The relationship between actual and predicted re-entry maximum dynamic pressure is characterized using a probability density function and a cumulative distribution function derived from sounding rocket flight data. This paper explores the properties of this distribution and demonstrates applications of this data with observed sounding rocket re-entry body damage characteristics to assess probabilities of sustaining various levels of heating damage. The results from this paper effectively bridge the gap existing in sounding rocket reentry analysis between the known damage level/flight environment relationships and the predicted flight environment

Lanzi, R. James

Vincent, Brett T.

English

NASA Technical Reports Server

N94-23653
DEVELOPMENT AND APPLICATION OF AN EMPIRICAL
PROBABILITY DISTRIBUTION FOR THE PREDICTION
ERROR OF RE-ENTRY BODY MAXIMUM DYNAMIC PRESSURE
R. James Lanzi
National Aeronautics and Space Administration
Goddard Space Flight Center
Wallops Flight Facility
Wallops Island, VA 23337
Brett T. Vincent
Computer Sciences Corporation
Wallops Flight Facility
Wallops Island, VA 23337
ABSTRACT
The relationship between actual and predicted re-entry maximum dynamic pressure is
characterized using a probability density function and a cumulative distribution function
derived from sounding rocket flight data. This paper explores the properties of this
distribution and demonstrates applications of this data with observed sounding rocket re-entry
body damage characteristics to assess probabilities of sustaining various levels of heating
damage. The re_!,_ from this paper effectively bridge the gap existing in sounding rocket re-
entry analysis between the known damage level/flight environment relationships and the
predicted flight environment.
1.0 INTRODUCTION
Figure 1 shows a schematic of a typical re-entry configuration for a Terrier-Black
Brant sounding rocket payload. Historically at NASA/GSFC/Wallops Flight Facility, 5
degree-of-freedom trajectory simulations (Reference 1) have been used to predict sounding
rocket re-entry body flight environment parameters such as maximum dynamic pressure,
Math 1 altitude and parachute deployment conditions. The aerodynamic characteristics used
in these simulations have been computed using the CDCG computer program which
incorporates a hybrid theoretical / empirical nonlinear crossflow aerodynamics model.
As the NASA Sounding Rocket Program has shifted toward larger payload size and
increased vehicle performance capability, concern over aerodynamic heating during re-entry
has become more than just an academic issue. Re-entry heating damage to sounding rocket
recovery systems has been a source of refurbishment cost, and a cause for apprehension over
possible recovery system failure. The level of heating damaged sustained by any given
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https://ntrs.nasa.gov/search.jsp?R=19940019180 2020-06-16T15:23:50+00:00Z
sounding rocket payload during the re-entry leg of the flight has been found to be directly
related to the peak level of dynamic pressure encountered and the static margin of the re-
entry body. A crude characterization of the relationship between the aerodynamic heating
damage and maximum dynamic pressure was determined in Reference 2. In this report, a
"damage line" is defined which separates values of flight dynamic pressure and re-entry body
static margin that constitute a "cool region', within which no heating damage has been
experienced, from values of flight dynamic pressure and static margin which constitute a
"hot* region where varying degrees of heating damage have been experienced.
.Flight levels of maximum dynamic pressure can vary significantly from the predicted
values. In Reference 2, the limitations of the analysis tools used to make predictions of the
re-entry enviro/_ent parameters were investigated, and several sources of prediction error
were isolated. In particular, a statistical characterization of the prediction error in the
aerodynamic properties was determ_ed.
Before the study of Reference 2, NASA/GSFC/WalIops Flight Facility imposed a
fixed limit on predicted maximum dynamic pressure to prevent recovery system failure due
