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Vo l u m e I V O C T O B E R * 1 9 4 6 N u m b e r
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oxoxo* *a*ta ruiuiHixc com?axy, kxxasha, whcoxh*
Q U A R T E R L Y OF A P P L I E D M A T H E M A T I C S
K. L. N I E L S E N f (L o u isian a Stale U niversity) a n d J L. S Y N G E ( The Ohio State U n iversity)
1. In tro d u ctio n . N e x t a fte r th e problem of th e m otion of a p article in a resisting m edium , th e problem of the m otion of a spinning shell is the m ajo r problem of ex
terio r ballistics. M a n y crude tre a tm e n ts have been given, b u t th e problem w as first discussed exhau stiv ely by Fow ler, G allop, Lock an d R ich m o n d .1'2 R eference m ay also be m ade to tre a tm e n ts by C ran z3 an d M o u lto n .4
An ex act tre a tm e n t of th e m otion of a spinning shell as a hy d ro d y n am ical problem is obviously o u t of th e q uestion. T h e problem m u st be tre a te d aerod ynam ically . T h is m eans th a t th e forces exerted on th e shell by th e a ir m u st be regarded as d ep e n d en t only on th e in sta n ta n e o u s m otion of th e shell. T h e connection betw een th e ae ro d y n am ic force system an d th e m otion c a n n o t be deduced logically. I t m u st a p p e a r in th e m ath em atica l th e o ry as a hypothesis, p referably su p p o rted by experim ental ob
servations.
B u t alth o u g h m ath em atica l th e o ry c a n n o t su p p ly th e aero d y n am ic forces, it does give us som e inform ation a b o u t them . T w o basic ideas are im p o rta n t here.
F irst, the shell has an axis of sy m m etry . T h is fa c t has been used in all existing theories.
T h e second idea is a little m ore sub tle. I t concerns th e connection betw een the position of th e m ass c e n te r (or ce n te r of g ra v ity ) of th e shell an d th e aero d y n am ic force system . In one m an n er of sp eakin g ,-th ere is no such connection. F o r tw o shells, m oving w ith id entical m otions b u t w ith different m ass-d istrib u tio n s, th e aero d y n am ic forces are th e sam e. B u t we c a n n o t in tro d u ce th e ae ro d y n am ic force system in to the m a th e m a tic a l a rg u m e n t w ith o u t expressing th a t force s}rstem m a th em atica lly as a force an d a couple (or som ething eq u iv ale n t). T o do this, we m u st use a base-point,
* Received January 22, 1946. T h is paper w as w ritten in 1942, when one o f the auth ors (J. L. S.) was at th e.U n iversity of T oronto. It w as issued as a restricted report in January 1943 by the B allistic Research Laboratory, Aberdeen P roving Ground, with perm ission of the N ation al Research C ouncil o f Canada.
Later work by other authors, issued in restricted reports, has improved on som e of the theory, b u t it has been th ou gh t ad visab le to publish the paper in its original form.
f On leave w ith U. S. N a v a l Ordnance P lant, Indianapolis, Ind.
1 R. H . Fowler, E. G . G allop, C. N . H . Lock, H . W . R ichm ond, The aerodynam ics of a spin n in g shell, Phil. Trans. R oy. Soc. London (A) 221, 295-387 (1920).
2 R. H . Fowler, C. N . H. Lock, The aerodynam ics of a sp in n in g shell, Part II, Phil. T rans. R oy. Soc.
London (A) 222, 227-249 (1921).
3 C. Cranz, Lehrbuch der B allislik, J. Springer, Berlin, 1925, p. 358.
4 F. R. M oulton, N ew methods in exterior ballistics, U n iversity of C hicago Press, Chicago, 1926, chap. 6.
V ol. IV O C T O B E R , 1946 N o . 3
O N T H E M O T IO N O F A S P IN N IN G S H E L L *
BY
202 K . L. N IE L S E N A N D J. L. S Y N G E (Vol. IV, N o. 3
a n d reduce th e force system to a force a t t h a t base-point, to g eth er w ith a couple.
I t is well know n th a t, for a given force system , th e force is in d ep en d e n t of the base- point, b u t th e couple is not.
Also, to describe th e m otion of th e shell m ath e m a tic a lly , we m u st use a base- point. T h e m otion is described by th e velocity of t h a t b ase-p o in t a n d an an g u la r velocity. T h e an g u lar velocity is in d ep en d e n t of th e choice of b ase-p oint, b u t the velocity is not.
Now it is n a tu ra l to use th e m ass ce n te r as base-point. If th ere are tw o shells, Si an d S 2, w ith m ass centers Oi an d 02, we m ay use Oi as b ase-po in t for Si an d 02 as b ase-po in t for Si. Suppose t h a t th e tw o shells are of id entical geom etrical form (b u t Oi an d 02 are n o t geom etrically corresponding points) a n d th a t th e ir m otions a t th e in s ta n t are th e sam e. (T his m eans th a t geo m etrically corresp on din g po in ts h av e equal velocities; th e velocities of Oi an d 02 are not th e sam e.) T h en th e force system s on th e tw o shells are th e sam e. B u t th e m om ents a b o u t Oi an d 0» are not th e sam e.
If we set o u t to fo rm u late aero d y n am ic laws, using th e m ass ce n te r as base-point, we m u st exercise g re a t care. W e m u st ensure invariance with respect to shift of mass center. W e m u st m ake sure, in th e case described above, th a t when we ap p ly our law, first to Si an d th en to S 2, we g et eq u iv ale n t force system s.
U n fo rtu n ately , Fow ler e t a l .1 paid no a tte n tio n to th is fa c t in fo rm u latin g th e ir aero d y n am ic laws (pp. 302-305), alth o u g h th e y d raw a tte n tio n to th e necessity for invariance (p. 305), a n d in fa c t m ake use of it. By considering a special case, it is easy to see th e fallacy in th e ir basic laws.
C onsider th e tw o shells described above. L e t th e velocity of 0 i be d irected along th e axis of th e shell, an d le t th e shell h av e an a n g u la r v elocity represen ted b y a v ec to r p erpen d icu lar to th e axis (plane m otio n). T h e yaw is zero, an d th e effect of th e a ir is a d rag along th e axis. B u t now consider S 2. On ac co u n t of th e a n g u la r velocity, th e velocity of 02 is n o t along th e axis; th ere is a yaw , a n d hence a cross w ind force in ad d itio n to a drag. I t is easy to see t h a t th e force system s on th e tw o shells are n o t eq u iv alen t, as th e y o u g h t to be since th e m otions are the sam e.
T h u s th e th eo ry of Fow ler e t al. co n tain s a logical co n trad ictio n . I t is very diffi
c u lt to discuss critically a th e o ry co n tain in g a logical co n tra d ictio n , for from incon
s iste n t h yp otheses we m ay a rriv e alm o st an y w h ere (a t 1 = 0 , for exam ple.) I t m ay well be, how ever, t h a t th e logical co n tra d ictio n does n o t in v a lid a te th e physical con
clusions of th e ir paper. In th e exam ple given ab ov e, th e yaw of S2 m ay well be very sm all indeed in cases of p ra ctical in tere st, an d th e logical inconsistency m ay be no m ore serious th a n t h a t involved in w riting 7r = 3.14. Used in one w ay, th is statem en t leads to 1 = 0 ; used in a n o th e r w ay, it leads to im p o rta n t p ractical results.
