14);

*,4 ,

TECHNICAL REPORT ARBRL-TR-02521

(Supersedes IMR No. 771)

Cv") LIQUID PAYLOAD ROLL MOMENT INDUCED BY A

SPINNING AND CONING PROJECTILE.4

Charles H. Murphy

September 1983

US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMANDBALLISTIC RESEARCH LABORATORY

ABERDEEN PROVING GRCUND, MARYLAND

Approved for publc volems distrl•ution uanilmted.

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* E

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Detroy this report when it is no lne eddDo. altrtrn-it to the originator.

, Aditional copies of this report may be obtainedfrom the National Technical Information Service,U.S. Department of Commerce, Springfield, Virginia

22161,

Thbe findings in this report are not to be construed asan official Departmont of the Army Position, unlessso desi3 nated by other authorized docMinets.

lASd "w7 r~Ne or 'Uiafm~wtremw nass in thi r~e

am t4 I.w f y" rtlP dt

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4. TITLE (And SubJdo) S. TYPE OF REPORT & PERIOD COVERED

LIQUID PAYLOAD ROLL MOMENT INDUCED BY A FinalSPINNING AND CONING PROJECTILE 6. PERFORMING ORG. REPORT NUMBER

7. AUTHONIO() . CONTRACT OR GRANT NUMBE-R(s)

Charles H. Murphy

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT PROJECT, TASK

US Army Ballistic Research Laboratory AREA & WORK UNL, JMBERS

ATTN: DRDAR-BLLAberdeen Proving Ground, Maryland 21005 RDT&E 1L161102AH43

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IS& SUIOLZ.MENTARY MOTES

This report supersedes IMR-771, dated March 1983.

It. KEY WORDS (Cmallneso foeo eWds it mwos~ey Ad Identity by block numlbor)

Liquid Payload Liquid-Filled ProjectileLiquid Roll Moment Spinning ProjectileLiquid Side Moment

a A•IAcr (:h,.... - N , - . (bja)

The linear theory for a coning and spinning liquid payload has been ex-tended to the calculation of a quadratic roll moment. The resulting quadraticroll moment coefficient is shown to be exactly the negative of the linear sidemoment coefficient for constant amplitude coning motion. This relationshipbetween side moment and roll moment has been observed for projectiles withmoving solid components and has encouraging experimental support for liquidpayloads.

W Palo160OWOFI10V ISOM4.ETg UNCLASSIFIEDSECUNTY CLASSI PICATION OF TNtS PAGE (.Nb D&ta "1e0ed)

•',(••:,•, ,. •., •, . . _• .. ", ',. . .- ,=.•°,.•.. . .. .. . .°. ..- -.. .......... .... . . . . . .... .... . ... . ..... ... ....V~r - ' > :'. * P * * , ~

TABLE OF CONTENTS

Page

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . 5

I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . 7

II. LIQUID ROLL MOMENT . . . . . . . . . . . . . . . . . . . . . . . . 9

III. LIQUID ROLL LOMENT. O . .................... 12

Table 1. Experimental Values of ..... ... 13

IV. DISCUSSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

V .SUMMARY . . . . . . . . . . . . 0 * * 0 0 0 . . . 0 4 * * 6 19iV'REFERENCES" .".. .. .. .. .. .. .. .. .. .. .. .. . .. 26

APPENDIX A . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

APPENDIX B . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

APPENDIX C . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . 43

DISTRIBUTION LIST . . . . . . . . . . . . . . . . . . . . . . . . 47

Aooess1on ForUTIS ORA&IDTIc TABUnarnnoun•od2 T111•: a to__

Dist ributi 1on/

Availability Codos 40i Ava.l and/or'

Dist spoaJ.o

3 PREVIOUS PAGEIS BLANK

LIST OF FIGURES

Figure Pg

1 CLRN From Gyroscope Tests (Ref. 15), c/a 4.291 . . . . . . . . 20

2 CLSM vs T for Re = 106, c - 0, and the D'Amtco-MillerParameters: c/a a 4.291, f - 1 . * . . . * w . * . . . . . 21

3 CL vs T for Re a 105, C = 0, and the D'Amico-MillerParameters: c/a = 4.291, f = 1 . . . . . . . . .. .. .. . . 22

4 svS T for Re 104 * - 0, and the D'Amico-MillerParameters: c/a -4.291, f = 1 . .0. . . .* . . # .. . . .. 23

5 CLs vs r for Re - 103, c = 0, and the D'Amico-MillerParameters: c/a - 4.291, f = 1 . . . . . 0 . .. *. . .. .. 24

6 CLSN VS T for Re = 102, = 0, and the D'Amco-M°llerParameters: c/a 4.291, f - 1 . . . . . .... . 25

5PRVIOUS PAGEIS ILANK

5

4

I. INTRODUCTION

In 1958 Karpov and Bradley1 observed short flights of shell, accompaniedby substantial reductions In spin rates. An eleven percent range reduction,for example, was accompanied by a fifty percent reduction in spin over thirtyseconds. Later tests showed that the spin loss occurs after the coning motionexceeds ten degrees and is not the cause of growth in the coning motion. Itwas shown2- 3 in 1977 that this coning growth and spin decay could be explainedby the motion of loose internal components. It should be emphasized that inall monitored flights of projectiles with large coning motion and no looseinternal parts, no large despin moments have been observed. Thus, thecombined presence of a large coning motion and a large despin moment issymptomatic of a loose internal component.

