The asymmetric nature of the fluid film stiffness and damping properties in rotors supported on fluid film bearings causes a forward or a backward whirl depending on the bearing parameters and the speed of the rotor. A rotor was designed to exhibit backward synchronous whirl. The rotor-bearing system exhibited split criticals, and a backward whirl was observed between the split criticals. The orbital diagrams show the whirl pattern

Rao, J. S.

Rhat, R. B.

Sankar, T. S.

Subbiah, R.

English

NASA Technical Reports Server

BACKWARD WHIRL I N  A SIMPLE ROTOR SUPPORTED ON HYDRODYNAMIC BEARINGS 
R. Subbiah, R.B. Bhat, and T.S. Sankar 
Concordia Un ivers i ty  
Montreal, Canada 
J.S. Rao 
Ind ian  Embassy 
Washi ngton , D. C . 
The asymmetric nature o f  the f l u i d  f i l m  s t i f f ness  and damping proper t ies i n  
ro to rs  supported on f l u i d  f i l m  bearings causes a forward o r  a backward w h i r l  depend- 
i n g  on the bearing parameters and the speed o f  the ro to r .  
e x h i b i t  backward synchronous w h i r l .  The rotor-bear ing system exhib i ted s p l i t  c r i t -  
i ca l s ,  and a backward w h i r l  was observed between the s p l i t  c r i t i c a l s .  
diagrams show the w h i r l  pat tern.  
A r o t o r  was designed t o  
The o r b i t a l  
INTRODUCTION 
The design o f  a r o t o r  system must consider several aspects such as c r i t i c a l  
speeds, peak unbalance response, regions o f  change o f  w h i r l  d i rect ions,  and ins ta -  
b i l i t y .  I n  general, la rge  r o t o r  systems i n  continuous operat ion are supported on 
hydrodynamic bearings. 
s t i f f ness  and damping proper t ies t h a t  vary w i t h  the speed o f  operation. 
property inf luences the dynamic behavior of the ro to rs  s i g n i f i c a n t l y .  
The dynamic behavior o f  such r o t o r s  can be predicted by using t rans fer  matr ix  
methods, f i n i t e  elements, modal analysis, etc.  [l-61. Rao [ 7 ]  and Rao e t  a l .  [ a ]  
used a simple ana ly t i ca l  technique t o  p red ic t  the dynamic behavior o f  such r o t o r s .  
They studied a s ingle-disk r o t o r  on f l u i d  f i l m  bearings and observed t h a t  f o r  spe- 
c i f i c  rotor-bear ing parameter combinations the system may e x h i b i t  two d i s t i n c t  peaks 
i n  the  response but  sometimes i t  may show only one peak i n  the response. Kollmann 
and Gl ienicke [9 ]  have shown experimentally the existence o f  the s p l i t  c r i t i c a l s  i n  
a simple r o t o r  supported on f l u i d  f i l m s  bearings. Kellenberger [ l o ]  der ived equa- 
t l ons  f o r  double frequency accelerat ions i n  turbogenerator r o t o r s  r e s u l t i n g  f r o m  
anisotropy i n  the p l a i n  c y l i n d r i c a l  bearings and showed the occurrence o f  backward 
w h i r l  between the c r i t i c a l s .  These studies p red ic t  a forward w h i r l  before the f i r s t  
c r i t i c a l  and a f t e r  the second c r i t i c a l ,  whereas the r o t o r  executes a backward w h i r l  
between the two c r i t i c a l s .  Also, the simple r o t o r  w i t h  on ly  a s ing le  peak i n  i t s  
response i s  known t o  w h i r l  i n  the forward d i r e c t i o n  a t  a l l  speeds. 
These hydrodynamic bearings e x h i b i t  asymmetric cross-coupled 
Such a 
Sitveral experimental inves t iga t ions  are reported i n  the l i t e r a t u r e  regarding 
the dynamic behavior o f  d i f f e r e n t  types o f  r o t o r  systems Supported on hydrodynamic 
bearings. A few notable works are by Yamamoto [ll], Hul l  [12], and Lund and 
Orcutt  [13]. 
Most of the re fe r red  work on p r a c t i c a l  ro to rs  supported on hydrodynamic bear- 
ings does no t  s a t i s f y  condi t ions f o r  a c lear  backward w h i r l  and hence the phenomenon 
