A linear analysis was performed to determine the dynamic response of a turbine blade to intermittent rubbing against a tip seal. The response analysis consisted of a parametric study where the rubbing friction force was assumed to vary as a half sine wave over a preselected contact arc. The length of the contact arc, as well as the pump speed, was varied to determine the effects of each. Results show that for a given contact arc there are distinct critical speeds at which the blade response becomes a maximum

Brusher, Gerald A.

Davis, Gary A.

Shea, Daniel M.

English

NASA Technical Reports Server

N90-28615
BLADE TIP RUBBING STRESS PREDICTION*
Gerald A. Brusher**, Gary A. Davis, and Daniel M. Shea
Rockwell International/Rocketdyne Division
Canoga Park, California
ABSTRACT
A l_near analysis was performed to determine the dynamic response of
a turbine blade to intermittent rubbing against a tip seal. The response
analysis consisted of a parametric study where the rubbing friction
force was assumed to vary as a half sine wave over a preselected contact
arc. The length of the contact arc, as well as the pump speed, was
varied to determine the effects of each. Results show that for a given
contact arc there are distinct critical speeds at which the blade
response becomes a maximum.
INTRODUCTION
Due to the requirement for high efficiency in modern turbomachinery,
clearances between turbine blade tips and the turbine housing are de-
sired to be as small as is practically possible. An opposing design
requirement is that the clearances be large enough to prevent rubbing
of the blade tips on the housing due to rotor flexibility. In practice,
a compromise is reached, and a small amount of rubbing is tolerated. For
example, in the Space Shuttle Main Engine (SSME) high pressure fuel
turbopump (Fig. i), the first-stage turbine blades experience light
rubbing over approximately 20 to 30 degrees of arc. To minimize tip
rubbing while keeping the clearances as small as possible, the turbine
is mounted eccentrically in the pump housing. Under steady-state opera-
tion, sideloads created by the pumping action of the impellers move the
rotor to the centered position and no rubbing occurs. During startup,
however, the rotor blades contact the pump housing for a short period
of time; this has been observed in the turbine hardware during posttest
inspections of the tip seal area. A typical seal segment showing the
light rub patterns is presented in Fig. 2.
The purpose of the analysis presented herein was to assess the
effects of tip rubbing on the dynamic response of turbine blades. In
particular, the turbine blades of the SSME high pressure fuel turbopump
were analyzed as a representative case.
*Work reported herein was sponsored by NASA/Marshall Space Flight
Center under Contract NAS8-36361
**Presently attending the University of Michigan.
43
https://ntrs.nasa.gov/search.jsp?R=19900019299 2020-03-19T21:41:11+00:00Z
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Fig. i. High Pressure Fuel Turbopump
Fig. 2. HPFTP First-Stage Tip Seal Segment
ANALYTICAL MODEL
To investigate the problem of
intermittent rubbing against a tip
seal, an analytical model was de-
veloped that utilized a finite ele-
ment representation of the turbine
blade. The model (Fig. 3) addressed
a single blade fixed at the shank
root and free at the tip. The
boundary conditions at the root
simulate the fixity of the blade
due to the centrifugal loads acting
through the firtree during high
speed operation. Since the flexi-
bility of the disk actually admits
motion of the blade shank, the
assumption of fixity results in
slight increases in both natural
frequencies and stresses. However,
these effects are inconsequential
in regard to understanding the dy-
namic behavior of the blade due to
to tip rubbing.
Rotation
: /" I I i
, ,_'_';,"3-_t-'-'__' Load _-._
:I'''''Lri'"11'IfJ __j--_i---_-'_=
Fig. 3. HPFTP First-Stage Turbine
Blade Finite Element Model
44
To I:_[>res_._[lll the rotation of a rotor about an eccentric axis, it
wds a:_.sul_led that the blade would be in contact with the tip seal over a
•_JelL-defined arc once per revolution.
Only tangential frLcti_orl effects were F(t)
considered; therefore, a tangential
load was applied to a ._LngLe point on
blade tip. The amplitude of the forc-
ing f,mct io[[ '_/dS set equal to the
product of the normal load and the
friction coefficient. The friction
coefficient was assumed to be con-
stant, however, the normal load was
assumed to vary as a hale s_ne wave
over the duration of the rub (Fig.
4). This assumption was reasonable
