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FROMDistribution authorized to U.S. Gov't.agencies and their contractors;Administrative/Operational Use; JUN 1957.Other requests shall be referred to NavalAir Systems Command, Washington, DC 20362.

AUTHORITY

USNOSC ltr dtd 5 Sep 1969

THIS PAGE IS UNCLASSIFIED

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Hydro~naato LaboratryQaiorna Institute of Tecunoloy

Nesadena C..3if a'k WiA

RepwtI No. L-73.3 ApW*O"d hr

June 1957 Ihtsmks hawio

%MW4jsi VWoped4 iti b AkboratOCy pasit. th drt, of

ei~ft Of tA* dy=Aic coeffiints f l MI dsdCrIbig t orCO aMd

moment reactiew on a aubmwget bo4 movin~g in water,, Tibme coeffiq~eaU

'Uis the patltl 6wvivativs of moment (about, the yaw axis) vnd of

~ (in the horiscatl plirn "nd pewpWe-toual" to tko 1orituinsXi W4

with respect to~ veloeity ar ! acceleration 40faponants in specified utreo.

tioe. So long as the instlltanowus eagleb of attack e~amn and ac4*

Offects are absect, thin COefficiOU hV bsCOnstgnt V&1W23.v A complete

list of coefficients is given in We. WS) as wre defli2tione, sipn cork

ventious Simi fOrmulas for making the coefficients nondinen iorial * The

eight coefficients tabulated below awe thoue, pertiient to lateral trams.

latias and rotatiOmiabout tha W e4 forboiof vltian:

S oeffticient of Votga sameteivaiv

V,3 virtual oMeni of iniertia cofficient (angular acceleration)

N, cetficiuit, of static moment dervatt"

virtoisl twment of inertia coefficient (lateral acceleration)coafficient of rotary tone derivativer

Tf virtua inri coefficient (angulr aoceleration)

T,' co*eiset of static force derivative

yvlr+.Aawnetia coefficiatt (lateral accelraton~)

j wji.ere the £wia Udcates; that the coefftcienti are in dimmniolesIt is Mis purposo of the e intal prop&* widrtak-c'n &zt 'is

Utwrxtory to detemliw the nwmxe±ial valuss of the abom va#-nftti for

J r 2

the %sic Fier missile (Fin. 1). Because of the r~ired differences

in the Aperadmentl methods, however, the program was divided into two

parts. This report &&3 only with Part 1, and is restricted to the

following quantities z N -INIY-I TI#&Ate'~a miainr y v j~!' rM Vh iercmiaiN I -Nan d& Y uY'# .Ngiquantities wlbe thesi bject ofr V r Vanother report.

?he procedure used to obtain the dynamic mommn t coefficiento is based

on the uasuresmt of the saplitude ratio and pnse argl* o,. the model while

it is oscillated at a known frequency in the aov~ag atreaa of water. For

this purpoee the Angulr Dynamic Balance was used, in conjunction with the

High Speed Water Twmul at this Laboratoz7. A couplete discussion and des-

oription ot the equipent is Siven in Ret. 2,* bt a brief *=ary is given

here:

The balance can be reparded as consisting of several separate fimction.

a cponanta (Fig. 2). The first o- these is the platform - platform

drive asseably. This uimt consists of a platform to which can be imparted

a smal) angular oscillation of knowa,,amplituds and controllable frequen .

To it is copled the splUih-modol asusAbl which is free to rotate on ban

bearinp and which is pbovided with a Wtew sea to prevent leaWka. Of tunne

watir to the outsid. The 4cupling member is a calibrated torsion spring

of controllable stiffns. Two mirror-les system, one of vhich is carried

by the driving platform and the other br the spindle, enable the observer to

mearure the angular deflections of oach of thoso cononents with respect to

ground. By projecting the imap of a Pulsing ight Source (tri ed by

contacts on the drive assembly) the a ilier denec*tn at a gven in t

3

in the cyc) can be maurod. The oontactor body is rotated tbrough known

a lainor nt and the deflection taken. IV plotting those data and

by reproonting the observid motion in Fourer series, the aaplitwh and

phae angles of the wn ts motion of both rivr and espindle can be

deternind*

A "U-4ontand lateral foare balenoe, using bonded 9traIn gae,

seve as the center section of the nodel wwblyi This unit measures the

insntanoua i ie force acting an the model and relates it to the iitan-P

toous position of the driving platt o.