In the summer of 1974, a 155mm shell with a liquid payload was fired with

a yawsonde telemetry unit 4 and this projectile developed a coning motion withamplitude in excess of forty degrees. 5 When the coning motion exceeded fortydegrees, the telemetry record showed a very rapid despin of forty percent Inless than five seconds. Since 1974, six more observations of liquid-filledl5Swmi projectiles with large coning motions and large despin rates have beenmade. 6 -8 In addition, thirty-six observations have been made of large coning

1. B.O. Xarpov and J.W. Bradley, "A Studty of Causee of Short Ranges of the8" T31? Shells" ERL Report 1049s May 1958 (AD) 377548).

2. C.M. Murphy, *Influence of Moving Internal Parts on AngZlar Motion ofSpinning Projectile," Journal of Guidance and Control, Vol. 1, March-April 1978, pp. 117-1*2. Se also BRL-MI?-2731, February 1977(AD A037338).

3. W.0. Soper, "Projectile Inatability Produced by rnternal Friction," Arouraml, Vol. 15, Jan 1978, pp. 8-11.

4. W.H. Mer-gen, "Meastremente of the Dynamical Behavior of ProjectilesOver &ung Flight Path.," Jouma ot Spaeearaft and Rockete Vol. 8, April

j 1 ,pp. 3.0-385. See also BRL MR-2"O78 N nne P97AD 717002).

5. V.P. D'Amioo, V. Oakay, W.N. Clay, "Flight Teets of the 255rm XM68?Binary Pro•jctie and Adeociated Deein Modification Prior to the NioleteWinter Teet 1719,"BRI.-IIR-27484 May 197? (AD B019969).

6. V.P. D'PAkio, W.H. C'lay, and A. Mark "7awaonde Data for *57-TypeProjetilee with Application to ..Rpi Spin Decwa and Steaartaon-TypeSpin-Up Isetabilitie," ARBRL-R-03027, June 1-980 (AD A089646).

7. M.P. D'"Amno, M.H. Clay, and A. MarkA "Diagnoatic Teats for Wiok-TypePayloade ad High-oVieocity Liquidas. ARBRL-MR-042,91 April 1979(AD A072812).

5. V.P. D'Amlcoo and M. H. Clay, "igh Viscosity Liquid Payload Yauaonde Datafor Smzll Launch Yas," AR5RL-4-3089, June 1980 (AD A088412).

7A. PREVIOUJS-PAGE

IS BLANK

° . .. . . .. . • -. . ". 4 ,'

motions and despin moments for projectiles with payloads of liquid-solidmixtures; namely, twenty-six XN61 projectiles containing liquid plus cottonwicks?*9wll and ten XH825 projectiles containing liquid plus feltwedges. 12"14 Indeed, all observed flights of projectiles with liquid orliquid-solid payloads that performed large coning motion exhibited largelosses in spin.

This characteristic association of a large despin moment with a largeconing motion for a projectile with a moving payload can become a diagnostictool. Miller 1 s suggested using this tool to determine the existence of smallamplitude unstable liquid-induced side moments. Spin measurements made duringconing motion on a gyroscope predicted flight pitch instabilities caused byvery viscous liquids, and these were observed in flight. 16 The original M687flight instability was caused by a large external Magnus moment, even thoughthe liquid caused a rapid despin. Thus the assumed connection between largeconing motion roll moments and small coning motion side moments may have beenfortuitous for the case of very viscous payloads.

The linear liquid-induced side moment was first computed by Stewartson 1 7

for an inviscid liquid payload by use of eigenfrequencies determined by the

9. V.P. D'ýAmioos "Early Fli~ght Expereionoes With the XN7SZ,0" ORL-M-Z781j,SeptwIbev 1877 (AD 80207?5L).

10. W.P. D#Anniooa "M fld rests of the XWOJ: First Di~agnoetiao Teat T m BRL-ME-27928 September, 18?? (AD B0249?,).

11. V.P. D'Amtoo, "Field Teast of the XArs 1: Seooud Diagnostio Tests" A•RRL-MR-0806s, JZamx¶ 1078 (AD B025305L).

1,. V.P. D'Awioo, .AeroZobaetio Testing of the XN&25 Projeotile: Phaee 1.0ARBRE.-E-O*011,, ,ah 1979 (AD B037880).

25. V.P. D'Antoos #Aeroballitia restinwg of the 0~825 Pzrojeottile: Phaee r1"ARBRL-N-03075. Joum.d 1981 (AD A090808).

14. W.P. D'Aiuutoo #Aerobatletio ?#st~ing of the XV825 Pirojeotileor Phase rII,Hi~gh Nusale Vetooity and High Qua4wpt El~evation,,* ARBRL-04-05 196,0SptebUr 198J (AD BO68U512).

4 1is. M.C. NiteZ Law Fugiht I'setab tt" of spinning Projeotile. Hav~ing ROI-rigid Pavtoid.,* Wow',l of Guiddunoo ControL,, anw Eknamia, Vol* 6ANarah-4PaP~? 19sa, pp. 151-157.

16* V.P. DAmtiOO and N.C. Miller, "fl~ht Inatabilty ProdUced by 4 RapidlySpiming, Highly Visousa Liquid,v Joumwnpt of Saeoreft and Roaketa, Vol.180 Jan-Feb 1790, pp. 65-64.

1?, X. Stmton, won the Stablity of a Spinning 8op Containing Lquds*Josw~ut Of Fluid Neahanioa Vol. 5, part d, September 2959,0 pp. 577-950.

fineness ratio of the cylindrical container. Wedemeyer 18 introduced boundarylayers on the walls of the container and was able to determine viscous correc-tions for Stewartson's eigenfrequencies, which could then be used in Stewart-son's side moment calculation. Murphy 19 then completed the linear boundarylayer theory by including all pressure and wall shear contributions to theliquid-induced side moment. The Stewartson-Wedemeyer etgenvalue calculationshave been improved at low Reynolds number by Kitchens et a1 20 through thereplacement of the cylindrical wall boundaqy approximation by a linearizedNavier-Stokes approach. NextGerber et al Z12Z extended this linearized NStechnique to compute better side moment coefficients for Reynolds numbers lessthan 10,000.

