o n H i gh Speed R o t o r Vihra t ion;;," ASME Paper No. 67-VlBH-2(), l%7.

15 . Falkenhagen, G. and E . J . Gun ter, ' 'Non l inea r Tran s ient Ana lys i s of a R igid Hotor Supported by Non Circular Bearings," Research Labora tori es for the Engineer ing Sciences, L:n ivers i ty of Virgi n i a, Charl o t tesvi l le, Repor t No . M E--W l0-1 02 -/0U, J u n e 1 910, 1 9 1 p .

1 6. Gross, W. A ., ' ' I nves t iga t ion of Wh ir l i n Externall y Pressurized A ir-Lubr ica ted J ourna l Bear in!!s." ] . o f Basic Eng., Trans . ASME, Series D, Vol: 31, pp. 1 :32 -l::lll, 1 962.

1 7. Gun ter, E . J ., Dynamic Stability of Rotor-Bearing Systems , NASA SP- 1 1 :3, Office of Techn ica l Lt i l i za t i on, U . S . Government Pr in t ing Office, Washing ton , D .C ., 22B pages, 1 966.

1 3. Gunter, E . J ., "The I n fluence o f I n terna l Fr ic t ion on the S tab i l i ty o f H igh-Speed Rotors," J o u rn a l of Engineer ing for I n d us t ry, Trans. AS.ME, pp. 68.3-688, November 1 967 .

1 9. Gun ter, E . J . a n d P . De Choudhury, Rigid Rotor Dynamics, Par t V, "Stabi l i ty and General Transient

ROTOR BEA.RING STABILITY

1 4 -

1 2

UN S TA B L E

1 0

8

4

2

K8 = 5 7 7 2 :t;'in . � � l x !65 REYN S PRELOAD = 0.0 w = 4 0� 2 w. = 5 # 3 w " 2 #

4

ci = . 0 0 5 i n . R � L = ! i n . R8 = ! .2 in .

STAB L E

UN STA B L E

5 0 0 , 0 0 0 5 , 0 0 0

��--�--�--b-�--�---�--�-��---L---�--��--�--�--��---�L---L--�--�--4--J 5 1 0 1 5

B U S H C L E A R AN C E 2 0

( M I L S ) 2 5 3 0

Figure 2.). Stability of a f ourn al Bear£ng £n a Floating Bus h Damper.

A nalysis ," p . 63, NASA CR-l:VJl, l.Jniversi ty of V i rgi n i a for NASA Lewi s Research Cen ter, Washi n gton , D.C. , Augus t I 969.

20. Gunter, E. J. and P . R. Trumpler, "The I nf luence o f I n terna l Fr ic t ion on the Stabi l i ty o f H igh -Speed Rotors wi th Ani;;otropic Supporb, • · Journal of Eng. for I ndu"try , Trans . ASM E, Vol . () ] , Series B , N . 1·, pp. 1 105- 1 1 1 3, November } <)()<).

2 1. Gunter, E. J . , ' · In fluence of Flexibi l i ty Mounted Rol l ing Element Bear ings on Hotor Response, Par t l : Linear Analys is, " Journa l of Lub . Tech .. Trans . ASME, pp. 59-7:1, Jan . 1 (J70.

22. H agg. A. C . , "The In fluence of Oi l-Film Journa l Bearin gs on the S tabi l i ty of Hotal ing Machines ," J . o f AppL Mech . , Trans . ASJ\'lE, Vol . 6ll, p . 2 1 1 , 1 946.

23. H agg, A. C. and P. C. \Varner , "Oi l \Vhip o f Flexi-b le Rotors ," Trans . ASME, Vol . pp . 1 339- 1341, October 1 �)53.

2-t Hor i , Y. , "A Theory of Oil Whip," J . o f Appl . Mech. , Trans . ASME, pp . Ul'J - 198 , June 1 959.

25. Kellenberger, W., "The Stabi l i t y o f H i gh Speed S hafts S upported by An i so tropic .Bear ings w i th External and I n ternal Dampin g," The Brown BoH�ri Heview, Vol . 50, No. 1 1 / 12 , p . 750-766, Nov ./Dec. 1 963 .

26. Kimbal l , A . J . , " I n ternal Fri c t i on Theory o f Shaf t Whir l ing," General Elec tr ic Heview , Vol . 27 , p . 2lt, l 1)2:L

2 7. K i rk , H . G . and E . J . Gun ter , "Trans ien t Journal Bear ing Analysis ," NASA CR- l S W, l!J70, p . 1 97 .