given the kinemat ics of rotation
about an eccentric axis.
-- Fpeak
_-TRu b =I= TFree
-- TRevolution _1 t
Half Sine Wave
Fig. 4. Friction Force Time
History
The model was designed to assess linear behavior only. It was ini-
tially postulated that the blade would display stick-slip behavior dur-
ing contact with the seal. However, this was shown to be highly unlikely
because the blade tip speed due to rotation is approximately 2000 times
greater than that due to vibration in the first-tangentlal bending
mode. Therefore, this nonlinearity was not included in the model.
M_THOD OF SOLUTION
The modal analysis technique was used to solve the equations of
motion for the blade. The first i0 natural frequencies and mode shapes
were extracted from the finite element model and then used to express
the uncoupled equations of motion in terms of principal coordinates.
These uncoupled equations were solved through time integration using
the central difference method. Modal displacements and stresses were
then summed to compute the physical displacements and stresses at
selected points on the blade.
PARAMETRIC ANALYSIS
Four parameters were expected to have a significant effect on the
blade response: normal force, friction coefficient, shaft speed, and
contact arc. Since it was difficult to specify exact values of these
parameters, it was deemed preferable to determine the response over a
range of values and then to rely on future empirical results to assist
in pinpointing their actual magnitudes. Ranges for these parameters are
presented in Table 1 and reflect experience with dynamic simulations of
the rotor as well as known operational parameters of the pump.
A typical response analysis consisted of calculation of the steady-
state solution for a specific normal force, friction coefficient,
contact arc, turbine speed, and point of application of the rubbing
force. The linear nature of the analysis was used as much as possible
45
Table i. Parametric Analysis
Parameter Minimum Maximum Increment
Normal Force (Ib)
Friction Coefficient
Shaft Speed (rpm)
Contact Arc (deg)
0
O.l
28,000
lO
2OO
0.3
38,000
3O
50
O.l
500
I0
to reduce the number of cases run. For example, the response results
for a given friction coefficient and normal force were ]inearly scaled
to different values of friction coefficient and normal force. The
sensitivity of the response to the point of application of the load was
investigated by forcing the blade tip near the center of the airfoil,
the leading edge, and trailing edge. Finally, turbine speed and contact
arc were varied by changing the frequency and shape of the forcing
function.
DISCUSSION OF RESULTS
The most significant results obtained from the linear parametric
analysis were the discoveries of critical speeds and contact arcs that
maximized the response of the blade. In addition, it was found that
regardless of whether the tangential friction force was applied to the
blade tip near the center of the airfoil, or at the leading or trailing
edge, the blade primarily responded in the lowest bending mode. Before
discussing the causes for the occurrence of critical speeds and contact
arcs, it will be instructive to consider the general aspects of the
blade behavior in the time domain. Then the problem will be character-
ized in the frequency domain, yielding further insight into the
response.
To begin, consider the behavior of a single blade as it moves
through several revolutions of the turbine disk. This is shown in Fig.
5, which illustrates a typical time history of stress in the root region
of the blade. The time between peak stresses corresponds to one revolu-
tion of the disk. For a given revolution, the maximum stress occurs at
the end of the contact arc because the friction load causes the blade
to deflect. At the end of the contact interval, the blade is released
and then it experiences free, damped vibration for the remainder of the
revolution. As can be seen from the Fig. 5, the blade response during
the time of free vibration is a function of the damping ratio which was
chosen as C - 0.01 throughout this analysis. An alternate method of
showing the motion is with a state space plot of the blade tip displace-
ment as presented in Fig. 6. This graph, which is plotted for a single
revolution of the turbine, clearly shows the disturbance from equilib-
rium due to rubbing and the subsequent damped response.
46
Blade Contact-* I _--
Period //
700
A 500
100
_ -100
l I
-5001 " l "I 1 " I I
0 1 2 3 4 5 6
Time (msec)
Fig. 5. Typical Stress Time History (Pump Speed
- 36,000 rpm)
5
3
o -2
>