The only theoretical work on liquid-Induced roll moments was done byVaughn 23 in 1978. Although fair agreement with Miller's data was obtained,the work was. m.arred by samie hard-to-justify algebraic steps. It is thepurpose of this report to compute the roll moment associated with the linearperturbation flow of Reference 19.

II. LIQUID ROLL [IWENT

The pitching and yawing motion of a symmetric spinning projectile can berepresented as the sum of two rotating exponentially growing two-dimensionalvectors. In terms of complex variables, this relationship assumes the form24

18. R.H. Ardeeyjer, MVisoous Correotion to Ste•wrtaon's Stability Criterions,"DRD-R--158j Jwe 1966 (AD 489887).

19, C.H. NMwhs "An gular Notion of a Spinning Projectile with a VisoousLiquid Payoad• , Nemorandiv Report ARBRL-MR-03194, Au•uet 1982 (ADA18876). See also Jourrnal of Guidance, Control. and Dynamnioe, VoL. 6,Ju/,y-Auguet 1853, pp. 280-868.

20. C.Y. Xit•hene, Jr., N. Gerber, and R. Sidney, "Oaoillatione of a Liquidin a Rotating Cylinder: Solid ody Rotation," M TeahnioaL Report ARNR-70R-•*081, Jaw. 1978 (AD A05?760).

2e. N. Gerbers R. Sedney, and J.M. Bartos, "Preseue mo•wnt on a Liquid-Fitted P•ojaotit.: Solid Body Rotation,* BRE TeohnioaZ Report ARBRL-TR-024*2, October 298* (AD A120567).

82. N. Gerber and R. Sednty, Nomnnt on a Liquid-Fitl•ed •pinning and Mutating

ProjeottiL: Solid Body Rotation," BRL 7eohniaazl Roport ARORL-TR-O027O,SFebwuar 1985 (AD A226332).

23. M.R. Vaughn, "Flight Dynawio Instabitities of Fluid-Filled Projeotilee,"Sandia La~bor'atori.es, Atuiquorque, Nil, SAND ?8-02999., June 2978.

*4. C.H. Marphy, "Free Flight Motion of Syetrio Ni.,ilea," BRL-R-1216,July 1"S (AD 44*76?).

9.-j;

% ,%:•%~ t%~.2s ~

+ 4 $,•" i K 1 e- + Ke (2.1)

where In (Kj/Kjo) -, Tjl;IIt

#j a#j + T ;t and

.4. ~ is the roll rate.

The transverse moment exerted by a liquid payload was expressed inReference 19* in the following form.

L?+'M1I NIKl + T2C jq2 K2eJ (2.2)

.4.' where

mL a mass of liquid in fully filled container

and

a a maximum radius of liquid container.

For linear fluid motion, CL1 should depend on t%, cl, time, Reynolds number,

fill ratio, shape of cavity, and direction of spin. A similar remark appliesto CUI2o

The CL a are complex quantities whose imaginary parts represent in-plane

moments causing rotation in the plane of exp (i.j) and whose real partsrepresent side moments causing rotation out of the plane of exp (i.1). Thus,CULX with its dependence on the direction of spin can be written as

C y iCl 4 (2.3)J j NJ1

wihere C" , CLIN , are real and Y I

4..'." •S im lat aw fn m Ref.w . •19 ov used tU•WImt t-. mpoit. m ermtafor' Refomww 19 is gt~. ia Appmudft A.

10

. • + • o - •+ . i+ -. -. -. -+ . • ,• ,

The liquid roll moment can now be defined in a similar way. Symmetryconsiderations imply an even dependence on coning amplitudes and an odddependence on spin rate.

aL mIL a, $131 ' T1 K12 CLaM1 + T2 K22 CLRM21 (2.4)

m where CLRM are functions of Reynolds number, fill ratio, shape of cavity, and

time. In addition, for j - I and 2, the CLRMj may be functions of Tj, ej and

K 2. C is the roll moment coefficient exerted by the liquid during spin-mLRMeup and is zero when steady state is reached. For the steady-state coningmotion considered in the Stewartson-Wedemeyer theory (K2 - 0), only one roll

* moment coefficient is present and we will omit the subscript on T, and CLRM:

N: U La ; I- K 1 CuM. (2.5)

Throughout this report, we will consider only roll moments that can beapproximated by the one-term Eq. (2.5) rather than thq more complete three-

. term Eq. (2.4) and will consider only positive spin, #.

"The roll rate of a projectile in flight is controlled by the sum of theinternal liquid roll moment and the external aerodynamic roll dampingmoment. For pure coning motion

V D2 2 2 K 2 C 26

For the liquid payloads, the large increase in roll moment usually occurssuddenly when the coning motion amplitude exceeds 300. On the other hand, theliquid/solid X9761/X4825 payloads produce large roll moments at a much lowerconing amplitude of 10-151. For all payloads, the coning motion increasesrapidly to a steady-state amplitude of 40-450.

Since definitions (2.1, 2.2, 2.4) were for small coning angles, moreprecise expressions are required for angles in excess of ten degrees. The yawplane is the plane containing the missile axis and the velocity vector. Theangle between the two vectors is the total angle of attack, at. The amplitude

of the complex variable ' is sin a and the argument of * is the phase angle

of the yaw plane about the missile axis. For the pure coning ,•tion of Eq.(2.5)6 the exact definition of K, is

K, "I •sin at (2.7)

*. .

