Contents Journal of Ship Research, Volume 2-3

lm

HIP RiESEARCH

ment Acting on a Rankine

rvoia Moving Under a Free Surface

By H. P,. P o n d i

The moment acting on a Rani<ine ovoid moving under the free surface of a fluid of infinite depth is calculated. The solution is carried through two approximations. The results of the second approximation are shown to be in excellent agreement with available experi-mental data.

T H E D a v i d T a y l o r M o d e l B a s i n is c o n d u c t i n g an i n -v e s t i g a t i o n of t h e forces a n d m o m e n t s a c t i n g o n sub-merged bodies m o v i n g near a f r e e sm-face. A s a p a r t of t h i s i n v e s t i g a t i o n t h e h y d r o d y n a m i c a l t h e o r y of t h e p r o b l e m s i n v o l v e d is b e i n g . s t u d i e d . C a l c u l a t i o n s of resistance [ 1 , 2 ] ^ a n d v e r t i c a l f o r c e [ 3 , 4 ] h a v e been m a d e Avhich s h o w reasonable ( t h o u g h b y n o means e n t i r e l y s a t i s f a c t o r y ) agreement w i t h e x p e r i m e n t a l results. H o w -ever, c a l c u l a t i o n s of m o m e n t s h a v e s h o w n l i t t l e or no agreement w i t h e x p e r i m e n t a l results ( f o r example, see discussion o n page 6 of Reference [ 5 ] ) . I n t h e present paper t h e c a l c u l a t i o n f o r t h e m o m e n t a c t i n g o n a R a n -k i n e o v o i d m o v i n g u n d e r t h e f r e e surface of a f l u i d of i n f i n i t e d e p t h is c a r r i e d t h r o u g h t w o a p p r o x i m a t i o n s , a n d t h e r e s u l t s of t h e second a p p r o x i m a t i o n are f o u n d t o be i n excellent agreement w i t h a v a i l a b l e e x p e r i m e n t a l data. The First Approximafion

W i t h t h e u s u a l assumptions of s m a l l w a v e slope, a n d t h a t t h e v e l o c i t y (due t o t h e w a v e m o t i o n ) of t h e f l u i d particles is s u f f i c i e n t l y s m a l l so t h a t t h e square of t h i s v e l o c i t y can be neglected i n B e r n o u l l i ' s e q u a t i o n ( R e f -erence [ 6 ] page 1 ) , t h e v e l o c i t y p o t e n t i a l of fluid m o t i o n ( f o r a n incompressible, nonviscous fluid) due t o a source (Reference [ 6 ] , page 4 0 4 ) l o c a t e d b e l o w t h e f r e e surface of a u n i f o r m s t r e a m of i n f i n i t e d e p t h is (Reference [ 7 ] , page 3 )

T/2

1 Physicist, A i r Force C a m b r i d g e Researoli Center, B e d f o r d ,

Mass.; f o r m e r l y D a v i d T a y l o r M o d e l Basin, N a v y D e p a r t m e n t , W a s h i n g t o n , D . C.

^ N u m b e r s i n brackets designate References a t end o f paper.

y, Z) = sec^

m \

Jo

— 4/com I ^ 0 , 771 in 4/com „ f - I f Tl ?'2 TT

Jo

'^^'^''^^cos(/c.-v cos e)cos{ky s i n 6)

n/2 w h e r e a-, y, z ƒ (j,{x, y, z) — c /Co g m P k — ko sec^ 6 s e c ö e "^''•(^-"^'"''''*sin(fco:r sec 6) cos(fcoy s i n 6 sec^ d)dd ( 1 ) r e c t a n g u l a r co-ordinates, z p o s i t i v e u p w a r d s (the u n d i s t u r b e d f r e e surface is t h e xy-plane of F i g . 1) d e p t h of source b e l o w u n d i s t u r b e d f r e e sur-face, X' + y' + iz+ f y X' + y' + ( z - f y v e l o c i t y p o t e n t i a l yii = v = —

tv

_bzj velocities i n p o s i t i v e x, y, 2-directions u n i f o r m s t r e a m v e l o c i t y acceleration of g r a v i t y s t r e n g t h of source (a source of s t r e n g t h m e m i t s a v o l u m e émii per u n i t t i m e ) placed before an i n t e g r a l s i g n m e a n s t h a t t h e C a u c h y p r i n c i p a l v a l u e is t o be t a k e n (Reference [ 8 ] , page 1 2 8 ) M A R C H , 1 9 5 9 1

Aeroelastic Stability of Lifting

aces in High-Density Fluids

By Charles J . Henry,i John Dugyndji,! and Holt Ashley'

The large increases anticipated in speeds of vehicles towed or propelled underwater suggests a re-examination of the problem of stability of flexible lifting surfaces mounted thereon. Experimental and theoretical evidence is assembled which suggests that os-cillatory aeroelastic instability (flutter) is very unlikely at the structural-to-fluid mass ratios typical of hydrodynamic operation. It is shown that static instability (divergence) is the more important practical problem but that its occurrence can be predicted with greater confidence. Flutter data obtained in high-density fluids are reviewed, and various sources of inaccuracy in their theoretical prediction are analyzed. The need is expressed for more precise means of analytically representing both dynamic-elastic systems and three-dimensional unsteady hydrodynamic loads. For a simple hydrofoi with single degrees of freedom in bending and torsion, the theoretical influence of several signiflcant parameters on high-density flutter is calculated and discussed. Recommenda-tions are made for refinements to existing techniques of analysis to include the presence of channel boundaries, free surfaces, cavitation or separated flow.

A I R P L A N E designers have been p l a g u e d f o r m a n y years b y t h e p r o b l e m of i n s t a b i l i t y of elastic d e f o r m a t i o n s of l i f t m g surfaces, such as w i n g s a n d t a i l s , exposed t o a high-speed a i r s t r e a m . W h e n o s c i l l a t o r y or d y n a m i c i n character, t h i s i n s t a b i l i t y is called " f l u t t e r , " w h i l e s t a t i c aeroelastic i n s t a b i l i t y is k n o w n as " d i v e r g e n c e . " I n e i t h e r case, t h e p h e n e m o n is m o s t serious a t t h e highest

values of t h e flight d y n a m i c pressure q = E v e n a near a p p r o a c h t o t h e c r i t i c a l speed f o r f l u t t e r or d i v e r

-gence c a n p r o d u c e d e s t r u c t i v e l y large stresses i n t h e s t r u c t u r e or undesirable levels of v i b r a t i o n .

Since ships m o u n t i n g e l a s t i c a l l y r e s t r a i n e d l i f t i n g sur-faces s u c h as rudders, s t a b f l i z i n g flns, h y d r o f o i l s a n d b o w planes h a v e l o n g experienced d y n a m i c pressures i n w a t e r greater t h a n those a t t a i n e d b y a n y b u t t h e fastest m o d e r n a i r c r a f t , a n i n t e r e s t m g p o i n t f o r s p e c u l a t i o n concerns w h y t h e y r a r e l y , i f ever, m e e t w i t h s i m i l a r i n s t a b i l i t i e s . A l t h o u g h no t r o u b l e has arisen i n t h e p a s t , t h e e x t r a -o r d i n a r y speeds s-o-on t -o be reached b y s u b m a r i n e s a n d o t h e r vehicles t o w e d or p r o p e l l e d b e n e a t h t h e surface suggest t h a t perhaps t h i s p r o b l e m s h o u l d receive a f r e s h e x a m m a t i o n . T h a t is t h e p u r p o s e of t h e present paper.

F r o m t h e s t a n d p o m t of d i m e n s i o n a l analysis, t h e r e is o n l y one p a r a m e t e r w h i c h has d i s t m c t l y d i f f e r e n t n u m e r i -cal v a l u e s i n " h y d r o e l a s t i c " studies of ships o n t h e one h a n d a n d aeroelastic studies of a i r c r a f t o n t h e o t h e r . T h i s is t h e so-called "mass r a t i o , " or r a t i o of a t y p i c a l

d e n s i t y of s t r u c t u r a l m a t e r i a l t o t h e d e n s i t y of t h e fluid. T h e r a t i o is c h a r a c t e r i z e d here b y t h e s y m b o l

1 Massachusetts I n s t i t u t e o f T e c h n o l o g y , F l u i d D y n a m i c s R e

-search G r o u p , Cambridge, Mass. M r . H e n r y is n o w associated w i t h the E x p e r i m e n t a l T o w i n g T a n k , Stevens I n s t i t u t e ot t e c h -nology, H o b o k e n , N . J.

m

(1)

w h e r e m is t h e mass per u n i t spanwise d i m e n s i o n of t h e l i f t i n g surface, 2h is a r e p r e s e n t a t i v e c h o r d - l e n g t h , a n d p is t h e fluid d e n s i t y . Since w a t e r is r o u g h l y one t h o u s a n d t i m e s denser t h a n sea-level air, p t u r n s o u t t o be of o r d e r or less f o r h y d r o f o i l s b u t 50 or m o r e f o r a i r p l a n e w n i g s . T h i s d i f f e r e n c e w o r k s g r e a t l y t o t h e a d v a n t a g e of t h e n a v a l a r c h i t e c t , as w i f l be seen l a t e r , a t least m t h e case of d y n a m i c i n s t a b i l i t y .