to re-entry heating. Payloads were ballasted to meet this constraint. Once the study of
Reference 2 was published, NASA began using the observed properties of the prediction
error of re-entry body aerodynamic characteristics to analytically determine the dispersion of
re-entry environment parameters such as maximum dynamic pressure (q,_ and Math 1
altitude (References 2 and 3). Re-entry bodies are then ballasted to bring the envelope of
predicted levels of maximum dynamic pressure beneath the "damage line" given in Reference
2. Although this technique does address the variability in predicted re-entry dynamic pressure
levels, the inherent noise involved in the determination of the prediction error of the body
aerodynamics combined with the extreme sensitivity of the re-entry environment to variations
in body aerodynamic characteristics makes it difficult to interpret the results from this
approach.
During the post flight mission analyses of Terrier-Black Brant V Flights 36.079 and
36.089 it was observed that the predicted re-entry maximum dynamic pressures (qp) were
20.2% and 15.3 % lower, respectively, than the actual re-entry maximum dynamic pressures
(q,). Flight 36.089 was a refly of the 36.079 payload. Further analysis revealed that there
was an apparent center of pressure shift _ from the S-19 canards for both flights.
Reference 2 noted that for the 28 analyzed cases, the apparent center of pressure shift was
towards the S-19 canards. Approximately 9 Lbs. of re-entry ballast were incorporated on
these flights to bring qp below the "damage line" of Reference 2. These flights illustrate the
difficulty in applying the results of Reference 2 to payload ballast design.
In this report, the direct statistical relationships between predicted and actual
maximum dynamic pressure are developed, and the relationship between re-entry heating
damage and maximum dynamic pressure is refined. These results are combined into a
coherent method for relating results from simulation to probability of sustaining various
levels of aerodynamic heating damage during re-entry.
270
2.0 CHARACTERISTICS OF RE-ENTRY ENVIRONMENT PREDICTION ERROR
2.1 Prediction Error Determination
The re-entry environment prediction error is characterized by the percent error in
predicting _ (%q_), which is given by:
_ ( %-qa) , loo%. (1)
q.
The percent error was determined for the 49 flights listed in Table 1. As was done in
References 2 and 4, qo was determined by initializing a GEM 5-DOF re-entry simulation
(containing nonlinear aerodynamics from the CDCG program) with radar data at some time
prior to re-entry. This was done to eliminate trajectory dispersion effects.
2.2 Generation of an Empirical Probability Density Function
It can be seen in Table 1 that %q_ varies widely, from -54.5% to +232.5%.
However, closer inspection of the data reveals that the majority of cases fall within +30%
error range. A distribution for %q_ was determined by generating a normalized relative
frequency histogram h(x); that is, dividing the %q_. values into 10 %q_,, class intervals and
determining the frequency of %q_ values within each interval. The intervals are represented
mathematically by
[Co,q), [q, c2), ..., [ci,_1,cl,),
where k is the number of class intervals (10 in this case). The above notation denotes that
the class intervals are closed on the left; that is, a value on a boundary point is assigned to
the class interval that has this value as its lower boundary. The interval widths were chosen
271
to give a more or less continuous distribution. The normalized relative frequency histogram
is defined by
f_n
h(x) - for ci_ t _ x c i, i=1,2,...,10, (2)
C i - Ci_ 1
where n is the number of observations and fi is the frequency of the class [ci.i, cO. The
normalized relative frequency histogram is an approximation of the probability density
function for %q,., Probability values for the event that a random variable, x, falls within a
certain interval are given by the area beneath the probability density function, f(x), over the
interval. Figure 2 presents the nortrfalized rda_ive frequency histogram for the %q_ of the
49 subject cases. As was observed in Table 1, the distribution of %q=,. is skewed to the right