-6
-20 -15 -10 -5 0 5 10 15 20 25
Displacement (mils)
Degree of Freedom = 1: Node 681, DIR
Pump Speed = 36,000 rpm
30
Fig. 6. State Space Representation of Blade Tip Motion
Since the blade is subjected to a periodic load, it is expected
that certain forcing frequencies might excite a resonant condition. The
frequency content of the forcing function is dependent upon the shaft
speed and length of the contact arc; hence, there should be certain
critical speeds and contact arcs at which the blade response is maxi-
mized. The existence of critical speeds is verified by the plots of
steady-state rms blade tip displacement versus shaft speed presented in
Fig. 7 through 9. The friction coefficient and magnitude of the normal
load are identical in all cases, however, the contact arc varies from
10 to 30 degrees. In each graph, it is evident that extreme displace-
ments occur at 29,500, 32,500, and 36,000 rpm. Thus, for each contact
47
"_ 20
E 18
16
C
o lO
= 6
_5 4
, 1 J I i L i I L 1 L 1 J E h L , I L
28 29 30 31 32 33 34 35 36 37 38
Speed (krpm)
Degree of Freedom = 1: Node 681, DIR 1
Contact Arc = 10 deg
Friction Coefficient = 0.1
Fig. 7. Rms Blade Tip Displacement vs Speed
(Contact Arc =, l0 Deg)
_" 40
_ 35
3O
25
_ is
_" lO
a
_ 5
NI L I I I 1 i L J I L I B I I B I I I A i
rT -28 29 30 31 32 33 34 35 36 37 38
Speed (krpm)
Degree of Freedom = 1: Node 681, DIR 1
Contact Arc = 20 deg
Friction Coefficient = 0.1
Fig. 8. Rms Blade Tip Displacement vs Speed
(Contact Arc = 20 Deg)
_" 40
E 35
- 30
20
15
a 10,
cn 5
_28 29 30 31 32 33 34 35 36 37 38
Speed (krpm)
Degree of Freedom = 1: Node 681, DIR 1 I
Contact Arc = 30 deg 1Friction Coefficient = 0.1
Fig. 9. Rms Blade Tip Displacement vs Speed
(Contact Arc = 30 Deg)
48
arc, the blade _espon:;e displays the same dependence on speed. .Simi-
larly, at a given speed, the amplitude of the response is dependent upon
the contact arc. This suggests the existence not only of critical speeds
but also of critical contact arcs, and provides motivation for further
investigation into these observed patterns.
To explain the occurrence of the critical speeds, it is necessary
to consider the frequency spectrum of the forcing function; recall that
the time history of the forcinq function was shown Jn Fig. 4 for one
complete revolution of the shaft, b:xpret_sing the forcing function as a
Fourier .%cries arid plotting the magnitude of the various Fourier coef-
ficterlts as a function of harmonic number results in the plot of Fig.
[0. The hart[tonic numbers should be interpreted as multiples of pump
speed, thus the horizontal axis of Fig. 10 is actually a frequency axis.
This representation of the force reveals a broad band of frequency con-
tent. Consequently, when the speed is such that the frequency of one
force harmonic is equal to the frequency of a blade mode, then a reson-
ant condition occurs. This is clearly presented in the Campbell diagram
of Fig. ii, which indicates the speeds at which interferences occur
between various harmonics of the forcing function and the first mode of
the blade. Clearly, the ninth, tenth, and eleventh harmonics excite
this mode at speeds of 36,000, 32,500, and 29,500 rpm, respectively.
Therefore, the c[iti.cal speeds are simply those at which a resonant
condition exists between one harmonic of the forcing function and the
first bending mode of the blade.
0,6
0.5
0.4
n
0.3
CL
E 0.2
0.1
0.0
0
lllli , ......
10 20 30 40 50 60 70 80 90 100
Harmonic Number
Fig. I0. Fourier Transform of Force (Half Sine Wave History
Over Contact Arc Time Window = 1 Revolution)
Comparison of Fig. 7 through 9 indicates that although the depen-
dence of the response on speed is identical in all three cases, the
amplitude of the displacement varies with the contact arc. To inves-
tigate this observation, use the Fourier series representation of the
forcing function and vary the length of the contact arc. Thus the
magnitude of each harmonic may be expressed as a function of the contact
49
A
N
J¢
v
C
O"
¢D
M.
I0
8
7
6
5
4
3
2
1
0
Fig. Ii.
Harmonics
I
11N
ION
First Bending Mode F 9N
_j_ ........ _L. __ _ ....
10 20 30 40
Shaft Speed (krpm)
HPFTP First-Stage Blade Campbell Diagram
arc as shown in Fig. 12; here only the ninth, tenth and eleventh har-
monics are presented because they were identified as causing interfer-
ences within the operating range of the turbine. It is clear from the
Fig. 12 that there is a unique angle for which each harmonic attains
its maximum value. This explains the difference in blade response for
different contact arcs at a given speed. The magnitude of each harmonic
