Results of experimental measurements made on a small high speed, axial flow test apparatus are presented to verify the existence of Alford's force (that circumferential variation of blade-tip clearances in axial-flow turbomachinery will produce cross-coupled (normal to the eccentricity) aerodynamic forces on the rotor) and to investigate the validity of his mathematical prediction model

Laudadio, F. J.

Vance, J. M.

English

NASA Technical Reports Server

_-_Pb_IMb_Xrl ' MEASUR_k_NT OF AL_' S P'LI_CE
IN AXIAL-FIX._ _I#/:]-IINb2_Y
John _. Vance
Department of Ivlechanical Engineering
Texas A_l University
College Station, Texas 77843
Frank J. Laudaaio
Motorola Corporation
Fort Lauderdale, Florida 33322
ABSTRACT
In 1965, J. S. Alford published a theory and mathematical model [1] which pre-
dicts that circumferential variation of blade-tip clearances in axial-flow turbo-
machinery will produce cross-coupled (normal to the eccentricity) aerodynamic forces
on the rotor. Ever since then, the theory has been used (without experimental veri-
fication) by rotor dynamicists as one of the few mathematical models available to
calculate the cross-coupled aerodynamic stiffness coefficients required for rotor-
dynamic stability analysis.
This paper presents the results of experimental measurements made on a small,
high speed, axial flow test apparatus to verify the existence of Alford's force and
to investigate the validity of his mathematical prediction model.
INTRODUCTION
There are many mechanisms which cause rotordynamic instabilities in turbines
and compressors. An instability in turbomachinery occurs when the shaft precesses
about the bearing centerline (whirling) with unacceptable amplitudes, at frequencies
nonsynchronous with shaft speed. Instabilities can become a costly problem since
they limit the operation of the turbomachinery by reducing the allowable speed.
They can be destructive if the instability threshold speed is exceeded.
There are documented cases showing the severity of instabilities in turbo-
machinery. Doyle [2] presents two cases where an instability has required exten-
sive rotor modification, resulting in lost production and heavy maintenance expen-
ditures. One of the cases mentioned is the 1964 Ekofisk gas reinjection problem
which has become a classical example of how costly an instability can become. A
centrifugal compressor was designed to reinJect the gas by-product from crude oil
production. Oil production was stopped for several months due to the instability
problem with this compressor.
260
https://ntrs.nasa.gov/search.jsp?R=19830007375 2020-03-21T23:42:35+00:00Z
In mathematical terms an instability occurs when an eigenvalue of the equation
that describes the motion of the rotor has a positive real component. In engi-
neering terms, the nonsynchrono_s amplitude of rotor whirl (any amplitude which has
a frequency not equal to running speed) grows with time until the system exceeds
its operating limits. In many instances the rotor orbits can be large enough to de-
stroy bearings and seals.
Computerprograms are now available [3] which can predict whirling frequencies
and threshold speeds, provided that accurate values for the destabilizing coeffi-
cients are used as input. The destabilizing coefficients usually take the form of
a linearized cross-coupled stiffness term [4]_ but sometimes are expressed as neg-
ative damping [5], depending on the mechanismproducing the instability.
Onemechanism,which is a type of self-excited instability, is the aerodynamic
excitation due to circumferential variation of blade clearance in axial-flow com-
pressors and axial-flow turbines. An objective of the work reported here was to
quantify a term in the mathematical equation due to Alford which predicts the mag-
nitude of the aerodynamic cross-coupled stiffness, along with verifying the form of
the equation.
Afford actually presented two mechanismswhich can cause severe rotor whirl in
axial flow compressors and turbines. Oneis due to the circumferential variation
of static pressure action on the cylindrical surface of a rotor, particularly with-
in labyrinth seals. The other is due to circumferential variation of blade-tip
clearance (the topic of this paper). Afford presented both seal deflection criteria
and bladed-disk torque deflection criteria for use as design guides for a stable
rotor system.