T1ICAL ARL=28

Up to this point the balance han been reprded as harmonic oscillator

which permits complete ananysi of the motions of both the driving platform

and of the spindle-odel. assmbly. Vy rwning the balance in air, and 1t

performing tUe necaswar" computations, the spinridlo-odel assembly is found

to possess a certain wmnt of inertia and little or no dwping with respect

to ground. S ilad yr the internal (lateral force) balance shon that the

center of ravity of the model "embly coincides very close3y with the axis

of angular oscillation, This fact is Indicated tV the abumo of &

measure riodic lateral for** in the pmenoe at the angtiw mtion.

Thm the' tunnel Is #0led with ater which Is moving relative to the

models the behavior of the modal is str l different. AnM4yhs of the

motions of the platform and isindle ohws that the spindle Osese a larger

Ness, coujled to ground Uwou* a eprini, and possessing a degree of dmping

with reepect to ground, This spring can either aid or oppoe the model dis-w

4 pla ient and the damping can be positive cr negative. In smlar fashion,

the interAl balane ehms that periodic side forcos are, present which behaveEI

as though caused by oscillating a larger macs, coupled to ground by a

spring, and also posessig smas amount of daping.

The relationship between the inetantmaou8 forces and nmments exerted

by the spindle on the model ard the resulting notion of the model can be

given in terms of the effective pisioal parameters of the system and the

aw lr ispllaeent P. oZ the models

I.0o + B F 0. K (2)

where the sign convention has been chosen to facilitate final expression of

the desired b'rrokamic coefficients. In these equations

a = total apparen mass of tht model

b lateral force duping coefficient of model with respect to ground

k - effective lateral spr'ing rate with respect to ground

I = total effectiv moment of inertia of model in vater

B = torsional damping moment coefficient of model

K = effective torsional spring rate with respect to ground

Thse apparent phyical properties of the model and model-spindle assembly

awe a result of the tq trrnazi forces acting on the mode. For the amall

oscillations exeouted by the modol and for the anf instantaneous angles of

attack, the t roknaic reaotl on can be regarded an coa vising the sum of

the indiv c al reactions arising from disp~.acment, valwity, and accelaratin

in the direction of motion, Letting N equal the total bydronamic moment

acting on the boy about the yaw axis and Y equal the total % o-n-c

lateral force, then

Mu+ N;16+V+ V + Hr + W (3)v vZ,

B\

5

and

we subscripts indicate partial divaivmes in accordance with the list

of syAbols on pp 13 and

U V u velocity of bo axis relative to fl,4d at infinitr.

r= angular velocity of model about the yaw axis.

v V.U o compoernt of velocity of bo4" axis relative to fluidin the direction perpendicular to the loreitudinalaxis and in the horiostal plane.

+v 2-U P component of acceleration of bt: relative to fluidin the direction pwpendioular to the longitudinaLaxis and in tA hrisontal plane.

Since the bod in question possess rotation sysmetry, the following terms

mest anish for oonstant u.

Nu, N~i.9 Iii, andYeU .

Equation (3) can therefore be rewritten in us

N 0,+ (Nr - N U) 00 - Nv U

a.(Y~ U) 0 Y. Up (14)' iP,+ r v o v 0

For tho present casen, the center of Oravity of the test boc r w very nearly

at the center of rotation, so that all side force reactiors were hydro ynaxic

in origin, r

(5)

4sop the mnt reaction on the spindle due to dampUW and displacement in

air was zero; therefore the only nonky&'oe'nAxic rsaction mu due to the

mvwmt of inertia of the spindlo.noel e.sal, or

N" bo " ii (6) &

so that by cubining trnM i (1), (2) and (4i), and substituting in()

and (6), the bhidotmanic coefficients -"n be i4emtified with the effective

physical quantities characUistic of vibrating mis s sms:

Y" b= (Y -Y U) k = -T Ur r VV(7) .5

If- j B= (N NU) 1 0. NvU

where U is the constant relative fluid velocity at a great dist ,e from

the botV

The axperiuental problem then reduces to one of finding the numerical

values of these effective p17sical constants, a, k, If, B, and K, t

whatever meara available. The 4naoc blanc and its w cessci'y instru-

mentation provide th, aeas. If the equations of motion for the system

are formulatcd, the following pair of equations is obtained:

+ Bs . Kd(P 02P) . o

(8)

I P + B Po + K Ks 4

where the syabol* are as defined on page 12 and as shown in, the equivalent

linear analogy of the moment measuring apparatus in Fig. 2. From these ex-

pressions can be obtained the stead-state solation

-1 2 K~K 2 i-s + -OW +m %iDB

K. Ks Ice5"I I

S, co(2 - 1) si 2"L(9)

which redosa to

V J V

wtmm term aoc i the negligible quantities Io and B. are dropped*

The real part of this equation identifies the in-phase component and

Wh :lmglna'y part identifies the quadrature component. From this ex-

pjwssim the effect.ve pbpical constanta I, B, and K, can be found,

&ceo?'dng to the method to be described later. Similarly, the expreision

Fr~% +'d2 + " 'o "B

LMK+ iWAMDW L(,.82) in(61 82)jJ L

V ~rst reduces to (1o)

.. coo (8 2) - ± sinl (68 +E coo] (4O52) sin 0

amd cam be solved to sive the amplitude and phase angle of the model in ter=

o inasurabis quantities. This infc-ation Is required for subsequent evalu-

&tion of the lateral forces mea wed by the internal strain gpo balance

a;omb1 xince, with the present setup, the internal balance actuafl ,

UArw Plaea angle ralative to the otion of the driving platform. The

4" . .. .. - , - - -- -.--- -, r : ,,' _ :¥.r/fL _ +,-- .- ," .; --- -' . . .. - " -.--- - , K

S

l teral force it related to the efective yyii c oontant, a b# and

k, -trough the sa-u.tae relation

vj co 2) +±±L j= [z .2.± A. iihere F iv the peak valve of the lateral tioe and 6 is the phue agle of

this force relative to the notion of the model. -

RLDUCTIOK AMD AMMIS CP U

kqUation 9 is sen to be a aoplex expression of the ftam

- . -+ d (12)a+ib

(

hich can be rationalized to yield

o da -,2- d2 and b--2+ (13)

where

b nB

plting a&- against w2(Fig. 3) and wmesuing the slope (t/1 ) and

y~ntrcpt(i + NK)two of the eqi4 valet #lyascal constants are found

Vi

it tam of the knmw spirdl spring rate* Salryif b is plotted

aTi t c (Fig. 4) the slopS of the reaulting 31 is equal to WK.

hA~dAO trstSint, of the latexWfI foras sc atloa (11) in Pigs. 5 and6

holade the quantities a. b , and kf. Uwe equivalent pyiwa parmeters

are then equated to the appropriate J~droc~amic cost fioientU acowrding to

lq. (7) and mmian siowaliseds as indicated in Ref. *1I The romultdiv,

c~zmc oefficients haVe beft Plotted aiwt RqnOld nuller in Fiat. 7

to 12. The use of Renolds nber as JW-siailituds parameter does not

iqply an a jwrif judgent that visoitr effects awe paramount, in deter.

miin the magnituds of than coefficient**. Indeed, for the experimental

SetUPaaibe foar this tu*, whoe the model size and flid viscosity

were fied, th Re~ndSd muber obmp. indicates only a caaW in velocity.

The aff ect of the presence of the modl suppot spindle ad asxnd e

shield was estimated tr the impg method. That.is, sup- ~Wtar rum we

auk with a dwW support isptindl shield on the side of the model opposite

the rel support spindl* and shield. This nuE did not touch the moel

The dita points taken with this lagc in place as indicated W solid sym.

bolae in the final crve It is assumed that the proseoe of the dMw

shield produces the sa m effect as the original fmctoa spindle and

spindle held hence the fina cm is tan to be dplaced an amount,

Wa to the difference beaon the origina, rums ad the iage rns, and

in the direction of the origi l rnms. In sam cases the reence of the

ime strut, hA no effect on the values of the coefficiets; for these no

imaed runs wo plotted.

10

83 WARY CF DBILTS

(1) Ni, virtual Mmt of e coffiient (Fig. 7):

This coefficient is neary indepenzI"t of wiocity, a predicted

kV perfect fluid theory, and sows little wr no se itivity to the presence

of the support .