o f  backward whir l  has so f a r  not  been observed experimentally. The occurrence o f  
backward w h i r l  i s  a lso not  des i rab le i n  pract ice.  I n  the present work condi t ions 
are derived f o r  backward w h i r l  considering bearing damping. Further resu l t s  o f  the 
145 
https://ntrs.nasa.gov/search.jsp?R=19860020700 2020-03-20T14:40:32+00:00Z
experiments carrled out on a laboratory rotor model designed so as to exhibit 
backward whi rl are reported. 
NOMENCLATURE 
a disk eccentricity 
C shaft damping 
czz~cyy,cyz,czy fluid film damping coefficients 
Ei ,Ti forward and backward component, respectively, of ith modal force 
vector 
F 
K 
overall exciting force vector 
overall stiffness matrix 
k shaft stiffness 
k,,,kyy9kyz,kZy fluid film stiffness coefficients 
M overall mass matrix 
Ni ,mi forward and backward component of ith modal displacement vector 
9 displacement vector at disk location 
$1 
r maximum unbalance response of rotor 
displacement vector at bearing location 
2.Y displacement of rotor at disk location in Z and Y directions 
zo*yo displacement o f  rotor at bearing location in Z and Y directions 
'li 
xi 
7th modal displacement vector 
ith complex eigenvalue 
cp right eigenvector of  system 
cp" left eigenvector of system 
ANALYSIS 
A schematic of the rotor is shown in Fig. 1. Equations of motion o f  the rotor 
are 
( z  + a cos ut) + k(z - zo) + c(; - io) = 0 d2 mdt2 
146 
0 ' 0  
my - my = 0 
3 
- " . . . .  0 0 
Z Y Z Y  zO zO 
O O m O  0 0 
O O O m  0 0 
O m O c  0 -C 
m O c O  -C a 
d" 
d t  
m 2 ( y  + a s i n  o t )  + k(y  - yo) + C C i  - io) = 0 
and the cons t ra in t  equations are 
zo + 2cyz * ;o ( 4 )  
Y Z  k(y  - yo) + c ( i  - io) = 2kyy * yo + 2~ io t 2k YY 
Equations (1) t o  ( 4 )  can be w r i t t e n  i n  the form 
where 
CHI = 
CKI = 
-2c 
ZY 
o a c o - ( c + ~ c  
ZZ 
-( c+2c ) 
Y Z  YY 
o o o c  -2c - 
-m 0 0 0  0 0 
O - m O O  0 0 
0 O k O  -k 0 
0 O O k  0 -k 
0 0 k 0 -(k+2kzz) -2k 
ZY 
-( k t2k ) 
-2kyz yy- 
L O  O O k  
2 {F}  = L O ,  0, maw2 cos Ut, maw s i n  w t ,  0, 01 
The eigenvalues and eigenvectors o f  the system are obtained by so lv ing  the 
homogeneous form o f  equations (6) as shown below: 
147 
Assuming a so lu t ion  o f  the f o r m  
where (9) represent the system eigenvectors, and subs t i t u t i ng  equation ( 7 )  i n  
equation (8 )  the eigenvalue problem becomes 
AEMI(cp1 + [K lEcp l  = 0 (8) 
Because o f  the presence o f  asymmetric cross-coupled s t i f f n e s s  and damping c o e f f i -  
c ien ts  i n  the bearings, the matrices [ M I  and [ K ]  are nonsymmetric, r e s u l t i n g  -0n 
a non-sel f -adjoint  system. Hence a conventional normal-mode method i s  no t  possible, 
and it i s  essent ia l  t o  consider t h a t  the b io r thogona l i t y  property o f  the modes o f  
the o r i g i n a l  system are those o f  the transposed system t o  the uncoupled equations 
o f  motion. THe l e f t  eigenvectors are obtained by transposing matrices [ M I  and 
[ K ]  i n  equation ( 6 ) .  The eigenvalues o f  the o r i g i n a l  and transposed systems a r e  
i den t i ca l ,  bu t  the eigenvectors a re  d i f f e r e n t .  
The so lu t ion  o f  equation ( 5 )  i s  assumed i n  the f o r m  
where [cp] contains the eigenvectors o f  the o r i g i n a l  system (which are ca l l ed  
r i g h t  eigenvectors). In t roducing equation (9) i n  equation ( 5 )  and premul t ip ly ing 
the r e s u l t  by [cp* IT ,  which i s  the l e f t  eigenvector o f  the system, lead t o  the 
fo l low ing  uncoupled equations: 
where 
and 
Equation (10) resu l t s  i n  uncoupled equation o f  the form 
l&t) + ../nq(t) = a i w  
This i s  solved by assuming steady-state so lu t i on  o f  the form 
n i ( t )  = N i  exp ( jot)  + ?ji exp ( - jw t )  
and 
148 
Subst i tu t ing  equation (12) i n  equation (11) resu l t s  i n  