Investigations of rotordynamic problems in steam turbines have shownthat un-
balanced momentson wheels and shafts, caused by steam forces, are a source for non-
synchronous rotor whirl. Steamwhirl was first described by Thomas[6] in 1958 and
is similar to the phenomenonhypothesized by Afford. Winter [7] performed model
tests in 1968. He used ball bearings in his test rig to eliminate the shaft whip-
ping due to hydrodynamic bearings, and applied external damping elements to the
bearings since ball bearings have little inherent damping. Winter proved the pre-
sence of a load-dependent instability but was not able to quantify his results.
THETESTAPPARATUS
The test apparatus used in the experiment consists of a flexibly mountedbladed
rotor driven by a variable speed electric motor, and a movable shroud (see Figure
i). The blade O. D. is 6". The motor is rated i/7 HPat I0,000 rpm.
An offset of the bladed rotor within the shroud clearance circle is produced by
translating the shroud laterally. The shroud is mountedon slides, and is trans-
lated laterally by turning a threaded rod.
The lateral stiffness of the rotor bearing support is madelow enough so that
very low levels of Alford's force (3-10 grams) produce a measurable rotor deflec-
tion. The rotor deflection is measuredwith an eddy-current proximity probe and
261
calibrated to lateral force in grams/mil. Oneof the most challenging and time-
consuming aspects of the project was the measurementof rotor deflection (DC signal)
in the presence of a relative large AC signal produced by rotor runout, synchronous
vibration, and ball bearing roughness. This was finally accomplished by passing the
probe signal through a very effective low-pass filter.
The speed of the rotor can be varied independently from the torque, over a range
of 0 to 7000 rpm, by changing the voltage supplied to the electric motor. The torque
is varied by adjusting the air velocity at the rotor inlet with an auxiliary blower.
This method allows the rotor speed to be changedwhile holding the torque constant,
or vice versa. Manycombinations of speed, torque, and shroud eccentricity can thus
be obtained.
Rotor speed is measuredby feeding a once-per-revolution keyphaser signal into
an electronic counter.
The aerodynamic torque is measuredwith a strain gage dynamometer. The driving
torque of the electric motor is entirely reacted by a steel rod in torsion, with
torsional strain gages calibrated to torque.
EXPERIMENTALPROCEDURE
For each set of data taken, a base measurementwas madewith the shroud in the
center position (i.e. with the lateral eccentricity nominally zero). With the shroud
in this position, the rotor was run up to the selected measurementspeed and measure-
ments were madeof rotor deflection, speed, and torque.
Immediately after completing the base measurements,an eccentricity was imparted
to the shroud (adjusting screws translate the shroud laterally) and new values of
rotor deflection and torque were measuredat the samespeed as for the base measure-
ment. Since the direction, as well as the magnitude, of shroud offset was found to
have an effect on the measurements, the procedure was repeated for an equal shroud
eccentricity in the negative (opposite) direction.
Large quantities of data were taken and averaged, since the repeatability was
found to be relatively poor (typically ±20% for two successive measurements of the
same variable under the same conditions). The poor repeatability was partially due
to apparent aerodynamic instabilities (surging) under some conditions of speed and
torque.
RE SULT S
The results of the experiments are shown in Figures 2 through 13. The lines
connecting the points in the figures were derived using a least squares fit.
Figures 2 and 3 show how the measured Afford force varied with rotor eccentric-
ity. The only difference between the two graphs is in the direction that the
eccentricity was imparted. Figures 4 and 5 show the magnitude of the efficiency
facor, B, and how it varies with rotor eccentricity. Again, the differnce between
the two figures is the direction that the eccentricity was imparted. The efficiency
262
factor 8 was calculated using Alford's equation which gives the aerodynamic force as
proportional to a radial deflection which is normal to the force. The constant of
proportionality is called cross-coupled stiffness. Thus we have
_Sx
Fy - DpH= KyxX (i)
The cross-coupled stiffness is seen to be _x - _BDpH"
Equation (i) implies that 8 remains constant for varying eccentricity. Figure
5 confirms this. Therefore it was felt that the point on Figure 4 corresponding to
20 mils eccentricity was not representative, and should not be included in the least
squares fit.
