OFFICE OF SCIENT IFI C RESEARCH AND DEVELOPMENT NATIONAL DEFENSE RESEARCH COMM ITTEE

DIVISION SIX - SEC TI ON 6.1

MK 25 TORPEDO

WITH

VARIOUS EXHAUST PIPES

ROBERT T, KNAPP

OFFICIAL INVESTIGATOR

TH E HIGH SPEED WATER TUNNEL AT TH E

CA LI FORNIA INST ITUT E OF TECHNOLOGY HYDRODYNAMI CS LABORATORY

PASADENA, CALIFORNIA

Sect ion No . 6 .. i-sr207··· 2236 Report Prepared by Harold L . Dooli ttl e

Laboratory 'No . ND-30 . 3 Hydraulic Engineer

July 4 , i945

This do c ument oontains informatinn affecting the notional defense of the United States within the meaning of th9 Espionage Act , ~O u. s.c . , 31 and 32 r as amen~ed o The transmission or the revelation of its contents in any manner to an unauthori ze d person is prohibit e d . by law .

OOlti I 6214 I I AL

ABSTRACT

For a i0° port rudder setting (horizontal rudders neutral) and for all yaw angles between -4° and +io0 ,it was found that the moment coefficien t with 2 and 4 pipes was practically the same as with no pipes .

For a i0° down rudder (vertical rudders neutral) and zero pitch angle,the ruddir effect for the 4- pipe arrangement is 30% less than w~th no pipes, and for the 2-pipe arrangement the rudder effect is 45% greater . It is b~lieved that the greater effect with the 2 pipes is due to the shielding of the rud~ers by the pipes .

The drag coefficient for the model at zero yaw and pitch is practically O. i3 for the 4-pipe , 2~pip~ . and no- pipe arrangements.

The torpedo with a single exhaust pipe will have about i5% more drag than with any of the other exhaust arrangements . With a down setting of the horizontal rudders there will be less rudder effe c t with a single exhaust pipe than with the other exhaust arrangements due to the single pipe acting as an up rudder .

OOllFIBEllTI AI

""'8141 I DEllT I AL;

i----· 43 .69"·--------..j

i---- -- 5 3. 99" - - ------ -- 3 4. 38" - ---------- 7262~" ---------i

--------------- --- 16100" - - ---------------- - - -

F IG. 1 - OUT LIN E UF MK 25 TORPEDO

L'.QHF I QF NJ I Al

GENERAL

MK 25 .TORPEDO

WITH

VARIOUS EXHAUST PIPES

CORP I bl:A I I Xt

This report covers force tests of a Mk 25 Torpedo model fitted wi th various types of exhaust pipes . The tests were con­duc ted at the Hydrodynamics Laborato ry of the Cal ifornia Insti­tute of Technology and were a u thor ized by Dr . E . H . Colpitts , Ch i ef of Section 6 . i . National Defe~se Research Commi ttee , in a letter da t ed May 4 , i944 .

This report is the third supplement to the report, Section No . 6 . i - sr207--i275. The tests were for the purpose of determin·­ing the drag_, cross force , l i ft and moment coefficients for the projectile when fitted with several different exhaust pipe designs and for various rudder settings .

In the normal design of existing gas turbine·--driven torpedoes the exhaust gas is discharged through a hollow propeller ~haft In order to eliminate 'many of the objections to this arrangement" a new ~esign has been made providing for the gas discharge through a fin or through iJ 2, or 4 pipes attached to the ring 'tail . In the case of th e 4-pipe des i gn ; . the ri~g is hollow to provide for t he supply of gas to the pipes ,

Repor t s ., Section Nos . 6 , i --sr207·-i27 S, i640 , i642 , and i9 i6 cover studies of t he cavi tat ion effect , o~ rudders .and propellers , of these various means of exhausting the gas .

·All data hereiq refer to the proto t ype unless otherwise stated

T he appe n dix contains definitions of te rms used in this report and o t he r pertinent d ata .

DESCR I PTION OF PROJECT!LE

Following are the principal dimensions of the projectile :

Diameter 22 . 42 inches

Length Overall i6 :1...00 inches

Di s tan c e , No se to C. G. 7 3 . 82 inches

Scale Ratio of Model i to ii. 2i

F i gur e i shows a gen e rdl outline drawing ~f the pro jectile f itted with the standard nos e that was us e d for all tests .