2B. Kirk R . G . . "Nonl inear Trans ient Analvs is of Mul t i M a�� Flexible Hotors ," Ph. D . Thesis , Cn i v . o f Vir g in ia , J u n e 1 9 72 .

29 . K o rovehinsk i i , M . V . , "The Elemen tary Theory o f t h e Stabi l i ty o f a Journa l o n a n O i l Fi lm," Treni e i Fznos v Machinakh i. Fric t ion a n d \Vear i n Ma ch ines ) , Vol . 7, Academy o f Sc iences , USSR, 1 952.

.30. Larson, R. H. and H. H. Richa rdson , "A Prel imi n ary S tudy o f Whi r l I ns tabi l i ty for Pressurized Gas Bear ings ," Journa l o f Basic Eng. , Trans . ASME, Series D , Vol . B4, pp. 5 l l -520.

138

1a 1

(b )

I C l

PROCEEDINGS OF THE FIRST TURBOMACHINERY SYMPOSIUM

C2w.l , '' ... . .l az ' t: ..........

\, II .......... ......... __ ..... ' -.. \ ' b ..... ..._ I ,, z -.. .. ..,..,

(U��) I I

I \Cl I ', UNSTABLE ...

0. 5 \ ',,

/[

\ I ' , ...,_

\ I ........ .. STABLE

\ B ....... -......... ...L C'"io. I ... _ _ _ _ _

b, 1 -... _ _ _ _ _

I - - - - - - - - "" --- --c,

1.0'---------------------

w1 Zwo

I'PEQUEiiCY

� Wo

..

8ALL. ROTOR JOUIW.. BEARING D :� · :

.... I I fl&rr 8

� ' 0 - - -

r tu• r p.M , ............ ·- , ...... .., ·-

A __.., ool o- A - -- -- -

,i ..... 8

IOZOr� c • � · · · ·

15110-D • =a a a n n; q

,, ....... [

., a e a I ! t e' ·�

F * ................. (

, .. _

...

� uo .....

��VJVJl t

n·a qp••= A 1 uoo ....

� . .. Hlr.pm

t q�� q" Q\ r 1 11 m:ot.n ,

_ _ _ .. 0'7 . ... _

..

Figure 24. Theoretical and Experimental Synchron o us and Asynchronous Oil Whirl (Hori) .

3 1 . Landzberg, A . H . , "Stabi l i ty o f a Turbine-Genera tor Ro tor Inc lud ing the Effects o f Cer ta in Types of S team and Bear ing Exc i ta t ions ," Trans . ASME, J o u rna l o f Applied Mechan i cs , Vol . 27 , Ser ies E , No . 3 , p . 4 10-4 1 6, Sept . 1 960.

35. Lund, J . W. , "The Stabi l i ty o f a n Elast ic Rotor i n Journal Bear ings Wi th Flex ib le Damped Suppor ts ," J . o f Appl . M ech . , Trans . ASME, Vol. 87, Ser ies E, pp . 9 1 1 -920, 1965.

36 . Lund , J . W . and E . Sa ibel , "O il Whip Whir l Orb i ts o f a Rotor i n Sleeve Bear ings ," J . o f Eng. for Indus t ry , Trans . ASME, Vo l . 89 , Ser ies B , No . 4 , pp . 8 13 -823 , November 1 967.

32 . Lara t ta, R. , "Oil Whir l o f R otors , Part I ; Vert ical S hafts , "Meccanica , Vol . 4, p . 336, 1 969.

33. Larat ta , A., "O i l Whir l o f Rotors , Part I I : Hor i zon t a l Shafts ," Meccania , Vo l . 5, p . 1 26, 1 970.

34. Lemon , J. R., "Analyt ical and Experimenta l S tudy o f Externally Pressurized A i r Lubricated J o u rn a l Bear ings,' � J . Bas ic , Eng. V o l . 84, N o . 1 , M a rch 1 962, pp . 1 59-1 65 .

37 . Lund , J . W. , "A Theoretical A nalys is o f Whi rl I n s tabi l i ty and Pneumat ic Hammer for a R ig id Rotor in Pressur ized Gas Bear ings ," J . o f Lubr ica t ion Tech . Trans . ASME, Vol . 89 , Ser ies F . , No . 2 , pp . 1 54 - 166, Apr i l 1 967.