N evertheless it is sound policy, in building u p a th e o ry in app lied m ath em atics to m ake it logically co n sisten t as fa r as possible. In th e p resen t p ap e r we shall tak e care to s ta te th e ae ro d y n am ic laws in such a w ay as to avoid logical inconsistency.
A p a rt from th e th o ro u g h tre a tm e n t of th e th e o ry of th e ae ro d y n am ic force system in sections 3 an d 4, th e following featu res of th e p re sen t p ap e r m ay be sum m arized here.
T h e exact eq u atio n s of m otion of th e shell (in d ep e n d en t of a n y aero d y n am ic h y pothesis) are given a v ery co m p act form in (2.6). In section 5 it is show n how th e ae ro d y n am ic fu n ctio n s m ay be found from high freq u en cy p h o to g rap h s of a shell. Such o b s e rv a tio n s sh o u ld p ro v id e th e u ltim a te t e s t of th e v a lid ity of th e a e ro d y n a m ic
m e th o d . In view of th e success of th e c ru d e r ju m p c a rd m e th o d of F o w le r e t a l., it seem s p ro b a b le t h a t th e a e ro d y n a m ic h y p o th e s is is v a lid , a n d , if so, th e p ro posed m e th o d of o b s e rv a tio n sh o u ld g iv e us all in fo rm a tio n re q u ire d c o n c e rn in g th e a e ro d y n a m ic fu n c tio n s.
T h e re a r e th r e e c o n d itio n s fo r th e s ta b ility of a s p in n in g sh ell (se ctio n 7), b u t th e y a r e to o c o m p lic a te d to in te r p r e t in th e g e n e ra l case. If M a g n u s effects a re a b s e n t (se ctio n 8), th e y beco m e m u ch sim p le r, a n d in fa c t th e r e is th e n j u s t one s ta b ility c o n d itio n (8.19). In th is c o n d itio n th e effect of th e p o sitio n of th e m ass c e n te r is sh o w n e x p lic itly . T h e c o n d itio n is s tro n g e r t h a n th e u su al c o n d itio n (8.13b) b a se d on th e s ta b ility f a c to r; a shell w h ich is c o n sid e re d s ta b le on th e b a sis of th e u su a l c o n d itio n m a y in fa c t be u n s ta b le . W e a re v e r y m u ch in d e b te d to P ro fe sso r E . J . M c S h a n e fo r h is c ritic a l c o m m e n ts on th is p a p e r in its o rig in al fo rm . H e h a s in fo rm e d us t h a t th e e x iste n c e of seco n d s ta b ility c o n d itio n , s tro n g e r th a n th e u su a l one, h a s a lre a d y been p o in te d o u t b y R . H . K e n t (R e p o r t N o. 85, B a llistic R e s e a rc h L a b o r a to r y ) . T h is c o n d itio n is im p lic it in th e p a p e r b y F o w le r e t a l. (1.332, e q u a tio n 3.6234, a n d 4 .1 2 ); th is is d iscu ssed in sec tio n 10, w h e re th e ir m e th o d is b r o u g h t in to line w ith th e m o re g e n e ra l m e th o d of th e p re s e n t p a p e r.
S om e w ell k n o w n fa c ts a re con firm ed b y th e o ry in se c tio n 9. F o r a s ta b le shell, a f te r th e o sc illa tio n s h a v e been d a m p e d o u t, th e ax is of th e shell a lw a y s p o in ts a b o v e th e tr a j e c to r y a n d to th e r ig h t if th e sp in is rig h t-h a n d e d . T h e p h e n o m e n o n of tr a ilin g is e x p la in e d ; th e ax is of th e sh ell tu r n s d o w n w a rd a t a r a te a p p r o x im a te ly e q u a l to th e r a te of tu r n in g of th e ta n g e n t to th e tr a je c to r y .
D r if t also is d isc u sse d in se c tio n 9. A g e n e ra l c o n d itio n (9.17) is o b ta in e d for s ta n d a r d d r if t, i.e ., d r if t to th e r ig h t fo r rig h t-h a n d e d sp in . W h e n we specialize to subsonic velocity an d flat tra je c
tory, th is condition simplifies to (9.20).
W hen th e num erical values of Fow ler e t al. are inserted, th is in eq u ality is liberally satisfied, so th a t th e p resen t th eo ry is in ag reem en t w ith th e ob served facts.
2. E xact equ atio n s of m otion. We shall now develop th e eq u atio n s of m o
tion of a shell in conv en ien t form . No a s
su m p tio n is m ade here re g ard in g the aero d y n am ic forces, a n d th e only a s
su m p tio n regardin g th e shell is th a t it has an axis of d y n am ic sy m m etry (i.e., th e m om ental ellipsoid a t th e m ass cen
te r is a spheroid). T h u s our equ atio n s w ould apply, for exam ple, to a hom o
geneous projectile of square section or to a bom b w ith th ree or m ore fins, placed sym m etrically.
W e shall use th e following n o tatio n , Fi g. 1
th e m otion being referred to a N ew tonian reference sy stem :
204 K . L. N IE L S E N A N D J. L. S Y N G E [Vol. IV, N o. 3
0 = m ass c e n te r of shell,
>» = m ass of shell,
A , C — tran sv e rse an d axial m om ents of in ertia a t 0, q = velocity of 0,
oj = an g u lar velocity of shell,
h = an g u lar m om entum of shell a b o u t 0,
F = v ec to r sum of aero d y n am ic forces ac tin g on shell, G = m o m e n t of a e ro d y n a m ic forces a b o u t 0 ,
F ' = w e ig h t of shell.
T h e n th e e q u a tio n s of m o tio n are
;;zq = F -j" F ', li = G. (2.1)
W e in tro d u c e a r ig h t-h a n d e d u n it o rth o g o n a l tr ia d , i, j, k, fixed n e ith e r in sp ac e n o r in th e shell (F ig. 1). W e ta k e k alo n g th e a x is of th e sh ell, a n d i, j p e rp e n d ic u la r to k , b u t th e final choice of i, j is d e fe rre d fo r th e p re s e n t. L e t i î be th e a n g u la r v e lo c ity of th e tr ia d .
W e m a y now re so lv e th e v e c to rs a s follow s:
q = «i + + tek,
0) = oui + 0)2j + 0)3k,
a = Oii + + n 3k,
h = .4o)ii + /lo)2j + Co)3k, F = F ji + F é + Fz k,
G = Gii + C2j + C3k,
F ' = F i i + F i 3+ F i k.
(2.2)
C le a rly = fl2 = W2-
In sc a la r form th e e q u a tio n s of m o tio n (2.1) th e n re a d m{u — v&z + wco2) = F i + F{ , m(v — wioi + tiQz) = F 2 + F i , m(w — «to2 + do)j) = Fz + F I , A (¿01 — 03o$lz) T Cu30)2 = G1, A(¿02 T Wlilg) — Co)36)1 — G2, Cw 3 = G3.
I t is now c o n v e n ie n t to in tro d u c e co m p lex v a ria b le s . W e w rite
(2.3)
(2.4)
u + iv = f,
0)1 + 10)2 = V, F, + iFo = F, Ci + jGz = G, Fi + iFl = F'
( 2 . 5 )
W e m u ltip ly th e second e q u a tio n of (2.3) b y i a n d a d d it to th e first, a n d deal sim ila rly w ith th e e q u a tio n s (2.4). T h u s we re d u c e th e e q u a tio n s of m o tio n to th e f o r m :
T h e se e q u a tio n s a r e e x a c t; no a p p ro x im a tio n s h a v e been m ad e.