T h e a e r o n a u t i c a l l i t e r a t u r e is replete w i t h p a p e r s o n b o t h flutter a n d divergence. Selective reference l i s t s w ü l be f o u n d i n t w o recent books o n a e r o e l a s t i c i t y ( F u n g , R e f e r e n c e [1],= a n d B i s p l i n g h o f f - A s h l e y - H a l f m a n [ 2 ] ) . N o discussion a i m e d p a r t i c u l a r l y a t a p p h c a t i o n s t o t h e design of h y d r o f ofls a n d t h e l i k e has come t o t h e a u t h o r s ' a t t e n t i o n .

T h e present t r e a t m e n t is l i m i t e d t o l i f t i n g surfaces w h i c h are f u l l y s u b m e r g e d a n d u n a f f e c t e d b y w a v e m a k i n g or c a v i t a t i o n . I n t h i s case, t h e analysis of i n s t a b ü i -ties w o u l d seem t o be esseirtially t h e same regardless of w h e t h e r w a t e r or a i r is t h e l o a d - p r o d u c i n g fluid, p r o v i d e d t h a t n o c a v i t a t i o n occurs i n t h e f o r m e r n o r c o m p r e s s i b ü -i t y e f f e c t -i n t h e l a t t e r . T h e s -i m -i l a r -i t y -is r e m f o r c e d b y t h e f a c t t h a t s t r u c t u r a l c o n f i g u r a t i o n s are v e r y m u c h a l i k e ;

2 N u m b e r s i n b r a c k e t s designate References a t end o f paper.

A Study of the Transient

Fikhiiiiq Oieilfeiffioos af (

QI

By P a u l © o l e v a f e ^

T H E m o t i o n s of a deeply submerged b o d y w i t h v e r t i c a l -p l a n e s y m m e t r y , e.g., a s u b m a r i n e , are c o m m o n l y t r e a t e d i n a m a n n e r c o m p l e t e l y analogous t o t h a t used f o r a i r c r a f t m o t i o n s . T h e b o d y is assumed t o h a v e i t s l a t e r a l a n d l o n g i t u d m a l modes u n c o u p l e d . T h e s m a l l m o t i o n s are described b j ' a set of f o r c e - a n d - m o m e n t equations w h i c h are linear, second-order d i f f e r e n t i a l equations w i t h con-s t a n t coethcientcon-s. Thecon-se p r o p o r t i o n a l i t y concon-stantcon-s ( " s t a b i l i t y d e r i v a t i v e s " ) relate t h e forces a n d m o m e n t s t o t h e instantaneous values of t h e p o s i t i o n , v e l o c i t y , a n d acceleration of t h e b o d y . T h e y are g e n e r a l l y e x p e r i -m e n t a l l y d e t e r -m i n e d -m t h e w -m d or w a t e r t u n n e l , a n d t h e c o n t r o l l e d or u n c o n t r o U e d m o t i o n s of t h e c r a f t are p r e d i c t e d based t h e r e o n . H o w e v e r , w h e n t h e b o d y approaches t h e f r e e surface, w a v e s are generated b y t h e m o v i n g b o d y a n d t h e simple concept breaks d o w n . T h e forces o n t h e b o d y n o w also d e p e n d o n t h e past h i s t o r y of t h e m o t i o n ; t h a t is, o n t h e surface waves generated b y t h e b o d y . T h i s is a k i n t o t h e u n s t e a d y l i f t o n l i f t i n g surfaces w h e r e t h e w a k e m t e r -a c t s w i t h t h e w i n g t o -alter t h e inst-ant-aneous l i f t .

T h e m e t h o d c o m m o n l y used t o d a y f o r t h e p r e d i c t i o n of ship m o t i o n s resembles t h e accepted a e r o n a u t i c a l p r a c t i c e . I m p l i c i t i n t h i s p r o c e d u r e is t h e a s s u m p t i o n t h a t t h e forces are l i t t l e i n f l u e n c e d b y t h e p a s t m o t i o n h i s t o r y . I n t u i t i v e l y one feels t h a t such a n a s s u m p t i o n m u s t be approached c a u t i o u s l y f o r t h e surface ship. C e r t a i n l y t h e t r a c t a b i l i t y of t h e r e s u l t i n g equations j u s t i f i e s t h e a t t e m p t t o use t h e m b u t adequate e x p e r i -m e n t a l v e r i f i c a t i o n of t h e -m e t h o d is -m a n d a t o r y .

T h e a e r o d y n a m i c i s t has several decades of e x p e r i m e n -t a l w o r k -t o c o n f i r m -t h e f o r e g o i n g assump-tions b u -t -t h e h y d r o d y n a m i c i s t has b a r e l y b e g u n t o v a l i d a t e his as-s u m p t i o n . T h e coefficientas-s i n t h e equationas-s h a v e been c o m p u t e d using a n a l y t i c a l techniques w i t h m a n y s i m p l i -fications. T h e r e has been a p a u c i t y of e x p e r i m e n t a l v e r i f i c a t i o n of these c o m p u t a t i o n s . Some e x p e r i m e n t a l

1 Seaworthiness D i v i s i o n , D a v i d T a j d o r M o d e l Basin, W a s h i n g -t o n , D . C.

Nomenclature

hip

checks o n t h e o v e r - a l l t e c h n i q u e h a v e been m a d e ; t h a t is, t h e response of models t o sinusoidal waves has been measured a n d c o m p a r e d w i t h t h e p r e d i c t e d m o t i o n s [ 1 ] . ^ These results h a v e been e n c o u r a g i n g b u t s t i l l i n c o n c l u -sive.

W h e n successful, a c o n f i r m a t o r y e x p e r i m e n t is o f t e n i n t e r p r e t e d t o be a v e r i f i c a t i o n of t h e t h e o r y . T h i s m a y n o t a c t u a l l y p r o v e t r u e f o r , i n t h e present case, i t is possible t h a t t h e procedure f o r c o m p u t i n g several co-efficients is i n e r r o r a n d t h a t these errors compensate f o r p a r t i c u l a r ship f o r m s .

E x p e r i m e n t a l techniques can be used t o d e t e r m i n e these coeflficients. I n t h e m e t h o d of f o r c e d osciUations, t h e p e r i o d i c forces necessary t o p r o d u c e s i n u s o i d a l m o -t i o n s of -t h e ship are d e -t e r m i n e d . C o m p a r i s o n of -t h e phase r e l a t i o n s b e t w e e n t h e m o t i o n s a n d forces y i e l d s t h e desired coefficients. H o w e v e r , there is a d a n g e r m t h e b l i n d a p p l i c a t i o n of t h i s techniciue. Once t h e s i m p l e m a t h e m a t i c a l m o d e l is assumed, a fit t o t h e d a t a is " f o r c e d . " A n answer is a l w a y s o b t a i n a b l e f r o m such tests since an e q u i l i b r i u m p e r i o d i c f o r c e is set u p as a r e s u l t of t h e m t e r a c t i o n of t h e osciUating b o d y a n d t h e generated w a v e p a t t e r n . H o w e v e r , t h i s answer is n o t s u f f i c i e n t t o d e t e r m i n e w h e t h e r t h e forces d e p e n d solely o n t h e i n s t a n t a n e o u s m o t i o n a n d are i n d e p e n d e n t of t h e p r e v i o u s h i s t o r y . C o e f f i c i e n t s can be d e t e r m m e d b u t a p h y s i c a l i n t e r p r e t a t i o n of t h e i r significance m a y n o t be possible. F o r example, supposed h y d r o d y n a m i c m o m e n t s of i n e r t i a h a v e been f o u n d t o be n e g a t i v e a t l o w f r e -c^uencies [ 2 ] ,

T h e m e t h o d of f r e e oscillations is a n a l t e r n a t e t e c h -n i q u e w h i c h o f f e r s c e r t a i -n adva-ntages. B y r e s t r a i -n m g t h e ship so t h a t i t possesses one degree of f r e e d o m o n l y , t h e t r a n s i e n t oscillations can be observed a n d c o m p a r e d w i t h t h e c o m m o n l y k n o w n b e h a v i o r of t h e l i n e a r oscillat o r . W i oscillat h o u oscillat a n y I m o w l e d g e of oscillat h e c o e f f i c i e n oscillat s , oscillat h e o b -served o s c i l l a t o r y decay can be n o t e d a n d c o m p a r e d w i t h t h e t h e o r e t i c a l e x p o n e n t i a l decay. T h i s c o u l d be t h e

2 N u m b e r s i n brackets indicate References a t e n d of paper.

B = beam of t h e m o d e l C = d a m p i n g coefficient g = acceleration due t o g r a v i t j ' I = t o t a l mass m o m e n t of i n e r t i a K = r e s t o r i n g m o m e n t coefficient L = l e n g t h of m o d e l £ = L a p l a c e t r a n s f o r m s = v a r i a b l e i n t r a n s f o r m e d e q u a t i o n t = t i m e u = h o r i z o n t a l v e l o c i t y of a s h i p T = p e r i o d of o s c i l l a t i o n a = angle of a t t a c k A = displacement of m o d e l = p i t c h angle a n d i t s d e r i v a t i v e s

I/- = hydrod5rnamic mass m o m e n t of

i n e r t i a

p = mass density of w a t e r

At) = W a g n e r f u n c t i o n = circular f r e q u e n c y

Effect of Size of Hatches on

By James P. Bailey^

The objective of the test described in this paper was to study the effect of hatch size on the torsional strength and rigidity of ships' hulls. Six plastic models, simulating the mid-ship portion of a cargo mid-ship, with various sizes and arrangements of hatches, were tested by subjecting to torsion and determining the resulting angle of twist, the approximate intensity and distribution of stress, and the torque required to produce fracture. Tor-sional rigidity was found to decrease generally with total width of hatches and with length of hatches. The torque to failure showed a greater variation between models than did the torque required to produce a given strain. The highest stress concentrations were at the hatch corners, and in every case the models failed at a hatch corner.