'

.-Ollf I &Eh FI AL -2-

FIG. 2 - R IN G TAIL WITH SINGLE EXHAUS T PIPE

FIG . 3 - RING TAIL WITH TWO EXHAUST P I PES

In the prototype, the exhaust gas is passed through the hollow top vertical fin to th e single exhaust pipe (Figure 2). The gas is fed to the two exhaust pipes through the h ollow hori­zontal fins (Figure 3 ). In the 4-pipe arrangement , Figure 4,

-eotl5 I DEtlil I' Isa..

COM I DEN I I AE

- --- - - ---i

FIG. 4 - RING TAIL WITH FOUR . EXHAUST PIPES

the ring is hollow, thus providing passage for the gas from the hollow horizontal fins to the pipes. As the hollow fins have an increased thickness, they might have some rudder effect, especially in the case of the single hollow top fin. In all of the model ~ests, for simplicity, the top vertical fin only was increased in thickness, regardless of the number of exhaust pipes. This intro­duces some asymmetry in the model that is not present in the prototype.

RING TAIL WITHOUT PIPES

Figure 5 is a photograph of the model ring tail without a dis­charge pipe, the exhaust being passed through the slot in line with . the holl o w top ve rtical fin . .

In Figure 6 the force coef­ficients are shown f or o 0 and i0° port settings for t he vertical rudders and with neutral horizon­tal rudders. As would be expected, the cross force and moment curves for neutral rudders pass through

FIG, 5 - RING TAIL WITH NO PIPE. EXHAUST THROUGH

SLOT IN RING

zero as the projectile is symmetrical about the vertical center line . There is a slight stabilizing moment between o0 and io yaw for a i0° port setting o f the vertical rudders.

In Figure 7 are shown the force coeff ici ents for neutral vert i cal rudders and various settings o f the horizontal rudders.

· OlllF I Dffi.t H

nQOtlE I B Elli I 'L

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o 5

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o 3

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v I/ -....... v ~ PRfZC NTAL RUDI >-- JIE'' 'D - - ·- ~--

i,........--v '-

-10 ·8 .. 5 .. . · 2 0 6 YAW ANGLE, DEG REES

FIG . 6 - FORCE COEFFICIENTS WI TH NO EXH AU ST P I PE

Rudders : Ver ti cal o 0 and i0° Port Ho r izontal - Neutral

Coeur 1otN1 -I At s;.

f

ERS

004

' 002

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RUDDER

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SETTING (UP)

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FIG . 7 - FORCE COEFFICI ENTS WITH NO EXHAUST P I PE

Rudde rs : Vertical - Neutral Horizontal - o0 , 2° , 5°, i0° up and d own

. CORF I D¢HTI Al •

'-00:11 ilfEllTIJllL -·6 -

The o0 ~ 2° , 5° a nd :1.0° up rudder settings at o 0 pitch angle result in s t abilizing moment coefficients of 0 . 0004, 0 . 00 i 3 , 0 -0022 ~ and 0 . 0037 .. respectivel y .. As can be seen from the cu r ves, the stabilizing moments are somewhat less for corr esponding down

rudder sett i ngs . The slight stabilizing momen t at ze ro rudder setting is very likely due to the thick top fin .

TWO cX HA UST P ! PES

Figure 3 shows the ring ta il with two exhaust pipes adjacent t o the t wo ho rizontal fins

The f6rce coefficien t s f or the pro jectile fitted with a ring tail and. two pipes .. as shown i n Figure 8 " are practically the same as fo r the ring withou t pipes (Figure 6 ) fo r the o 0 and :1.0° port rudder s-ett ings In F igure 9 it is seen t h u t the :1.0° down setting of the ho r izontal rudders results in a · moment coefficient of .. -o 0 48 at o0 pitch _. wJ:t i ch is about 50% greater than with no ex--· h aust pipes . There . is practically no rudder effect due to th is exhaust pipe arrangement

FOUR EXH AUST P I PES

Figure 4 shows the ring tail wit h four exhaust pipes

Figures iO and ii s h ow the fo rce coefficients f o r this e~­h aust p i pe arrangement .