of the friction force is a function of the contact arc, therefore there
is a specific contact arc that yields the greatest response at each cri-
tical speed. These combinations of speed and contact arc, which can be
retrieved from Fig. Ii and 12, are listed in Table 2. It should be noted
that the critical contact arcs are within the range of wear observed in
the SSME first-stage tip seal.
0.32
0.28 9N
o 0.24 _J_10NE
_ 0.2
x
"_ 0.10
O
'O
_ o.12
C
• 0.08
0.04
0 20 40 60 80
Contact Arc (deg)
Fig. 12. Forcing Function Magnitude vs Contact Arc
5O
Table 2. Critical Speed and Contact Arc
Combinations
Resonant
Harmonic
9
I0
II
Critical
Speed (rpm)
36,000
32,400
29,455
Critical
Contact Arc
(deg)
27.3
24.6
22.4
In addition to considering different speeds and contact arcs, the
sensitivity of the response to the location of the applied load was
also investigated. Separate response analyses were completed with the
friction force applied at the airfoil tip near the leading edge,
midchord, and trailing edge locations. A summary of the results are
presented in Table 3, which shows that the highest stresses were found
when forcing the blade at the trailing edge.
Table 3. Normalized Radial Stress at Shank Root
(i Ib peak friction force)
Contact
Arc
(deg)
lO
lO
lO
20
20
20
30
30
30
Steady State rms Stress
Force
Applied
TE
MC
LE
TE
MC
LE
TE
MC
LE
29,500
rpm
128
I02
g7
190
148
145
I70
140
130
(psi)
32,500
rpm
129
I06
lOl
208
l?O
160
206
169
158
36,000
rpm
125
lO0
94
205
167
157
220
182
170
NOTE: TE - trailing edge, MC - midchord,
amd LE - leading edge
The stresses presented in Table 3 have been normalized to show the
blade response due to a a peak friction force of 1 pound. Using the
maximum stress (220 psi) given in this table, it is easy to calculate
the friction force necessary to create a given alternating stress. For
example, an alternating stress of 5000 psi rms would require a peak
friction force of 22.7 pounds. Assuming a coefficient of friction of
0.2 would translate this into 114 pounds of normal force, which is
entirely within the expected range. Therefore, significant alternating
stresses can be generated by blade tip rubbing.
51
In slmm_ary, the turbine blade parametric analysis provided several
interesting results. The significant dependence of the maximum steady-
state response upon both speed and contact arc is summarized by the
surface shown in Fig. 13. This plot provides a qualitative representa-
tion of the effects of shaft speed and contact arc, and clearly illu:;-
trates the existence of critical conditions. The speed axis shows the
frequency response of the blade, while the contact arc axis tells how
the magnitude of the driving force varies. Critical speeds and/or con-
tact arcs occur when these variables produce a maximum response as shown
in Fig. 13.
soo
300
Fig. 13. Normalized Stress as a Function of Speed
and Contact Arc
CONCLUSIONS
General conclusions, which can be drawn from the analysis, are that
for a turbine blade under tip rubbing excitation, there exist distinct
critical speeds at which the dynamic response of the blade is maximized.
Furthermore, at each speed, a critical contact arc exists which maxi-
mizes the response of the blade. The overall maximum response occurs
at the point where the critical speed and critical contact arc occur
simulataneously.
The parametric analysis also provided information for specific con-
clusions regarding the SSME turbine blade. From the results of the
analysis, three critical speeds were _ndentified within the range of
operating speeds. These critical operating points represent the speed
at which resonance occurs between the applied friction load and the
first bending mode of the blade. In addition, three critical contact
arcs that maximize the amplitudes of the resonant force harmonics were
52
identified within the range of observed wear in the SSMEfirst-stage
tip seal. Finally, the rms alternating stresses at the blade root were
found to be significant depending upon the value and rub location of
the friction force. No measurementsof friction force have been made,
however, the analysis shows that sizable stresses can be produced with
reasonable values of friction force.
It is intended that the results of the l_near parametric analysis
presented herein will provide guidance in the preparation of the full-
scale test program, and that the combination of analytical and empirical
results wilt contribute to a fundamental understanding of the tip
rubbing problem.
53


Blade tip rubbing stress prediction

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