T H E o b j e c t i v e of t h i s test was t o o b t a i n some q u a l i t a -t i v e idea of -t h e e f f e c -t of size a n d a r r a n g e m e n -t of cargo hatches o n t h e t o r s i o n a l s t r e n g t h a n d r i g i d i t y of a ship's h u l l , a n d t o show v i s u a l l y t h e m a n n e r of f l e x i n g a n d t h e l o c a t i o n of t h e areas of m a x i m u m s t r a i n .

T h i s was considered t o be desirable because of t h e cur-r e n t intecur-rest i n designs f o cur-r ships m t h lacur-rgecur-r a n d / o cur-r m o r e hatches t h a n h a v e been u s u a l l y f i t t e d , either t o i m p r o v e " s p o t t i n g " a b i l i t y i n h a n d l i n g cargo or t o acc o m m o d a t e speaccial accargo as, f o r example, large acc o n t a i n -ers. I n such designs i t has been recognized t h a t some loss of t o r s i o n a l s t r e n g t h a n d r i g i d i t y is i n v o l v e d , b u t i t is g e n e r a l l y assumed t h a t t h e loss is n o t great a n d t h a t i n a n y case t h e t o r s i o n a l l o a d i n g of ships' h u l l s is m i n o r . T h e r e is a need f o r m o r e d e f i n i t e k n o w l e d g e r e g a r d i n g t h e m a t t e r , a n d t h i s s t u d y is a s m a l l step i n t h a t direc-t i o n .

Models

Six models were c o n s t r u c t e d u s i n g Plexiglas I I - U V A a n d PS-18 C h e m i c a l D e v e l o p m e n t C o m p a n y cement. T h e models were 5.5-ia. deep a n d 10 i n . -wide, or v e r y r o u g h l y Hoo f u l l size f o r a n o r m a l cargo vessel, a n d 48 i n . l o n g . F i g u r e 1 shows t h e dimensions, thicknesses of m a t e r i a l , cross section, a n d h a t c h c o n f i g u r a t i o n s f o r each m o d e l . M o d e l 1 represents " u s u a l p r a c t i c e , " w i t h a single r o w of hatches whose w i d t h is 40 per cent of t h e b e a m . M o d e l 2 represents a t w i n - h a t c h design, w i t h t h e t o t a l Avidth of b o t h hatches 70 per cent of t h e b e a m a n d ' w i t h a centerline l o n g i t u d i n a l b u l k h e a d . M o d e l 3 a p p r o x i m a t e l y represents a t r i p l e - h a t c h design, w i t h a n aggregate w i d t h of t h e three hatches also equal t o 70 per cent of t h e beam, b u t w i t h s h o r t e r h o l d s a n d

1 H y d r a u l i c L a b o r a t o r y , N e w p o r t N e w s S l i i p b u i l d i n g a n d D r y

D o c k Compan}', N e w p o r t N e w s , V a .

hatches t h a n N o . 2, a n d w i t h t w o l o n g i t u d i n a l b u l k h e a d s . M o d e l 4 is s i m i l a r t o N o . 3, b u t w i t h l e n g t h of h o l d s a n d hatches t h e same as N o . 2 t o p e r m i t d i r e c t c o m p a r i -son. M o d e l 5 has e x t r e m e l y large hatches, 90 per cent of t h e beam, t o explore t h e effect of such a n e x t r e m e . M o d e l 6 has single hatches 70 per cent of t h e b e a m i n w i d t h , t h e same as N o . 2 a n d N o . 4, t o p e r m i t d i r e c t c o m p a r i s o n w i t h N o s . 2 a n d 4.

T h e cross-sectional area of t h e deck is t h e same f o r a l l models, b u t t h e t o t a l cross-sectional area of t h e models varies, o w i n g t o t h e v a r y i n g n u m b e r of l o n g i t u d i n a l buUdieads a n d h a t c h coamings. A d o u b l e b o t t o m is n o t fitted b u t t h e thickness of t h e b o t t o m is t w i c e t h a t of t h e rest of t h e shell t o s i m u l a t e r o u g h l y t h e e f f e c t of t h e d o u b l e b o t t o m .

A l l hatches h a d r o u n d e d corners i n t h e deck p l a t i n g , w i t h a r a d i u s of }4 i n . , corresponding t o a r a d i u s of a b o u t 2 f t i n t h e fuU-size ship.

Testing

Rig. T h e l o a d i n g r i g is s h o w n i n F i g s . 2 a n d 3. T h e models were secured a t one end a n d a t o r q u e a p p l i e d a t t h e o t h e r end. T h e ends of each m o d e l w e r e m a d e u p oi n}4 X 7 X Vs-m. Plexiglas a n d secured t o steel face plates of t h e same w i d t h . T h e fixed-end steel face p l a t e was secured t o t h e steel t e s t b e d o n t h e t a b l e . T h e o t h e r steel face p l a t e was fitted w i t h a M - i n . needle b e a r i n g . T w o 10-lb w e i g h t pans were h u n g f r o m each side of a lever a r m , 18in. f r o m t h e center. T h e P l e x i -glas models were annealed a f t e r c o n s t r u c t i o n .

Procedure. T e s t i n g consisted of a test of c o m p a r a t i v e t o r s i o n a l r i g i d i t y , t w o b r i t t l e - l a c q u e r tests of stress u n d e r c o m p a r a b l e loads, a n d a test t o f a i l u r e .

Test of Torsional Rigidity. D i a l i n d i c a t o r s w e r e p l a c e d a t b u l l d i e a d s aloirg each side of t h e m o d e l , as s h o w n i n Figs. 2 a n d 3. T h e m o d e l was loaded i n i n c r e m e n t s

A d d e d Mass of a ThreeParameter Family of T w o

-Dimensional Forces Oicilioting in a Free Surface'

By

I.

L a n d w e b e r ^ a n d M ö ï i l d e M a e a g n © ^

The added-mass characteristics of shiplil<e forms, oscillating vertically or horizontally in a free surface, are derived for a three-parameter family of forms. The parameters varied are the draft-beam ratio, the section-area coefficient, and the ratio to the draft of the radius of gyration about the transverse axis in the free surface. The added-mass coefficients are presented as a series of curves for about 70 members of this family. It is suggested that the added masses of arbitrary shiplike sections may be obtained, with only small error, from these curves by interpolation at the parametric values of the given section.

T H E present w o r k continues t l i e a p p l i c a t i o n of a general t h e o r y f o r t h e addedmass coefficients of t w o d i m e n -sional f o r m s o s c i l l a t i n g h o r i z o n t a l l y or v e r t i c a l l y i n a free surface.^ I n t h e p r e v i o u s paper the t h e o r y was a p -p h e d t o a -p r a c t i c a l , t w o - -p a r a m e t e r f a m i l y of shi-p sec-t i o n s , called sec-t h e L e w i s f o r m s . I sec-t has been s h o w n b y Prohaska,5 however, t h a t t h e t w o p a r a m e t e r s used, t h e d r a f t b e a m r a t i o a n d t h e sectionarea coefficient, are i n -s u f f i c i e n t t o defuie an a d d e d ma-s-s. F u r t h e r m o r e , i t ha-s been s h o w n t h a t t h e L e w i s f o r m s cannot g i v e ship sec-tions h a v m g area coefficients close t o u n i t y . ' ' T h u s i t appeared t h a t t h e d e t e r m i n a t i o n of the added-mass co-efficients f o r a generalized, t h r e e - p a r a m e t e r f a m i l y of f o r m s w o u l d be v e r y u s e f u l . A f t e r t h e d r a f t - b e a m r a t i o a n d section-area coefficient, t h e m o s t s i g n i f i c a n t t h i r d p a r a m e t e r seemed t o be t h e r a t i o of t h e r a d i u s of g y r a t i o n a b o u t t h e t r a n s v e r s e axis i n t h e f r e e surface t o t h e d r a f t . C o n s e q u e n t l y , t h e n e w f a m i l y of f o r m s , w h i c h , i t w i l l be seen, is a n a t u r a l exten-sion of t h e L e w i s f o r m s , will be expressed a n a l y t i c a l l y i n t e r m s of these t h r e e p a r a m e t e r s . A s was f o u n d f o r t h e L e w i s f o r m s , t h e extended f a m i l y also gives r e a l or s h i p -like sections f o r o n l y c e r t a i n ranges of values of t h e p a r a m e t e r s w h i c h are i n v e s t i g a t e d a n d d e f i n e d .