3. T h e g e n e ra l aero d y n am ic h y pothesis. W h a t is here set dow n is p ro b a b ly a little m ore g e n e ra l a n d e x p lic it th a n p re v io u s s ta te m e n ts a b o u t a e ro d y n a m ic force sy ste m s. T h e re is n o im p lic a tio n t h a t th e h y p o th e s is is p h y s ic a lly a c c u ra te in all cases. All we ca n ho p e is t h a t d e d u c tio n s from th e s e a s s u m p tio n s le a d in s u it
a b le cases to re s u lts in fa ir a g re e m e n t w ith o b s e rv a tio n . B u t it seem s b e s t to m a k e th e h y p o th e s is m a th e m a tic a lly clear.
F ir s t w e c o n sid e r a fluid, a t r e s t o r in m o tio n . W e a re n o t p a r tic u la r ly con- e rn e d w ith th e p ro p e rtie s of th e fluid. T h e im p o r ta n t th in g is t h a t i t defines
(i) a s c a la r field of d e n s ity p;
(ii) a s c a la r field of local so u n d v e lo c ity c;
(iii) a v e c to r field of v e lo c ity W.
T h is la s t field d efin es tw o o th e r v e c to r fields, v o r t i c i t y ( V = l / 2 r o t W ) a n d a c c e le ra tio n ( a = d W /r f i) .
U su a lly in b a llis tic s we d ea l w ith th e s t a t i c case in w h ich W = 0 a n d p, c a re fu n c tio n s of h e ig h t o n ly . A m o re a c c u r a te m o d e l is t h a t in w h ich W is h o riz o n ta l, b u t in d iffe re n t d ire c tio n s a t d iffe re n t h e ig h ts to allow fo r ch a n g es in th e d ire c tio n of th e w in d w ith v a r ia tio n of h e ig h t.
N ow su p p o se we w ish to in v e s tig a te th e m o tio n of a solid th ro u g h th is fluid.
T o t r e a t th e p ro b lem a d e q u a te ly we sh o u ld of co u rse co n sid er th e d is tu r b a n c e p ro d u c e d in th e fluid b y th e solid. B u t we d o n o t do th is . W e use th e fluid m e re ly to c o m p u te fro m its u n d is tu r b e d m o tio n th e a e r o d y n a m ic fo rc e s a c ti n g on th e s o lid .
L e t 0* be th e c e n tro id of th e so lid , i.e ., th e p o s itio n its m ass c e n te r w ou ld o c c u p y w ere th e solid of u n ifo rm d e n s ity . L e t th e m o tio n of th e solid be d e sc rib e d b y th e v e lo c ity q* of 0* a n d th e a n g u la r v e lo c ity o>*.
T h e b a sic h y p o th e s is is th e n a s follow s:
A e r o d y n a m ic hypothesis: T h e a e ro d y n a m ic force sy ste m e x e rte d on th e solid b y th e fluid c o n sists of
(i) a n a e r o s ta tic fo rce;
(ii) a n a e ro k in e tic force s y s te m . The aerostatic force a c ts a t 0* a n d e q u a ls
f + ¿£123 - i m = (F + F')/m,
1) + ¿7,i23 - ¿CV,7) = G / A , (C' = C /A ),
w — jiW 2 T vo)i = (F3 T Fi )/m,
d>3 — Gs/C.
(2 .6)
pFo(a - P ) (3.1)
w here p is th e d e n s ity of th e fluid a t 0 * , V 0 is th e v o lu m e of th e solid, a n d P is th e b o d y force p e r u n it m ass a c tin g on th e fluid a t 0*. (N o te t h a t if a = 0 a n d P is g r a v ity , th is is sim p ly th e A rc h im e d e a n b u o y a n c y .) The aerokinetic force s y s
206 K. L. N IE L S E N A N D J. L. S Y N G E [Vol. IV, N o. 3
tem is re p re s e n te d b y a force F* a t 0 * a n d a c o u p le G*; th e se a re fu n c tio n s of p a n d c a t 0* a n d of th e v e c to rs
q* - W, <o* - V. (3.2)
If q* = W a n d w * = V , th e n F* = 0 a n d G* = 0.
H e n c e fo rth w e sh all a ssu m e W = 0, a n d so F*, G* d e p e n d o n ly on p, c, q*, w*, w hile th e a e r o s ta tic force is —p F oP . If we w ere d isc u ssin g th e a e ro d y n a m ic s of a d irig ib le , th e a e r o s ta tic force w o u ld be v e ry im p o r ta n t. F o r a sh ell it is q u ite tr iv ia l a n d we sh all o m it it a lto g e th e r .
T h u s fo r o u r p u rp o se s th e a e ro d y n a m ic force sy ste m c o n s is ts of th e force F*
a t 0* a n d th e co u p le G*; th e y a re fu n c tio n s of p , c, q*, a n d <o*.
I t w ill be o b se rv e d t h a t o u r b a s e -p o in t 0* h a s been ch osen in a d e fin ite w a y w ith re s p e c t to th e geometry of th e solid, a n d n o t w ith re s p e c t to its mass-dis- tribution. T h is frees o u r law s from th e o b je c tio n ra is e d in th e In tr o d u c tio n to th e law s of F o w ler e t al.
I t is to be n o te d t h a t i t is b y no m e a n s e s s e n tia l to se le c t th e c e n tro id as base p o in t. B u t i t is le a s t co n fu sin g to cho ose, once a n d fo r a ll, a p o in t sim p ly re la te d to th e g e o m e try of th e solid, a n d th e c e n tro id seem s th e m o st n a tu r a l p o in t to ta k e .
4. T h e aerod ynam ic force sy stem for a shell w ith an axis of sym m etry . W e now c o n sid e r a shell w ith a n ax is of a e ro d y n a m ic s y m m e try . B y th is we m e a n t h a t its e x te r io r is a s u rfa c e of re v o lu tio n . W e m ig h t p ro c eed fo r th e p re s e n t w ith o u t in tro d u cin g th e m a ss-d istrib u tio n of th e shell, b u t it seem s sim p ler to proceed a t once to th e case of c o m p le te s y m m e try . W e sh all th e re fo re s u p p o se t h a t th e shell h a s a co m m o n a x is of a e ro d y n a m ic a n d d y n a m ic s y m m e try . All t h a t is s ta te d in se c tio n 2 is th e n v a lid a n d we sh all use th e sam e n o ta tio n .
T h e m ass c e n te r of th e sh ell is a t 0 a n d its c e n tro id a t 0 *. L e t us w rite 0 0* = rk,
a n d
q* = velocity of 0*,
<o* = angular velocity of shell,
F* — vector sum of aerodynam ic forces, G* = m om entofaerodynam icforcesaboutO *.