.'A 1.0° p ort rudder setting with the 4-p ipe arra ngement results i n a moment coefficient or r udder effect at zero yaw of 0 . 007 or only about 60% of that of the rin g tail without pipes . The rudder effect for i0° down rudders with the 4- pipe design is about the same as for the 2--pipe design ( Figure 9) wit h o nl y a 5° d o wn rud­der settin g , This symmetr ical design results in p ractically no rudder effect with a ll rudders n eutral

S ! NG LE EXHAUST P l PE

Tes ts were run on the mode l with a ring tail fitted wi th a single exhaust pipe as shown in Figure 2 . Figures 1.2 and i3 s how the results of these tests which indica t e t h at the moment coef­ficient with this single pipe is about the same as that for the other exhaus t arrangements over a wide range of yaw angles . With a down setting of t he horizontal rudders there is le ss rudder effect with a single pipe 3/4 :: lon_g than with any of the o ther exhaust .pipe arrangements This is due to the singl e projecting pipe ac t i ng as an up rudder . This woul d t end to inc r ease th e rudder effect wit h an up rudder _. as is seen i n Figure ~3 .. The var1at i on i n moment with pitch angle for a 5 .. .. :1./2 ::· si n gl e pipe was n ot at a ll co n sis t e nt wit h the re s ults obta i ned wi th the other exhau~t arrangements Th i s discrepancy was probably due to rudde r effect caused by some unnoticed malali gnment of the long: single pipe

. . .. tJ;;; IDEt!I!O!

"' . .... , CORF I DEN I I AL

- 7-

It is apparent t hat the s ingle exhaust pipe wil l produce a grea t er des tabi l izin g momen t wit h va ryin g pi t ch angles than is ob taine d with the other exhaust arrangemen ts . It also appears that the single pipe desi gn inc r eases the dra g about i 5%.

COMPARATIVE RESULTS

In Figure i4 have been comb ined the force coe fficient curves fo r the four exhaust a rrangements d i scussed herein . The plain ring t ail with no e xhaus t pipe gives the highest des tabilizing moment wi th all rudders neu t r al ; and the 4 ·-p i pe design gives the l owest..

The moment coeffici e nt curves for i0° por t and also i0° do wn rudd er sett.ings have be.en combined; for comparison , in Fi gure iS .. For the i0° port rudder setting there is li ttle variation in the moment coefficient for · the three exhaust arrangements for yaw angles between -4° · a n d +io 0 . Wi th t he vert i cal rudders neutral , a io0 se tti ng of th e hor izontal rudders results in a stabi lizing moment for s~all pit ch angles with al l exhaust arran gements, al­though there are wide variations in the rudder effect . It is suggested tha·t the greater rudder effect with the 2·-pipe design is due t o the shielding of the end of the rudder by the pipe, thereby increasing the effectiveness of the rudder .

The following table gives data taken from Figure iS :

Ver tical Rudders Neut ral ·- Horizontal Rudders i0° Down

Exhaust Arra ngement

4 pipes

Rudder Effe ct CM at o 0 Pi t ch

-0.023

No pipes ·-0 033

2 pipes -0 . 048

1 pipe 3/4" long - O .Oi2

The dra g coeffi cient was determined. for the no- pipe, 2--pipe; and 4-p ipe arrangements ; and in al l cases t he drag decreased with increasing yaw an d pi t ch angles . It is believed th a t this con­di t ion is no t in accord with t h e fac t s a lthou gh the cause has n ot yet been investiga t ed . The drag observat ions made at zero yaw a n gle are thought to be sub stantially correc t for comp arison of model performance . The se are tabulated below, and i t is seen that the addition of 2 oi 4 pi~es increases the drag very little . These drag measurements were made on a 2- inch d i ameter model a t a water veloc ity of 32 . 4 ft/sec io t he value for th ~ pro totyp e would b e appreciably lower

Exhaust Arrangement Drag Coefficien t for Ze r o Yaw and P it ch

No pipes O . i2

2 pipes o . i3

4 pipes O. i3

1 pipe 3/4" lon g O . i S

CURI I DEIH I AL

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YAW ANGLE, DEGREES

F IG . 8 - FORCE COEFF IC IENT S FOR R I NG TAIL WI TH TWO P I PES

Rudde rs : Vertical o0 and i 0° Port . Horizontal - Neut r a l

CUICI IDEtlT! 'I.