T h e g r a p h i c a l p r e s e n t a t i o n of a f u n c t i o n of t h r e e variables is i n h e r e n t l y a tedious t a s k . T h u s i t was neces-sary t o select o n l y a f e w values of each p a r a m e t e r i n order t o r e s t r i c t the n u m b e r of shapes of t h e n e w f a m i l y

1 Worlc sponsored b y T l i e Society of N a v a l A r c h i t e c t s a n d M a r i n e Engineers.

' Research Engineer, I o w a I n s t i t u t e of H y d r a u l i c Research, a n d Professor, D e p a r t m e n t of Mechanics a n d H y d r a u l i c s , State U n i -v e r s i t y of I o w a , I o w a Citj^, I o w a .

' Research Associate, I o w a I n s t i t u t e of H y d r a u l i c Research, State U n i v e r s i t y of I o w a , I o w a C i t y , I o w a .

^ L . Landweiier a n d M a t i l d e Macagno, " A d d e d Mass of T w o -D i m e n s i o n a l F o r m s Oscillating i n a Free Surface," J O U R N A L OP

S H I P R E S E A R C H , v o l . 1 , 1 9 5 7 .

' C. W . Prohaska, " V i b r a t i o n s verticales d u n a v i r e , " B u l l e t i n de L'Association Technique M a r i t i m e et A e r o n a u t i q u e , 1 9 4 7 , p .

w h i c h are e x h i b i t e d t o a b o u t , 70, T h e added-mass coefficients are presented as f u n c t i o n s of t h e t h r e e p a r a m e -ters i n a series of curves.

The Three-Parameter Family of Forms

A t h r e e - p a r a m e t e r f a m i l y of f o r m s m a y be d e r i v e d f r o m t h e u n i t circle i n t h e f - p l a n e b y t h e t r a n s f o r m a t i o n

2 = f + ^ + p + «1, as, at r e a l (1) where z a n d f are t h e c o m p l e x v a r i a b l e s

z = X + iij f = re''

Since t h e coefficients a i , a^, are r e a l a n d o n l y o d d powers of f occur, t h e c u r v e i n t h e 3-plane o b t a i n e d b y s u b s t i t u t i n g f = ré' h i t o E q u a t i o n (1) is s y m m e t r i c a l w i t h respect t o b o t h t h e x a n d J/-axes. T h e e q u a t i o n s of t h e f o r m s , f r o m E q u a t i o n ( 1 ) , are g i v e n b y

X = /(I -|- aO cos

0 -\-

Ch cos 3Ö -\- cos

56

y

= (I —

tti) sin

6 —

Ui sin 3Ö — as sin

50

w h i c h m a y also be expressed i n t h e f o r m s X = (1 - f a i — 3a3 - f

5ai)

cos

6

+ 4(a3 — 5aö) cos^

6 + IGa^

cos^

8

y = -{1 — ÜL — 3az — Sas) sin 6 + 4(a3 + 5ai) s i n ^ ö - 16

as

sin^

6

L e t b denote t h e h a l f - b e a m of the f o r m a t t h e w a t e r l i n e a n d H t h e d r a f t a t t h e keel line. Since x = b w h e n 0 = 0 a n d y = H w h e n 6 = 7r/2, we o b t a i n f r o m E q u a t i o n s (2) •

b = I + tti + az + an

H = 1 ~ ai + az — at

T h e sectional area of t h e f o r m , o b t a i n e d f r o m E q u a t i o n (12) or t h e earlier paper,'' is S =

I

(1 - « 1 ^ - 3a3^ - 506^) (5)

36

J O U R N A L O F SHIP RESEARCH

lournal of

SHIP R E S E A R C H

The Damping and W a v e Resistance

of a Pitching and Heaving Ship

By J . N. Newman^

This paper considers the damping and wave resistance of a thin ship which is moving in calm water with constant velocity and oscillating in pitch and heave. The velocity poten-tial is obtained from Green's theorem after a process of systematic linearization in terms of perturbation parameters representing the belength ratio and the oscillation am-plitude. An asymptotic expansion of the Green's function is derived from which the energy radiation is obtained. The coefficients of damping and increased wave resist-ance are then found by separation of the energy components. No separation of the two cross-coupling damping coefficients is obtained, however. Calculations are" presented for a polynomial hull and compared with experimental data.

T H E analysis of energy r a d i a t i o n i n surface waves is a c o n v e n i e n t m e t h o d of e v a l u a t i n g t h e d a m p i n g a n d w a v e resistance of a ship. T h i s m e t h o d was d e v e l o p e d i n t h e t h r e e - d i m e n s i o n a l case b y H a v e l o c k f o r w a v e resistance i n steady m o t i o n a n d f o r d a m p i n g i n n o n t r a n s l a t i n g m o t i o n . A n e f f o r t t o generalize t h e d a m p i n g p r o b l e m t o t h e case of a m o v i n g ship was m a d e b y t h e present a u t h o r [9]^ b u t these results are i n f a c t o n l y v a l i d f o r

1 Worlc done under the sponsorship of the Office of N a v a l

Re-search, C o n t r a c t , N o . N o n r - 1 8 4 1 ( 3 1 ) .

" Research Assistant, D e p a r t m e n t of N a v a l A r c h i t e c t u r e a n d M a r i n e E n g i n e e r i n g , Massachusetts I n s t i t u t e of Technology, Cam-bridge, Mass.

' N u m b e r s i n brackets designate References a t end of paper.

Nomenclature

zero speed. U s i n g a d i f f e r e n t procedure, H a v e l o c k [6] has considered t h e p i t c h d a m p i n g of a m o v i n g t h i n p l a n k , a n d f o u n d s i g n i f i c a n t speed-dependence. T h i s f a c t . h a s been e x p e r i m e n t a l l y c o n f i r m e d f o r a Series 60 m o d e l [ 7 ] .

T h e present analysis is a n a p p l i c a t i o n of t h e energy m e t h o d t o t h e general p r o b l e m o f a m o v i n g ship w h i c h is p i t c h i n g a n d h e a v i n g i n otherwise c a l m w a t e r . B y separating t h e energy components, t h e w a v e resistance a n d d a m p i n g coefficients are o b t a i n e d b u t s e p a r a t i o n of t h e cross-coupling d a m p i n g coefficients is n o t achieved. I r r o t a t i o n a l f l o w a n d s m a l l oscillations are assumed, b u t no a s s u m p t i o n is m a d e r e g a r d i n g t h e source d i s t r i b u t i o n . I n s t e a d i t is assumed t h a t t h e b e a m - l e n g t h r a t i o is suf-c ₌ f o r w a r d velooitj^ E = energj-G = Green's f u n c t i o n B = g r a v i t a t i o n a l acceleration h

₌

e q u a t i o n of h u l l beam i , ) , K

=

u n i t vectors K = d u m m y v a r i a b l e of i n t e g r a t i o n M = p i t c h m o m e n t N = coefficients of d a m p i n g w o r k n = u n i t o u t w a r d n o r m a l {P, Q) = h u l l f u n c t i o n s defined b y equa-t i o n s (69), (86) and (87) V = pressure R = polar co-ordinate

=

wave-resistance coefficients

s

= c o n t r o l surface I = t i m e M = d u m m y v a r i a b l e of i n t e g r a t i o n ^ f O f o r r ^ V4 \ o o s - i ( l / 4 r ) f o r r ^ 1/4 v = fluid v e l o c i t y v e c t o r 1 V„ = n o r m a l v e l o c i t j ^ of the c o n t r o l surface Wo = w o r k done b y d a m p i n g

TFje = w o r k done b y w a v e resistance

Xo, 2/0, = fixed co-ordinate S3'stem X, y, z = ' m o v i n g co-ordinate sj^stem x',y',z' = co-ordinate system fixed i n ship

Z — heaving force «s = free-surface e l e v a t i o n

a = parameter representing oscil-l a t i o n a m p oscil-l i t u d e s

13 = beam-length r a t i o

5 = phase lag of heave b e h i n d p i t c h

f = heave displacement 6 = p i t c h displacement; p o l a r co-ordinate X = d u m m y v a r i a b l e of i n t e g r a t i o n Xj = f u n c t i o n s defined b y e q u a t i o n ( 5 5 ) (?) Vi D = source-point co-ordinates P = fluid density 2 = c o n t r o l surface r = coc/ff $ = absolute v e l o c i t y p o t e n t i a l 0 = v e l o c i t j r p o t e n t i a l r e l a t i v e t o ship <p = t i m e independent f o r m of 0 CO = c i r c u l a r f r e q u e n c y of oscilla-tions J U N E , 1 9 5 9 1

Effect of the Free Surface on the

Flutter of Submerged Hydrofoils'

By Wen-Hwa Chu^ and H. Norman Abramson^

Expressions for tfie unsteady lift and moment acting on an oscillating hydrofoil submerged under a free surface are derived by an extension of classical unsteady thin-airfoil theory. The method of images is employed to account for the free surface, but the additional contribution resulting from surface waves, considered small, is neglected. The results of flutter computations are presented for a hypothetical example, using a simple repre-sentative station analysis, for ratios of depth of submergence to semi-chord of 2 and infinity; it is found that the lower fiutter speed corresponds to infinite depth of sub-mergence.

R E C E N T d e v e l o p m e n t s i n N a v a l a p p l i c a t i o n s h a v e suggested t h a t increased speeds a n d t h e c o n s i d e r a t i o n of n o v e l c o n f i g u r a t i o n a l f o r m s m a y g i v e rise t o n e w p r o b lems, m a n y of w h i c h f a l l w i t h i n t h e r e a l m of " h y d r o -e l a s t i c i t y " [ 1 ] . * A m o n g th-es-e is t h -e p r o b l -e m of f l u t t -e r of s u b m e r g e d l i f t i n g surfaces.