(2) N , CoefOfciet of Static kmwt Derivative (ng. 8)l

This coefficient is a Measure of the aceent compoent due to

velocity ppeicular to the lO~gitudlual bocb' axis ror small amgles

of yaw thi ie sen to be identical to the static moment rate, d Cabk

and for this reaso it serves as a good check on the experimntal and cam-

putational proCOdur. The areeimnt betwen the values reported in Fig. 8

and those reported in Ref. (3) arevery good# even as to the effect of the

presence of the support spindl&.

(3) N,-R; o ment c 9 )t (i. ),

Actual1y a linear cobination of the coefficient of Rotary Momnt

Derivative and the Virtual Moment of Lurtia Coefficient (to lateral

acceleration), cannot be resolved withoA furtber experixents, which are

to be coduted at a later tii.

(4) Yi ', Virtual Inertia Coefficient (anglar acceleratiom) (Fig. 10):

This quantity is rrtgigibly mal in tw ;resece of the pondarable

value of YT"

(5) YV' Coefficient of Static Force Derivative (Fg, 11):

The analog to itea (2) above, can e idwtijfed with the slope of

the static lift coefficient curv 0 CL/da found tV conjntitna static

balance mas nt As with ite (2), the aVpement with previous static

11

aeoi~emust (Ref. 3) 1. @xcef.unt.

(6) t T1 Combined Lateral rorce Coeffilent (ig. 12)x

As with item 3) this coefficient aamot be ccmpletey

auaysed until the Becond phase of this iinestigation :is a td.

C -

LIIT CF $!D(

Th Wtion of the test boy it rsatricted to to plM of y b The IF-

bole used to describe thi sotion. and ,the asociated IVc rolU msa on

tho boV are in the greater part dentical to those rwo'nded in the ?echR.

nidal and Research Bulletin 1. 1-5 of the Society of lkawm Architects and

Narine En4inoers titled "Noen wlatwu* fcr Veating the Lotion of a Butmared

BAY(r throug a Fluid", These symbols re mrked with an asterisk(*) in the

follwing lst.

The coefficient. have been made non-dImn iosal in accorftw with p ;-

oecren outlined in the ab publication. DiensiaoleS qUantit* a&r

primed all other quantities are in the pomd-foot-second ystem of units.

A cross seotion area or *Aotl

Aos A:L* A 2 pe a L -~ au1 t action of bo*, suppait spindleand drive platfors

b d coefficient in Zq. (1)

B torsional deqing coefficient in Sq, (2)Be torsioal dauping rate of spindle seal

d dimeter of model

peak maprtude of lateral force in the horivntal plane

I= Ib + It total effective wm nt of inertia of model in water (Eq. (2))

1 b m of inertia of model

I apparent uwmt of inrtia of model assembly dueto by*odimio roactione

I s mment of inertia of model-support spindle

k efiaetivw sprir4 rate dm to tdron&a ic reactions (Eq* (1))

K eff.ev1w txsorial spring rate due to b4*ocrnaxic reactions (Eq.2)

Kd torsx=l sprinp rate of driv sW ing ich cowples thedriving platfow to the spindl -modol assembl

Ks torsimal spring rat of odl aupport spindle

'a apparont *Aix of moda~l d t ~o kwdrobr~z4udro~adlom (Ev f-1))

13

LM tr S!D (oont'd)

* JRydo!mlc moment acting an~ tba moftl

N, N ,oimnt on bo4r supplied to support spindle

* N' /' A4/U) coefoient of rotary mment dwiatv*v4Is

* I.'(3/2p 3) virtu&, =ot of ioa'Viz coeffcientN- Ii#' 12~ (angular acce eltioii}

S = , (1/ U) coefficient of satic moment derivative

* N,, NjI (1/24?) vitu a m ent Of inwtia Coeffricient(lateral acceleration)

ra aula vvlocity about the $w axi

* Vvelocity of bo axle relative to fluid at itftntt

u om0t of elacity of bomb axi relative to fluid

in the direotion pral l to the l t o iUna axis

v at of V lOof bor a3d relativ to- Mid

in the direcon perpendcular to the longitudinalaxis and in the hwlsatal plan

Y ~c~ai lateral force acting on the model

T a Y latutl faree on the nodal supplied tb the support spindls

*Y r ' (1/pAdu) coefficient of rotay force derivative

* , (1/M) virtual net a eoeffiosnt (angular aoceleration)