S,, ,.,,-

Unfortunately, the yawsonde data for the XM761/XM825 flights were toopoor to yield numerical values of the liquid roll moment coefficient. Sevenflights of the M687 payloads did have sufficiently good data and their liquidroll moment values could be computed from Eq. (2.6). The change in roll

acceleration at high cone angles, A *, can be easily obtained from the yaw-sonde data. From Eq. (2.6)

S(a 2/I x ) 2 CILRM (2.8)

Table I gives the seven M687 values of the roll mowient coefficient. For"very low Reynolds number, a value of about -. 04 seems representative while ahigh Reynolds number value of -. 01 is indicated. Since Round 7254 was notcompletely filled, it is an isolated value. Miller's gyroscope data's isgiven in Figure 1 and indicates a range of -. 001 to -. 65 for CLRM. Two

recently obtained values of roll moment for eight-inch shell 25 are also

given in Table 1. Since the flight data were poorly resolved during spin-down, the liquid roll moment coefficients of Table 1 should be consideredestimates with accuracies no better than 30%.

MI1. LIQUID ROLL MOINT THEORY

Two coordinate systems will be used in this report: the nonrolling

aeroballistic XYZ system whose I-axis is fixed along the missile's axis ofsymetry and the inertial XYZ system whose X-axis is tangent to the trajectoryat time zero.* Both coordinate systems have origins at the center of thecylindrical payload cavity, which is assumed to be at the center of mass ofthe projectile. Location in the cavity can be specified it the aeroballistic

system by the cylindrical coordinates *, I, I and in the inertial system by x.

r, 0. The boundary of the cavity is given by X a cand F a awhere 2c is theheight of the cavity. Linear relations between cylindrical coordinates in thetwo coordinate system% were derived in kference 19:

x a x - r K, cos ( -) (3.1)

F r + x K cos (41 - 0) (3.2)

I*# trpat.ioiu oar. he wtte appr.oazteJ by, a otvaight iuse for a wamber ofp.$od of o~w dmgutwanion~i.

23. V. P. D'4Avio and R. 4. YatamnMoil "*Ymeooe Tet of Me &-J,,oh VS??5IwM PMXW e: Pun low am *.WYu16a Rapoft in pepamaio".

12

m CA C4N N -v4j 4 0

0 0 N n n I,.00 In C 0 0m

IA * * 0

S S S 5 * * S S

tv k * %n 00 a

0 0

US *k 0 b -

~13

- e + (x/r) K1 sin (1" e) (3.3)Since the gradient vector for the surface = f (x, r, 0) is a vector inthe direction of increasing x, the unit vector in the direction of increasingITM 0ý is

""ý" r r' / af + k Tr-) + (73-)/ - (ex, er, e (3.4)

In a similar way, unit vectors in the direction of increasing r and 'e can becomputed in inertia system cylindrical coordinates. These unit vectors alongthe aeroballistic cylindrical axes are:

(ex, eý.r, ej.0)

(1,i - K, cos ( 1 " e), - K1 sin ( 1 " e)) (3.5)

e= (e-x, e.r, e-e)

=(K cos ( 1 " o), 1,K1 (x/r) sin )(3.6)

.•,• =(e~x, e~rs o•e)4 ~i

(3.7)

(Kj in f, e)$ ,- (x/r) sin (*- 1),

'A Eqs. (3.5 - 3.7) are correct to terms of order K12

Since compressive viscous shear is neglected in boundary layer flows,the components of the shear tensor in cylindr'cal coordinates have thevery simple fom 26

Txx Trr Tee a -p (3.8)

0 r (3

Sxr Trx + Lx r3

28. N. SoJiZohting,, "Boundary Layera nlow," MoG•rw-MiZ Book Company, NewYoz*, 1860, P. 54,

14

;;tg*~ ...~

2 ave 1xe 't' j = + 1

-x 0e Lax r a 5 (3.10)A r

I8 [ra(V+/r) aVr]

ar r r (3.11)

The stress vector on an area element of the interior lateral cylindricalsurface of the liquid container is

dit= (dF , dFtnr dFzo) (3.12)

where

dF = [e=-x Txx + e-r Trx + e-e x a dx d•

r=a

dFr [ e-. rxr + er Trr + e eri aTrx rrr a

dF = [er'x Txe + err Tre + ero sel~ a d^ d'rua

The corresponding roll moment differential is

(3.13). { Tre + [cxr sin (1 " +O) +rxeCos 01 e)] K1 } a2 d-dW

, For the boundary layer on the lateral wall, only derivatives normal to theA wall need be considered.

aV~Txr ar (3.14)

Ki ' xe •0 (3.15)

15

P r(V /r) (.6

"rr

tre = ii r -v-- (3.16)

The roll moment differential on the forward and rear end walls (j = 1, 2)can be computed in a similar manner.

dMxej X x Exr (x/r) sin (*,-e) (3.17)

+-recos (ý,-e)] K, r2 dr deX=tc

On the endwall only derivatives with respect to x are important.

Axr a ax (3.18)S~av0

8= -(3.19)

!re = 0' (3.20)

We will now compute the quadratic liquid roll moment under the restric-tive assumption that only linear fluid mechanics terms need be considered. InReference 19, the velocity components were expanded in terms of perturbationvelocities that were linear in K.

V " R {us esf" i0 }(a;) (3.21)

Vr aR vs esO - }(a;) (3.22)

V. a ;+ R {w5 e5 , (a;) (3.23)

where

Sm (C +)

16

i!U%,,

and where us, vs, ws are functions of r and x and are linear in K. Eqs.