T h e p r e v i o u s discussions of t h e flutter of submerged surfaces h a v e been l i m i t e d t o those w h i c h are deeply submerged a n d u n a f f e c t e d b y t h e presence of t h e f r e e surface or w a v e - m a k i n g [2, 3 ] , a n d o n l y v e r y l i t t l e e f f o r t has been g i v e n t o t h e effects o f c a v i t a t i o n [ 4 ] . T h e present t r e a t i n e n t considers t h e e f f e c t of t h e f r e e surface o n t h e flutter characteristics of f u l l y submerged l i f t i n g surfaces.

A b r i e f r e v i e w of p r e v i o u s w o r k c o n c e r n i n g forces o n h y d r o f o i l s i n steady m o t i o n is g i v e n m [ 5 ] , a n d a correc-t i o n f a c correc-t o r f o r submerged h y d r o f o f l s is o b correc-t a i n e d b y a n extension of l i f t i n g - l i n e t h e o r y a n d t h e i n t r o d u c t i o n of a generalized image m e t h o d , t h e l a t t e r h a v i n g first been proposed b y W e i n i g [6] t o a c c o u n t f o r t h e i n t e r f e r e n c e e f f e c t of t h e free surface. T h e e f f e c t of surface waves f o r finite F r o u d e n u m b e r was i n c l u d e d l a t e r b y K e l d y s c h a n d L a v r e n t i e v [ 7 ] .

I n t h e case of u n s t e a d y m o t i o n s , T a n [ 8 ] , m o d i f i e d t h e image m e t h o d t o a p p l y t o a source a n d v o r t e x of fluctuat-i n g s t r e n g t h b e n e a t h a f r e e surface a n d was successful fluctuat-i n o b t a i n i n g a n expression f o r t h e v e l o c i t y p o t e n t i a l , i n c l u d -i n g t h e w a k e . K a p l a n [ 9 ] c a l c u l a t e d t h e forces o n a n o s c i l l a t i n g h y d r o f o i l , as s i m u l a t e d b y a v o r t e x a n d a d o u b

-' T l i e results presented i n t h i s paper were o b t a i n e d d u r i n g t h e course of research carried o u t under t h e B u S h i p s F u n d a m e n t a l H y d r o m e c h a n i c s Research P r o g r a m a d m i n i s t e r e d b y the D a v i d T a y l o r M o d e l Basin.

' Research Engineer, Southwest Research I n s t i t u t e , San A n -t o n i o , T e x .

' Manager, Engineering A n a l y s i s Section, Southwest Research I n s t i t u t e , San A n t o n i o , T e x .

•* N u m b e r s i n brackets designate References a t end of paper.

let. Since t h e s t r e n g t h s of t h e v o r t e x a n d t h e d o u b l e t , a n d t h e i r images, are assumed t o h a v e t h e same values as i n a n i n f i n i t e m e d i u m , t h e i n t e r f e r e n c e e f f e c t of t h e f r e e surface is o n l y p a r t i a l l y a c c o u n t e d f o r ; h o w e v e r , t h e p r i m a r y e f f e c t of g r a v i t y is i n c l u d e d .

I n t h e analysis of t h e present paper t h e i m a g e m e t h o d is a g a i n e m p l o y e d t o a c c o u n t f o r t h e f r e e surface, b u t the a d d i t i o n a l c o n t r i b u t i o n a r i s i n g f r o m surface waves is neglected. T h e results t h u s o b t a i n e d s h o u l d be q u i t e a c c u r a t e f o r s u f f i c i e n t l y large values of the F r o u d e n u m -ber.

A n a l y s i s

Differenfial Equations and Boundary Conditions

F o r t w o d i m e n s i o n a l , incompressible, nonviscous, i r r o -t a -t i o n a l flow, a p o -t e n -t i a l &l-t;j&g-t; exis-ts w h i c h sa-tisfies -t h e e q u a t i o n T h e c o m p l e x v e l o c i t y is = 0 u + iv = W(t> = ~ +

Ox

d<j) . Ö0 dy ( 1 ) (2) T h e p o t e n t i a l f u n c t i o n m a y be decomposed i n t o a steady p a r t a n d a n u n s t e a d y p e r t u r b a t i o n p a r t . A s t h e p e r t u r b a t i o n s are expected t o die d o w n a t i n f i n i t y , t h e p e r t u r b a t i o n pressure p is g i v e n b y t h e l i n e a r i z e d r e l a t i o n

d 0 Ö0

èt ^ ^ bx (3)

where U is t h e s - c o m p o n e n t of v e l o c i t y a t i n f i n i t y ( w h e n t h e co-ordinates are fixed o n t h e h y d r o f o i l ) , p i s t h e fluid d e n s i t y , a n d 0 n o w denotes t h e p e r t u r b a t i o n v e l o c i t y p o t e n t i a l ( u n s t e a d y p a r t ) .

B y a s s u m i n g t w o modes of sinusoidal osciUations, p i t c h i n g a n d flapping, t h e c o n d i t i o n o f zero r e l a t i v e n o r

Note on Propeller-Excited Hull Vibrations

By A . J . Tachmindji' and R. T. McGoldrick'

This note presents a summary of the information and techniques which are available to the designer for predicting the levels of service vibration of a ship in the design stage. This Involves the two phases of estimating the exciting forces and the vibratory response of a hull to given forces. It is emphasized that at present the reliability of such predic-tions depends on the availability of data on the levels of service vibration or the exciting forces for ships actually in operation as a standard of comparison. It is pointed out that the concept of mechanical Impedance at the stern of a ship is useful in predicting forced vibration.

W H I L E n u m e r o u s p u b l i c a t i o n s have appeared i n re-cent years i n t h e general field of ship v i b r a t i o n , t h e designer is u s u a l l y t o o p r e o c c u p i e d w i t h i m m e d i a t e p r o b -lems t o keep u p w i t h t h e e v e r - e x p a n d i n g v o l u m e of l i t e r a t u r e i n t h i s field. T h i s n o t e is a n a t t e m p t t o m e e t t h e need f o r a summar3^ of t h e i n f o r m a t i o n w h i c h is n o w a v a i l a b l e i n h e l p i n g t o p r e d i c t t h e levels of service v i b r a t i o n of ships. I n order t o p r e d i c t t h e l e v e l of service v i b r a t i o n , t h e designer m u s t first be able t o e s t i m a t e t h e forces a n d t h e n t h e response of t h e h u h t o such forces. T h i s n o t e is a n a t t e m p t t o i n d i c a t e t h e techniciues w h i c h m a y be used a n d t h e i n f o r m a t i o n w h i c h is a v a i l a b l e i n o r d e r t o a c c o m p l i s h these goals. T h e details of c a l c u l a t i o n s a n d m e t h o d s c a n n o t be g i v e n i n a s h o r t n o t e a n d m u s t be o b t a i n e d f r o m t h e o r i g i n a l references. T h e p r a c t i c a l u t i l i t y of t h e m e t h o d s suggested here w i l l , t o a large e x t e n t , d e p e n d o n t h e a v a ü a b i l i t y t o t h e i n d i v i d u a l designer of i n f o r m a t i o n o n e x i s t i n g levels of v i b r a t i o n o n ships n o w i n service. T h i s is p a r t i c u l a r l y t r u e f o r t h e p r e d i c t i o n of c e r t a i n e x c i t i n g forces. F u r -t h e r m o r e , c e r -t a i n es-tima-tes of f o r c e d response do r e q u i r e t h e a v a i l a b i l i t y of d i g i t a l c o m p u t e r s or analogs.

T h e m a t e r i a l presented here applies: o n l y t o s t e a d y -s t a t e v i b r a t i o n a n d p a r t i c u l a r l y t o p r o p e l l e r - e x c i t e d v i b r a t i o n a n d does n o t t a k e i n t o c o n s i d e r a t i o n t h e severe t r a n s i e n t v i b r a t i o n s t h a t m a y be generated u n d e r s l a m -m i n g c o n d i t i o n s i n a seaway. T h e r e are t h r e e d i s t i n c t t y p e s of p r o p e l l e r - e x c i t e d v i b r a t i o n : Unbalance Vibration. T h i s t y p e of p r o p e l l e r v i b r a t i o n w i l l be generated i n t h e h u l l b y u n b a l a n c e d forces or m o m e n t s , i r r e g u l a r i t i e s b e t w e e n t h e blades i n t h e m a n u -f a c t u r e o-f t h e p r o p e l l e r , a n d i m p e r -f e c t i o n s due t o b e n t s h a f t i n g . H e r e are i n c l u d e d t h e effects of b o t h mass u n -balance a n d p i t c h u n b a l a n c e . These v i b r a t i o n s w i l l a l l

1 I n s t i t u t e f o r Defense Analyses, Weapons Systems E v a l u a t i o n , T h e Pentagon, W a s h i n g t o n , D . C. F o r m e r l y , D a v i d T a y l o r M o d e l B a s i n , N a v y D e p a r t m e n t , W a s h i n g t o n , D . C . 2 D a v i d T a y l o r M o d e l Basin, N a v y D e p a r t m e n t , W a s h i n g t o n , D . C . occur a t a f r e q u e n c y e q u a l t o s h a f t f r e q u e n c y , a n d t h e y w i n not be considered i n t h i s n o t e .

