w. a. kilgore, s. balakrishna, and d. h. butler nasa ... · w. a. kilgore, s. balakrishna, and d....

F

AIAA-2001-1162

Reduction of Tunnel Dynamics at the NationalTransonic Facility (Invited)

W. A. Kilgore, S. Balakrishna, and D.NASA Langley Research CenterHampton, Virginia

H. Butler

39th AIAA Aerospace Sciences Meeting & Exhibit8-11 January 2001

Reno, Nevada

For permission to copy or republish, contact the American Institute of Aeronautics andAstronautics

1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344

https://ntrs.nasa.gov/search.jsp?R=20010013155 2020-07-01T13:57:01+00:00Z

AIAA-2001-1162

REDUCTION OF TUNNEL DYNAMICS AT THE NATIONALTRANSONIC FACILITY

W. A. Kilgore', S. Balakrishna t, and D. H. Butler t

Aerodynamics, Aerothermodynamics, and Acoustics Competency

NASA Langley Research Center

Hampton, Virginia

ABSTRACT

This paper describes the results of recent effortsto reduce the tunnel dynamics at the National

Transonic Facility. The results presented describe

the findings of an extensive data analysis, theproposed solutions to reduce dynamics and the

results of implementing these solutions. Theseresults show a 90% reduction in the dynamics

around the model support structure and a smallimpact on reducing model dynamics. Also

presented are several continuing efforts to furtherreduce dynamics.

INTRODUCTION

The National Transonic Facility (NTF) is the firstlarge scale closed circuit fan drive cryogenic

pressure tunnel designed to provide flightReynolds number testing at transonic conditions in

a ground based facility. Throughout the NTF'shistory several testing programs were cut short, or

severely limited, because of dynamics. Theselimitations manifest themselves as excessive

balance dynamic loads, large modeldisplacements and increased data scatter.

Excessive model and/or tunnel movements (i.e.dynamics) can produce questionable results and

concern over the safety of the model,

instrumentation and the facility. Because the NTFis capable of generating large dynamic pressures(up to 7000psf) and high Reynolds numbers (up to

146million/ft) the dynamics are greatly amplified.Figure 1 shows the NTF operational envelope.

Dynamics and Control Engineer, Research Facilities Branch,Senior Member, AIAA

:l:SeniorResearch Scientist, ViGYAN Inc.

Tsenior DesignEngineer,ViGYAN Inc.

Copyright © 2001 by the American Instituteof Aeronautics andAstronautics, Inc. No copyright is asserted inthe United Statesunder Title 17, U. S. Code. The U. S. Government has aroyalty-free license to exercise all rights under the copyrightclaimed hereinfor Governmental Purposes. All other rights arereserved by the copyright owner.

Several efforts to understand and minimize the

dynamics were conducted and documented overthe history of the NTF. 1'2'3'4 These efforts had

limited opportunity to implement any proposedsolutions. In FY-2000 a concentrated effort was

established to "take action" and implement severalsolutions. The results of this effort are presented

in this paper.

A

M

PT

p'qRe

tT

NOMENCLATURE

Area

measured fluctuating acceleration, (RMS)Mach number

tunnel total pressure

measured fluctuating pressure, (RMS)dynamic pressureReynolds number in million/chordtime, s

tunnel flow total temperature

Subscripts and AbbreviationsX

Y

ZAFARC

DSSFFF

HSDMSFNF

PMRM

RMS

SFTSYM

axial direction, tunnel flow axis

lateral direction, orthogonal to X and Z

vertical direction, orthogonal to X and Ybalance axial forcearc sector

down stream support frame

fixed fairinghigh-speed diffusermodel support framebalance normal force

balance pitching momentbalance roll moment

root mean squarebalance side forcetest section

balance yaw moment

NTF STRUCTURE

The NTF dynamics can be easily divided in to

three separate areas (internal support structure,model support structure and model/balance/stingstructure). Each area has its own unique

1American Institute of Aeronautics and Astronautics

AIAA-2001-1162

dynamics, and when the dynamics in one areacoalesce with another area, excessive dynamicsOccur.

Internal Support Structure

Because the NTF is a cryogenic pressure tunnel,several unique features are inherent in its designto accommodate the large changes in

temperatures (150°F to -250°F) that preclude any

fixed mountings or supports. These uniquefeatures are most prevalent in the area of theinternal support structure for the test section.