(3.21-3.23) give the velocity components in terms of x, r, 0. In computingthe linear liquid side and in-plane moments, x, r, 0 could simply be replaced

by x, r, 8. The liquid roll moment, however, is a quadratic function of K1and Eqs. (3.1-3.3) must be used. A useful approximation for this change ofvariables is

h (r,x) e' = h (F,i) et)ý + (r -)

h h 'ý)(x - i) - ih (r,i) (ax- ( )] e'(2

The roll moment induced by the lateral wall can now be computed byintegrating Eq. (3.13) with the aid of Eqs. (3.14-3.16, 3.21-3.24).

C (T m a2 2 K12) 1 c 2dML-C Jo (3.25)

(2Tr)' cc R{ If '1 di

-cr=aX=X

a 2w a2 wauwhere f ax - a 2 - - ia..s. Re-1 (3.26)

K =K1 e (3.27)

Similarly, Eq. (3.17) can be integrated to yield

CLRMe1 CLRe (1/2) CLRMe

(3.28)

.(2,ac)-, RI 2e ^K'1 r0 •r-nr

XM{

whre f a 2w s a Rea(.9where fe -r' - 7 +c~ (wT + iv5 r r- IIRs 3.9

17

4$ h $ * . ... . ,- a i *I*i" *,* a .*i i . - --

FinallyCLRM 0 CLRMe + CLRM+ (3.30)

In Appendix B it is shown that

CLRM a-CLSM + (Te/2)[1 - (4/3) (c/a) 2] (3.31)

This equation is valid for linear fluid mechanics and states that for thesimple case of constant amplitude motion (c = 0) the liquid roll moment is thenegative of the liquid side moment. If the complete nonlinear fluid equationsare used, additional contributions to the quadratic liquid roll moment whichare large enough to affect Eq. (3.31) may be present.

IV. DISCUSSION

Reference 2 considers the effect of two types of internal componentmotion: (a) a mass moving in a circle in a plane normal to the missile'saxis; (b) forced coning motion of a spinning component. For these motions theside moment, MS, and the roll moment, MR,, were computed and for both cases

S-K1 Ms. (4.1)

This is precisely Eq. (3.31) for constant amplitude motion. Equation (4.1) isequivalent to the statement that the liquid moment about the trajectory iszero, i.e.,

MLX 2 0 . (4.2)

This essentially follows from the linear assumption that the azimuthalvelocity about the trajectory varies as exp (-18). A nonlinear theory couldwell have terms that are independent of e.

The liquid side moment was computed in Reference 19 for c/a = 4.291 andRe from 102 to 106; the results are given in Figures 2 - 6. Miller 1 s observedthe roll moment to be independent of roll rate. This means that the liquidroll moment should be proportional to T. As can be seen from Figures 2 - 6,this is roughly the case for the liquid side moment coefficient. SinceMiller's measurements were taken for T varying from .12 to .25, we will usethe liquid side moment coefficient for T - .18 to compare with Miller'smeasurements in Figure 1. The agreement is encouraging when we recall that weare applying a linear fluid mechanics calculation to measurements taken at a20* cone angle and a boundary theory to Reynolds numbers as low as 100.

18

4 .. 1 W '4 ' .. -. - * ... .. , " - .,

V. SUMM*ARY

1. The linear theory for a coning and spinning liquid payload has beenextended to the calculation of a quadratic roll moment.

2. The resulting quadratic roll moment coefficient is shown to beexactly the negative of the linear side moment coefficient for constantamplitude coning motion.

3. This relationship between side moment and roll moment has beenobserved for projectiles with moving solid components and is reasonably wellsupported by experiment for liquid payloads.

19

N.4

S U, •o to

Lnj

- JJ-j

0 z.

I- I,

WI

CL o

00 'C%

o 4C0

000

0

0 0S.0

4. 0

06

0 CP

.49

00Vo0

ciI

UjW

210

'.44

Ojc

*41

C;

* U.

cii

22,

40

100

OD0

U.

ci1In.

0 ci ci C

230

* �1* 1

'4

LN 0 43*

E* 9043LI'

4

S..03

0* C

>1 U

CC.

CcI,

9�.

N c0* 90

a0w

* 9m** a4,;4

1) -- U -

4 o 43W

.9 II. 9.-

b' �%J

9A q

o aN

_ U U*

.4'4.

0'

4.-

g�.C,

ELmmm�J

4�4

0- - 5 84 4o 0 00

.4

'S

.4.

.4

Z4.9�

4�

CC>,. � 4�t -. - -. -, C... 4 .. -� -* -i.C.C .4. -- �.. -� p. C � '. **.* � * * C CCC C...... 4 - 4

______ * * . * *

.4

.4

4. a

_____________________________ I.GD4.8

�ED 35-* 4

* . 5-

ED

q

.4,

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C

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.ME."

4..' 0U) -* , U �

o ED.5.o

I 's-p.''.4u 0'

.4' 4�A 4

2* tJu

0* 4-. 4 '0.4 4,;

ED£

us.4

4,, - IL'4 1.1

'4 I I U U I U I I 1 � � , �0

U) 0 S0 0

*4

S.'..4,' L)JQ�E4,:.

'4

9,

-�

4f4��-4..

4�*4*

'.4,

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i .4. 4-44

.'* **�? a � �*4� � 4��'t�4*.4� ;.....' - 4*

� *44 4 * -- - 'a - - 4 -.. - -, - . 4

REFERENCES

1. B.G. Karpov and J.W. Bradley, "A Study of Causes of Short Ranges of the80 T317 Shell," BRL Report 1049, May 1958 (AD 377548).