T h e n
q* = q + o X rk, <■»*=' u , " 1 F* = F , G* = G
+
F X rk.j
In th e n o ta tio n of (2.5) w ith a s te ris k s a tta c h e d to th e sy m b o ls re fe rrin g to 0*, we h a v e in co n se q u en ce
(4.1)
(4.2)
N ow F* a n d G* d e p e n d on q* a n d <o*. I t follow s from th e a e ro d y n a m ic s y m m e try t h a t if th e p a ir of v e c to rs q*, to* is g iv en a rig id b o d y r o ta tio n a b o u t th e a x is of s y m m e try , th e n th e p a ir of v e c to rs F*, G* is also r o ta te d rig id ly a b o u t th e ax is th ro u g h th e sam e an g le. H e n ce th e follo w in g te n s c a la r q u a n titie s a re u n a lte re d by su ch a r o t a t i o n :
F3, G 3,
u*Fl* + v*Ff, v*Fi* - u*Fo*, m*Gi* + v*G*, v* G f -:u*Gt*, )
«1*F}* + co2*P2*, W.//G* - o * F * , u f G f + u * G * } to 2* C ,* - CO!*(.,*. J '
B u t, to w ith in su ch a ro ta tio n , th e v e c to rs q*, co* a re d e te rm in e d b y th e q u a n t i tie s
W, CO,, «** + »«, co,*a + CO,«, » V * + I» W , l l V - ' A ' , * , (4.6) b e tw e e n w h ich th e r e e x ists th e id e n tity
( U * - + + C02* 2) - (if*C O i* + 1»*C02* 2) 2 = (lt*C02* - V * i0 ! * ) 2. ( 4 . 7 )
T h e re fo re th e q u a n titie s (4.5) a r e fu n c tio n s of th e q u a n titie s (4 .6 ); in fa c t, for a shell of g iv en size a n d sh a p e , (4.5) a re fu n c tio n s o n ly of (4.6) a n d th e a ir s c a la rs p, c a t 0*.
W e now w rite
u * F * + v * f 2* = s u v * F * — u * F i* = s 2. (4.8) M u ltip ly in g th e second e q u a tio n b y i a n d s u b tr a c tin g i t from th e first, we g e t
| *p* = s1- i s i , (4.9)
th e b a r d e n o tin g th e co m p lex c o n ju g a te . D e a lin g s im ila rly w ith th e o th e r q u a n t i tie s in (4 .5 ), w e see t h a t
f*F*, |*G*,
V*F*, n*G*, (4.10)
a re co m p lex fu n c tio n s of th e re al q u a n titie s in (4.6).
W e c a n n o t p ro c eed f u r th e r w ith o u t a n a d d itio n a l h y p o th e s is . We shall as
su m e that
F*, F 2*, G*, C2*
are linear f u n c t i o n s of
« * , V * , « x * , Ci)2* .
T h is is c e rta in ly a re a so n a b le a s s u m p tio n w h en th e l a t t e r q u a n titie s a re sm all.
W e can th e n w rite
F* = aiu* + a 2o* + / W -f- ¿ W , ]
r (4-11)
G* = 7 lie* + 7 2w* + 5io>i* + 52o>2*, J
w h e re th e e ig h t com plex coefficients a re fu n c tio n s of iv, W3, p a n d c. W h en we form th e q u a n titie s (4.10) a n d use th e fa c t t h a t th e se m u s t be fu n c tio n s of th e q u a n titie s (4.6 ), w e find a i= fia i, p2 = i&i, e tc ., a n d so
F* = £*T* + 1 G* = f P '* + r,*Q'*> I
w h ere P * , Q*, P '*, Q1* a r e com plex fu n c tio n s of w, w3, p, c.
(4.12)
208 K. L. N IE L S E N A N D J. L. S Y N G E [Vol. IV, N o. 3
T h e c o m p o n e n ts F3, G3 a re fu n c tio n s of th e q u a n titie s (4.6 ). We shall assum e that they are f u n c t i o n s only o f w, oj3, p, c. T h is also is a p la u sib le a s s u m p tio n w hen u*, v*, co*, io* a re sm all.
T o sum u p : There are ten real aerodynamic fu n c t i o n s of w, co3, p, c, contained i n the set
P*. Q*,
Q'*.
F i C' * C/3 . (4.13)L e t us see w h a t th e s e a s s u m p tio n s a m o u n t to in th e case of a sh ell in a w in d - tu n n e l. W e t h i n k of th e shell a s m o v in g a n d th e a i r a t r e s t. W e p u t
V* = 0, « ! * = CO 2* = C03* = 0, F f + i F f = u*F*, G* + iG* = u*P'*.
In th is sim p le ca se w e m u s t h a v e , b y s y m m e tr y sin c e w3* = 0, /?,* = G * = G3* = 0,
F * = u*P*, iG* = u*P'*. (4.14)
a n d (4 .1 2 ) g iv e s
an d so we have
I t is easy to see th a t these equ atio n s im ply th a t (for sm all yaw ), the cross w ind force an d th e m o m ent are propo rtio n al to the y aw . T h is is th e usual assu m p tio n .
W e now pass from th e centroid 0* to th e m ass c e n te r 0 by th e tran sfo rm atio n (4.4). W e get for th e force system F, G on th e shell
6 3 — G3*, (4.15)
(4.16) F = Fy + iF, = {P + nQ,
G - Gi + iGo = £P' + VQ \ u 3
w here P, Q, P ' , Q' are com plex functions of w, co3, p, c, given by P = P*, Q = Q * - irP*,
P ' = P'* + irP*, Q' = Q'* - irP'* + ir(Q* - irP*).
T h is gives th e tran sfo rm atio n of th e aero d y n am ic functions w hen w e pass from th e cen tro id 0* to th e m ass ce n te r 0 . A ctu ally th is is th e tran sfo rm atio n for passage from a n y b ase-point to a n y o th er, provided of course th a t b o th lie on th e axis.
T o show th e real an d im ag in ary p a rts of th e aero d y n am ic functions, we shall w rite (w ith sim ilar eq u a tio n s in asterisk ed form )
P = P i + i P 2, Q = (?! + iQt , P ' = P l + i P i , Q' = Q{ + iQJ.
T h e tran sfo rm atio n (4.16) th en gives P i = P i * ,
P-, = P 2*, Qx
=
Q i *+
r P2*,
(4.17)
&
P I = P{* - rP 2*
rPx*, V
i >
Qi = Qi* + rP l* + r ( - Q t + r P f ) , rPi* + r(Q* + rP *).
H =
Qi = Qi*
(4.18)
T h e m ethod used above for the resolution of the aero d y n am ic force system is n o t the usual one. T h re e im p o rta n t v ectors are involv ed: k the axis of th e shell, q the v e
locity of th e m ass center, o th e a n g u la r velocity. In resolving vectors, it is necessary to pick o u t one of these th ree as a fu n d a m e n tal v ecto r an d build a basic tria d on it.
T h e tra d itio n a l plan is to pick o u t q as fu n d a m e n tal an d ta k e k as a seco n d ary vector, so th a t q a n d k to g e th e r give one of th e planes of th e basic triad . R esolution of F along q an d perp en d icu lar to q in th is plane gives the usual d ra g an d lift. H ow ever convenient th is m ay be for w ind-tunnel w ork in w hich q is fixed w hile k is altered , it certain ly ap p e ars less convenient th a n th e m ethod of th e p resen t p ap e r for a sim ple m a th em atica l form ulation of th e problem of the sp inn in g shell. T h ere is a fu rth e r objection to th e usual plan ; the direction of q depends on th e m ass center.
T h e conventional term inology does n o t su it th e p resen t resolution. T h e following is suggested. T h e a ste risk in d icates t h a t th e centroid is used as base-point. T h e sam e n o tatio n w ith o u t asterisk s refers to th e m ass center.
u*i + u*j = cross velocity, wk = axial velocity, coji + w2j = cross spin,
w3k = axial spin.
P1* | £* | — cross force due to cross velocity ( —),
£* | = M agnus force due to cross velocity ( + ) ,
(4.19)
Q* I v* I = M agnus force due to cross spin ( + ) , Q* | V* | = cross force due to cross spin ( + ) ,
F3 = axial force ( — ).