Blade-Frequency Vibrations. These r e s u l t f r o m t h e w a k e d i s t r i b u t i o n i n t o w h i c h t h e p r o p e l l e r is o p e r a t i n g a n d t h e presence of r i g i d surfaces i n t h e v i c i n i t y of t h e p r o p e l l e r o p e r a t i n g i n u n i f o r m i n f l o w . Higb-Frequency Vibration. I n a d d i t i o n t o t h e f o r e g o i n g t y p e s , c a v i t a t i o n c o n d i t i o n s m a y g i v e I'ise t o a n i r r e g u l a r v i b r a t i o n n o t r e a d i l y classified o n a f r e q u e n c y basis. A d d i t i o n a l v i b r a t i o n s m a y be g e n e r a t e d l o c a l l j ' ' i n t h e p r o p e l l e r blades b y t h e h y d r o d y n a m i c forces. These types, of Anbration w i l l 7iot be discussed i n t h i s n o t e .

T h e a m p l i t u d e of t h e v i b r a t i o n generated i n t h e h u l l f r o m b l a d e - f r e q u e n c y forces can be a p p r o a c h e d f r o m several v i e w p o i n t s . T h e f o l l o w i n g w i l l be considered: (a) T h e h y d r o d y n a m i c c o n d i t i o n s of t h e p r o p e l l e r w h i c h i n d u c e t h e b l a d e forces, i.e., forces o c c u r r i n g a t a f r e q u e n c y e q u a l t o t h e r p m t i m e s t h e n u m b e r of blades or t h e i r m u l t i p l e , w i t h a v i e w t o d e t e r m i n i n g t h e h u l l h y -d r o -d y n a m i c characteristics f o r -decreasing these forces.

(b) T h e h u l l s t r u c t u r a l c h a r a c t e r i s t i c s w i t h a v i e w t o p r e d i c t i n g t h e frequencies a n d n o r m a l modes. (c) T h e p r e d i c t i o n of t h e f o r c e d response of t h e h u l l w h e n s u b j e c t e d t o p r o p e l l e r e x c i t e d forces whose a m p l i t u d e s are k n o w n , e i t h e r f o r resonant or n o n r e s o n a n t c o n -d i t i o n s . Propeller-Exciting Forces T h e p r o p e l l e r - e x c i t e d v i b r a t o r y forces are p r o d u c e d b y h y d r o d y n a m i c r e a c t i o n s o n t h e p r o p e l l e r a n d o n t h e h u l l i n i t s v i c i n i t y . T h i s f o r c e s y s t e m consists o f t h r e e forces a n d t h r e e couples a c t i n g i n t h r e e m u t u a l l y per-p e n d i c u l a r per-planes.

These f o r c e s can be considered [ 1 , 2, 3 ] ^ as c o m p o s e d o f a p a r t Avhich acts b y d i r e c t pressure o n t h e s u r f a c e of a h u l l , i n c l u d i n g s t r u t s a n d r u d d e r s , a n d a p a r t w h i c h acts on t h e sui'face o f t h e p r o p e l l e r a n d is t r a n s m i t t e d t o t h e

3 N u m b e r s i n brackets designate References a t e n d of paper.

O n Unstable Ship Motions

Resylfi

&Dg

From Nonfaear Coupling

By J.

R. Paullingi and R.

M .

Rosenberg^

Nonlinear equations of motion of a ship having the three degrees of freedom of heave, pitch, and roll are investigated. The nonlinear terms are all of second order; they repre-sent couplings between the degrees of freedom and, if they were ignored, coupling be-tween some of the equations would be eliminated. It is shown that unstable motion may occur in any one of the degrees of freedom through excitation by one of the other two. Instabilities occur when the natural frequency in the unstable motion is nearly one half of that of the exciting motion, or when these natural frequencies are nearly equal. Experi-ments have been carried out for the roll-heave systems, and these have confirmed the analysis.

I T IS w e l l k n o w n t h a t some of t h e degrees of f r e e d o m o f a ship, i n i t s m o t i o n a b o u t a state of s t a t i c o r d y n a m i c e q u i l i b r i u m , couple w i t h others, i.e., m o t i o n i n one degree of f r e e d o m w i l l i n general be accompanied b y , a n d m i g h t i n d u c e or suppress, m o t i o n i n another. W h e n t h e co-o r d i n a t e s are sco-o chco-osen t h a t c co-o u p l i n g appears co-o n l y i n t h e acceleration t e r m s , t h e c o u p l i n g is called " d y n a m i c . " I n a d i f f e r e n t co-ordinate system, c o u p l i n g m a y exist o n l y because a displacement i n one degree o f f r e e d o m produces displacement i n a n o t h e r ; t h e c o u p l i n g is t h e n called " s t a t i c . " W h e n the c o u p l i n g does n o t f i t e i t h e r of these cases, n o special name is a t t a c h e d t o i t .

C o u p l i n g exists w h e n e v e r the e q u a t i o n of m o t i o n i n one of t h e degrees of f r e e d o m contains one of t h e t i m e d e r i v a t i v e s , or t h e displacement itself (the z e r o ' t h d e r i v a -t i v e ) of a n o -t h e r degree of f r e e d o m . T h e -t e r m i n a n ecjuation o f m o t i o n w h i c h contains t h e " f o r e i g n " degree of f r e e d o m is t h e c o u p l i n g t e r m ; t h i s m a y be a hnear, or a n o n l i n e a r , t e r m . I f a l l c o u p l i n g t e r m s are n o n l i n e a r , t h e n t h e " l i n e a r i z e d " s y s t e m is u n c o u p l e d . I t is w e l l k n o w n t h a t t h e l i n e a r i z e d analysis of a m u l t i d e g r e e o f -} A c t i n g Assistant Professor of N a v a l A r c h i t e c t u r e , U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y , C a l i f . ^ Professor of E n g i n e e r i n g Mechanics, U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y , Calif. f r e e d o m s y s t e m m a y y i e l d i n c o r r e c t results w i t h respect t o s t a b i l i t y , a n d t h i s becomes, i n f a c t , a s t r o n g p o s s i b i h t y w h e n l i n e a r i z a t i o n removes a l l c o u p l i n g , o t h e r w i s e present, f r o m t h e e q u a t i o n s of m o t i o n . One of t h e e a r l y i n v e s t i g a t i o n s of c o u p l i n g b e t w e e n heave a n d r o l l is t h a t b y F r o u d e [1 ]^ w h o observes t h a t ships have undesirable r o l l characteristics i f t h e n a t u r a l frequencies i n heave a n d r o l l are i n t h e r a t i o of 2 : 1 , a n d i f t h e r o l l axis does n o t l i e i n t h e plane o f t h e w a t e r sur-face.

R e c e n t l y G r i m [ 2 ] a n d K e r w i n [3] h a v e e x a m i n e d t h e s t a b i l i t y of r o l l i n g m o t i o n of ships iir l o n g i t u d i n a l waves. I n t h e i r w o r k t h e r e s t o r i n g m o m e n t i n r o l l is a p e r i o d i c f u n c t i o n of t i m e because o f t h e p e r i o d i c i t y of w a v e e n -c o u n t e r a n d , i n -consequen-ce, t h e (single) e q u a t i o n of m o t i o n i n r o l l is a M a t h i e u e q u a t i o n , e v e n w h e n o n l y hnear t e r m s are i n c l u d e d i n t h e forces. T h e y f i n d t h a t , a t c e r t a i n frequencies of w a v e encounter, r o l l o s c i l l a t i o n s m a y become u n s t a b l e .

I n t h i s paper, we examine t h e s t a b i l i t y of ships i n a c a l m sea, a n d w e consider t h e e f f e c t of n o n l i n e a r , second-order, c o u p l i n g t e r m s i n t h e equations o f m o t i o n . W e d e m o n s t r a t e t h a t i n s t a b i l i t i e s ( i n a v a r i e t y of degrees of

' N u m b e r s i n brackets designate References at end of paper.

. N o m e n c l a t u r e .

co-ordinate axes f i x e d i n ship components of v e l o c i t y along

^, y, z

r o t a t i o n s of ship about iC, y, z angular v e l o c i t y components

a b o u t y, z general displacement

components of force along components of m o m e n t a b o u t t h e X, y, z-axes ( r o l l , p i t c h , y a w m o m e n t s ) 7j,, / „ , / , = mass m o m e n t s of i n e r t i a of X, y, z U, V, 11} v, 9, Q P, 1, r f X, Y,Z K, M, N ship a b o u t S, y, z ji-axis •m = m a s s . - ó f ' s h i p 00 = a m p l i t u d e o f p i t c h i n g m o t i o n Q = general force or m o m e n t Zo = a m p l i t u d e of h e a v i n g m o t i o n

= area of ship's w a t e r p l a n e of Xj = generahzed m o t i o n a m p l i t u d e

• f l o t a t i o n 5, e = parameters i n t h e M a t h i e u = .f-co-ordinate of c e n t r o i d oi e q u a t i o n

G M = transverse metacentric h e i g h t T = dimensionless t i m e

A = displacement of ship u = circular f r e q u e n c y Zl, —12 = K-oo-ordinates o f f o r w a r d a n d p = mass d e n s i t y of w a t e r

a f t e r perpendiculars g = acceleration of g r a v i t y

2*

= z-co-ordinate of baseline D o t s denote d i f f e r e n t i a t i o n w i t h respect t o = m o m e n t of i n e r t i a of A „ a b o u t t i m e .

