CASCARAS DINAMICA

8/13/2019 CASCARAS DINAMICA

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S iD 67-498

STUDY OF APOLLO WATER IM P A C TFINALREPORT

VOLUME 2D Y N A M I CR E S P O N S EOF SHELLS OF REVOLUTION DUR ING

VERTICAL IMPACTINTO WATER - NO INTERACTIONContract NAS9-4552, G.O. 5264)

May 1967

P r e p a r e d b y

A. P . C a p p e l l iJ . P . D. Wilk i n son

Autho r s )

P r o g r a m M a n a i e r

S t r u c t u r e s a n d M a t e r i a l s

L . A . H a r r i s-

k i e n c e a nd T echno logy

N O R T H A M E R I C A N A V I A T I O N , I N C .SPACE DIVISION


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.. PRECEDING PAGE BLANK NOT FILf;'ED.

FOREWORD

T h i s r e p o r t w a s p r e p a r c d by N o r th A m e r i c a n Av ia ti on , Inc . , SpaceDivis ion , und er NASA Con t ra c t NAS9-4552, fo r the Nat iona l Ae ron au t ic s andS p a c e A d m i n i s t r a t i o n , M a n ne d S p a c e F l i g ht C e n t e r , H o u s t on , Te x a s , w i thD r . F . C . H u ng , P r o g r a m M a n a g e r a nd M r. P. P. R a d k o w s k i , A s s i s t a n tP r o g r a i n M a n ag e r.S t r u c tu r a l Me c h an i c s D iv i s i o n , M S C, Hous ton , T exas w ith D r. F . St ebb i n sa s t h e t e c h n i c a l m o n i t o r ,

T h i s w o r k w a s a d m i n i s t e r e d u n d e r t he d i r e c t i o n of

T h i s r e p o r t i s p r e s e n t e d i n e l e ve n v o l u m e s f o r c o n v e ni e n c e i n h an d li n ga nd d i s t r i bu t i on . A l l v o l u m e s a r e u n c l as s i fi e d .

The o b j ec t i v e of t he s t u dy was t o dev e lop m e th od s and F o r t r a n IVc o m p u t e r p r o g r a m s t o d e t e r m i n e by th e t e c h n i q u e s d e s c r i b e d be l ow, t h eh y d r o -e l a s t i c r e s po nse of r ep r e sen t a t i o n of t he s t r uc tu r e of t he Ap o llo Com-ma nd Modu le imm ed ia t e l y fo ll owing imp ac t on t h e wa t e r . The de ve lopmen tof t h e o r y, m e t h o d s a n d c o m p u t e r p r o g r a m s i s p r e s e n t e d a s Ta s k I Hy dr o -d y na m ic P r e s s u r e s , Ta s k I1 S t r u c t u r a l R e sp o n s e a nd T a s k 111 H y d r o e l a s t i cR e s p o n s e A n a l y si s .

U n d e r Ta s k I - C o m p u ti n g p r o g r a m t o e x t en d f l e x i b l e s p h e r e u s i n g t h eA n a l y t i c a l f o r m u l a t i o np e n c e r a n d S h i f fm a n a p p r o a c h h a s b e e n d e v e l o pe d ,

b y D r. L i u s i n g n o n l i n e a r h y d r o dy n a m i c t h e o r y on s t r u c t u r a l p o r t i o n i sf o r m u l a t e d . I n o r d e r to c o v e r a w i d e r a n g e of i m p a c t co n d i ti o n s , f u t u r ee x t e n s i o n s a r e n e c e s s a r y i n t h e fo llo win g i t e m s :

a . U s in g l i n e a r h y d r o d y n a m i c t h e o ry t o i n c l u d e h o r i z o n t a l v e l o c it yan d ro t a t i on .

b. N o n l i ne a r h y d r o d y n a m i c t h e o r y t o d ev e l o p co m p u t i ng p r o g r a m o ns p h e r i c a l p o r t i o n a n d t o d ev e lo p n o n l i n ea r t h e o r y o n t o r o i d a l a n dc on i c s ec t i ons .

U n d e r Ta s k I1 - C o m p ut in g p r o g r a m a n d U s e r ' s M a n u a l w e r e d e v el o pe df o r n o n s y m m e t r i c a l l o a d in g o n u n s y m m e t r i c a l e l a s t i c s h e l l s .d e v e lo p t h e t h e o r y a n d m e t h o d s t o c o v e r r e a l i s t i c A p o l l o c o nf i g ur a t io n t h ef o ll o wi n g e x t e n s i o n s a r e r e c o m m e n d e d :

To fu l ly

a. Modes of v ib r a t i o n and m oda l a n a l y s i s .

b. E x t e ns i o n to n o n s y m m e t r i c s h o r t t i m e i m p u l s e s .