P l* | £* \ — M agnus torque due to cross velocity ( —),
P i * | s* I = cross torque due to cross velocity ( — ), 7j* | = cross torque due to cross spin ( —),
= M agnus torque due to cross spin ( + ) , G3 = M agnus axial torque ( —).
(4.20)
Q l * \ Q i *
I
v *(4.21)
It is a consequence of s y m m e try th a t w here th e w ord “M ag n u s” is included above, th e q u a n tity in question changes sign w ith a>3; w here th e w ord “M ag n u s” does n o t occur, th e q u a n tity in question does n o t change sign w ith m3. F o r u n iform ity, we h ave called th e axial (viscous) to rq u e “M ag n u s” ; th ere is ju stification for th is in th e fa ct th a t it is th e viscous to rq u e th a t sets up the circulation which is responsible for the o th e r M ag n u s effects. T h e signs in paren th eses in dicate probable signs of th e v ario us q u a n titie s w hen <a3 is positive, assum ing a ce n te r of pressure in fro n t of th e cen troid .
Since
| r | = 9* sin (q*. k ). I V* | = “ sin (to, k), (4.22) it is clear th a t the usual sine law of v aria tio n is im p licit in (4.20), (4.21). B u t since we suppose th e angles in question to be sm all, th e sine, ta n g e n t an d circu lar m easure are n o t distinguishable.
I t is conv en ien t to in tro d u ce positive dim ensionless aero d y n am ic functions, a s is done by F ow ler e t al. So we w rite, pay ing a tte n tio n to dim ensions an d signs,
210 K . L. N IE L S E N A N D J. L. S Y N G E [Vol. IV, N o. 3
p * = - pfl2w/i*, A, II p a sii>3f * , Q * = p uW * , rO II p a 3w gt*,
il*
- pa'o)3f { * , P i* = — p a 3w f i * , Q(* = - p a 'w g ! * , Qi* = ptPoJsgi*
(4.23)
H ere p is th e air-d en sity and a the rad iu s of the cross section of th e shell. T h e fu nc
tions (/*, g*) depend certain ly on w /c, an d possibly also on ao>3/ c an d th e R eynolds n um ber. T h e above eq u atio n s m ay be regarded as definitions of th e eig h t aero d y n am ic functions (/*, g*), w hich arc analogous to th e / t , f.v, etc. of Fow ler e t al. T o th e above eq u a tio n s we m ay add
F3 = — pa2w2f 3, G3 = — pa4w u3g3, (4.24) w here f 3 an d g3 are dim ensionless; f 3 is th e usual d rag except for th e slight difference th a t we resolve along the axis of the shell an d tre a t w as basic instead of 3*.
As the n o ta tio n is necessarily som ew hat com plicated, let us su m m arize as follows:
A skerisked q u a n titie s refer to the centroid, un astcrisked to the m ass center.
T h e ae rodynam ic force system is d enoted by
F* = F ? + if'-*, G* = G * + iG*, F3, G3.
T h ere are ten real aero d y n am ic functions con tained in the set
p *, Q*, P ’*, Q’*, f„ G3i
an d these m ay be expressed in term s of th e ten positive dim ensionless ae ro d y n am ic functions
/t
1ya >
S1>
,? 2>
J1>
J2j
g1,
g? , j3,
g3- T h e sam e n o tatio n m ay be used w ith reference to th e m ass center, b u t since th e aero dy n am ic force system has n o th in g to do w ith th e m ass cen ter as such, th e asterisk ed q u a n titie s are th e m ore fu n d a m e n tal. If we wish to pass from 0* to 0 , we m u st tra n s form b y (4.18) an d (4.23). T h u s f * = f u f * = / 2, f 3* = f 3, g3* = g 3, b u t the o th e r functions change.One m ore n o ta tio n will be intro d u ced for convenience in (6.4).
I t is clear from (4.20), (4.21), (4.23) th a t if the dim ensionless ae ro d y n am ic fu n c
tions (/*, g*) are co n stan ts, we h ave th e following p ropo rtio nalities, 5 d eno ting th e sm all y a w :
cross force due to cross velocity « w25, cross torque due to cross velocity <x w28, axial force « w2, axial torque «
(4.25)
T h e first th ree of these are in ag reem en t w ith experim en t for subsonic velocities-—the effects v ary as th e sq u are of th e velocity. T h e last (axial torque) requires com m ent.
T h e form of G3 in (4.24) agrees w ith F ow ler e t al., b u t one m ay ask w hy (a p a rt from th e th eo ry of dim ensions) th e fa c to r w should be present. T h e following is a pos
sible exp lan atio n . T h e ro ta tio n of th e shell g en erates a ro ta tin g w ake. If th is w ake has, th ro u g h o u t, the sam e spin as th e shell, it has an g u lar m o m entum %Trpa4u3 per u nit length. In u n it tim e a length w of w ake is g en erated , an d so, by the conserv atio n of
an g u lar m om entum , th e ra te of loss of an g u lar m o m entu m of the shell is
— G 3 = Wpcï'woos.
T h is a rg u m e n t n o t only confirms the form G3 of (4.24) ; it gives
g s = k - ( 4 . 2 6 )
A crude a rg u m e n t of this so rt m u st be accepted only provisionally in th e absence of experim ental check.
5. D e te rm in atio n of th e aerodynam ic fun ction s by ob serv atio n. Fow ler e t al.
stressed th e im p o rtan ce of avoiding th e sim ple em pirical assu m p tio n s previously em-
ployed. As in th e case of th e d ra g function, it is necessary to d eterm in e th e ae ro d y n am ic functions experim entally . W h a t follows is a refinem ent an d generalization of th e ju m p card m eth o d of Fow ler e t al. Unless th e re are technical difficulties, o r unless the basic aero d y n am ic hyp o th esis is w rong, th e following m eth o d should yield all th e aero d y n am ic fu nctions q u ite sim ply, except perh ap s g3, and no d o u b t a m ethod could be devised for it also.
L et a shell be fired h orizontally and ob serv atio n s m ad e of it not long a fte r it leaves the m uzzle. T hese o bservation s consist of high-frequency p h o to g rap h s, one se t of p h o to g rap h s being tak en v ertically an d the o th e r set ho rizo n tally from th e side. T hese p h o to g rap h s show successive positions of th e shell a t sh o rt in te rv a ls of tim e.
W e now tu rn to th e exact eq u atio n s of m otion (2.6). T h e re is som e in determ inacy in these because we have n o t y et chosen the v ec to r i definitely. L et us choose it in the v ertical plane thro u g h th e axis of th e shell (k), pointin g dow nw ard (F igure 2).
212 K . L. N IE L S E N A N D J. L. S Y N G E [Vol. IV, N o. 3
T hen
F' = mg cos 0, (5.1)
an d th e first tw o eq u atio n s of (2.6) m ay be w ritten
F = ;«(£ + t£il3 — iwri) — mg cos 0, j G = A {t} -{- 23 — i C'033). J
(5.2)
T hese eq u atio n s are exact. W e m ay p u t cos 0 = 1, since th e axis of th e shell is ap p ro x i
m ately h orizontal. T h en il3 = 0 by (6.2).