("1 + joFri) N i  exp ( j o t )  + ("1 - j w i )  w i  exp ( - j o t )  
= E l  exp ( j o t )  + Ti exp ( - j o t )  (13) 
Equating the coe f f i c i en ts  o f  the forward and backward r o t a t i o n  terms,  we obtain 
Hence the displacements are determined f r o m  equations (10) t o  (14) and then the 
unbalance response o f  the r o t o r  i s  obtained nondimensionally as 
L L i Y  
a r =  
where a i s  the d isk  eccent r i c i t y .  The complex eigenvalues are obtained f r o m  
equation (6). 
Depending on the bearing parameters the r o t o r  w i l l  have e i t h e r  s p l i t  c r i t i c a l s  
o r  j u s t  a s ing le  peak i n  the fundamental c r i t i c a l  speed region, corresponding t o  
synchronous wh i r l .  I n  the case o f  the s p l i t  c r i t i c a l s  i n  the synchronous wh i r l ,  the 
backward component ?ji i s  la rger  than the forward component N i  between the 
s p l i t  c r i t i c a l s .  
EXPERIMENTAL RESULTS 
The d e t a i l s  o f  the t e s t  r o t o r  are given i n  t a b l e  1. I t  consists o f  a c i r c u l a r  
shaf t  w i t h  a c i r c u l a r  d isc a t  the center and supported on two iden t i ca l  hydrodynamic 
bearings a t  the ends. These bearings are mounted on cast  i r o n  pedestals a t  the t w o  
ends, and i n  t u r n  these pedestals are r i g i d l y  fastened t o  the support, which i s  made 
o f  s tee l  angles. The pedestals were impact exc i ted both i n  the hor izonta l  and ver- 
t i c a l  d i rect ions,  and the resu l t i ng  acoust ica l  response was measured close t o  the 
pedestals. A frequency analysis o f  the measured sound showed t h a t  the f i r s t  peak 
occurred a t  a much higher frequency than the c r i t i c a l  speeds o f  the ro to r ,  i nd i ca t -  
i n g  t h a t  the pedestals were r i g i d .  
g rav i t y  feed. The unbalance response o f  the shaf t  i s  measured i n  both the x and y 
d i rec t ions  by two prox imi ty  pickups. The signals from the pickups are fed t o  a 
twin-channel FFT analyzer, and the o r b i t  diagrams are obtained w i t h  a x-y p l o t t e r .  
The d i r e c t i o n  o f  p l o t t e r  pen motion ind icated the d i r e c t i o n  o f  r o t o r  wh i r l .  
The bearings are supplied w i th  o i l  through a 
I n  designlng the r o t o r  such t h a t  i t  exh ib i t s  a backward wh i r l ,  the bearing 
coe f f i c i en ts  t o  be used i n  equations ( 3 )  and (4)  were taken from the resu l ts  o f  
Lund [14]. Lund's resu l t s  agree we l l  
w i t h  the experimentally determined bearing coe f f i c i en ts  by Gl ienicke [9 ] .  
These resu l t s  can a lso be found i n  [ l 5 ] .  
149 
The w h i r l  o r b i t s  obtained experimentally are shown normalized w i t h  respect t o  
the disc eccen t r i c i t y  i n  Figs. 2 t o  4 f o r  d i f f e r e n t  r o t o r  speeds. Orb i ta l  diagrams 
are shown f o r  a bearing clearance o f  1 . 8 7 9 6 ~ 1 0 - ~  m, since a backward w h i r l  could 
be i d e n t i f i e d  only f o r  t h i s  case. The two c r i t i c a l  speeds are 2200 and 2600 rpm. 
The r o t a t i o n  o f  the r o t o r  i s  i n  the counterclockwise d i r e c t i o n  and hence Figs. 2 
and 4 show forward wh i r l s  and Fig. 3 shows backward wh i r l .  
t o  the backward w h i r l  i n  F ig .  3 i s  2500 rpm, which f a l l s  between the two c r i t i c a l s  
a t  2200 and 2600 rpm. Since the backward w h i r l  between the  c r i t i c a l s  i s  o f  i n t e r -  
est, a photograph o f  backward w h i r l  motion o f  the r o t o r  a t  2500 rpm was taken f r o m  
the FFT analyzer screen. 
there t s  a q u a l i t a t i v e  agreement between theory and experiments. 
The speed corresponding 
Theory a lso pred ic ts  these w h i r l  d i rect ions,  and hence 
REFERENCES 