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.-- v vm ICAL RUD ERS

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0.12 - 10 -8 - G - 4 - 2 0 2 4 6 10

PITCH ANGLE , DEGREES

FIG. 9 - FORC E COEFFIC IENTS FOR RING TAIL WfTH TWO PIPES

Rudders: Verti ca l - Neutral Ho rizontal - o0 , 2°, s0 , i0° up and down

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.... e111r I BEN I.I AL

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llUI IDEA SETT ~G llPOR ti .I

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RUDI ER KTTI ~G (I k!RT)

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.OJ 0 ·IO - 8 - 6 -4 - 2 0 2 4 6 8 10

YAW ANGLE, DEGREES

F IG. 10 - FORCE COE FFI C IENTS FOR RING TAIL WITH FOUR P I PES

Rudders: Vertical Oo and iOO port Horizontal - Neutral

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PITCH ANGLE, DEGREES

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F 1G. 11 - FORCE COEFF I C ~ENT S FOR R I NG TA I L WITH FOUR P I PES

Rudders: Vertical - Neutral Horizontal - o0 , 2° , s0 , 10° up and down

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YAW ANGLE, DEGREES

FIG. 12 - FORCE COEFFIC IENTS FOR SINGLE P I PE 3/4" LONG

C•IF I BEtlT I Hs

Rudders: Vertical - o0 and i0° Port Horizontal - Neutral

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PI TCH ANGLE , DEGREES

F I G . 13 - F ORCE COE FFICIE NTS F OR S I NGLE P I PE 3/4 11 LON G

Rudders : Vertical - Neutra l Ho rizont al - o0 , 2° , s0 , i0° up a n d down

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APPENDIX

DEFINITIONS

YAW ANGLE, t./J

The angle : in a horizontal plane.: which th e axis of _the pro-­j ec t i le makes with the di r ec t io n o f mo ti on . Looking down on th e projectile, yaw angles in a clockwise direction are posi tive (+) and in a counterclockwise di rection , ne ga t ive (-) .

PITCH ANG LE, a

Th e angle _, in a vertica l p l ane , which the jec t ile m~kes wi t h the direc t ion of motion . positive ( +) wh en t h e nose is up and negative is down .

axis of the pro­Pi tch a n g les a r e (' - ) when the nose

The force in pounds : exer ted on the projectile normal to the direct i on of mot i on and in a ver t ical plane . The lif t is positive (+) when acting upward and negative r-) when acting downw·ard .

CROSS FORCE 1 · C

"The f orce . in pounds _; exerted o n t he projectile n ormal to "the direction of motion a n d in a horizontal plane . "The cross forc e is positi ve when acting i n the same direc t ion as the displacemen t of the projectile nose for a positive yaw angle . i . e ., to an o bse rver facing in the direc t io n of travel J a posit i ve croe s force act s t o the right

DRAG, D

The force in pounds . exerted on the projectile parallel wi th the direction of mo~ion . Th e drag is positive when acting in a direct ion opposite to the d i re c tion of mo tion

MOMENT . M

Th e to rque . in foot pounds, tendi ng to rotate the projectile about a t ran sverse axi s Yawi n g mome~ts tend i n g t o rotate th e projectile in a c l ockwise d irec t ion (when looki n g down on th e p r o­jectile) . a r e positive (+) _, and those tending t o cause counter--­clockwise rotation ar e n egative ( -) . Pitching moments t endi n g to r ota t e th e projecti l e in a c l ockwise direction (when look i ng at the proje c t il e from th e port side ) a re positive ('+ ), and those t e n d ing t o cause counterc l ock wi se ro t atio.n ·are n ega t ive · (- ) .

• fOllFIBEllTIPI

~- Al 614 F 11J@HT PJt t ·-b-

In accordance with this sign conventi o n a moment has a de­stabilizing effect when it has the same sign as the yaw or pitch angle .

In all model tests the moment is measured about the point of support . Moments about the center of gravity of the projectile have the symbol , Meg ·

NORMAL COMPONENT, N

The sum of the components of the drag and cross force actin~ normal to the axis of the projectile " The value of the normal component is given by the following:

N = D sin~ +.c cos~ ( i)

in which

N Normal component in lbs .

D Drag in lbs

c Cross force in lbs

-~ Yb.w angle in degr ees

CENTER OF PRESSURE, CP

'The point in the axis o~ the projectile at which the resul­tant of all forces acting on the projectile is app l ied .