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C. L i n e a r b u c k l i n g a n d e l a s t o - p l a s t i c a n a l y s i s

T h e s e t e c h n i c a l e x t e n s i o n s w i l l n o t o nl y b e u s e f u l f o r A p o ll o a n df u t u r e Apol lo g rowth c on f i gu ra t i ons , bu t they w i l l a l so be of va lu e t o o th e ra e r onau ica l and s pac e c r a f t p r og r a m s .

T h e h y d r o e l a s t i c r e s p o n s e of t h e f le x i b l e s h e l l i s obta ined by then u me r i c a l s o l u t i on of t he c om bi n ed h yd rody n amic and s he l l e quat i ons .r e s u l t s o bt ai ne d h e r e i n a r e c o m p a r e d n u m e r i c a l l y w it h t h o s e d e r i v e d b yn e g l ec t in g t h e i n t e r a c t i o n a n d ap p ly i ng r i g i d b od y p r e s s u r e s t o t he s a m ee l a s t i c s h e l l .i m p a c t of t h e p a r t i c u l a r s h e l l s t u di e d , t h e i n t e r a c t i o n b e t w ee n t h e s h e l l a n dt h e f lu i d p r o d u c e s a p p r e c i a b l e d i f f e r e n c e s i n t h e o v e r a l l a c c e l e r a t i o n of t h ece n t e r of g r av i t y of t he sh e l l , and i n the d i s t r i bu t i on of t h e p r e s s u r e s andr e s p o n s e s . H o we ve r t h e m a x i m u m r e s p o n s e s a r e w it h i n 1 5 % of t h o s e p r o -duc ed whe n th e i n t e r ac t i o n be t wee n t he f l u id and t he sh e l l i s ne g l ec t ed . A

b r i e f s u m m a r y of r e s u l t s i s shown in t he abs t r ac t s o f i nd iv i dua l vo lumes .

T h e

T h e n u m e r i c a l r e s u l t s s ho w t h at f o r a n a xi a ll y s y m m e t r i c

T h e v ol u m e n u m b e r a n d a u t h o r s a r e l i s t e d o n t h e f ol lo w in g p a ge .

T h e c o n t r a c t o r ' s d e s i g na t io n fo r t h i s r e p o r t i s SID 6 7 - 4 9 8 .

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.INDEX FOR FINAL REPORT DRAFT

"Apo ll o Wa te r Impa c t "

Volume No,

1

Volume Ti t le A u t h o r s

Hydrodynamic Analysis of ApolloWa t e r I m p a c t

T. L i a nd T. Sug i m u ra

2 Dyn ami c Respo n s e of Shells ofR evo l u t i on Dur ing Ve r t i c a l Impac tI n t o Wa t e r - No In t e r ac t i on

A . P. Ca ppe l l i , andJ . P. D. Wilk in son

3 D y n a m i c R e s p o n s e of Sh ell s of

R evo l u t i on Dur i ng Ve r t i c a l I mp ac tI n to Wa t e r - H y d r o e l a s t i cI n t e r a c t i o n

J . P. D. Wilkinson,

A. P. Cappe l l i ,R . N . S a l z m a n

Co mp a r i s o n Wi th E xpe r ime n t s J . P. D. Wilkinson

U s e r ' s M a n u a l - No In t e r ac t i on J . P. D. Wilkinson

U s e r ' s M a nu al - I n t e r a c t i o n J . P. D. Wilkins on andR .N . Sa l zman