with the majority of cases being between ±30_. _ -,. ....... _ _ ._ -
The relativefrequencyOf_ interval[a,b),where Co < a < b < Ck,can be
computed by the integral
f : h(x)dx.
Since the relative frequency is an approximation to probability, this integral can be
thought of as an approximation to the probability that X, the random variable under
consideration (%q_), is in the interval [a, b). It follows that a cumulative distribution
function F(x) can be derived from
f':F(x) = P(X < x) = P(X < x) = w)dw, (3)
where f(x) is the limit of h(x) as n increases and the lengths of the class intervals go to zero.
2.3 Discussion of the Cumulative Distribution Function
The cumulative distribution function (c.d.f.) in Equation 3 gives the probability of
having a value of the variable X (in this case, %q=,.) less than some given value x. Figure 3
presents the c.d.f, for %q,,_. The c.d.f, shows that while there is a 47.7% probability of
272
having a negative %q_. (that is, q_ greater than qp), there have been no recorded cases of
%q_, exceeding -60%. Qualitatively, the c.d.f, shows that most of the time the re-entry q,_,
prediction is fairly accurate, and is about as likely to be under the flight value as it is over,
but the over predictions are of a much larger magnitude than the under predictions. Of
course, it is the under predictions of re-entry q,_, that are of greater concern, in terms of re-
entry heating damage.
3.0 COMPUTING DAMAGE RISK USING THE EMPIRICAL ERROR
DISTRIBUTION
Figure 4 shows the re-entry flight envelope of maximum dynamic pressure and re-
entry static margin for the Terrier-Black Brant V family. The different symbols on the plot
represent the different levels of heating damage sustained during the flight according to the
following classifications:
Re-entry Heating Damage Classifications
0 - No damage. (36.096)
1 - Exterior discoloration. (36.053)
2 - Minor exterior erosion, especially near surface
discontinuities. (36.049)
3 - Melting at surface discontinuities and erosion making
its way into interior. (36.034)
4 - Major exterior damage. Skin eroding and peeling away
from joint. (36.057)
5 - All above exterior damage plus significant interior
damage. (36.086).
6 - Failure due to re-entry heating (no known cases)
The lower line in Figure 4 separates the region in the flight envelope of Level 0 and 1
heating damage from the region of Level 2 and 3 heating damage. This line was previously
published in Reference 2 and is given by the equation
273
q(l)
max < 4200 - 17.5 ,SM 2, (4)
where
q,_ = Flight maximum dynamic pressure (psf)
SM = [xcg - xcla[ (% Total Body Length).
The upper line in Figure 4 separates the Level 2/3 region from the higher level
heating d_age regions in the flight envelope. _s line is defined by several borderline =
flights and is given by the equation
q(2)
mx _ 4600 - 10.5 *SM 2. (5)
For a re-entry body with a fixed static margin, values of q,_) and q,a) can be
computed using Equations 4 and 5. A value of q, is determined using the nonlinear
aerodynamics of the CDCG program and the GEM 5-DOF simulation. At this point the
probability of falling into the three regions of the flight envelope can be computed using the
cumulative distribution function derived in Section 2.2. This is done as follows:
. Calculate the prediction errors associated with flight values of maximum
dynamic pressure equal to q,_(_) and q,_):
_ (1)
,_, (I) qp - qmx
7oqerr = * 100%
q(1)
max
_ (2)
o/_(2) qp - q._
-/o%_, = * 100%
q(2)
. Compute the following probability values from the cumulative distribution
curve (CDF) of Section 2.2:
274
p_ = 1 - CDF(%q_ _))
P2 = CDF(%q ¢2))
P3 = 1 - P l - P2
where
p!
P2 --
P3 ----"
Probability of flight q_, (q_) falling into the level 0/1 region.
Probability of q_ falling into the higher level damage region
(3/4/5/6).
Probability of q_ falling into the level 2/3 damage region.
The procedure outlined above has been automated in a FORTRAN program called
REPROB. The damage demarcation curves and the empirical cumulative distribution function
will be updated as new cases are added to the re-entry database.
CONCLUSIONS
An empirical probability density function describing the re-entry environment