Now m, A , C’ are know n for th e shell; w m ay be found from the o b servatio ns or otherw ise (m uzzle velocity), an d o>3 deduced from th e rifling. T o find £, 77 as fun ctio ns of t, it is m erely necessary to m easure on the p ho to g rap h ic p lates th e linear displace
m ents of the m ass cen ter an d th e an g u lar displacem ents of th e axis of th e shell, cor
responding to th e sh o rt in terv a ls betw een successive pho to graph s. S m ooth g raphs m ight be m ade show ing u, v, «i, co2 as functions of t o r th e com plex q u a n titie s £, 77 m ig h t be p lo tte d on an A rgand diagram w ith the values of t m ark ed in. In a n y case it should n o t be difficult to o b ta in £ an d 17 also as fu n c tio n s of t from th ese graphs.
W hen these fu n ctions are in serted in th e rig h t-h an d sides of (5.2), we have F an d G as fun ctions of t. By (4.15) we have
If we use tw o values of t, each of these equ atio n s yields tw o com plex equ atio ns, an d from them P , Q, P ', Q' can be found. H ere we h ave a good te s t of th e aero d y n am ic hypothesis, for th e valu es of P , Q, P ', Q' should be in d ep en d e n t of th e p a rtic u la r in
sta n ts chosen.
I t m ay be advisable, as a refinem ent, to allow for th e decrease in w betw een the tw o in sta n ts in question. T his can easily be done from o u r know ledge of th e d rag function.
B y re p eatin g th e experim en t on th e sam e shell, b u t using different m uzzle veloci
ties an d riflings, we o b tain P , Q, P ', Q' as functions of w an d w3.
T h e next step is to transform from th e m ass ce n te r to th e centroid. T h is is done b y (4.16), an d we o b tain P*, <2*, P '* , Q'* as functio ns of w an d w3. F in ally , th e dim en- sionless ae ro dynam ic functions (/*, g*) are found from (4.23).
I t should be stressed th a t these last functions are c h a rac te ristic of th e fo rm of the shell an d co m pletely in d ep en d e n t of th e m ass d istrib u tio n . Indeed, to a certain ex
te n t th ey will be in d ep en d en t of the size of th e shell, b u t th is m u st be accepted w ith 6. P la n of solution an d partial lin earizatio n of th e eq uation s. W e now in tro d u ce fixed axes OoX0yoZt>, O0Z0 being directed v ertic ally upw ard. L et 9 be th e inclination of k to the ho rizontal (F igure 2), an d 4> th e inclination of th e ho rizon tal projection of k to Oox0. W e h ave alread y m ade th e v ec to r i definite in section 5. W e hav e th en
( P + vQ = F, £P ' + vQ' = G. (5.3)
caution.
F ' = mg cos 9i — mg sin Ok, j
£2 = — <j> cos 0i — 0j + <f> sin 0k. J (6.1) H ence
■t] = — cos 6 + iO), = <j> sin 0. (6.2)
W e su b stitu te from (4.15) in (2.6), an d th e eq u atio n s of m otion become
| 3 — iwy — £ W + i]Y -f- g cos 8, 17 + i’9^3 — iC'oi3t] — £X' +
rjY',
where
w — nut + nwj = F z/m ■
¿>3 = Gs/C,
g sin 8,
(6 .3)
X = P / m , Y = Q/m, X ' = P ' / A , Y ' = Q ' / A . (6 .4) If we s u b stitu te from (6.2) for 17, il3 an d regard X , Y, X ' , Y ' , F3, G3 as known fun c
tions of w, W3, p, c, we have six real eq u atio n s for th e d ep e n d en t v ariab les u, v, w, 6, <j), a>3. B u t unless we assum e p, c to be co n stan ts, we m u st b rin g in fu rth e r eq u ation s.
L et us assum e them to be functions of height (s0) only. By resolution of velocity we h ave
-t'o + iyo = («■ sin 6 + iv + w cos 6)6**,
¿0 = — M cos 8 + w sin 6. (6 .5)
W hen th e la s t of these eq u atio n s is associated w ith (6.3), we h ave seven real e q u a tions for seven unknow ns, nam ely, those sta te d ab o v e an d z0. W hen th e y have been solved, th e tra je c to ry of th e m ass c e n te r is given by (6.5).
W e now m ake th e following tw o assu m p tio n s: (i) th e v ertical plane throu gh the axis of th e shell tu rn s slow ly; (ii) th e angle of yaw is sm all. T h e first assum p tio n im plies t h a t <j> an d hence is sm all; the second im plies th a t £/w is sm all. On ac co u n t of th e sm allness of we re je ct th e second term s in th e first tw o eq u a tio n s of (6.3), an d on ac co u n t of th e sm allness of !-/w we re je ct th e second an d th ird term s in the th ird eq u atio n .
O ur p a rtia lly linearized eq u atio n s now read
where
Î — iwij = £X + 7)7 + g cos 8, v - ic'wsv = + v r ,
w = F 3/ m — g sin 6,
«3 = g3/ c ,
j) = — cos 8 + id).
(6.6)
(6 .7) 7. T h e stability of a spinning shell. In discussing rapid oscillations of th e shell, we tr e a t w an d o>3 as c o n sta n ts in th e first tw o eq u a tio n s of (6.6). C on sequ ently X , Y, X ' , Y ' are co n stan ts. In rap id oscillations differentiation w ith respect to t g re a tly increases th e im p o rta n c e of a term . H ence w e shall t r e a t cos d as a c o n s ta n t in th e first eq uation of (6.6); th e term corresponding to a sm all change in 9 will be neg
ligible in com parison w ith th e term s in 77.
W e h av e th e n lin ear eq u atio n s w ith c o n sta n t coefficients, w hich hav e solutions of th e form
where aa, a 2 are th e roots of the eq uation
214 K. L. N IE L S E N A N D J. L. S Y N G E [Vol. IV, N o. 3
a2
-(iCV, +
X + Y ')a + ¿(C'co3X - w X ' ) + X Y ' — X ' Y = 0, (7.2) andA , = - — cos e(iC'w3 + 7 ') , E
Bs =
E cos 0 ■ X ' , (7.3)
£ = i(C'u3X - w X ') + X Y ' - X ' Y .
T h e condition for sta b ility is th a t b o th roots of (7.2) should h ave n on -p ositiv e real p arts.
If we w rite
K i = AT + Y { , AT = C'ctiz + AT + IT ',
AT = - C'o>3AT + wAT' + ATIT - ATF2' - X I IT + AT'IT, AT = CV,AT - w.Y( + ATIT' + ATIT' - AT'IT - AT' IT, then (7.2) becomes
a 2 - (AT + fAT)a + (AT + ¿AT) = 0.
T h e condition for sta b ility m ay be w ritte n
A i -}- f cos x ^ 0- w here f , x arc defined by
(7 .4)
f = (A i — A 2 — 4AT) + 4(ATAT — 2A 3) f à 0, f sin 2X = 2(AiAT - 2AT),
f cos 2x = A i — K l — 4A 3, ¿rr â X S è rr.
(7.5)
(7.6)
(7.7)
I t is im m ediately ev id e n t th a t th ere is in sta b ility if AT > 0 . If AT SO, th en th e con
d itio n (7.6) is e q u iv a le n t to
AT à f cos' x.
or
2A j ^ f (1 + cos 2X).
On s u b s titu tin g for f 2 cos 2x from (7.7), th is becom es
AT + AT + 4AT S r2-
(7.8)
(7.9)
(7.10) T h u s th ere is in stab ility if AT SO, A T + A T + 4 A 3 < 0 . If AT ¿ 0 , K } + K £ + 4 A 3S 0 , th e condition (7.10) is eq u iv ale n t to
(AT + AT + 4AT) à i , (7.11)
an d , on su b stitu tio n from (7.7), this becomes
K i K3 + K \ K iK i - K \ à 0. (7.12) T o sum up, the motion of the shell is stable if, and only if, the following three condi
tions are all satisfied :
K i S 0, K \ + k \ + 4 K3 à 0, k \ k3 + K lK2K i - k \ â 0.