2. C.H. Murphys "Influence of Moving Internal Parts on Angular Motion ofSpinning Projectiles," Journal of Guidance and Control, Vol. 1, March-April 1978, pp. 117-122. See also BRL-IIR-2731, February 1977(AD A037338).

3. W.G. Soper, "Projectile Instability Produced by Internal Friction,"AIM Journal. Vol. 16, Jan 1978, pp. 8-11.

4. W.H. Mermagen, "Measurments of the Dynamical Behavior of Projectiles

Over Long Flight Paths," Journal of Spacecraft and Rockets- Vol. 8, April1971, pp. 380-385. See also BRL-,R-2079, November 1970 (AD 717002).

S. W.P. D'Amico, V. Oskay, W.H. Clay, 'Flight Tests of the ISSam XM687Binary Projectile and Associated Design Modification Prior to the NicoletWinter Test 1974-1975,;BRL-'-2748, Nay 1977 (AD B019969).

6. W.P. D'Amico, W.H. Clay, and A. Mark, *Yawsonde Data for M687-TypeProjectiles with Application to Rapid Spin Decay and Stewartson-TypeSpin-Up Instabilities," ARBRL-MR-O=7, June 1980 (AD A089646).

7 7. W.P. D'Amico, W.H. Clay, and A. Mark* Diagnostic Tests for Wick-TypePayloads and High-Viscosity Liquids,& ARBRL-MR-02913, April 1979(AD A072812).

8. W.P. D'Amico and U.H. Clay, "High Viscosity Liquid Payload Yawsonde Datafor Small Launch Yawss, AOL-M-03029, June 1980 (AD A06811).

9. V.P. D'AIizo "Early Flight Experiences with the 9M761," BRL-MR-2791,September 19h7 (AD 8024975L).

S10. M.P. D'Amico, "Field Tests of the YX761: First Diagnostic Test," -RLMR-2792, Septar 1977 (AD 8024976L).

11. V.P. D'Amico, "Field Tests of the XM761: Second Diagnostic Test," ARBRL-MI-020, January 1978 (AD M8050).

12. V.P. D'mico Aeroballistic Testing of the M S Projectile: Phase 16"AIRRL46-M l. Namh 1979 (AD B037680).

13. V.P. D'Amico, "Aeroballistic Testing of the OW Projectile: Phase I,"NkUL-M-=372., January 1961 (AD A098036).

- 14. M.P. D'Amico *Aeroballistic Testing of the " ,Projectile: Phase IllsHigh Hazle feloclty and High Quadrant Elevation," AQU•1Rt-03196,Septemer 1962 (AD MM6U11).

261'

'p

.1

REFEREWCES (Continued)

15. M.C. Miller, "Flight Instabilities of Spinning Projectiles HavingNonrigid Payloads," Journal of Guidance. Control, and Dynamics, Vol. 5,March-April 1982, pp. 151-157.

16. W.P. D'Amico and M.C. Miller, "Flight Instability Produced by a RapidlySpinning, Highly Viscous Liquid," Journal of Spacecraft and Rockets, Vol.16, Jan-Feb 1979, pp. 62-64.

17. K. Stewartson, "On the Stability of a Spinning Top Containing Liquid,"Journal of Fluid Mechanics Vol. 5, Part 4, September 1959, pp. 577-592.

18. E.H. Wedemeyer, "Viscous Correction to Stewartson's Stability Criterion,"BRL-R-1325, June 1966 (AD 489687).

19. C.H. Murphy, "Angular Motion of a Spinning Projectile With a ViscousLiquid Payload," BRL Memorandum Report ARBRL-MR-03194, August 1982 (AD

. A118676). See also Journal of Guidance. Control, and Dynamics, Vol. 6,July-August 1983, pp. zwU-zo.

20. C.W. Kitchens, Jr., N. Gerber, and R. Sedney, "Oscillations of a Liquidin a Rotating Cylinder: Solid Body Rotation,* BRL Technical Report ARBRL-

'. TR-02081, June 1978 (AD A057759).

21. N. Gerber, R. Sedney, and J.M. Bartos, "Pressure Moment on a Liquid-. Filled Projectile: Solid Body Rotation," BRL Technical Report ARBRL-TR-

"02422, October 1982 (AD A120567).

22. N. Gerber and R. Sedney, "Moment on a Liquid-Filled Spinning and NutatingProjectile: Solid Body Rotation," BRL Technical Report ARBRL-TR-02470,February 1983 (AD A125332).

23. H.R. Vaughn, "Flight Djynamic Instabilities of Fluid-Filled Projectiles,"Sandia Laboratories, Albuquerque, MR,, SAND 78-0999, June 1978.

4

24. C.N. Murphy "Free Flight Motion of Symmetric Missiles," BRL-R-1•16,July 1963 (AD 442757).

25. V.P. D'Amico and R.J. Yalamanchili, "Yawsonde Tests of the 8-Inch XM877Binary Projectile: Phase 1," BRL Memorandum Report in preparation.

26. N. Schlichting, "Boundary Layer Flow,," McGraw-Hill Book Company, NewYork, 1960, p. S.

2

"12

0.

.•,• APPENDIX A

;!!i!Errata for Reference 19

"N

.. 4

,'p

APPENDIX A

ERRATA FOR REFERENCE 191. Third term of Eq. (5.20) should contain (s -1), not (s -1).2. Second line of Eq. (6.4) should read:

12

I (c/2a) x c Ic1 K- dx + (h/c) 2 m- r-a P•th (6.4)

3. Eq. (7.1) should read:

'1mv" a (2v K Re)"1 es i 1 2ee *do dx (1.1)-1 0

4. Eq. (7.4) should read:

"(w v x rdr (%e - 2 K Re) 3b ax F ~ 'SO'. veh (.4

where4..'