7 Mo dific ation of She ll of Revol utio nA n a l y s i s

A . P. Cappel l i andS . C . F u r u i k e

8 U n s y m m e t r i c S h e l l of Revolu t ionA n a l y s i s

A. P. Cappel l i ,T. Ni s h imo to ,P. P. Radkows k i a n dK. E. P a u l e y

A. P. C a p p e l l iMode Shap e s and Na tu r a lF r e q u e n c i e s A n a ly s i s

10 U se r ' s Man u a l f o r Mo d i f ic a ti on ofShel l of Re volu t ion Analys i s

A. P. Cappel l i andS . C . F u r u i k e

11 U s e r ' s M a n u a l f o r U n s y m m e tr i cShe l l of Re volu t ion Analys i s

E. C a r r i o n ,S . C. F u r u i k e a n dT. N i s h i m o t o

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ABSTRACT

A g e n e r a l m e th o d i s d e ve l op e d f o r p r e d i c t i n g t h e

r e s po nse of t h in e l a s t i c she l l s of r evo l u t i on sub j ec t e dt o a r b i t r a r y t i m e -d e p e nd e n t l o a d s ,the dynamic s he l l equa t ions by m ea ns of a def in i te -d i f f e r en ce f o rm u l a t i o n i n both spa c e a nd t ime . A s ana p p l i c a t io n of t h e g e n e r a l n u m e r i c a l m e th o d , t h e r e s -pons e of b lun t sh e l l s of revo lu t io n dur in g a v e r t i c a li m p a c t i n to a n i n c o m p r e s s i b l e f l ui d i s s tud i ed ,f o r m u l a i s d e r i v e d f o r t h e p r e s s u r e s o n a r i g i d bo dyof r ev o lu t i on du r ing ve r t i c a l impac t .t he p r e s s u r e s a r e m i n i m u m a t th e i m p a c t po in t, a n dv e r y l a r g e n e a r t h e e dg e of t h e w e t t e d s u r f a c e of t h ebody. T h e s e p r e s s u r e s a r e a p pl ie d a s a f o r ci n gfunc ti on t o a sh e l l r ep r e s en t i n g a t y p ic a l r e - e n t r yv e h ic l e . T h e n u m e r i c a l r e s u l t s sh ow t h a t t h e r e s -p o n s e i s l a r g e s t a t t h e i m p a c t p oi nt , a n d t h e th ei n t e r a c t i o n of t h e f l ex ib l e s t r uc tu r e a nd t he f l u idm a y h a v e a n e f f ec t o n t he p r e s s u r e d i s t r i b u t i o n s .

T h e m e t h o d s o l v e s

A

I t i s shown tha t

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C O N T E N T S

L I S T O F I L L U S T R AT I O N S .

N O M E N C L AT U R E .I N T R O D U C T I O N .

P R E S S U R E P R O F I L E S .

D YN AM IC R E S P O N S E O F S H E L L S O F R E V O L U T I O N

N U M E R I C A L A N A LYS I S .N U M E R I C A L R E S U LT S

CONCLUSION .

I L L U S T R AT I O N S .R E F E R E N C E S .

A P P E N D I X A - A U X I L I A RY R E S U LT S .

P a g e

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LIST O F ILLUSTRATIONS

Fig u r e

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Model of Re- En t ry Vehic le Dur ing Im pac tAcce le ra t ion of Center o f Gravi ty of a Sphere Dur ing

P r e s s u r e P r o f i l esS h e l l E l e m e n tDeflect ion W a t A pex .Me rid i ona l Bending Mom ent MC a t ApexM e r i d i o n al M e m b r a n e F o r c e NE, a t Apex

Veloci ty of She l l ap ex WDeflect ion W .M e r id i on a l S t r e s s a t th e E x t r e m e F i b e rDeflect ion W a t A pe x f o r Va r i o u s Ti m e I n c r e m e n t s .M e r i d i o n a l B e nd in g M om e nt a t A pe x f o r Va r i o u s Ti m e

I n c r e m e n t s .Reduct ion in Mer id iona l Bending Moment a t Apex Due

TO R e duced Edg e P r e s su r e .M e r i d i on a l Bend ing Mom en t Mf .S t r e s s Ug On Ou te r F ib e r a t A pex .S t e r e o g r a p h i c P a i r s of t h e Ri gi d -B o dy P r e s s u r e s

Ave ra ge P r e s s u r e - Pull Sc a l e Mode l .Ave ra ge P r e s s u r e - Q u a r t e r S c a l e M o d e lD i sp l a c eme n t W a t Apex I l lus t r a t ing Effec t of She l l

Mer id iona l Bending Moment M 5 a t Apex I l l u s t r a t i ng

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r e f e r e n c e s t r e s s l ev e l

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index denoting sh e l l s ta t i on poin t

d i m e n s i on l e s s m e r i d i o n a l s p a t i a l i n c re m e n t

num ber of sp a t i a l s t a t i ons

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INTRODUCTION

T h e d e s ig n of e l a s t i c s t r u c t u r e s w h ic h i m p a c t i n t o f l u id s r e q u i r e s akno wledge of t he man ne r i n w h ich t he s t r u c t u r e r e spo nd s t o t he im pa c tf o r c e s , a n d a d e t e r m i n a t i o n of th e im p a c t f o r c e s t h e m s e l v e s .o f t h i s r epo r t i s t wo -f o ld : f i r s t , t o p rov ide a me an s of c a l cu l a t i ng t he p r e s -su r e p ro f i l e s wh i ch ac t on a sh e l l of r evo lu t i on du r ing f l u id impac t , ands e c o n d , t o p r e s e n t a n u m e r i c a l m e th o d f o r t h e d e t e r m i n a t i o n of t he d y n a m i cr e s p o n s e of e l a s t i c s h e l l s of r e v o l u ti o n s u b j e c t e d t o a r b i t r a r y t i m e - v a r y i n gl o a d d i s t r i b u t i o n s .t h o s e o c c u r r i n g d u r i n g a v e r t i c a l i m p a c t a t a po i nt o n t h e s h e l l w h e r e th era te of change of cu rv a t ur e i s sm a l l in t he ne ig hbo rhood of t he i m p ac t .

c a l c u l a t i o n of d y n a m i c r e s p o n s e , h o w e v e r, i s q u i te g e n e r a l , a n d c a n b eap p l i ed t o an y t h in e l a s t i c sh e l l s of r evo lu t i on w i th in the f r a m e w o r k of l i n e a rf i r s t - o r d e r s h e l l t h eo r y.