Figures 6 and 7 showhow Alford's force and B vary with torque. Again B was
calculated using equation (i). Figure 8 showshow the measuredAlford force varied
with speed holding the rotor torque constant. Figure 9 was derived from figures 6
and 8. The curve corresponding to 5000 rpm was derived using a least squares fit to
the data from all 3 tests in Figure 6. The four remaining curves on Figure 9 were
derived from this one curve and Figure 8, using extrapolation techniques. Figures
i0, ii, 12 and 13 showhow the measuredAfford force varied with rotor speed and
torque. These figures help support the empirical equation of Alford's force to be
presented below. They showhowthe developed rotor torque varies with the speed.
AN IMPROVEDPREDICTIONMODEL
Oneof the remarkable features of Alford's theory is its simplicity. Figures
2, 3, and 6 verify the predicted linearity of Alford's force with rotor eccentricity
and stage torque. However, Figure 6 shows that Alford's force does not appear at all
until a certain level of stage torque is reached: then it suddenly appears and in-
creases steeply with torque. Alford mentions, in reference [I], that "large power
inputs to the compressor rotor appear to increase the hazard of whirl. The vibration
problem was encountered only at the full i00 percent power rating of the engine."
Figure 6 is compatible with this observation.
Figures 7 and 8 also reveal certain limitations of equation (i), principally
that the efficiency factor 8 actually varies with the aerodynamic load torque and
that the developed Alford's force is speed-dependentas well as torque-dependent.
Also, extrapolation of the curves for Alford's force versus torque (e.g. Figure 6)
shows that the y-intercept is not zero as equation (i) predicts.
To obtain an improved prediction model, the speed must be included as a perti-
nent variable. Dimensional analysis shows that if the speed is pertinent, then the
inlet velocity is also pertinent. In dimensionless groups, the prediction equationtakes the form
--= f X V
' H ' NDp (2)
263
where N ffi speed, rpm
V = inlet velocity, fps
and the other variables are as previously defined.
In the experiments reported here, measurementsof the inlet velocity V were not
made. Therefore the improved prediction model at this stage must be purely empiri-
cal, to fit the available test data.
Equation (i) is modified to include a y-lntercept, which gives
Fy = T_____xxDpH+ C (3)
where the constant C is evaluated as C = -17.37 from Figure 9.
In addition, Figure 9 showsthat the slope of the curve is affected by the rotor
speed. A relationship can be calculated between the rotor speed and the slope
x/(DpH), as follows:
8x____= -10.54 x 10-3 (N) + 89.33 (4)
DpH
Substituting equation (4) into equation (3) gives
Fy = (-10.54 x 10-3 (N) + 89.33) r - 17.37 (5)
In a more general form; factoring out x/(DpH) where x ffi40 mils, Dp ffi4.875 in., and
H = 0.875 in. gives
X
Fy = I(-I.12(N) + 9526.42) T - 1852.35} D_
where
Fy = Alford force, grams
T = developed rotor torque, in-lb
N = rotor speed, rpm
x ffieccentricity, in
Dp = blade diameter, in
H = bucket height, in
(6)
If the units of Alford force are taken to be pounds (ib), the equation becomes
X
Fy = {(-2.47 x 10-3(rpm -I) x (N) + 21.0) T - 4.08(in-lb)} D_
(7)
Equation (6) reasonably approximates all of the curves presented in this paper
that show Alford's force as a function of developed rotor torque_ rotor speed, and
eccentricity. It is a purely empirical equation which fits the measured data for
the described test rig.
264
CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH
Due to the size of the test rig and the lack of measurement capability for aero-
dynamic velocities and pressures, the conclusions which can be made from this re-
search are quite limited in scope. They are as follows:
I. The aerodynamic force postulated by Alford for axial flow turbomachinery
does exist, and can be measured experimentally.
2. Alford's force is speed-dependent as well as torque-dependent, and is
probably also a function of the gas velocity entering the stage.