Figure 2 shows the original build-up of the NTFtest section outside the plenum and indicates the

support points and the direction of constraint.

Unique to the NTF, the entire test section supportstructure is free to move; and constrained

vertically and laterally using pads and guides

between the test section and the plenum pressureshell as shown in figure 3. The upstream and

downstream portions of the test section areseparate and independently secured withaccommodations for movement via Bellevilles

spring washers at the front and back of the plenumbulkheads.

Model Support and Model/Balance/StinqsStructure

The model support system is mounted within the

test section support structure as shown in figure 4.The model is supported on the end of a 13ft stingand attaches to the roll drive that is installed in the

arc sector. The entire arc sector, roll drive, sting

and model (+350001bs) rotates to pitch the model

assembly.

The arc sector rides between four I-beams as

shown in figures 2 and 5 (one set of four on topand one set of four on the bottom). These I-beamsconnect between the down-stream support frame

(DSSF) and the model support frame (MSF)

providing stiffness to the model support structure.

The arc sector maintains its alignment and loadcarrying capability through four sets of lateral

bearings (two upper pairs and two lower pairs) andtwo thrust bearings (one upper unit and one lower

unit) as shown in figure 4. These bearings aremounted just below the innermost I-beams

adjacent to the arc sector. Each bearing providesa sliding surface on the arc sector that is springloaded to 20001bs with Bellevilles washers. All the

loads generated from testing are distributedthrough the bearings into the I-beams, DSSF andMSF. As shown in figure 2, the DSSF is

constrained vertically and laterally, while the MSFis cantilevered off the DSSF with no directconstraints.

Immediately behind the arc sector is a fixed fairing(does not move with the arc sector) as shown in

figure 4. The fixed fairing is a hollow aero-faringthat consists of horizontal flow liner panelsconnected to a solid vertical beam at the aft end of

the fixed fairing and extends forward to the aft

edge of the arc sector. This fixed fairing vertical

beam spans the test section, is secured to theceiling, and passes down through the floor for

several feet to where it is vertically pinned.

ANALYSIS

An extensive amount of dynamic data was

collected over several test programs covering theentire operational envelope of the tunnel. This

data included 128 channels (sampled at 320samples/second, simultaneous sample and hold)

of accelerations, positions, loads and fluctuatingpressures located throughout the high-speed legon the tunnel structure and model. 5

This current effort concentrated on reducing thetunnel structure dynamics in the lateral directionalong the high-speed leg as a first step in reducing

the dynamics.

Analysis of Tunnel Structural ResponseThe analysis of the fluctuating pressure data

shows that the all the fluctuating pressures arebroadband white noise without any specific

frequencies and the level varies with the tunneldynamic pressure. The model support area (arc

sector and fixed fairing) experienced the largestfluctuating pressures of the entire high-speed leg.

This relationship between the fluctuating pressureand dynamic pressure is shown in figure 6 for the

fixed fairing. The fixed fairing fluctuating pressures

(P'FF-Y) varies linearly with the tunnel dynamicpressure except for the exponential vertical spikes.These spikes occur when there is supersonic flow

in the test section, and an unsteady normal shockexists in the arc sector/fixed fairing area causing a

local increase in p'. Thus the tunnel can

experience high fluctuating pressures even at very

low dynamic pressures.

The analysis of the acceleration data indicated asimilar relation between the local RMS

accelerations (_.) of a structural component andthe dynamic pressure. Again, using the fixedfairing as an example, figure 7 shows the fixed


AIAA-2001-1162

fairing RMS accelerations in the Y direction (_I.FF.Y)

along the centerlJne of the tunnel versus the

dynamic pressure. This relation varies linearlyexcept for the exponential vertical spikes

associated with supersonic conditions.

The data for the fluctuating pressures and RMS

accelerations were found to be mostlyindependent of temperature (and Reynolds No.).

There is some indication that the high-speeddiffuser response changes with pressure because

of the increase in diffuser gas mass.