T h e p u r p o s e

T h e c a lc u l a t i o n of p r e s s u r e p r o f i l e s i s r e s t r i c t e d t o

The

T h e d y n a m i c a n a l y s i s of t h e e l a s t i c s h e l l i s b a s e d o n S a n d e r s ' s h e l lt h e o r y 1 , an d i nvo lve s a n e x t ens io n of t he n um er i c a l p roc ed u re deve loped by

2B ud iansky and Ra dkowski f o r t he s t a t i c ana l y s i s of sh e l l s . An impl ic i tn u m e r i c a l m e t h od of t i m e w i s e i n t e g r a t i o n i s u s e d i n t h e s o l ut i o n of thed y n a m i c p r o b l e m . T h i s i m p l i c i t m e t ho d w a s f i r s t u s e d s u c c e s s f u l l y byHo u bo l t i n s t ud yi ng t he r e s po nse of a i r c r a f t t o g us t l oads 3 . n T he Houbol tm e t h o d w a s r e c e n t l y e m p l o y e d b y J o hn s on a n d G r e i f i n d e t e r m i n i n g t h ed y n am ic r e sp on s e of a cy l i n d r i ca l she l l 4 .ge n e ra l i z a t i on o f t ha t u s ed and su gges t ed i n Re f e r ence 4 .

T h e m e t h o d p r e s e n t e d h e r e i s a

T h e H o ub ol t m e t h o d f e a t u r e s t he u s e of b a c k w a r d d i f f e r e n c e e x p r e s s i o n st o r e p r e s e n t t h e i n e r t i a t e r m s i n th e d y n am i c s h e l l e q ua t i on s .J o h n s o n 5 , t h e m e t h od i s n u m e r i c a l l y s t a b l e a n d t h u s o f f e r s a n a d v a n ta g eo v e r v a r i o u s o t h e r p o s s i b l e f i n i t e - d i ff e r e n c e m e t h o d s .t i m e i n c r e m e n t , h o w e v e r, m u s t be s e l e c t e d s m a l l e no ug h t o g u a r a n t e ea c c u r a t e r e s u l t s 6 .p r o c e d u r e d e v e l o p e d i n R e f e r e n c e 2 , and involves the exp ans i on of thed e p e n de n t v a r i a b l e s i n F o u r i e r s e r i e s , w it h t h e s u b s e q u e n t r e p r e s e n t a t i o n of

t h e s p a t i a l d e r i v a t i v e s i n f i n i t e - d i ff e r e n c e f o r m .t e ch n iq u e ( P o t t e r s ' m e t h o d 7 ) i s use d t o so l ve t h e s e t of a l g e b r a i c e q ua t i o n sr e s u l t i n g f r o m t h e fi n i t e - d i ff e r e n c e a p p r o x i m a t i o n s t o t h e r e d u c e d s h e l ld i f f e r en t i a l eq ua t i on s of mo t ion .

A s shown by

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A d i r e c t m a t r i x e l i m i n at i o n

The s t u dy of t he im pac t of s t r u c t u r e s i n to f l u id s f i n ds app l i c a t ion i nt h e p r o b l e m s a s s o c i a t e d w it h t h e s l a m m i n g of s h i p s i n h e a v y s e a s , t h e w a t e r

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en t r y of t o r pedo e s , and t h e wa t e r l an d ing of sp ac e c r a f t .l i t e r a t u r e c o n c e rn e d w it h s l a m m i n g a nd w a t e r i m p a c t h a v e b e e n m a d e b y

t h e m a x i m ui ii o v e r a l l f o r c e s o c c u r d u r i ng a p e r i o d w h e n th e m o t i o n o f t h e

f l ui d a b ou t th e s h e l l i s a d e q u a t e l y d e s c r i b e d by i n c o m p r e s s i b l e p o t e n t i a l

Su rve ys of the

Chu and Abram son8 and by Szebeh e ly and Ochi 9. I n 'the p r e s e n t p r o b l e m ,

flow t .

The c l a s s i c a l pa p e r s of von K 'a rm 'an'' a nd Wagne r " h ave e s t a b l i s he dme t ho ds of t r e a t i n g t he two-d im e ns iona l imp ac t of r i g id V-w edges i n to ani n c o m p r e s s i b l e f lu i d.t h e f lu i d co u ld b e u s e d t o a d v a n t a g e , a n d d e t e r m i n e d t h e p r e s s u r e s o n r i g i dw edges by a pply ing the known s t e ad y - s t a t e so lu t i ons f o r t he f low abou t a f l a tp l a t e to t h e u n st e ad y h y d r o d y n a m i c p r o b l e m . T h e t e c h n i q u e s d ev e l o pe d f o rw edges a l so app ly t o t he s t udy of b od i e s o f r evo lu t i on .bod i e s h a s b een i n ves t i ga t ed by Sh i ffma n and Spencer13 , who a l so g ave t he

14s o l u t i on fo r the po t e n t i a l a s s oc i a t e d w i th t he f low abou t a pene t r a t i n g sp he reT h e i r s o lu t io n f o r t h e s p h e r e h a s b e e n u s e d by K o r k e g i 1 5 t o p r e d i c t th e p r e s -s u r e h i s to r y f o r a n i m p a ct i n g s p h e r e by c o n s i d e r i n g t h e f lo w a b o u t a c i r c u l a rd i s c .