(7.13a) (7.13b) (7.13c) T h e K ’s are given by (7.4).
T hese conditions are m ore general th a n an y given previously.
If th ere is stro n g s ta b ility (i.e., if the real p a rts of a it a2 are neg ativ e an d large), then th e first term s in (7.1) die aw ay quickly. In fact, th e rap id oscillations are d am p ed ou t, an d we arc left w ith
an d also v ~ — cos 6-\-i9).
In (7.14), (7.15) and th e last of (6.5) we have seven real eq u atio n s for the seven q u a n titie s u, v, w, 6, </>, o>3, z0- it is a function of w a n d co3 as in (7.3); it also involves z0, since th e properties of th e a ir depend 011 z0 an d ae ro dy nam ic functions X , Y, X ' , Y' depend on th e prop erties of the air. T h e above eq u atio n s determ in e th e m otion of the stab le shell.
W e no te th a t the equ atio n s (7.14), (7.15) are sim ply (6.6) w ith the term s £, 1? d e
leted. T o te s t w h e th e r this tre a tm e n t is valid, we should solve (7.14), (7.15) for £, V*
calculate £, r) by differen tiatin g these solutions, an d com pare these calcu lated values w ith th e o th e r term s in (6.6). T h ey should, of course, tu rn o u t to be sm all.
8. S tability in th e ab se n ce of M ag n u s effects. If we ac cep t the linear law (4.11), th e aero d y n am ic force system (4.13) is the m ost general possible. As w e shall see in section 10, the force system of Fow ler e t al. is a special case. T h e system (4.13) co n
tain s ten real functions, an d it ap p e ars im possible to m ake a n y d ed u ctio n s of physical in te re st w ith o u t in tro d u cin g some sim plifications. W e shall re ta in a force system a little m ore general th a n th a t of Fow ler e t a l.; o u r system satisfies th e fu n d am en tal condition of in variance w ith respect to shift of m ass center, w hereas th eirs does not.
L et us refer to (4.20), (4.21), an d assum e th a t all M agnus effects vanish, except this m eans th a t
t = - — cos 0 - ( iC u i + Y')
E (7.14)
t] = - c o s e - x ' . E
W ith these we associate th e last two eq u atio n s of (6.6), viz.
w = F 3/ m — g sin 6, ]
¿3 = G3/C, j (7.15)
P 2* =
Q*
=Pl* = Q±* =
0. (8.1)216 K . L. N IE L S E N A N D J. L. S Y N G E [Vol. IV, N o. 3
T h is leaves us w ith fo u r real aero d y n am ic functions, in add itio n to F3 an d 6V P i* < 0, 0 2* > 0, P i * < 0, Q{* < 0. (8.2) T h ere can be no d o u b t th a t these in eq u alities are physically valid.
W e now tran sfo rm to th e m ass ce n te r 0 by (4.18). W e find
p 2 = Qt = p i = Q> = o. (8.3)
T h u s th e M ag n u s effects do n o t re a p p e a r w ith change of b ase -p o in t; in fa ct, the vanishing of M agnus effects is an invariant condition. F o r b ase-p oint O th ere are again ju s t four real aero d y n am ic functions in ad d itio n to F3 an d G3:
P i = Pi*, Qt = 02* - rPi*, P ’ = P i * + rP * ,
Q[ = Q{* + rP-i * + r ( - 0 2* + rPi*).
T h en by (6.4), (7.4) an d (8.3),
AT = X , + Y{ = P f f m + Q U A , A*2 = C'oj 3,
A , = w X i + X i Y l + X IF 2 = — + — (PrQi + P i Q i) ,
A mA
(8.4)
C'cojAT = wi
(8.5)
T h e s ta b ility conditions (7.13) read
X i + I T 'g O , (8 .6a) (C'oj3) 2 + i w X i + (Xx + T / ) 2 + 4 (X 1F i + X 2' Yi) â 0, (8.6b) X i F / (C'oi3y + (X i + F iJ ^ w X / + X i F / + X 2'F 2) è 0. (8.6c) T hese are th e s ta b ility conditions in th e absence of M agn us effects. N ow b y (4.23), (6.4), (8.4), we h ave (since A * = m r2-\-A)
X i =
F , =
X i = - pa^w
m />*, pa3w
m pa3w
( f i * + /.*)
AT + Y { =
X iF ,' + X i Y2 =
p a 4» r r r 2
V i = - - — «(* + - (g2* + /,'* ) + — /.* ,
A L a a- J
4 t e r r T*
— \ g i * + — f a * + / i * m - —
1 L a ma-
p2a6w-
mA (fi*gl* ~ ft*/*'*)-
(8.7)
If we s u b stitu te these expressions in (8.6) we g et sta b ility conditions in term s of the functions (/*, g*). H ow ever, these conditions are som ew hat co m plicated, an d we shall m ake appro xim ations.
T h e f ’s of F ow ler e t al. h ard ly exceed 10 in value. O ur (/*, g*) functions are defined in a slightly different w ay, b u t it certain ly seem s leg itim ate to asse rt th a t th e dim en- sionless q u an titie s
p a 3
e = — / (8.8)
m
are m uch less th a n u n i t y , / stan d in g for an y one of th e (/*, g*) functions. T h en it is clear th a t
(AT + I T ) 2, AT 17 + AT' IT
are b o th sm all relativ e to w X { . C onsequently o u r s ta b ility conditions (8.6) m ay be sim plified to
AT + 17 ^ 0, (8.9a)
(C 7 3)- + 4wA7 7 0, (8.9b)
A T I'/(C V ,)- + (X a + Y { y -w A7 ^ o. (8.9c) I t will be noticed th a t IT has d isappeared from the sta b ility conditions in the last approxim atio n. T h is aero d y n am ic function corresponds to cross force d u e to cross spin relative to the m ass ce n te r [cf. (6.4) an d (4.20)]. T h u s it m ig h t be asserted th a t, for th e discussion of s ta b ility in th e absence of M agnus effects, cross force due to cross spin m ay be neglected. B u t th is s ta te m e n t is n o t e n tire ly co rrect, because this cross force c o n trib u te s to th e m om en t Y { , an d IT' rem ains in th e sta b ility conditions.
L et us exam ine th e first s ta b ility condition (8.9a). On su b s titu tio n from (8.7) it reads
— (g2* + // * ) + gl* + ~ f i * ^ 0. ( 8 . 10a)
a map-
If r is po sitive (so th a t th e m ass c e n te r lies behind th e centroid), th is in eq u ality is ce rtain ly satisfied; it is also satisfied for som e neg ativ e range of r. B u t an in tere stin g q uestion arises: C an we m ake th e shell u n stab le by pushing its m ass cen ter forw ard tow ards th e nose? T h is is h ard ly to be expected on physical grou nds, a n d it m ay well be th a t (8.10a) is satisfied for all perm issible values of r, i.e., all v alu es w hich place the m ass ce n te r inside th e shell.