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
4. 
10 0 
Lund, J.W., 1965, Mechanical Technology Inc., Latham, N.Y., U.S.A., 
AFAPL-Tr-65-45, "Rotor Beari ng Dynamics Design Techno1 ogy , Part  V : 
Computer Program Manual f o r  Rotor Response and S tab i l i t y " .  
Kramer, E., 1977, ASME Publ icat ion 77-DET-13, "Computation o f  Unbalance 
Vibrat ions o f  Turborotors". . 
Nelson, H.D. and McVaugh, J.N., 1976, Journal o f  Engineering f o r  Industry, 
ASME, Vol, 98, p. 593, "The Dynamics o f  Rotor Bearing Systems Using F i n i t e  
Elements". 
Lalanne, M. and Queau, J.P., 1979, Societe Nationale des Indus t r ies  
Aerospatial es , T a l  cu l  par E l  ements F i  n i  s du Comportement Dynami que des 
Chaines Cinematiques de Reducteur". 
Morton, P.G., 1967-68, Proc. Inst .  Mech. Engrs., Vole 182, Part  1, No. 13, 
" In f luence o f  Coupled Asymmetric Bearings on the Motion o f  a Massive 
F1 e x i  b l  e Rotor", 
Gunter, E., Choy, K.C. and A l l a i re ,  P.E., "Modal Analysis o f  Turborotors 
using Planar Modes - Theory", Journal o f  the Frank1 i n  I n s t i t u t e ,  Vol . 305, 
N o 4, Ap r i l  1978. 
Rao, J S . ,  1982, Mechanism and Machine Theory Journal, Vol. 17, No. 2, 
"Conditions for  Backward Synchronous Whirl o f  a F l e x i b l e  Rotor i n  
Hydrodynamic Beari ngs" . 
Rao, J.S., Bhat, R.B. and Sdnkar, T.S., 1980-81, Transactions o f  the 
Canadian Society of Mechanical Engineers, Vol. 6, No. 3, pp. 155-161, "A 
Theoretical Study of the  Effect o f  Damping on the Synchronous Whirl o f  a 
Rotor 
Kollmann, K. and G1 ienicke, J., JSIYI.5, 1361, Semi-International Symposium 
4th-8th Sept. 1967, Tokyo, "The behaviour o f  tu rb ine  bearings w i th  respect 
i n  Hydro-Dynami c Beari ngs" 
t o  i n s t a b i l  
Ke l l  enberger , 
(London), 
t y  predict ion".  
W., Proceedings o f  the I n s t i t u t i o n  o f  Mechanical Engineers 
1980, Paper No. C312/80, P.415-Fj20, "Doubl e f requency 
150 
accelerat ions i n  turbogenerator r o t o r s  r e s u l t i n g  from aniSOtrOphy i n  t h e  
bedrings". 
11. Yamamoto, 1954, Memoirs o f  the  Facul ty o f  Engineering, Nagoya U n i v e r s i t y  
V 0 1  . 6, No. 2, p. 755, "On the C r i t i c a l  Speeds o f  a Shaf t " .  
12. Hu l l ,  E.H., 1961, Journal o f  Engineering f o r  Indust ry ,  Vol. 83, p.  219, 
" S h a f t  Whir l ing as Inf luenced by S t i f f n e s s  Asymmetry". 
13. Lund J.W. and Orcutt ,  F.K., 1967, Transactions o f  ASME, Journal o f  
Engineering f o r  Indust ry ,  Vol . 89, No. 4, "Calcu lat ions and Experiments on 
t h e  Unbalance Response o f  a F l e x i b l e  Rotor". 
14. Lund, J.W., 1965, Mechanical Technology, Inc., AFAPL-Tr-65-45, "Rotor 
B e a r i n g s  Dynamic Design Technology, Par t  111: Design Handbook f o r  F l u i d  
Bearings". 
15. Rao, J.S., 1983, Wiley Eastern Limited, New Delhi ,  Rotor Dynamics. 
TABLE 1: DETAILS OF TEST ROTOR 
Shaft Diameter 
Shaft  Length 
Weight o f  Disk 
Shaft  S t i f f n e s s  
Bear i  ng Diameter 
Bearing L/D Rat io  
O i l  V iscos i ty  a t  25.5OC 
Unbalance o f  Rotor 
Bearing C1 earance 
0.0222 m 
0.5080 m 
89 N 
8.9 x l o 5  N/m 
0.0254 m 
1 
0.96 x Pascal sec. 
1.084 x kg.m 
1.8796 x m (0.0074 i n )  
151 
Figure 1. - Schematic diagram o f  single mass rotor. 
2 
r l  
t z 
OL 
9- z 
Y u
\ 
W 0
Y o  
P 
3 -' 
- 
-I a. 
-2 
- 2  -1 0 1 2 3 -3 
AnPCIlUDE/ECCENTRICITY 
Figure 2. - Rotor uhirl Orbit at  2150 rpm (counterclockwise direction COtTeSpoRdS to forward whirl). 
152 
Figure 3. - Rotor whirl orbit at 2500 rpm (clockwise direction corresponds to backward whirl). 
-3 -2 -1 0 1 2 3 
AMPLITUDE/ECCE~TRICITY 
Figure 4. - Rotor whirl orbit at 3000 rpn [counterclockwise direction corresponds to forward whirl). 
153 


Backward whirl in a simple rotor supported on hydrodynamic bearings

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