Th e y showed how t he c o n cep t o f t h e v i r t u a l m a s s of

The impac t of co n i c a l

I n t h i s r e p o r t , a m o d i f i c a t i o n of K o r k e g i ' s a n a l y s i s i s m a d e f o r a r i g i ds p h e r e . T he p r e s s u r e s d e r i v e d f r o m th i s m o d if ic a ti o n a r e a p pl ie d a s af o r c in g func ti on t o a sh e l l of r evo lu t i on .t h e d y n a m i c r e s p o n s e of a s h al l o w s p h e r i c a l s h e l l d u r i n g w a t e r i m p a c t . T h es h e l l s t r u c t u r e i s c o n s i d e r e d t o be c l a m p e d b el ow a h e a v i e r r ig i d m a s s( F i g u r e 1 ) s o t h a t t h e i r c o m b in e d m a s s s i m u l a t e s a t y p i c a l r e - e n t r y v e h i c l e .I n t h i s s t u d y, t h e e f f e c t s of t h e h y d r o e l a s t i c i n t e r a c t i o n b e t w e e n t h e f l e x i b l esh e l l and t he f l u id have be en neg l ec t e d .t h e ba s i s of a r i g id -bo dy ana ly s i s , an d t he r e s po nse of t he sh e l l t o t h i sforc ing func t ion i s s t ud i e d .p o s s i b l e a n e v a l u a ti o n of t h e e f f ec t o n th e h y d r o e l a s t i c i n t e r a c t i o n w h ic h w i l lf o r m t h e s u b j e c t - m a t t e r of Vo lu m e 3 of th i s re po r t .

N u m e r i ca l r e s u l t s a r e p r e s e n t e d f o r

The fo r c i ng f unc t i o n i s de f ined on

It i s i n t en d ed t h a t t h e s e r e s u l t s w i l l m a k e

t h e p r e s e n t p r o b l e m , t h e m a x i m u m f o r c e s a r e f ou n d t o o c c u r w i th in t h em i l l i - s e c o n d r a n g e .t o p r ed o m i n a te o nl y i n t h e m i c r o s e c o n d r a n g e a n d a r e t h e r e f o r e n e g l ec t e d i nt h i s p a p e r .im pa c t of f la t p la tes .

T h e e f f e c t s of c o m p r e s s i b i l i t y a n d a i r b u b b l es a p p e a r

T h e s e e f f e c t s a r e d i s c u s s e d by C h ua ng lO i n c o n n e c t i o n w i th t h e

-2 -


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P R ES S U RE P R O F I L E S

Accord ing to in co mp res s ib le po ten t ia l theo ry, the flow of f lu id a ro un dthe sh e l l i s def ined by a ve loc i ty po ten tia l I w h i c h s a t i s f i e s L a p l a c e ' se q u at i on ( s e e , f o r i n s t a n c e , R e f e r e n c e 1 6 ) . T he p r e s s u r e i n t h e f l u i d i sr e l a t e d t o I t h r o u g h B e r n o u l l i ' s e q ua ti o n

T h e f l u i d i s d i s p l a c e d b y t h e p e n e tr a t in g s h e l l , and a th in shea th of f lu ide n v e l o ps t h e s h e l l , a s s h ow n in F i g u r e 1, w h e r e t h e f r e e s u r f a c e of t h er e s u l t i n g s p l a s h i s denoted by Fn.b e e q u a l t o t h e a m b i e n t p r e s s u r e .w i ll be a s s u m e d t h a t th e s e co n d t e r m in B e r n o u l l i ' s e q u at io n i s s m a l l c o m -p a r e d w i th t h e f i r s t , w h ic h m e a n s t h a t t h e e f f e ct of t h e s p r a y r o o t i sn e g l e c t e d , a n d t h e f r e e s u r f a c e r e m a i n s p l a n e d u r i n g i m p a c t .p l a n a r s u r f a c e i s denoted in F igu re 1 by Ft h a t

On t h i s f r e e s u r f a c e t he p r e s s u r e m u s tFol lowing Shi ffman and Spe ncer14 , i t

T h e r e s u l t i n gI t fo l lows f rom Equat ion ( 1 )

P'

P= O o n F

D u r in g t h e i m p a c t , t h e p e ne t r at i n g v e h i c l e i m p a r t s s o m e of i t s

m o m e n t u m t o a v i r t u a l m a s s of f l u id a d j a c e n t t o i t .m a g n i t u d e of t h i s v i r t u a l m a s s , t h e w e t te d s u r f a c e of th e p e n e t r a t i n g s h e l l i srep lac ed by a f l a t d i sc of rad iu s c . Consequently, t h e v i r t u a l m a s s m of t h ef l u i d i s t