If one assumes a white noise (broadband)

fluctuating pressure (p') is acting on an exposedarea of the structure (A), it will exert a force ontothe structure. It can be theorized that the structure

exposed to this force will then linearly respond at

its natural modes. This relation can be simplyexpressed as:

NTF Simplified Structural Dynamics Model

r

Area of - ,_exDosureJ Linear multimodal _ &FF

- '_,_1 multidegreefreedom _ _

Broadba_dd NTF model support _ aARC

excitation structure _ _'xx...... I

This simplified expression indicates that there is a

relation between p' and _. that is unique to each

specific structural component and this relationshipis characteristic of the component.

The data show a first order linear relation between

the local fluctuating pressure (p') and the localstructural acceleration (&) for all structuralcomponents. Again using the fixed fairing as an

example, figure 8 shows this relation.

From figure 8 the slope (A&/Ap') provides aparameter that is a measurement of the inversestiffness of the structure. This stiffness parameter

can then be determined and compared fordifferent structural components to identify areas of

relative weakness. Table 1 provides a summary ofthe stiffness parameter for structures of the NTFhigh-speed leg from measured data.

Structure, Direction

Downstream Support Frame, Y

Model Support Frame, YArc Sector Centerline, Y

Fixed Fairing Centerline, Y

High Speed Diffuser Lip, Z

StiffnessParameter

(in/s2/psi)

3,500

5,300

22,000

60,000

240,000Table 1 - NTF Structural Stiffness

This table shows that the most flexible and

responsive structure is the high speed diffuser

followed by the fixed fairing and arc sector.

Surprisingly, the MSF and DSSF are only about 3-4 times stiffer laterally than the arc sector, when

one can reasonably expect such support structureshould be significantly stiffer.

Analysis of Model Response

The RMS values of six components of the model

strain gauge balance (NF, SF, AF, PM, YM, RM)show a similar relation to the local fluctuating

pressure (P'Ts). The analysis shows that the NF,

SF, PM, and RM have a low response to P'TS, andthe YM and AF have a larger response. Figure 9

shows an example of this relation between the

RMS value of the YM to P'TS-

Frequency Domain AnalysisA frequency domain analysis was completed to

identify the modes, and their characteristics,associated with different structures. As expected

the amplitudes of each mode increased withincreasing dynamic pressure as shown in figures10 and 11.

Figure 10 shows the spectra of the arc sectorcenter Y direction. The dominant mode

frequencies are at 11-12Hz, 17-19Hz, 22-23Hz,34-42Hz and 70+Hz. Although not the largestmode the 34-42Hz mode shows the characteristics

of an aero-elastic mode where the frequency

decreases from 42Hz to 34Hz as the dynamicpressure increases. In a detailed analysis it wasdetermined that this mode changes frequency with

dynamic pressure but does not show evidence of

decreased damping. This mode behavior isexplained by aerodynamic loading of the elasticarc sector.

Figure 11 shows the spectra of the fixed fairingcenter Y direction. The dominant mode

frequencies are at 23Hz, 30-32Hz and at 70+Hz.

The 30-32Hz mode is by far the most dominantfrequency of the three modes. The smaller 23Hz

3

American Institute of Aeronautics and Astronautics

AIAA-2001-1162

mode is an arc sector mode transmitting from thearc sector due to contact between the fixed fairingand arc sector.

Summary

The analysis provides an understanding of the

dynamics problems at the NTF by first identifyingand understanding the behavior of the local

fluctuating pressures (p') as the excitation source

for the dynamics. The analysis also identified therelative stiffness of each structural component

along the high-speed leg and, based on spectraanalysis, identified the behavior and contributions

of each structural component to dynamics.

Based on this analysis, the reduction of lateral

dynamics response at the NTF can be

accomplished by reducing p' and/or increasing thestiffness of the model support structure in the load

path.

SOLUTIONS

It was established early that any proposed solution

could not change the tunnel flow lines; therefore

any efforts to minimize p' were not considered atthis time.

From the data analysis it was clear that changes

made in the model support structure (FF, ArcSector, MSF and DSSF) were the best way to

reduce the lateral dynamics. The solutions weredivided into two areas:

1) Increase support structure stiffness2) Change the structure boundary conditions

Increase Support Structure StiffnessTo increase the support structure stiffness (Arc

Sector, MSF and DSSF), large 0.75in aluminumshear plates were installed horizontally securing

together the MSF, outermost I-beams and DSSF.