prediction error has been derived by comparing predicted re-entry maximum dynamic
pressure (qp) to flight maximum dynamic pressure (ct0 for 49 flights. From this, a
cumulative distribution function has been generated relating _ to q,; allowing the mission
analyst to predict, to a given probability, if _ will be in a possible heating damage region.
The re-entry error distribution and the damage region definitions can easily be updated as
new cases are flown and as previous flight data is found.
\
The methods given in this report utilize the properties of the re-entry environment
prediction error to give a more direct means of assessing possible re-entry heating
difficulties. Future work in refining the aerodynamic models should address the effect of
changes on the prediction error distribution.
275
REFERENCES
1. 841.1 Wallops Flight Facility Computer Programs:
WFF CDCG - Crossflow Drag Computer Program
WFF GEM - 6-DOF Simulation Program
2. Lanzi, R. J.: A Quantitative Evaluation of the Mission Analysis Tools Used for the
Prediction of Reentry Body Aerodynamic Characteristics and Reentry Trajectory
Parameters. NASA Internal Memo, 9/91.
3. Lanzi, R. J.: A Procedure for the Determination of Sensitivities of Reentry Trajectory
Predictions with Known Error Bands in Predicted Aerodynamic Coefficients. NASA
Internal Memo, 12/91.
4. Vincent, B. T.: PreIim_ Findings of the Comparison Between the Current CDC
Code and a Modified CDC Code (CDCG) With Flight Data. CSC Memorandum,
November 30, 1990.
5. Hogg, R. V.; Ledolter, J." Engineering Statistics. Macmillan Publishing Co.,
1987.
276
STATION
(in. TNT)
0.00
35.66
54.20 --_--
56.70
78.70
87.20
94.65'
106.65
120.84
(CG) 127.62
135.84
137.09 _
(CLA) 138.76
152.13
163,39 --__
166.59 __
167.84
233.84
238.71
L 17.26"
_" 18.26"
RE-ENTRY GRAVIMETRICS
Weight = 875.00 Ibs.
Length = 203.05 in.
CG = 127.62 in. TNT
CLA = 138.76 in. TNT
[CG-CLA] = 11.14 in.
Ix = 8.61 slug-ft2
ly = 665.47 slug-It 2
Figure 1. Terrier - Black Brant VC 36.068 GG
Re-entry Configuration
277
Table I. CDCG Comparison with Flight Data
Pred. Flight
Flight Qmax Qmax %Qcrr
27.075 3565.6 2481.5 43.69%
27.106 2502.1 3071.5 -18.54%
27.121 2599.4 781.7 232.53%
27.122 945.6 821.0 15.17%
27.123 1888.3 1317.0 43.38%
27.124 2335.9 2388.0 -2.18%
27.129 1057.0 1116.0 -5.29%
27.130 1160.0 894.0 29.75%
27.133 1056_5 305.5 245.83 %
36.016 2305.0 5061.6 -54.46%
36.025 3740.1 4271.6 -12.44%
36.027 _2703.4 1505.4 79.58%
36.030 2616.5 4103.5 -36.24%
36.032 2609.4 3090.7 -15.57%
36.034 2067.7 4110.0 -49.69%
36.035 3317.0 2673.2 24.08%
36.041 2874.5 3778_3 -23.92%
36.043 2730.0 1408.5 93.02%
36.047 1808.0 1230.9 46.88%
36.048 2800.0 2700.0 3.70%
36.049 3607.7 4334.9 -16.78%
36.052 1536.0 91_.4 67.07%
36.053 1965.0 2938.5 -33.13%
36.054 1635.0 2344.1 -30.25%
36.057 4949.0 4739.7 4.42%
36.058 4643.0 4285.0 8.35%
36_059 2749.9 1277_ 115.31%
36.060 1633.0 622.2 162.46%
36.062 3958.0 4744.6 -16.58%
36.063 2342.0 3888.7 -39.77%
36.066 3383.1 3760.3 -10.03%
36.067 2715.3 1946.0 39.53%
36.068 4907.0 5238.2 -6.32%
36.069 3443.3 3352.5 2.71%
26.070 3365.0 3701.2 -9.08%
36.072 1916.3 1888.9 1.45%
36.073 4964.0 4267.6 16_32%
36.074 3146.4 2608.9 20.60%
36.077 5020.6 3268.3 53.62%
36.078 4584.0 4341.1 5.60%
36.079 3794.8 4757.0 -20.23%
36.085 4190.0 4230.0 -0.95%
36.086 4919.0 4413.5 11.45%
36.087 3153.2 3500.0 -9.91%
36.088 2124.7 2600.0 -18.28%
36.089 3921.0 4631.0 -15.33%
36.090 3800.0 3940.0 -3.55%
36.096 1907.7 812.7 134.74%
36.098 2740.0 2740.0 0.00%
278
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RE-ENTRY BODY DYNAMIC PRESSURE PREDICTION ERROR
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0.0 50.0 100.0 150.0 200,0 250.0 300.0
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RE-ENTRY BODY DYNAMIC PRESSURE PREDICTION ERROR
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Prediction Error- %
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250.0 300.0
279
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Terrier-Brant Reentry Study
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280


Development and application of an empirical probability distribution for the prediction error of re-entry body maximum dynamic pressure

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