To d e t e r m i n e t h e

T h e r a d i u s c of t h e w e t t e d s u r f a c e i s r e l a t e d t o t h e c u r v a t u r e of t h e s h e l lR ( f ) , a n d t o t h e d e p t h of p e n e t r a t i o n b. I f b / R


14/52

F o r a s h e l l w ho se c h a n ge i n c u r v a t u r e i s sm a l l i n t he ne ig hbo rhood of t h ei mp a c t po in t ,

s t h a t , app r ox ima t e ly,

1 2c = (2bR)

Acco rdi ng t o t he p r i nc ip l e of co ns e r va t i on of mom e n tum , t he i n s t an -taneo us ve loc i ty V of the cen te r of g r av i ty of the vehic l e i s

dbdt

= - = Vo ( 1 t m / M ) - '

T h e d i s t a n c e s c an d b a r e a p p r o x i m a te d by:C::C

b = V o t

c = (2R Vo t ) 1 / 2

T h u s , t h e a c c e l e r a t i o n A of the center of gravi ty i s given by

A = - 4 f l p Vo / 2 R3j2 t '" /M) (1 t Y)-

w h e r e

T he t i m e a t w hich t he a c c e l e r a t i o n i s a m a x i m u m i s

7 )

9 )

The v e loc i ty p o t en t i a l on t h e su r f ac e of t he d i s c a t a n y t i m e i s given by( R e f e r e n c e 16, p . 144)

+ = - c v ( 1 - y zC:$:::Actually th e dep th Oi p e n e t r a t i o n m a y b e o b t a i ne d f r o m E q u a t i o n ( 7 ) i n t h e

p a r a m e t r i c f o r m

b [ 1 t 1 6 6 p ( R b ) 3 / 2 / 1 5 M ] = Vot

-4 -


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T h e p r e s s u r e o n t h e s u r f a c e of t h e d i s c ( a t x = 0 ) i s f o un d f r o m E q u a t i o n ( 1 )to be

4 r 2

2 1tY ) 2 ['; il2 -5 ]Pv:

T h e s eco nd t e r m on t he r i gh t - han d s i de of Equa t i on ( 1 2 ) a r i s e s f r o m t her e t en t i on of t he t e r m -- P (V9)2 i n Be rno u l l i ' s Equa ti on (1 ) .

In o r d e r t o i l l u s t r a t e s o m e ty p ic a l n u m e r i c a l r e s u l t s , a s t udy was m a de

12

of the im pa ct of a sp he re of r ad iu s 175 in ch es and weight 10000 lb s. in towa te r of den s i t y (62. 5 l b s . / c u . f t . In F i g u r e 2 i s s ho wn t he a c c e l e r a t i o nh i s t o r y of t h e c en t e r of g r av i t y f o r va r i ous i n i t i a l im pac t ve lo c i t i e s .m a x i m u m a c c e l e r a t i o n f o r Vo = 30 f p s . o c c u r s a t t = 9. 5 m i l l i s e c o n d s a f t e ri m p a c t .veloci ty of 30 f p s .s e c o n d t e r m o n t h e r i g h t - h a n d s i d e .

T h e

I n F i g u r e 3 p r e s s u r e p r o fi l es a r e p r e s e n t e d f o r a n i n it i al i m p a c tT h e y a r e c a l c u l a t e d f r o m E q u a t i o n ( l 2 ) w h il e i g n o r in g t h e

T he p r e s s u r e s a r e m i ni m um a t t h e po int of i n i t i a l impac t , an d i n c r e a s eto an i n f i n i t e peak a t t he e dge of t he we t t ed s u r f ace . If t h e s e c o nd t e r m i nEquat ion ( 1 2 ) w e r e t o be r e t a i n e d , i t s e f fe c t w ou ld b e s m a l l e v e r y w h e r ee x ce p t a t the e dge of t he we t t ed su r f a ce , w h e r e i t s s i n g u l a r t e r m w o ul d b e

n ega t i ve a t r = c.w ou ld m o d i f y an y s i n g u l a r i t i e s o c c u r r i n g i n t h e p r e s e n t l i n e a r h y d r o d y n a m i cthe o ry.p r e s s u r e a t t h e e d g e of t h e w e t te d s u r f a c e , o n ly t h e f i r s t t e r m of E q u a t i o n ( 1 2 )w i l l be u s e d t o p r e d i c t t h e p r e s s u r e p r o fi l es .f e a t u r e s of t h e e x p e r i m e n t a l l y o b s e r v e d p r o f i l e s t t . At t he i n s t an t of im p a c t ,t h e p r e s e n t i n c o m p r e s s i b l e f lo w t h eo r y p r e d i c t s t h a t th e p r e s s u r e on th e s h e l li s i n f i n i te , a lt ho u gh t h e t o t a l f o r c e i s a lway s f i n i t e. I n p r a c t i c e , of co u r se ,t he e f f ec t s of c om pre s s ib i l i t y a nd a i r b u b bl e s w i l l m o d if y t h e s e e x t r e m e l yl a r g e p r e s s u r e s . It shou ld a l s o be no te d th a t a t s o m e t ime tn a f t e r t h ei m p a c t , t h e p r e s s u r e g i v en by E qu a ti on ( 1 2 ) m a y b ec o m e n e g a t i v e n e a r t h e