Y I? (2/2pAU) coefficient of static force derivative

T+ 1.'s (1/2pMd) virtual inertia, coefficient (3.atweral acceleration)

Pot P3. 02 ag-lar dIsp LA-mnt about the ymw ai of the model,spindle, and &riv p~Latfor

$o' 8is 82 phan costant of the motion of the model, spindle, anddriw platform

a f ph"* angl of the Iate1 force relative to model motion

P -s2 dcnsir or =4aIp

asz 2"rf eircu~i frequsnty; f = frequency in cycles pr owand

LIST (W REFNWMKS

1. Kicewiuk, Ta", "Dyramic Coefficienits of the Uk-13 Tmreda'6 ,Clifonia nstitut* of Tecbmoo~, VI oc~aics LaboratorYReport No* E -, 12*200 April 1957

2. St&U'ksap Jobn Asp "Manurment- of DyrAmic Coefficients OfEflipaids", California Institute of Technoqo amodnmicaLabot'to:7 BapWI oat, z. - 35.149 Septesbs 1956

3* Xeren Re 1., "Sttic Foce Coefficients of the BasicFirmer 1wissile in fully Afetted Flow", California Instituteof Techwooy, Uydr*muics Laboratory. Report 9 73.2,Jura 11957.

I _ __ _ -o

r-

C9C

Fig. I Important geometrical dimensions of the Basic Finner missile.

0 0

I K

IIII

L0000 b n f --

DRIVING PLATFORM SUPPORT SPINDLE MOOEt.

K5 TORSIONAL SPRiNG RATE 1Ff SPiNDC/3 ./J D SV4.ACEMENT OF PLATFORM, SPINDLE, MODELB$~ TORSIONAL DAMPING RATE OF SPIND)LE SEAL A,* A, A0 iWO ,u~M AMPLITUJDE OF PLATFORM, SPINDOLEMCOELIs$ MOMENT OF INERTIA OF SuPPORT SPINDLE hrS, & o Ft CONSTANT OF PLATFORM, SPINDLEMOOELI b MOMENT OF INERTIA OF MODELIl APPARENT MOMENT OF lNERTIA -A, SIN (wt-6,1X EFFECTiiE TORS40NAL SPRING RATE OuE 10 -A SIN (wit -hi

HY0PODfNAMiC REACTIONS LoXg$Nit-o8 EFFECTIVE TORSION DAMPING RATE DuE To

,H-YRODYNAMIC REACTIONS.

Fig. 2 - Schematic diagram of dynamic blance driving-platform nti

* moment measuring components.

-0 '4 Mh C . 45-t' FO %'S

40

FeC

10 20 0 40 50 bO60 4

SQUIAE OF FRECKUENCY IN

Fig. 3 -Moment data reduction plot: In-phase component(real part Eq. 9).

7 .4 j LEGEND JI_'0 NO IMAGE SHIELD,9-FINS-

0*WITH IMAGE SH-IELED. 90- FINS1)( NO IMAGE SH4IELD, 45- FINS

t~10

.~t2

FRE~uEPNCYv N(yCLSECOND,

Fig, 4 -Moment data reduction plot, Quadrature component(imaginary part Eq. )

0!0-, , vkE0iTv 30 P5i __ _

-.. - _ _ - -. ,: - , - ... ........-.. 4

IEI

0 -

E* _____20

o - O MAT 5MAGILD, 90" FN5_0 W AGE h40LD, 450 FiN5

0 10 20 30 40 o0 90 100

SQUAI OF FREQUENCY IN I

Fig. 5 - Lateral force data reduction plot: In-phase component(real part Eq. H). -

20 -

1-. 10.zLL. 0

70-

6 0z

0t- 50

30~SOLID YWIOL INDICATES IMAGE RUN30 DSED. LI1N INDICATES VLUE OF COEFFICIENT

I-- 0 CORRECTED FOR SUPPORT SPINDLE SHIELO.IX

> 0

12 3 4 5 6 7 a 9 10

REYNOLDS NUMBER x 10~

Fig. 7 *Virtual moment of in~ertia coefficient (angular acceleration)as a function of Reynolds number.