Elacent Contributions Under the Bureau of Ships

Fundamental Hydromechanics Research Program'

B y M . St. Denis^ and J . P. Craven''

The third main nautical objective, Seakeeping, refers to that aspect of ship performance in which the seaway enters in a dominant manner and affects fundamentally the character of the problem. In the section on Control [3],' a pattern was introduced according to which seakeeping was related to the uncontrolled as well as the controlled oscillations which take place with reference to a ship's inertial system. But, more broadly speaking, seakeeping includes all of the following subjects: (a) Description of the seaway, (b) Determination of the forces imposed by the seaway on the vessel—the excitation (hydro-dynamic loadings, wave bending moment, slamming forces, and soon), (c) Determination of the response of the ship in her six degrees of freedom (ship motions), (d) Prediction of the perils to which a ship may be exposed (capsizing, foundering, safety at sea), (e) Prediction of the loss in speed she will sustain in heavy weather, (f) Evaluation of the amount of stabilization necessary to prevent unacceptable or undesirable displacements and accelerations. The elastic response and strength of a vessel's structure, though not included under seakeeping, depend, nevertheless, in an essential manner thereon.

4 — S e a k e e p i n g

I N SEAKEEPING studies, t w o basic approaches h a v e

been u t i l i z e d . I n t h e f i r s t — w h i c h m a y b e t e r m e d k i n e t i c — a causal r e l a t i o n s h i p is s o u g h t b e t w e e n t h e sea-w a y a n d t h e m o t i o n s t h a t t h e vessel experiences, t h e l a t t e r b e i n g o b t a i n e d as f u n c t i o n s of t h e forces imposed b y t h e seaway o n t h e vessel. T h i s was t h e a p p r o a c h used b y F r o u d e (1861) a n d K r i l o f f (1896) (1898a) i n t h e i r m o n u m . e n t a l w o r k s w h i c h established t h e f o u n d a -t i o n s o f -t h e s c i e n -t i f i c s -t u d y o f -t h e s u b j e c -t . H o w e v e r , because t h e i r a p p r o a c h d i d n o t y i e l d p r a c t i c a l s o l u t i o n s , n a v a l a r c h i t e c t s were f o r c e d t o a d o p t a m o r e p r a g m a t i c m e t h o d , w h i c h m a y be t e r m e d t h e k i n e m a t i c a p p r o a c h . H e r e n o causal r e l a t i o n s h i p is sought, t h e p u r e l y e m -p i r i c a l f o r m u l a t i o n t h a t is d e r i v e d states w h a t t h e f u t u r e m o t i o n s o f a ship w i l l be based s i m p l y o n a k n o w l e d g e of t h e p a s t h i s t o r y o f t h e m o t i o n s themselves. T h e r e is n o reference t o t h e forces b y w h i c h t h e y are caused.

Section [ 1 ] — I n t r o d u c t i o n , a n d [ 2 ] — P o w e r i n g , appeared i n the October, 1 9 5 8 issue, pages 1 - 3 6 ; Section [ 3 ] — C o n t r o l , appeared

i n t h e D e c e m b e r , 1 9 5 8 issue of T H E J O U R N A L O F S H I P R E S E A R C H ,

pages 1 - 2 2 . References t o the f o r e g o i n g sections w i l l be designated b y n u m b e r s i n brackets as shown.

^ I n s t i t u t e f o r Defense Analyses, W e a p o n Systems E v a l u a t i o n D i v i s i o n , T h e Pentagon, W a s h i n g t o n , D . C .

' P h y s i c i s t , D a v i d T a y l o r M o d e l Basin, N a v y D e p a r t m e n t , W a s h i n g t o n , D . C.

N O T E : Because of t h e l e n g t h of Section [4], i t w i l l be published i n t w o i n s t a l l m e n t s .

B u t e v e n i n i t s u l t i m a t e f o r m , t h e k i n e m a t i c a p p r o a c h can o n l y y i e l d r e s u l t s of l i m i t e d usefulness w h i c h w i l l n o t s a t i s f y t h e s c i e n t i f i c designer. I t has become necessary, t h e r e f o r e , t o e x t e n d t h e k i n e t i c a p p r o a c h a n d o b t a i n m o r e r e a l i s t i c solutions. T h e essential f i r s t step i n t h i s d i r e c t i o n w a s t o develop a n a d e q u a t e d e s c r i p t i o n of t h e seaway.

T h e seaway b e i n g k n o w n , one m a y t h e n proceed t o d e t e r m i n e t h e ship d y n a m i c s , i.e., t h e forces o n t h e vessel t h a t r e s u l t f r o m t h e a c t i o n of t h e sea a n d , f i n a l l y , t h e s h i p k i n e t i c s , i.e., t h e sea b e h a v i o r o f t h e vessel, v i z . , h e r m o t i o n s , loss i n speed a n d s t a b i h z a t i o n r e q u i r e m e n t s . T h i s p a t t e r n is a d o p t e d f o r t h e p r e s e n t a t i o n h e r e i n .

THE SEAWAY

Survey Since k n o w l e d g e o f t h e seaway is t h e o b v i o u s , f i r s t necessary p r e r e q u i s i t e , i t is q u i t e s u r p r i s i n g t h a t t h i s k n o w l e d g e has l o n g r e m a i n e d a n elusive i t e m . U n t i l q u i t e r e c e n t l y , t h e seaway w a s i n t r o d u c e d i n t o p r o b l e m s o f s h i p m o t i o n s i n a f o r m r e p r e s e n t i n g a n ex-t r e m e i d e a l i z a ex-t i o n ; ex-t h e waves were a l l l o n g - c r e s ex-t e d sinusoids each i d e n t i c a l i n p r o f i l e w i t h a l l t h o s e p r e -ceding o r f o l l o w i n g i t . F o r some t i m e , t h i s i d e a l i z a t i o n ( u p o n w h i c h t h e classical w a v e t h e o r y has b e e n erected) w a s a c o n v e n i e n t one, f o r i t p e r m i t t e d t h e d e v e l o p m e n t of c e r t a i n a n a l y t i c a l a n d e x p e r i m e n t a l aspects o f t h e

lournal of

SHIP R E S E A R C H

An Analysis of Ship C@[lisi@rnis \MA

to Pr@t®€rt@n of Nyclaar Power [Plooifs

By V. U. Minorskyi

W I T H t h e a d v e n t of nuclear p o w e r p l a n t s , n a v a l a r c h i -tects as w e l l as m a r i n e engineers find themselves con-f r o n t e d w i t h a n u m b e r ocon-f new problems. One ocon-f these concerns t h e e x t e n t of collision p e n e t r a t i o n s i n connec-t i o n w i connec-t h safeguard r e q u i r e m e n connec-t s .

Of a l l t h e m a j o r accidents w h i c h a ship can i n c u r , c o l l i -sions are t h e m o s t f r e q u e n t , a n d w i t h t h e i n c r e a s i n g l y h i g h e r speeds a n d displacements of m o d e r n ships, i t appears t h a t i n t h e f u t u r e , collisions w i l l be, o n t h e average, m o r e serious t h a n i n t h e past unless m o d e r n n a v i g a t i o n a l aids are u t i l i z e d t o t h e i r u t m o s t . A serious c o l l i -sion i n w a y of a n u n p r o t e c t e d nuclear p o w e r p l a n t c o u l d cause t h e release of "radioactive m a t e r i a l t o t h e outside e n v i r o i m i e n t . Since a h i g h percentage of collisions occur i n h a r b o r channels a n d i n t e r r i t o r i a l waters, i t is necessary t o p r e d i c t w i t h some degree of accuracy t h e c o n d i -t i o n s u n d e r w h i c h -t h e r e a c -t o r space of a nuclear ship w i l l r e m a i n i n t a c t a n d , consequently, w h a t s t r u c t u r a l s t r e n g t h s h o u l d be b u i l t i n t o t h e h u l l of a nuclear ship o u t b o a r d of t h e r e a c t o r p l a n t i n order t o absorb safely a g i v e n a m o u n t of k i n e t i c energy i n a collision.

Method of Solution

T h e r e is available a large a m o u n t of p h o t o g r a p h i c a n d d e s c r i p t i v e evidence t o t h e e f f e c t t h a t t h e collision of m e r c h a n t ships is a case of a l m o s t w h o l l y inelastic i m p a c t . I t i s also r a t h e r o b v i o u s t h a t t h e resistance forces w h i c h d e v e l o p d u r i n g t h e i m p a c t p e r i o d do n o t l e n d themselves t o a n a l y t i c a l m e t h o d s of c a l c u l a t i o n ; i.e., t h e s t r u c t u r e of b o t h ships is s u b j e c t t o complex progressive f a i l u r e s b y b u c k l i n g of panels, shearing, t e a r i n g , c r u s h i n g , b e n d i n g a n d t w i s t i n g of plates a n d shapes. M o s t of t h e w o r k done b y t h e forces exerted u p o n t h e s t r u c t u r e takes place b e y o n d t h e elastic range f o r steel.