i m pac t po in t , p r e s u m a b l y b e c a u s e a t t h a t t i m e t h e a s s u m p t i o n s of l i n e a r i t y

N ea r t h i s p o int t h e e f fec t of t he n o n l i n e a r f r e e s u r f a c e

B e c a u s e t he s e c on d t e r m i n t r o d uc e s a p h y s i c a ll y u n r e a l i s t i c n e g a t i ve

It p r e d i c t s t h e e s s e n t i a l

tt E x p e r i m e n t a l w a t e r i m p a c t s t u d i e s h av e b e e n c a r r i e d o ut by S . Stub bs a tt he NASA L ang l e y Re se a r ch Ce n t e r on a 1 / 4 - s c a l e r i g id mode l of t heApo l lo Co mmand M od u le, wh ich ha s a sp he r i c a l ba s e . Exce p t a t t hes i n g u l a r i t y, t h e p r e s s u r e p r of il es of Equat ion ( 1 2 ) c o m p a r e f a v o r a bl y w i tht h e u n p u bl i sh e d d a t a o b t a in e d d u r in g t h e s e i m p a c t t e s t s .


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a r e no longer va l id ,of t he ii i ax imum rig i d-b ody acc e l e r a t i on , when t he max imu m to t a l f o r c e s a c t011 t h e she l l , a n d a s w i l l b e sh ow n l a t e r , i t i s n ot u n r e as o n a b l e t o a s s u m et h a t t h e i na xi ii iu in s t r e s s e s i n t he s h e l l w i l l h a v e a l r e a d y b e e n a t t a i n e d b yth i s t i i ne .

H o w e v er , t n al w a ys o c c u r s a p p re c i a b ly a f t e r t h e t i m e

-6-


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D Y N A M I C R E S P O N S E O F S H E L L S O F R E V O L U T I O N

T h e d e t e r m i n a t i o n of t h e dy na mic r e sp o ns e of she l l s of r evo lu t i oninvolves a n ex tens i on of the work of Budiansky and Radkowski2 deve lopedf o r t h e s t a t i c a n a l y s i s of s h e l l s .i n R e f e r e n c e 2 wi l l be ado p t ed he r e un l e s s o t he rw i se no t e d . F u r t h e r m o r e ,t h e c o o r d i n a t e s y s t e m , s i g n c o nv e nt io n , a n d f o r c e d e s c r i p t i o n s ( s e eF i g u r e 4) a r e g i ve n i n R e f e r e n c e 2 , a nd w il l n ot be r e p e a t e d h e r e i n d e t a i l .

F o r c o n ve n i e n c e , t h e n o m e n c l a t u r e u s e d

The equa t i ons of mo t ion f o r a sh e l l of r e v ol u t io n b a s e d o n S a n d e r s 't h e o r y ' a r e g i v e n by

a2w- a p ( " ~ N C w e e ) a'pq - phEo7 = 0i3T

w h e r e t h e n o n d i m e n si o n al ti m e a nd s p a c e v a r i a b l e s T andt h e r e s p e c t i v e d i m e n s i o n a l q u a nt i ti e s t a n d s by t h e e x p r e s s i o n s

a r e r e l a te d t o

T = ( E o / r n o ) 1 / 2 / a , c = s / a

-7-


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I n t h e s e f o r m u l a s , h i s th e s h e l l t h i c k n e s s , m O i s t he m a s s d e n s i ty, E o i s are f e r en ce Young 's modu lus , and a i s a r e f e r e nc e l eng th .t h e e f f e c t s of t r a n s v e r s e s h e a r d i s t o r t i o n h av e b e e n n e g l e ct e d .v i s c o u s d am p in g t e r m s h a v e b e e n n eg l e c te d b ut t h e i r i n c l us i o n p r e s e n t s n oinhe ren t d i f f i cu l t i e s ( e . g . , s e e R e f e r e n c e 1 7 ).