Additionally, bulkheads with extending lowerbrackets were installed between the top I-beams. 6

Figures 12, 13, and 14 show simple graphical

representations of the shear plates (upper andlower), bulkheads and brackets.

Chanqe the Boundary ConditionsThere were two solutions for changing the

structure boundary conditions. One solution was toinstall wedges between the fixed fairing and the

adjacent tunnel wall, and the second was theinstallation of new active bearings.

The fixed fairing wedges are designed to fill the

gap between the fixed fairing and the tunnel floor.

By filling this gap the wedges relocate the lateralsupport points of the fixed fairing from the vertical

pin up to the tunnel floor. This change reducesthe fixed fairing's effective length from 160in to

125in as shown in figure 15, thereby increasing itsnatural frequency and reducing its displacement.

The installation of new active bearings was based

on the data analysis that indicated the top of thearc sector experienced large lateral accelerations

and deflections. Additionally data from a previous

report indicated tightening the existing bearingsreduced the yaw dynamics of the model. 1Therefore two pairs of active bearings (a front pair

and an aft pair) were installed just above the

existing bearings as shown in figure 15. Thesenew active bearings can be remotely controlled to

set up to 100001bs load against the arc sector andstill allow the arc sector to freely move.

Operationally these new active bearings wereloaded prior to tunnel operations and remainedloaded at all times.

RESULTS

Installing the shear plates, brackets, bulkheads

and the fixed faring wedges greatly reduced thedynamics of the tunnel model support area (MSF,DSSF, arc sector and fixed faring).

Figures 16 and 17 show the significant reduction

in the stiffness parameter before and after thestructural modifications for the fixed fairing and thearc sector. These results are summarized in Table

2 showing the increase in the tunnel model

support structure stiffness.

Structure

DSSF, Y

BeforeStiffness

Parameter

(in/s2/psi)3500

Affer

Stiffness

Parameter

(in/s2/psi)200

MSF, Y 5300 470

ARC Centerline, Y 22,000 1300

FF Centerline, Y 60,000 3700

Table 2 -Structural Stiffness (Before and Affer

The increased structural stiffness has resulted in a

significant reduction of the amplitudes of thefrequency spectra for each model supportstructure component. Figures 18 and 19 show this

reduction for the fixed faring and arc sector.

For the fixed fairing spectra shown in figure 18 the

previously dominating 30-32Hz mode has beensignificantly reduced and shifted to 45Hz.

4


AIAA-2001-1162

In the arc sector spectra shown in figure 19 there

remains four inherent modes at greatly reducedamplitudes. The first two modes around 8Hz and

12Hz are highly model/balance/sting dependent.The other two modes at 15-16Hz and 23Hz are

associated with the model support structure and

are inherent to the design of the arc sector.

With this significant decrease in the structuraldynamics, there is also a small decrease in the

model dynamics as shown in figure 20. This smallreduction indicates, as expected, that the model

dynamics are dominated by the

model/balance/sting modes. Future work inreducing the model dynamics will be to avoidmode coalescence between the

model/balance/sting and the modes inherent to the

model support structure.

Active Bearinqs

The use of active bearings showed that the Ydirection acceleration (and thereforedisplacement) at the top of arc sector was

considerably reduced. However the Y directionaccelerations of the centerline of the arc sector

showed no discernable change. This suggests

that the existing bearings are providing sufficientboundary conditions for the arc sector in their

current configuration. The active bearings dohowever, provide a limited ability to move the

frequencies of the arc sector modes. Furthertesting is required to fully document the results of

using the active bearings.

FUTURE IMPROVEMENTS

Several efforts are currently underway to furtherreduce model dynamics. The NTF has finishedfabrication of a new balance (NTF-116A) that is

4.3 times stiffer in pitch, 5.7 times stiffer in yawand 12 times stiffer in roll without sacrificing dataquality. 7 This balance should have significant

impact in reducing model dynamics by avoidingcoalescence with the modes inherent to the model

support structure.

The NTF is also working to develop new stings

using passive and active damping approaches toreduce the model dynamics and avoid any modalcoalescence.

CONCLUSIONS

Providing additional stiffness to the model supportstructure has significantly reduced the problems of

dynamics at the NTF. Efforts continue to furtherreduce the dynamics with future solutions focusing

on reducing the model dynamics.