3. Although the relationship of Alford's force to the stage torque is fairly
linear at a given speed, the y-intercept is not zero. That is, no aerodynamic force
is generated until a certain level of stage torque is reached.
The authors believe that these conclusions are significant, since rotordynamics
engineers have used Alford's theory for years with no experimental verification.
However, it is easy to see why rotordynamic stability thresholds cannot be
reliably predicted at present by computer analyses which use Alford's equation as a
model for the destabilizing coefficients. In order to do this, it will be necessary
to develop more accurate prediction models for the aerodynamic force, which will re-
quire experimental measurements on full scale turbomachinery components, such as an
axial flow stage from the compressor in a turbojet aircraft engine, or an industrial
process compressor.
REFERENCES
ii
.
.
Alford, J. S., "Protecting Turbomachinery from Self-Excited Rotor Whirl,"
Journal of Engineering for Power, Vol. 87, Series A, No. 4, October 1965,
pp. 333-334.
Doyle, H. E., "Field Experiences with Rotordynamic Instability in High-Perfor-
mance Turbomachinery," Proceedings of a Workshop held at Texas A&M University,
May 1980, NASA Conference Publication 2133.
Murphy, B. T., "Computer Programs for Critical Speeds and Stability Analysis,"
Rotordynamics of Turbomachinery Short Course, Texas A&M University, May 18-20,
1981.
4. Vance, J. M. and Laudadio, F. J., "Rotordynamic Instabilities in Centrifugal
Compressors - Are all the Excitations Understood?", ASME Journal of En$ineer-
inS for Power, April 1981, pp. 288-293.
5. Vance, J. M., "Instabilities in Turbomachinery", Proceedinss of the 5th Annual
Vibration Institute Seminar on Machinery Vibration Analysis, April 9, 1981,
New Orleans, La.
265
oB
Thomas, H. J., "Unstable Oscillations of Turbine Rotors Due to Steam Leakage in
the Clearances of the Sealing Glands and the Buckets," Bulletin Scientifique,
A. J. M. 71, 1958, pp. 1039-1063.
Winter, C. J., "Instabile Bewegungen von Turbinenlaeufern infolge lastabhaengiger
Erregerkraefte," Energie and Technik, January 1969, pp. 7-13.
266
Yt tl_ / Y!-cr,__J-,-
t
.I_/.,--."hat"
X,.l_______fj ' /
"_1-o ',1 /
/L_ ._ _." Y . .
Figure 1. - Dra,ing of top of test rig showing inlet configuration and
coordinate system.
267
8 _m
6" m
4 m
Z --
Q
S-ROJ) ¢Flr_'T, +X
ROTOR _ ° _
TCR¢_ o B.S53 IPH.B
le ;_ 3B ,we _ t:a
EZC_rRI_'r_ (mils)
Figure 2, - Alford force as function of rotor eccentricity, showing ±2 standard
deviations (shroud offset: +X),
A
L
r
0
.R
D
r
o
R
c
r
(
s
r
)
m
8
6 -
4
Z -
a
ROTOR SP[ZD * _ RPM
_ - B._3 I)+'LB
I I I I I II 4 I | | I I I I I I I I 4 1 | I I I I I I I I
19 ZO _ID 48 _ 60
[:CC][_TRZCZ7_ (mils)
Figure 3. - Afford force as function of rotor eccentricity, showing ±2 standard
deviations (shroud offset: -X).