B R U COMPRESSOR AND COUPL I NG AMPL I TUDE vs SPEED

(/) ..J ::iE

0.5

0.4

0.3

w 0. 2 0 :::> 1-..J a.. � 0. 1

a:: � 0

..J � ' :e U a.. I- a:: cr 0 () 0

10 � � � � �

I I I I

I 0 ..J 0 :X: (/) ..J w � :!! a:: () :X:: I a.. I-!:: I a:: 5 8

Ol ... .n c > N r<"l I-

�I m �I

a:: 0�----------�------------�----------�------------�------------�------1 0 w � � 00

1 0 COU P L I NG MOT I ON

5

o�-----------L------------�----------�------------�------------�------1 0 2 0 30

R P M X 1 0- 3 40 50

Figure 25. NASA BRU Compressor and Coupling A mplitude vs . Speed.

38. Lund, J . and Chiang, T . , "Rotor-Bear ing Dynamics Design Technology, Par t V I . The I n fluence of Elecomagnetic Forces on the S tabi l i t y and Response of a n Alternator ," MTI -67TR3-t., Mechanica l Tech· n ology Inc. 1 967 .

39. Lund, J . W. , "Rotor Bearin g Dynamics Design Tech· nology, Part V I I : The Three Lobe Bear ing and Floa t ing Ring Bear ing," AFAPL TR-65-45, M .T. I . , 2 6 7 p ages, 1968.

40. Newkirk , B . L . , "Shaft Whipping," Genera l Elec t r ic Rev iew, Vol . 2 7, p . 169, 1 924.

41. Newkirk, B . L. and H. D . Taylor , "Oil Fi lm Whi r l -An I n vest iga t ion of Dis turbances D u e t o O i l Films in Journa l Bear ings," General Elec t r ic Review, 1 925.

42 . Newkirk, B. L . and L. P. Grobe! , "Oi l Fi lm WhirlA Non-Whir l ing Bearing," Trans. ASME, Vol . 56, No. 8, p. 607, 1 934.

43 . P inkus, 0., "Note on Oil Whip ," J. o f Appl. Mech. , Trans . ASME, Vol . 2 1 , p . 450, 1 953 .

44 . Pinkus , 0. , "Experimenta l I nves t igat ion o f Resonan t Whip," Trans . ASME, Vo l . 78 , p . 975 , 1 956.

45. Por i tsky, H. , "Con t r ibu t ion t o the Theor y o f O i l Whip ," Trans . ASME, V o l . 7 5 , p p . 1 153- 1 1 6 1 , A u · gus t 1 953.

46. Por i tsky, H . , "Rotor -Bear ing Dynamics D es i gn Technology, Par t I I ; Ro to r S tab i l i ty Theory ," A F APL TR-64-45, Par t I I, MTl -64-TR-34.

47. Por i tsky, H . , "Rotor S tab i l i t y ," Proceedi n gs of the Fif th U. S . Nat iona l Congress o f Appl ied Mechan ics , ASME, pp . 3 7 -6 1 , 1 966.

-t8. Robertson , D. , "The Whir l i ng o f Shafts ," Eng ineer , London, Vol . 1 58, pp . 2 1 6, 228, 1 934.


NASA B RAYTON CYCL E TEST ROTOR AT VAR I OUS POWER LEVELS N = 52 ,000 RPM S = 100 p i n/em

2

3 4

5 6

Figure 26. NASA Brayton Cycle Test Rotor at Various Power Levels.

49. Robertson , D. L., "Hys teret ic I n fl uences on the Whir l i ng o f Rotors ," Proc . lns t . Mech . Engrs . , Lon · don , Vol . 1 3 1 , p . 5 1 3 , 1 935 .

50 . Ruhl , R . L. and J . F. Booker, "A Fin i te Element Model for Dis t r ibu ted Parameter Turbomotor Systems," Presented ASME 1 9 7 1 Vibrat ions Conf. , Sept . 1 971 , Paper N o . 7 1 -Vibr-56.

5 1 . Shawki, G. S . A . , "Whir l ing o f a Journa l Bear ing Experiment Cnder No Load Condi t ion ," Engineer· i n g, Vol . 1 79, pp . 243-2-1.6, 1 955.

52 . Smith , D. M . , "The Mot ion o f a Rotor Carr ied by a Flex ib le Shaft in Flexible Bear i ngs ," Proceed ings o f the Royal S oc i et y o f London, Series A , Vol . 142 , p p . 92- 1 18 , 1 933 .

53 . Schul ler , F. T. and W. J . Anderson, " Experiment s on the Stabi l i ty o f Water-Lubricated Three Sector H ydrodynamic Journa l Bear ings at Zero Load," Lewi s Research Cen ter, NASA TN D -5752, Apri l 1 970.