I t was decided t h a t , since a n a t t e m p t t o p r o d u c e a n a n a l y t i c a l s o l u t i o n w o u l d of necessity rest o n m a n y d o u b t f u l assumptions, i t was p r e f e r a b l e t o f o l l o w a s e m i a n a l y t i c a l a p p r o a c h based o n t h e f a c t s of a c t u a l c o l l i -sions. D a t a were p r o v i d e d f o r 50 recent colhsions b y t h e U . S . Coast G u a r d . T h e d a t a i n c l u d e d speeds, angle of

1 N a v a l Ai-chiteot, George G. Sharp, I n c . , N e w York, N . Y .

encounter, displacements, d r a f t s , a n d e x t e n t a n d l o c a t i o n of damage. T o t h i s w a s added t h e p a r t i c u l a r s of t h e Stockholm-Andrea Doria collision, a n d those of t w o o t h e r recent collisions where i n each case a t a n k e r was s t r u c k b y a passenger ship a t h i g h speed.

Of these collisions a f e w were e l i m i n a t e d because o f l a c k of precise s t r u c t u r a l d a t a . Collisions w h e r e t h e angle of e n c o u n t e r was t o o sharp were also e h m i n a t e d because i t was decided t h a t t h i s w o u l d render t h e selection of sign i f i c a sign t s t r e sign g t h members evesign m o r e d i f f i c u l t ; i sign a d d i -t i o n , -t h e p e n e -t r a -t i o n s i n o b l i q u e collisions were m u c h smaller a n d t h e r e f o r e these collisions were n o t as p e r t i -n e -n t t o t h e p r o b l e m as r i g h t a-ngle colhsio-ns. T h e t w e n t y s i x collisions t h e n r e m a i n i n g w e r e a n -a l y z e d f o r t h e k i n e t i c energy -absorbed i n t h e c o l h s i o n -a n d f o r t h e e x t e n t of damage. T h e procedure is as f o l l o w s : D e f i n i n g : MB, VB as t h e mass a n d v e l o c i t y a t i m p a c t of s t r i k i n g vessel.

MA, VA as t h e mass a n d v e l o c i t y of t h e s t r u c k vessel, i j as t h e final c o m m o n v e l o c i t y i n t h e d i r e c t i o n of t h e s t r i l d n g vessel, dm as t h e v h t u a l increase i n mass of t h e s t r u c k vessel due t o w a t e r e n t r a i n e d , we h a v e U{,MB + MA + dm) = MAVA + MBVB f r o m t h e c o n s e r v a t i o n of m o m e n t u m . T h e k i n e t i c en-ergy r e m a i n i n g i n t h e system a t t h e e n d of i m p a c t is 1 / 2 {MB + MA + dm) W

Since, as i n d i c a t e d , w e are p r i m a r i l y i n t e r e s t e d i n penet r a penet i o n s n o r m a l penet o penet h e s penet r u c k ship's c e n penet e r l m e , i n a penet -t e m p -t i n g -t o find a c o r r e l a -t i o n be-tween k m e -t i c energies a n d d a m a g e t o t h e c o l l i d i n g ships o n l y t h e v e l o c i t y c o m -p o n e n t s n o r m a l t o t h e s t r u c k shi-p's centerline need be considered. B y selecting collisions a p p r o x i m a t e l y a t r i g h t angles i n t h i s s t u d y t h e e r r o r due t o n e g l e c t i n g c o m p o n e n t s of k i n e t i c energy paraUel t o t h e s t r u c k ship's axis is m i n i m i z e d .

W i t h these considerations, VA of t h e s t r u c k s h i p is

Model Experiments With Fixed

B o w Antipitchihg Fins

By George P. Stefun^

The purpose of this paper is to provide additional information relative to the problem of designing efficient antipitching devices. The results of seakeeping experiments are pre-sented for a model fitted with several alternate designs of fixed bow fins. The bulk of the experiments is concerned with the effects that fins of different aspect ratios have on the pitching and heaving motions, the phase angles between pitch and heave, the vertical accelerations, and the speed reductions in waves. Results also are presented showing the effects on pitching of fin area and of fin-tip fences.

T H E effects of f i x e d b o w a n t i p i t c h i n g f i n s o n t h e seak e e p i n g characteristics of s h i p models have been i n v e s t i -g a t e d a t t h e D a v i d T a y l o r M o d e l B a s i n as p a r t of t h e f u n d a m e n t a l h y d r o m e c h a n i c s research p r o g r a m , a n d f o r specific a p p l i c a t i o n t o i n d i v i d u a l I I . S. n a v a l ships

[ 1 , 2 ] . 2 T h e present paper gives t h e results of e x p e r i -m e n t s designed t o show t h e effects of v a r i o u s f i n con-f i g u r a t i o n s o n t h e p i t c h a n d heave, speed loss, phase angles, a n d v e r t i c a l accelerations of a 6 . 6 - f t m o d e l i n r e g u l a r h e a d seas. T h e t e s t c o n d i t i o n s i n c l u d e d a speed range corresponding t o F r o u d e n u m b e r s f r o m 0 t o 0 . 3 0 , a n d a range of w a v e l e n g t h s corresponding t o w a v e l e n g t h - s h i p l e n g t h r a t i o s f r o m 0 . 7 5 t o 1 . 6 1 .

I n general, t h e results i n d i c a t e t h a t f i x e d b o w fins p r o -duce m a x i m u m p i t c h r e d u c t i o n s f o r ship-speed a n d w a v e l e n g t h c o m b i n a t i o n s t h a t correspond t o near s y n -chronous c o n d i t i o n s . F o r t h e p a r t i c u l a r models a n d t e s t c o n d i t i o n s of t h i s i n v e s t i g a t i o n , m a x i m u m p i t c h r e d u c t i o n s u p t o 3 7 per cent w e r e o b t a i n e d w i t h fins of t o t a l p l a n area e q u a l t o 2 . 0 per cent of t h e w a t e r p l a n e area a n d aspect r a t i o e q u a l t o 2 . 0 . S i m i l a r fins w i t h a n aspect r a t i o of 0 . 5 0 were a b o u t 3 0 per cent less e f f e c t i v e

1 D a v i d T a y l o r M o d e l Basin, N a v y D e p a r t m e n t , W a s h i n g t o n , D . 0 .

^ N u m b e r s i n brackets designate references at t h e e n d of t h e paper.

t h a n t h e fins of higher aspect r a t i o , f o r corresponding tests c o n d i t i o n s .

T h e results of a brief s t u d y o f t h e effects of fin area o n p i t c h r e d u c t i o n s i n d i c a t e t h a t 2 . 0 p e r cent fins were t w i c e as e f f e c t i v e as 1.0 per cent fins w i t h t h e same aspect r a t i o . F i n s of 4 . 0 per cent area, h o w e v e r , were o n l y a b o u t t h r e e t i m e s as e f f e c t i v e as t h e 1.0 per cent fins. T h u s , t h e increase i n eft'ectiveness was d i r e c t l y p r o p o r -t i o n a l -t o -t h e increase i n area, o n l y i f -t h e areas d i d n o -t become t o o large.

T h e v a r i o u s fin c o n f i g u r a t i o n s caused a 1 0 t o 1 5 per cent increase i n t h e c a l m - w a t e r resistance of t h e m o d e l , I n waves however, t h i s increase was m o r e t h a n o f f s e t b y a decrease i n t h e m o t i o n - i n d u c e d resistance. I n general, t h e r e f o r e , t h e use of fins r e s u l t e d i n i m p r o v e d a b i l i t y t o m a i n t a i n speed, especially i n s y n c h r o n o u s c o n d i t i o n s .

Table 1 Fin Particulars

F i n N o . 1. 2. 3. 4. 4 ( a ) 5. Span, i n . 5 . 0 2 . 5 3.15 4 . 5 0 4 . 5 0 6.37 C h o r d , i n . 2 . 5 5 . 0 2 . 1 3 . 0 3 . 0 4 . 2 5 P l a n area 0 . 0 2 0 . 0 2 A „ 0 . 0 1 A„ 0 . 0 2 A „ 0 . 0 2 A „ 0 . 0 4 A,„ Aspect r a t i o T i p fences ( l e n g t h , w i d t h ) N o n e N o n e 2 . 1 , 0 3 . 0 , 1 N o n e 4 . 2 5 , 1 . 5 6 25

-Nomenclature-A-w —

w a t e r p l a n e area

h

M =

aspect r a t i o

K

B =

b e a m

. L

CB = b l o c k c o e f f i c i e n t V F = F r o u d e n u m b e r

g =

acceleration of g r a v i t y

H =

d r a f t = l o n g i t u d i n a l r a d i u s of g y r a t i o n = l e n g t h b e t w e e n p e r p e n d i c u l a r s = m o d e l speed = a m p l i t u d e of heave w i t h fims i n s t a l l e d = a m p l i t u d e of heave w i t h o u t fins A = d i s p l a c e m e n t

5 = phase angle b y w h i c h heave lags p i t c h X = w a v e l e n g t h xpf. = a m p l i t u d e of p i t c h w i t h fins i n s t a l l e d iZ-o = a m p l i t u d e of p i t c h w i t h o u t fins 14 J O U R N A L O F SHIP RESEARCH