4cah aai rMveh (2 K e)-1 (c/ah) rsv- Vrrb x* -1

_4.'

5. Third term of Eq. (9.4) should contain (s - 1), not (s-1).

31ft.

qgf

.1!

APPENDIX B

Relationship Between C and CL

I

33

APPENDIX BRELATIONSHIP BETWEEN CLRM AND CLSM

In Reference 19 the pressure parts of the side moment coefficient for thelateral and endwalls are given in Eqs. [6.4 - 6.5].* The viscous shear con-tributions are given in Eqs. [7.2, 7.4]. These can be combined to yield thecomplete side moment coefficients for the lateral and endwalls, respectively:

:,:CLSM• = (2Tc)"1 (R {g• -IC d• (B-l)

-1

C• -S 2 c Cc r=a

:,? • X = X

CLSM = (Tac)' R{gKI} K dF (B-2)ea

X=C

where9 + ws +ia-2. Re-1(x/a) Ps L

(8 r ar

IB .rjRe' (8-3)

S1e (r/a) p5 + c [( - i 5i ) Re(Be:: = a(s Vs) (B-4)

As is shown by Eqs. (3.25 - 3.29), the roll moment coefficients have avery similar form:

CLRM (2c)" - R f. K-} d' (B-5):• -cr=a

xax

CLRM (K} (B-6)e a 0 Rr=F

xac

* Bracket;, identifjj equationa from Reference 19.

.5• 35

4S•'- • .• • .. . .• , . • :% _", . ", •? = . 3 : .. .

where2w 2 a 5 s -w fs x7 2 WS R"

[ax a - ia Re (B-7)

S 2w + ( 32ws aws avf= er22Ws + r _ s j5 . e1 (B-8)axar a

2[ ax Tx'•32ws (a,x)

wSince -ax)is the dominant term in CLRM it must be computed more

carefully than by the usual boundary layer approximation. The linear circum-ferential momentum equation as given by Eq. [B6] is

2 -1 [2ws+ a w 2w 21va Re- + 7 =r5ar rar =ax

SlaPs (B-9)= (s - i) ws + 2vs_-

On the boundaries r = a, x = c, or x = -c, vs and ws must satisfy the boundaryconditions as given by Equations [3.8 - 3.9]:

ws = i (s-i) (x/a) K = -ivs (B-1)

Thus, on the boundary surfaces of the liquid container2 + aWs + : -2 [sap 2+a w+ + i 2) (x/a) K Re (B-11)

2 a2On the lateral boundary (F = a), -x- is zero and ws in Eq. (B-7) can

Ux ar2be computed by use of Eq. (B-11). The result can be integrated ands',iplified by use of Eq. (B-10).I"-

CLRM a - CLSM- (2/3) (c/a) 2 Te (B-12)it

36

a2ws Iws 32wSimilarly, ar2 and -ar are zero on the endwalls (x ic) and U2 in Eq. (B-8)

can be eliminated by use of Eq. (B-11)

CLRMe -CLSMe + "102

(B-13)

+ (r Re)'lR a - K

a x ) =cX=C

The derivative of ws normal to the endwall at the junction of the walls can be

computed from Eq. (B-0) and is the same size as the derivatives omitted in theboundary layer approximations of Eqs. (3.18-3.20). Thus, the last term of Eq.(B-13) can be neglected. Eqs. (B-12 - B-13) can then be added to yield the com-plete liquid roll moment coefficient:

CLRM = -CLSM + (tc/2) [1 - (4/3) (c/a) 2) (B-14)

For constant amplitude coning motion, then, the liquid roll moment is thenegative of the liquid side moment.

37

ii 37I ; • T. • • • . . ., . . . , . .- . . . - , , , • .. , " , . ' . ,k T . ., • -. . - . "I ,, .- , - • - . . . . . .

-;•¥n;"i,••h ," - -,• , 0 , ,0'"'

APPENDIX C

LIQUID ROLL MOMENT FOR PARTIAL FILL

9.,9

'2

APPENDIX C

LIQUID ROLL MOMENT FOR PARTIAL FILL

If the cylindrical container is only partially filled with liquid, afully spun-up liquid will fill the space between the outer cylindrical walland an inner cylindrical free surface with radius b. The ratio of the volume

of this inner cylinder to the volume of the complete payload cavity is b2 /a 2 .

The fill ratio for the payload is, therefore, 1 - b2/a 2 and is denoted by f.

In the presence of small amplitude coning motion, this cylindrical freesurface is deformed slightly. The equation for this surface in aeroballistic

coordinates is

r= b (1 + n) (C-i)

where n R ns(x) e5O - (C-2)

and the perturbation function ns is linear in K.