5-t.. S o u thwel l , R . V . and Barbara S . Gough , "On the S tab i l i t y o f a Rota t ing Shaft , Subj ected S imu l tane ous ly t o End Thrus t and Twist ," Report , Br i t i sh Assoc . Adv . o f Science, 89th Meeti n g, pp . 345-355, 1 92 1 .

55 . S tern l i gh t, B . , " I n vest igat ion o f Trans la tory Flu i d W h i r l o f Vert ical Shafts ," Ph . D . Thesi s , Co lumbia Univers i ty , November 1 954.

56. Stodola , A . , "Kr i t ische Wellenstorung I nfolge der Nachgiebigkei t des Oelpolsters i m Lager" Schwei· zer ische Bauzer tung 1 925, Vol . 85, p . 265.

57 . Sweet , J . and J . Gen i n , "Squeeze Fi lm Bearin g for Elim ina t i on o f O i l Whip ," ASME Trans . J . o f Lub . Apr i l 1 9 7 1 , pp . 252-261 .

58. Tatara , A . , "An Experimental Study o f the Stabi l i z ing E ffec t o f Floa t ing-Bush Journal Bear ings ," Bul l e t in o f JSME, Vo l . 1 3 , No. 61 , p . 858-863, 1 970.

ROTOR ORB I TS OF T URBO COMPRESSOR BEFORE AND AFTER STA B I L I ZATION

N • l l ,300 RPM, F I RS T CRI T I CAL N1 • 4 , 300 RPM

UNSTABLE ROTOR ORB I T D I SCHARGE PRESSURE

P0 • 175 psi g L A RGE N 1 COMPONENT

STA B I L I ZED ROTOR ORB I T X 2 AMPLI F I CATION

STAB I L I ZED ROTOR ORB I T P o • 650 psi g

S MALL N 1 COMPONENT

STABI L I ZED ORB I T F I LTERED SIGNAL

UNBALANCE ORB I T ONLY N • 1 1 , 300

Figure 27. Rotor Orbits of a Turbo Compressor Before and After Stabilization .


1 00 z 0 i= <( ::> 5 0 a LLI \) i= 1:1) - 0 a:: LLI .... u <( a:: <( :r: - 5 0 u LL 0 .... 0 0 a:: - 100 I

.... a:: <( 0.. ..J - 150 <( LLI a::

1 0 5 0 1 0 0 200 500 1000 2000

Q = 1 0 0 , 0 0 0 l b/ in N = 10 ,0 0 0 R P M

UNSTABLE STABL E

ROTOR C HA R ACT ER I S T I C S W2 • 6 7 5 1 b. 5 K5• 2 . 8 x 1 0 l b/in

WJ • 3 1 2 l b. C5 • 1 . 0 l b- s e c /i n

w, • 1 5 l b • C I • 0.0

BEARIN G C HA R A C T E R I ST I C S l<xx 1 .2e1 ,. 1 cf l b/j n

6 Krf 1.4 2 8 x l 0 l b/jn

cxx· 1 20 0 l b - &eC/In

Crt 1 2 90 lb - se C/in

5000 10 0 0 0

S U PPOR T DAM PI N G ( l b - r.eC/inl

Figure 28. Stability of a Flexible Rotor With A erodynamic Cross Coupling (Q == 100,000 lb/in . , N == 10,000 RPM) .

59. Tondl , A . , "Experimental I nves t iga t ions o f SelfExcited Vibrat ions of Rotors Due to the Act ion of Lubr ica t ing O i l F i lm i n Journal Bear ings," Wear, Vol . 5 , pp . 1 36 - 147, 1 962 ( The Nat iona l Research Ins t i t u te o f Heat Engineer ing, Prague. )

60. Tondl , A . , Som e Problems of Rotor Dynam ics, Publ i sh i ng House o f the Czechslovak Academy of Sc i · ence, Prague, Chapman and Hal l , London , 1 965.

6 1 . Wi l l iams, R., and R . Trent , "The Effect s o f Non · l i n ea r Asymmetr ic Supports on Turb ine Eng ine Rotor S tabi l i ty , SAE Paper No . 700320, 1 970.

62. Yamamoto, T. and 0. H i roshi , "On the Uns tab le Vibra t ions o f a Shaf t Carryi n g on Unsymmetr ic Ro tor," J . o f Appl . Mech . , Trans . ASME, Paper No . 64-APM-32, 1 964.

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ROTOR-BEARING STABILITY