There will always remain some fundamental

frequencies that are inherent to the supportstructure that cannot be eliminated. The design of

future model/balance/stings system must avoid the

frequencies of the support structure to preventcoalescence.

ACKNOWLEDGEMENT

The success of this work was made possiblethrough the efforts of several groups working

together for a common goal of improving the NTF.

The authors would like to especially recognize T.Arboneaux, P. Bauer, J. Barry, J. Bledsoe M.Chambers, M. Hilleren, L. Rash, and J. Zalarick fortheir hard work.

REFERENCES

1. Young, C.; Popernack, T.; and Gloss, B.:

National Transonic Facility Model and ModelSupport Vibration Problems, AIAA Paper No.90-1416, 1990

2. Buehrle, R. D.; Young, C. P., Jr.; Balakrishna,

S., and Kilgore, W. A.: ExperimentalInteractions Between Model Support Structure

and High Speed Research Model in theNational Transonic Facility, AIAA Paper No.

94-1623, 1994

3. Igoe, W. B.: Analysis of Fluctuating StaticPressure Measurements in the National

Transonic Facility, NASA TP 3475, March1996

4. Edwards, R. W.: National Transonic FacilityModel and Tunnel Vibrations, AIAA Paper No.97-0345, 1997

5. Balakrishna, S.: Data Analysis Report, NTF

Operational Data from Tests 100, 107 and111, Final Report - Part A, ViGYAN ReportR00-05, NASA Langley Research Center,

Contract NAS1-96014, Task RF01, February2000

6. Butler, D. H.: NTF Structural Modification

Proposal Based on NTF Operational Data

Analysis, Final Report - Part B, ViGYANReport R00-05, NASA Langley ResearchCenter, Contract NAS1-96014, Task RF01,

February 2000

7. Parker, P.: Cryogenic Balance Technology atthe National Transonic Facility, AIAA PaperNo. 2001-0758, 2001

5


AIAA-2001-1162

7000...._,.140.0 million

" 6000 ,_D"_ ".,..

50001 /'+" ;'..100.0 million

+oot ...._ ::::i _60.Omillion5"_ 10 "

_>. 000i ,_c__ 20"0 milli°n

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Mach Number

Figure 1 - NTF Operational Envelope

I-Beams Arc Sector

MSFDSSF

Figure 5 - Model Support Structure

Figure 2 - Test Section Support Structure

Vertical Constraint Lateral Constraint

Figure 3 - Test Section Support Pads

Figure 4 - Model Support Structure

04

035

_ o3

n

9o2,5E

_ 02

o15

E

.-- OO5

- TI,_3-01

," TIOO-02 _ "

TI00 03 ..... _ ....' T100-04

TI_ 05

, T100-06

• T100-07

TI00-08

• T100+09

o T100+10

o II00q2

T101J-14

o T100-16

- T100 21

, T100-2rJ

5LIJ I000 1500 2000 2500 31_7(.i

Dynamic Pressure, psi

Figure 6 - Fixed Fairing Fluctuating

Pressure vs. Dynamic Pressure

35OO

._=E>

t-O

---_ 12OO

09

_; 08lie

##6

_ 04LL

"O_ O2

._xI.L

0

2 • Tl1_}-01 1 ; ' _'<' "5;uperson c + _i;_8 TmO-o.I ' -° _ . ,tit

• +mo-o+l • + + il .,_lll" T100-04I° + _ _../ _ "I,

1 4 , T100-07 _" I ' .: ........ :._illi_._

[1O009 <l _ 7_:!:7:"!_:_:" I_l[ ll"o T0040] Jl+<i.,_+':_ .... _iJ_ +_+ TI00+12I lli_ :++_"Ill .,+,ili_" ,ioo+i+1 : illl :_ ili .t/.;>Lt_t+ TI00-16i - _ ';+ilri -I._'_;

TIO0._CI " * ; +i ..... "*'"++,o+++•+4+

+ i

5oo _coo _oo 2ooo 2_ 3000 3_

Dynamic Pressure, psf

Figure 7 - Fixed Fairing RMS Accelerationvs, Dynamic Pressure


AIAA-2001-1162

t--

O:>

==O

>-(/3

CE

16000

14000

12000

10000

8000

6000

4000

2000

00

--_:' Y' I1c, M=n_,7_, •

,m, ij., ! 1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45p'Y-FF, psi

Figure 8 - Fixed Fairing RMS accelerationsvs. Fluctuating Pressure

0.9

0.8

0.7

0.6

0.5

0.4

0.3!