I t is tedious (and p erh ap s of little physical in tere st) to discuss th e o th e r s ta b ility conditions for sufficiently large neg ativ e values of r. W e shall th erefo re assu m e e ith e r t h a t r is positive, or, if it is negative, it is such th a t (8.10a) is satisfied a n d also
T h is is essentially the sam e as th e usual sta b ility fa c to r.5 T h en the second sta b ility condition (8.9b) tak es the fam iliar form
5 ^ 1, (8.13b)
while the th ird condition (8.9c) m ay be w ritten (AT + F i ) 2
5 ^ --- (8.13c)
4 AT IT'
Since the fraction on the rig h t is never less th a n u n ity , this condition replaces (8.13b).
L e t us su b s titu te in (8.13c) from (8.7) an d sum up as follows:
S t a b i l i t y c o n d i t i o n . T h e following assu m p tio n s are m ade:
(i) M agnus effects are negligible (except th a t Gz m ay exist).
(ii) T h e q u a n titie s « of (8.8) arc v ery sm all.
(iii) T h e m ass cen ter is behind th e centroid, or, if in fro n t, its n eg ative co ordinate r is such th a t (8.10a) is satisfied and also
218 K. L. N I E L S E N A N D J. L. S Y N G E [Vol. IV, N o. 3
f i * + - h * > 0, a
gi* + - (g-T+ m + —j ? > o.
a a-
(8.14)
T h en th e m otion of the shell is stab le if, and only if,
* > — [g» * + (r/a)(gt* + f i * ) + {A*/ma*)h*Y S = 4/1 + ( r / a){g* + f i * ) + (r2/ o 2) / * ] ’ w here 5 is as in (8.12), or eq u iv ale n tly
r 2 - C a>3
J = ---p---A = A* - m r \ (8.16) 4p«L-l w~ [fi* + ( r /a )f* ]
T o show th e dependence on r m ore explicitly, we intro d u ce th e dim ensionless q u a n tity
C 0)32
p = --- > (8.17)
4 pasmw- so th a t
P = s ~ - ( f i * + - f A . (8.18)
ma* \ a /
T h en the sole condition fo r stability reads
> (f i * + (>•/«)/.*) [g/* + {r/a){gt* + / n + Q1*/»!«2)/.*]2 4 /.* [g i'* + (r/a)(gz* + /{*) + (f2/ a 2)/i*]
5 T . J. H ayes, Elements of ordnance, J. W iley and Sons, N ew York, 1938, p. 418.
Since A * is th e tran sv e rse m o m en t of in ertia a t the centroid, th e position of th e m ass ce n te r is involved in this form ula only in the sym bol r.
W e see therefore th a t the usually accepted criterion for s ta b ility (8.13b) is no t th e tru e one; it m u s t be replaced by one of th e in equ alities (8.13c), (8.15) o r ( 8 .19), which are of course eq u iv ale n t to one a n o th er. As we rem ark ed in th e In tro d u c tio n , th e existence of a second condition for sta b ility has been noticed b y R . H . K e n t. W e shall refer to sta b ility again in section 10.
9. The trajectory of a stable shell in the absence of Magnus effects. L e t us a s
sum e, as in th e preceding section, th a t M agn us effects are ab se n t, ex cep t t h a t G3 m ay exist. T h en , using (8.3) an d (6.4) w ith (7.14), we g et for th e tra je c to ry of a stable shell, a fte r the d istu rb an ce has been dam ped out,
£ = - — cos 0(fC'co3 + Y { ), E
g
.
it = i — cos 0 • X i , 71 = — <j) cos 0 — id.
E
(9.1)
H ere E is as in (7.3); let us m ake the appro x im atio n ind icated above (8.9), so th a t E = w X2 T iC'oi3X i.
S p littin g (9.1) in to real a n d im ag in ary p a rts we g et g
(9.2)
it =
E \ - g
e V
cos 0[AT(C'w3)'- + w X l IT
cos 0'CVlW X i ,
(9.3)
(w here we h ave d ropped a term X \ Y{ in com parison w ith w X i ) an d
= — E |
g E 2
C ' ^ X i X u
cos 0- w { X i ) 2.
(9.4)
W e shall assum e, as in section 8, th a t X i , X i , Y ( arc all negative. F u rth e r, since th e shell is stable, we have as in (8.9c)
B u t
an d therefore
x , i t (C'coj)2 + (Xi + Y iy -w X i a o.
(AT + Y i Y > I T 2, x i < 0,
A T F / i c v .) 2 + v r - w x i a o.
(9.5)
(9.6) I t follows a t once from (9.3) t h a t u is positive. T h is m eans th a t the nose of the shell points above the trajectory.
220 K . L. N IE L S E N A N D J. L. S Y N G E [Vol. IV , N o. 3
From (9.4) we see th a t <¿><0 if co3> 0 . T h u s fo r positive (right-handed) spin the vertical plane through, the axis o f the shell turns to the right? F o r n eg ativ e spin it tu rn s to th e left.
These tw o facts arc well know n to be tru e in practice.
T h ere rem ain tw o o u tsta n d in g physical facts to explain. T hese are (i) th e trailin g of the shell along th e tra je c to ry , (ii) th e d rift.
We see from (9.4) th a t 9 is negative, i.e., the inclination of th e axis of th e shell to the horizontal decreases stead ily . B u t does it decrease a t th a t ra te req u ired for trailing? W e m u st be careful to avoid a circular arg u m e n t. W e h av e assumed th a t trailing tak es place— otherw ise the yaw is n o t sm all, an d all o u r arg u m e n ts are based on th e sm allness of th e yaw . W e m u st now verify th a t 9, as given by (9.4), is ap p ro x i
m ately equal to th e ra te of tu rn in g of th e ta n g e n t to th e tra je c to ry of th e m ass cen ter.
T h e th eo ry of th e plane p a rtic le -tra je c to ry gives, on resolution along th e norm al, g cos do
90 = - , (9.7)
w
w here 60 is th e inclination of th e ta n g e n t to th e horizo n tal. T o estab lish th e req uired resu lt, we m u st com pare th is w ith (9.4), an d show th a t
I £ | 2
1— = 1, (9.8)
(■wxty-
appro x im ately . Now b y (9.2), (8.12), (8.7), th is fraction is 2 / C V jV 4s AT
1 + A T I ) = 1 --- V w X l J - w X i
pa3 A /,**
= 1 + 4 s o r •*/ i / / Y /■ - (9.9)v '
m ma2 f 2*' + ( r / a ) / :
T h e last expression here is of th e o rd e r of se, w here e is as in (8.8). H ence, unless th e s ta b ility fa cto r s is v ery g re at, th is expression is v ery sm all, an d the condition of trailin g is ap p ro x im a te ly fulfilled.
I t is in tere stin g t h a t if s is v ery g re a t th e verification breaks dow n, for th is is ju s t w h a t we w ould expect. If, b y som e m echanism , an enorm ous spin were im p a rte d to a shell, th e gyroscopic s ta b ility w ould be so g re a t t h a t th e d irection of th e axis w ould rem ain fixed an d th e shell w ould n o t trail.
T o discuss th e d rift, we w rite dow n (6.5) a g a in :
Xo + i y0 — (« sin 6 + iv + w cos 9)e{*. (9.10) T h is is th e horizontal velocity of th e m ass ce n te r in com plex form . C onsider the com
plex q u a n tity
¿o + iyo ,
a -j- ifi = — ' (9.11)
xa + iyo
I t is obvious th a t th e v ec to r x0+ i y0 tu rn s to th e left if j3 is positive, an d to th e right if d is negative. I t is o u r business to in v estig ate th e sign of /3.
‘ H ayes, op. cit., 420.