Ujj

20.-

0

0 10___ __ _

z SOLID SYt, 01. IND ICATES AME RUN

WU DAO, J INDICATES VALUE OF COEFICIENTCOF "C; SUPORTSPP',X'E SHIELD

Fi.8 Cefcefofsaimoetdrvtvasafunction of

Reynods nmber

24

200z

LL

w 'w

140--

Z 120

z40

2o-O 1,, .. .

2 - 4; 5 6 7 S 9 10

REYNOLDS NUM8ER x I0 "

Fig. 9 - Combined moment coefficient as a function of Reynolds number.

I- o

S 60-_

4 0

0 0 0

-20

21 a 3 4 5 6 7 P 9 r)REYTJOLD NIMKR x 10'4

Fig. 1 - Virtual inertia coefficient (angular acceleration) as afunction of Reynolds number.

r 'a -- -- - - -- -

I ' '~24

14-

12

3 4 5 6 '7 8 9 10REYNOLDS NUMBER x 10-4

Fig. 11 - Cuefficient of static force derivative as a functionof Reynolds number.

I

100 . . ..

90I-

. 0

0, 60

40

Z 30-1

20

tOI

2 3 4 5 6 7 8 9 10

REYNOLDS NUMBER x I0"'

Fig. 12 - Combined lateral force ccefficient as a functionof Reynolds number.

DISTRIBUTION LIST

Copy Noj

1-4 Chief, Bureau of Ordnance, Navy Dept., Washington I, D0 C.Attn: Code ReO-3

5S-8 Chief, Bureau at Or~ance, Navy Dept., Washington 25, D. C.Attn. Code iU. .

9-10 Chief, Bureau of Ordnance, Navy Dept., Washington 25, D. C.Attn: Code Ad3

11-13 Ciief, Bureau of Aeronautics. Navy Dept., Washington Z5, D. C.

14-18 Chief, Bureau of Ships, Navy Dept., Washington 23, D. C.

19-21 Chief. Office of Naval Research, Navy Dept.. Washington 25, D. C.Attn: Code 438

zz Commanding Officer, Cffice of Naval Research Branch Office,1030 East Green Street, Pasadena 1, California

23-24 Commanding Officer and Director, David Taylor Model Basin,Washington 7, D. C.

25-26 Commanding Officer. U. S. Naval Underwater Ordnance Station,Newport, Rhode Island

27-28 Commander, U. S. Naval Crdnance Laboratory. White (bk,Silver Spring, Maryland

29-30 Commander, U. S. Naval Ordnance Test Station, Pasadena,California

31 Commander, U. S. Naval Ordnance Test Station, China Lake,California

32 Director, Experimental Towing Tank, Stevens Institute ofTechnology, via: Bureau of Aeronautics Representativec/o Bendix Aviation Corp., Eclipse-Pioneer Division,Teterboro. New Jersey

33 DMsector, Ordnance Research Laboratory, Pennsylvanla StateUniversity, University Park, Pennsylvania

34 Alden Hydraulic Laboratory, Worcester Polytechnic Institute,Worcester. Mass.. via: Inspector of Naval Material,495 Summer Street, Boston 10, Mass.

35 -36 Librarian, U. S. Nava! Postgraduate School, Monterey, Calif.

DISTRIBUTION LZST (cont'd)

~No.

37-46 British Joint Services Mission, Navy Staff, via: Chief, Bureauof Ordnance, Navy Dept., Washington 25, D. C.,Attn: Code Ads

47-49 Commander, U. S. Naval Proving Ground, Dahigren, Virginia

S0-S1 National Advisory Committee for Aeronautics, Langley MemorialAeronautical Laboratory, Langley Field, Virginia

52 Natioial Advisory Committee for Aeronautics, Lewis Fligh.Propulsion Lab., Cleveland Airport, Cleveland, Ohio

53 Director, National Advisory Committee for Aeronautics,151Z H Street, N. W., Washington 25, D. C.

S4 Director, Nattona. , tisory Committee for Aeronautics, AmesLaboratory, Moffett Field, California

55-S6 Commander, Air Research and Development Command, PostOffice Box 1395, Baltimore 3, Maryland

57 ASTIA Reference Center, Technical Information Division,Library of Congress, Washington 25, D. C.

SB-63 Director, Armed Services Technical Information Agency,Documents Service Center, Knott Building, Dayton 2,Ohio. Attn: DSC-SA