268
3.9
° 2.5
2°0
t.5
1.0
m
oF-F-_-r: +x
ROTOR _ " 5ee6 RPI'I
TORGI.(., B.G53 ,TPPH..B
,,,,],,,,l,,,,I,,,.],,,,],, ,I
_9 29 3B 4B SO 60
F.O::_rTRZC13"Y (mS ]8 )
Figure 4. - Efficiency factor, B, as function of rotor eccentricity,
±2 standard deviations {shroud offset: +X).
showing
3.5
• 3.il
£ •
F"
F"
C 2.5
Z
£
H
¢
Y
Ir 2.8
R
C
1"
0
R
1.5
1.0
SHROL_ OF'F'sc'r: -X
Ro'roR s;=_3 - 5eea RPM
_ ,. 0.853 Ih_-k.9
l
,,,,],,,,I,,,,I,, ,I, ,,I,,,,I
18 29 38 40 50 60
EC_HI'RI CI"Pt (miLs)
Figure 5. - Efficiency factor, B, as function of rotor eccentricity,
±2 standard deviations {shroud offset: -X).
showing
269
R
L
r
.0
R
D
r
o
it
¢
[
(
s
P
)
e am
e.- D
4 --
Z
SHROI_ OffSET: ÷X
_zcTrY - 4 I'IILS
ROTOR S=_£D - SEm RPN
A -_MO. 2[3
ra -TI[3T NO, Z4
-'[13"r J._o. 25
.... I,,,,],,, ,],,,,I
9,4 B,5 il,G i!,? 9,8
TCRGI.J[ (lrr-lb)
Figure 6. - Alford force as functton of developed rotor torque
different tests).
lfor three
:,.s
?..8
:L.5
2.e
e.s
8.8
_z +X
(CC:]_TRZCZ'rY • 411 _ /
ROTOR S=r_= " SeEe RPrl II
- I_ -11[_r No. 24 / / /
0 -_-r .o. zs
,,,,I .... I,,,,1,,,,[
1,4 8,5 8,1; e,? e,e.
(ln--lb)
Figure 7. - Efficiency factor, B, as function of developed rotor torque
three different tests).
{for
270
_m_ I
S"
e.
-UI-
-15
Si4qO_ ClrF_'Tt 44(
_CI"rY - 4D KILS
\ o 8
B
0
- @
0
i
0 -TCI_U[ ° 0.691 IJq-LB
-TC_U[ - e._ Zff-tJB
,,,,I,,,,I,,,,I ,,,,I
aee8 5BS m
iaEZ]) (rl_)
Figure 8. - Alford force as functton of rotor speed (for two different va|ues
of torque).
/
/
/
TGq_J[ (tn-lb)
Ftgure 9. - Alford force as function of developed rotor torque (for different
values of speed).
271
18
A
L 8
F
0
R
D
r $
0
R
¢
r
(
S 4
P
)
SH_L_ (::I_r"SET: +X
_ICI"TY - 48
(rpm)
B.474 e.E33 8.B84
TCK_ILE (ln-lb)
Figure lO. - Alford force as function of developed rotor torque and rotor
speed, showing +2 standard deviations (shroud offset: +X).
t_um
S;-ROU90FF3L_" -X
_ZCZTY • ,_a n]3.s
t8
A
L
F
o
R
9 8
ir
o
R
c
£
(
s
P
U
)
B
m
G° m
4
8
/
/
/
/i
I t I II I I ,! [ I ! I I I J ! . ! ! ! I
(rpmm)
8.474 8.E53 0.E84
(In'-lb)
Figure ll. - Alford force as function of developed rotor torque and rotor
speed, showing +2 standard deviations (shroud offset: -X),
272
4 u _
SHarD MOTICrt: _.X
[¢£Dt7_CI_ - 48 MILS
B
, 3 °
C Z--
0
R
!
I I 1 II I I I I I I I I I I I I I I I
(rlm,m)
II.4"/'4 II._ II.e6.4
Fi gure 12.
and rotor speed, showing +2 standard deviations
Efficiency factor, B, as function of developed rotor torque
(shroud offset: +X).
3"
Z--
1--
_: -X
rJ::cIi_zczl'Y - _ $_
,,,,I,,,,I,,,,I,,,,I
S_ (r1_m)
B. 4'74 B.i_3 0.884
T_ (Irr-lb)
m
Figure 13. - Efficiency factor, B, as function of developed rotor torque
and rotor speed, showing ±2 standard deviations (shroud offset: -X).
273


Experiment measurement of Alford's force in axial-flow turbomachinery

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