CENTER-OF-PRESSURE ECCENTRICITY, e

The di .stance between the center of pressure (CP) and the center of gravity (CG) expressed as a decimal fraction of the length (1) of the projectile , The center-of·-pressure eccentricity is derived as follows :

e-(1 ·- lcglt - cp 1

in which

i .J'l!.c:og 1 N

e = Center- of-pressure eccentricity

1 Lerigth of projectile in feet

leg Distance from nose of proje~tile to CG in feet

lcp Distance from nose of projectile to CP in feet

UllF i D l!llT PM!•

( 2)

WNfi.BbiC I I Ab - c - -

COEFF I CIENTS

The three force and moment coefficients used are derived as follows :

in which

D

c

L

p

w

g

Drag coefficient ,

Cr oss force coefficient.

Lift coefficient i

Moment co~fficient ,

Measured drag force in lbs .

CD = __ __ D ___ ._

v2 p 2

AD

Cc = _, _____ _g_ __ _ 2

p V AD 2

Cr_, = ·---·---L--.--. 2 v

p AD 2

C1vi = - -·-·-.-Ji..~-- .--2

p V Aol 2

Measured cross force in lbs

Measured lift force in lbs

Density of the fluid in slugs/cu ft = w/g

·specific weight of t he fluid in lbs/cu ft

Acceleration of· gravity in ft/sec 2

AD= Area in wq ft at the maximum cross section of the pro­jectile taken normal to -the geometric axis of the pro­jectile

V Mean relative velocity between the water and the pro­jectile in ft / sec

M Moment , in foot-pounds , measured about any particular point on th e geometric axis of the projectile

1 Ove r all len gth of the projectile in feet

(3)

(4)

(4a)

(5)

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.. " ttt:E'Dsttl:!tt -d-

RUDDER EFFECT

The total increase or decrease in moment coefficientJ at a given yaw or pitch angle , resulting from a given rudder s ·etting . This ·increase or decrease in moment coefficient is measured from th~ moment coefficient curve for neutral rudder se tting.

REYNOLDS NUMBER

In inertia utility .

comparing hydraulic systems involvi ng only friction and forcesJ a factor ~alled Reynolds number is of great This is defined as follows :

R

in which

l V v

lVP µ

R Reynolds number

l Overall length of projectile, feet .. V Velocity of project ile , feet p~r sec

Kinematic viscosity of the fluidJ sq ft per sec

p Mass density of the fluid in slugs per cu ft

µ = Absolute viscosity in pound-seconds per sq ft

( 6)

µ/p

Two geometrically similar systems are also dynamically simi-1 ar when · they have the same value of Reyno lds number , For the same fluid in both cases, a model wit h smal l linear ~imensions must be used with correspondingly large velocities . It· is also possible to compare t wo c~ses with widely differing fluids pro­vided 1 and V are properly chosen to give the same value of R.

CAViTATI ON PARAMETER

In the analysis of cavitation phenome.na -' the cavitation parameter has been found very useful .. This is defined as follows :

in which

K = Pr,

p

K = Cavitation parameter

( 7)

PL= ·Absolute pressure in the undisturbed liquid, lbs/sq ft

p B= Vapor pressure corresponding to the water temperatureJ lbs/sq ft

V Veloci ty of th e projectile , ft/sec

lil@ll Fl Bl!IH I Atrt

't&IR \!36#1 I ilil! .. - -e-·-

p mass density of the fluid in slugs per cu ft w/g

w weight of the fluid in lbs per cu ft

g = accele~ation of g ravity

Note tnat any homogeneous tation of this parameter,. this parameter in terms of

set of units can be used .in the ·compu-­Thus , i t is often.convenient to express the head, i . e .. _,

hL ·-- hB K = v2 ( 8)

2g

where hL ~ Submergence plus the b~rom~tric head, ft of water

hB = Pressure in the bubble ~ ft of water

It will be s een that the numerator of both . expressions is simply the net pressure acting to collapse the cavity or bubble . The denominator is th e velocity pressure . 'Si nce the entire variation in pressure around the moving body -is n result of the velocity, it may be considered that the velocity head is a measure o f the pres­sure available to open up a cavitation void . From this point of view, the cavitation parameter is s i mply th e ratio of t he pressu r e available to collapse the bubble to the pressure available to open it . If the K for incipient cavitation, is considered, it can be interpreted to mean the maximum reduction in pressure on the sur­face of the body measured in . terms of the veloci ty head . Thus J if a body starts t o cavitate at the cavitation parameter o f oneJ it means that the lowest pressure at any point on the body is one velocity head below that of the undisturbed fluid .