021

0.1

0

":' r: P

ElalanceNTF113, S1S. ' , _', I_.'bo'l.l ,_*°

M=05-093. q=tS0O-2cjo0 p_ - =_ ',_IP_ 11_16_> ,.'0 !,

•++':, +',i+,+++__ gP ++';

//:J_;°*P_r _+"_:/'._. , " _ II . T1TT11111-[34Q3

/ ++ - ' z ; I " Tt + 1-06

• "< ...... I T111 07

"_ ' ' I ' T111-0B

"#"' '" ' T! 11-09

_° -" I Tl11-10

0 0.05 0.1 0.15 0.2 0.25

P'Y-TS,psi

Figure 9 - RMS Yaw Moment vs. FluctuatingPressure

.E 20C

_1800U-.£1600_1400

12008 1000< 800E

"E

•_ 4001200:

0 0 10 20 30 40 50 60 70 80 90

Frequency Hz

Figure 11 - Fixed Fairing Spectra forHigh and Low Dynamic Pressures

100

Bulkheads

Arc Sector

Upper Shear Plates

Brackets

DSSF

Figure 12 -Arc Sector/Support Structure

(Top)

Bulkhead Sector

c-O

.<sO

O3

<

8O

70

60

5O

40

3O

2O

10

10 20 30 40 50 60 70 80 90

Frequency Hz

Figure 10 - Arc Sector Spectra for High

and Low Dynamic Pressures

Shear PlateBrackets

Additional

Bearings

ExistingBearings

Figure 13 - Arc Sector/Support Structure

(Top)

100

7


AIAA-2001-1162

Lower Shear

Figure 14- Arc Sector/Support Structure

(Bottom)

MSF DSSF

ActiveBearings

60in

Wedges

Figure 15 - Arc Sector/Support Structure

(Sideview)

r-

u.

r

18000 a TIO0Z-14, BEFORE " M:0 2t , T=-25OF'_-20_80 _s,a

{ T131-000, ,a.FTEF_

16000 * TI00-20, BEFORE - • " .",''.,. " "

- T131-005, AFTER _'._:_",_'.-_-_** -" ,,',

14000 • T100-08 BEFORE I * ", "=_ l[_e_-'_ &

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8000 _'_" _!_ slope --E]0O0 ,r,sls,'ps,

"_ p'_ .... ..... After2000 '= ' : i i

005 01 015 02 0_ 03

p'Y-FF,psi

Figure 16- Fixed Fairing Results (Before

and After)

t-

5OOO

45O0

4O00

3500

3OO0

2500

2O0O

1500

1000

5O0

° _,_I_,BEFO_ _ ' ' /: TI31-O06, AFTER =: _=/-

- T100-/{} BEFORE _ " " - "

- T131.O05 AFTER _/*''* _ _,'" ._ J

< T131-O04 AFTER _,_ _'° &'L'-_._-._ " : t

@_t" o • "'_ - . " "

. tt'_"_'_' =;_m.t_'/_i_" Before

005 01 015 0"-' ON 03

p'Y-FF, psi

Figure 17 - Arc Sector Results (Before

and After)

140

120 - -

20

Before

After

i00 20 40 60 80 100 120 140 160

Frequency, HzFigure 18- Fixed Fairing Results (Before

and After)

8


AIAA-2001-1162

%d_

¢-.m

<,

600[ --I

i!

500 -

4OO

20O

100

Before !

!i

00 20 40 60 80 100 120 140 160

Frequency, Hz

Figure 19 - Arc Sector Results (Before and

After)

Pathfinder Model, 113C Balance

j

'°°°t- !_1400_ - - _ L_ Before

E 1200 _ - - _- ,lit

__looo_o 800

rr 600After

400 :I

2ool , l ;00 - 500 1000 1500 2000 2500

Dynamic Pressure, psf

Figure 20 - Model Yaw Moment Results

(Before and After)

30OO

9


w. a. kilgore, s. balakrishna, and d. h. butler nasa ... · w. a. kilgore, s. balakrishna, and d....

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