In earth-fixed variables the surface has the form:

F (x, r, e, t) R + b2 [I + nj2 x2- r2 =0 (C-3)

From Eq. [A12]

, -x -R {r Ke is (C-4)

.4"! and n must vary such that the .free surface moves with the liquid, i.e.,

rv L+ V LF + F+O 0 (C-5)

Equations 1C-3) - 5(-)with Equations (3.21 - 3.23) can now be used to obtain an

equation for that is linear in K.

avs(bx)'b% (x) - x K + s- (C-6)

According to Eq. [B14], at X -c

"41

•.. . .... ..e , .* .* .* . . . . .. , . . . -... , ... , ., .. .. . ,. -.. , -. , .- . . . , .. . ., . .. ,. .

vs - (s - i) (c/a) I (C-7)

""ns (e ns (-c)- 1(C-8)

On the free surface, the pressure is constant:

0 .(C-9)

The linear version of this is Eq. [5.83, which was used as the inner boundary-condition:

(s - i) Ps (bx) + (b/a) vs (bx) a 0 (C-10)

The side moment coefficient has the same integral relation for CLSi asEq. (B-1 but Eq. (5-2)becomes

CLsM -080- 1 SR {ge r }. F(-iXUC

Similarly, for the liquid roll moment coefficient Eq. (B-S)is unaffected butEq. (8-6) becomes

CL (,CDC)" Rif. K* ), F}r d (C-i2)C b xac

Eq. (l-13) then becomes

c We 0 -C We + (,c/2). E1 - (b.,)43

(C43)

+(Oo Re)I R J~r aw(roc)

.'. CULN •-CLs + (vr/2) E1 - (4/3) (c/a) 2 - (b/l) 4]

- b2 (Ta Re)"M R t sK.

42

LIST OF SYMBOLS

Sa radius of a right-circular cylindrical cavity containing liquid

c half-height of a right-circular cylindrical cavity containingliquidCLIM fast (Jul) and slow (J-2) mode liquid in-plane momentcoefficients; imaginary part of CL.

C fast (J-1) and slow (J-2) mode complex liquid moment.j coefficients defined by Eq. (2.2)

, aerodynamic roll damping moment coefficient defined by Eq. (2.6)

CCR for steady-state coning motion (K2 0)CR1

.1 CLRM fast (Jul) and slow (.J-2) mode liquid roll moment coefficientsdefined by Eq. (2.4)CLR that part of CLI, induced by the end walls of the cavity

CLRM that part of CLRM induced by the lateral wall of the cavity

• .CLN for steady-state coning motion (K2 - 0)

.CLS fast (Jul) and slow (Ju2) mode liquid side moment coefficients;1real part of C L

xjx ~ unit vectors along the aeroballistic cylindrical axesSeZxpe~r9eZG components of e*Z in the inertial x. r. 8 systemaf e -xe -re components of i- in the inertial x, ro a system

e&e components of eL in the inertial x. r. a systemorr! ejx.ejr,e;0 components of r-in the inertial x, r, 0 system

S&axial moment of inertia of the projectile

K .IO exp (i1io)

43

a,-. o --•, .-•,. -°°

-K K magnitude of the fast (Jul) or slow (J-2) yaw arm

•Kjo initial value of Kj

St reference length

Q L mass of the liquid in a fully filled cylindrical cavity

N -, N , N XYZ components of the moment exerted by a liquid payload.LX LY LZ

r, F radial coordinates in the inertial and aeroballisticsystems, respectively

Re Reynolds number

s (C +1) T

S reference area

t time

Us, Vs, Vs coponents of the liquid velocity perturbation in theInertial x, r, a system

V magnitude of the missile's velocity

Vxb Vr, V0 velocity comonents in the inertial x, r, s system

xe x axial coordinates In the inertial and aeroballistic systems,respectively

XYZ inertial axes; X-axis tangent to the trajectory at timezero*

' XYlZ nonrolling aeroballistic axes; I-axis fixed along themissiles axis of symmetry

a. I angles of attack and side-slip In the XYZ system

%•l nondmnsionalized grV th rate of Kj

4. azimuthal coordinates In the inertial and aeroballisticsystem, respectively

i p viscosity

Il . a ii i 1 I. iI , d.

!p atr density

T T for steady-state coning motion (K2 - 0)

STj the nondlmensiona~1zed frequency of the j-th yaw arm(J-I,2)

S4 roll rate

#j #J 0j + ,j ; t

#J initial orientation angle of the j-th yaw arm (J-1,2)1

I

-4

I

>4

S

44

9 45

S tS.I,*~t~~v~S~.~~~

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College Park, MD 20740 EngineeringATTN: J.C. Westkaemper

University of Maryland - Austin, TX 78712Baltimore County

Department of Mathematics 1 University of VirginiaATTN: Dr. Y.M. Lynn Department of Aerospace5401 Wilkens Avenue Engineering & EngineeringBaltimore, MI) 21228 Physics

ATTN: I.D. JacobsonSUniversity of Santa Clara Charlottesville, VA 22904

Department of PhysicsATTN: R. Greeley 1 University of Virginia ResearchSanta Clara, CA 95053 Laboratories for the

Engineering SciencesATTN: Prof. H. G. WoodP.O. Box 3366University StationCharlottesville, VA 22903

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4. 'r

i

DISTRIBUTION LIST

No. of No. ofCopies Organization Copies Organization

1 University of Washington Aberdeen Proving GroundDepartment of Mechanical

Engineering Director, USAMSAAATTN: Tech Library ATTN: DRXSY-DSeattle, WA 98105 DRXSY-MP, H. Cohen

1 University of Wyoming Commander, USATECOMATTN: D.L. Boyer ATTN: DRSTE-TO-FUniversity StationLaramie, WY 82071 Director, USACSL, Bldg. E3330, EA

ATTN: DRDAR-CLN3 Virginia Polytechnic Institute W. C. Dee

and State UniversityDepartment of Aerospace Director, USACSL, Bldg. E3516, EA

Engineering ATTN: DRDAR-CLB-PAATTN: Tech Library M. C. Miller

Dr. W. Saric DRDAR-CLJ-LDr. T. Herbert DRDAR-CLB-PA

Blacksburg, VA 24061 DRDAR-CLN

1 Woods Hole OceanographicInstitute

ATTN: J.A. WhiteheadWoods Hole, MA 02543

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