The shape and size of the cavi t ation bubbles for a specific projectile are functions of th e cavitation par~meter . If PB is taken to represent the gas pressure within t he bubble instead of the vapor pressure of the water_, as in normal investiga t ions, the v a lue of K obtained by the above formula wil l be applicabl e to an a i r bubble . In o th e r word s , the behavior of . the bubble will b e the s ame wh e ther t he bubbl e i s due to cavi_tation , the injection of exhaust gas , o r t he ent r a i nmen t o f a i r at t~e time o f latinching ..

The cav i tat i on p arameter for incipien~ cavitation has t h e symbol Ki ..

Th e follo wing chart gives values of the cavi tation parameter as a function of veloc i ty ·dn.d submergence in sea water ..

GENERAL DISCU SS I ON OF STAT !C STAB IL ITY

Water tunnel tests are made under steady flow c onditions , ~ons equ ently th e result s o nly indicate the t e nde ncy o f the steady state hydrodynamic c oupl es and forces to caus e the projec tile to re turn t o or mo ve away fro m its equilibrium position after a

QH I PENT'' Al

e'e14J . ID bi;. I AF -f--

dist urbance. Dynami c coup l es and forces including either positive or negative damping are not obtained . If the hydrodynamic moments are restoring the projectile, th en it is said to be statically stable , if nonrestoring, statically unstable . I n .the discussion of static stability the actual motion following a perturbation is not .considered a t all.. In fac t ., t he projectile may osci llate con­tinuously about an equilibrium posit i on without re~aining in it ,

· In this case it .wou l d be statically stable , but would hnve zero damping and hence , be dynamically unstable . With negat ive damping a projectile would oscillate with continually increasing ampli t ude follow.ing an initial perturbation even though it were statically stable ., Equilibrium is obtained if the sum of th e hydrodyna,mic , buoyant, and propuls i ve momen t s equa l zero , In general ,- propul­sive thrusts act through the center of gravity of t he project4le so only the first two items are important .

If a projectile is rotating from its equi l ibrium position so as to increase its . yaw angle pos i tively,, the moment coefficient must increase negatively (according to the sign conven t ion adoi'­ted) in order that it be sta tica lly stable " Therefore , for pro­jecti les without controls or with fixed control surfaces , a nega­tive slope of the curve of moment coefficient vs yaw gives static stability and a posi tive slope gives instabiliti . For a pro­jectile without controls , static stability is necessary for a successfu l fli gh.t unless stab ili ty is obtai;ned by spinning as in the case of rifle shells • For a projectile wi t h controls, stabil­izing moments can be obtained by ad just ing the control surfaces , and the s l ope of the moment coefficient , as obtained with fixed rudder position , need not give stat i c s tabili ty. ~here b u oyancy either acts at the center of gravity or can be n.eg'.lected , equil-­ibrium is obtained when the hydrodynamic moment coefficient equals zero . For symmetrical projectiles thi s occurs at zero yaw angle, i . e ., when the projectile mxis is parallel to the t rajectory . For nonsymmetrical Hr_oiectiles , such as a torpedo wh'e1:ft' ... the rudders are not neutral, the ·moment is not zero at zero yaw but vanishes at some defin it e angle of attack . Where buoyancy cannot be neglected equi librium is obtained when c1v. = - C,, , and the axis of the

1 '-'uoyancy projecti le is at s ome angle with the trajectory.

For symm.etrical.project.iles .the degree of s t abili t y , or in­stability can be .obtained f roin the." center .-of .. -pressure· curves . If the center of pressure,; falls ·behind the cent er of gravity ,, a re­stor ing moment e xists g ivin g static stabil ity . ' If tbe center of pressure falls ah-~ad of '.±1he ce.nter of gravity , the mo~ent is non­restoring , and the projectile 0il l be sta~ically unstable . The degree of stabil ity or instability is indicated approximately by the dis tance between the center of gravity and the center of p ressure . ' In general , for ncinsymmetrical projectiles, the cross force or lift is not zero when the moment vanishes so that the cen ter of pressure curve i s n o t symme\trical and the s ·i~l e r ul es just s t ated cannot be. used to determine'whether or not the pro­jectil e will be stable . In such cases careful interprela~ion of the moment c urves is a mo re satisfac t ory method of determining stabili ty relationsh i p .

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