In Equa t ion (13 )In addi t ion ,

A l l v a r i a b l e s i n E q u a ti o n ( 13 ) a r e now e x p an d ed i n F o u r i e r s e r i e s i nt h e c i r c u m f e r e n t i a l v a r i a b le 8 a s fo ll ows:

uohon= 0

m

= u 0 c= 1

c o s n e

s i n n e

w h e r e uo i s a r e f e r e n c e s t r e s s le v e l , a nd ho i s a r e f e r e n c e t h i c k ne s s .Fo u r i e r coe ff i c i en t s i n the above expans ions a r e func ti ons of 6 and r .Subs t it u t ion of the above F ou r i e r s e r i e s i n to Equa t ion (13 ) pe rm i t s t heu nc ou pl in g of t he s h e l l eq u a t io n s i nt o s e p a r a t e s e t s f o r e a c h F o u r i e r i n de x(n ) .that fol low.

The

F o r c o n v e ni e nc e , t h e s u p e r s c r i p t ( n ) w i l l b e o m i t t e d f r o m t h e e q u a t i o n s

-8


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5y e x p r e s s i n g f o r c e s an d m o m e n ts i n t e r m s of t h e f ou r v a r i a b l e s uu O , w , and i n t , a nd by i n t ro d u c in g a n a u x i l i a r y m o m e n t - d i s p l a c e m e n te q u a ti o n a s w a s d on e i n R e f e r e n c e 2 , the s he l l equa t io ns ca n now be -wr it tenin the fo l lowing form:

t a7w t a g m i t a9mS - hu = c1f

-..a16wl a 1 7 ~ a 1 8 m - hu = c 285

..3

~ w = c

1 I 1

a30 uc a 31 u t a32u0 a33wa34wI a 3 5 ~

w h e r e

h = h / h o

T he c o e f f i c i en t s a i a r e f u n c ti o n s of t he m a t e r i a l a n d g e o m e t r i c p r o p e r t i e s oft h e s h e l l an d o f t h e F o u r i e r i n d e x n.a p p l i ed l oad .in Appendix A o f R e f e r e n c e 2 .d e n o t e d b y p r i m e s a n d d o ts , r e spe c t i ve ly.t he m a t r i x f o r m

Th e t e r m s c i a r e f un c ti ons of t heT h e q u a n t i t i e s ( a i , a 2 , . . . a36 , c1 , c2 , c3 , c ) a r e g iv en4

D i f fe r e n ti a t io n w i t h r e s p e c t t o 5 a n d T i sE q u a t io n ( 1 6) m a y b e w r i t t e n i n


20/52

w h e r e

z =

5

u0

W

.m

D =

h O O O

O K 0 0

O O K O

0 0 0 0

a n d t he m a t r i c e s E , F , G, a nd e a r e g ive n by Equa t i o ns (41 ) of R e f e r ence 2.

T h e bo u nd a ry c o n d i ti o n s c a n be w r i t t e n i n n o n d i m e n s i o n a l m a t r i xf o rm a s

w h e r e t h e m a t r i c e s H, J , and f a r e g i v e n by E q u a t i o n ( 5 1 ) of R e f e r e n c e 2.A s exp l a ined by Bud iansky and Ra dkowsk i2 ,m a t r i c e s , a n d e i s a p r e s c r i b e d c o l u m n m a t r i x i n d i ca t i ng t h e t y pe of b o u n d a r yc o nd i ti o n u nd er c o n s i d e r a t i o n t t . F o r e x a m p l e , i f u i s known a t t h e b o u n d a r y,the f i r s t diagonal e lement of C i s z e r o , t h e f i r s t d i agona l e l em en t of A i su n i t y, and t h e f i r s t e l em e n t of e i s t he p r e s c r i b e d v al u e of u

C and A a r e p r e s c r i b e d d ia go na l

f

5'

t t A s p e c i a l m o d if ic a ti o n i n t h e a b o v e m a t r i c e s is n e c e s s a r y w he n t r e a t i n gp r o b l e m s w h e r e t h e s h e l l h a s a c l o s e d a p ex .s u c h s i n g u l a r p o in t s a r e d i s c u s s e d i n R e f e r e n c e s 1 8 a n d 19.

P r o c e d u r e s f o r h a nd li ng

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NUMERICAL ANALYSIS

In t h i s vo lume the d i f f e r en t i a l equa tions (18) and ( 2 0 ) w i l l b e e x p r e s s e di n f i n i t e - d i ff e r e n c e f o r m , a nd a n u m e r i c a l p r o c e d u r e f o r t h e i r s o lu t io n w i l lb e p r e s e n t e d .

D i s c r e t e va lues of t he nond imens ional space va r i ab l e a r e deno ted

b y

f . = i A i = O , 1 , 2 , . . . , N ( 2 1 )

w h e r e t h e n o n d i m e n s io n a l s p a c e i n c r e m e n t A i s d e f i n e d a s

A = s / a ( N - 1 )

S i m i l a r l y, d i s c r e t e va lues o f t he nond imens iona l t im e va r i ab l e T a r e d e n o t ed

by

= j c j = - 2 , -1, 0 , 1 , 2 , . . . (22)'jw h e r e i s t he nond imens iona l t im e inc reme n t . A s w i l l b e s e e n l a t e r , t h ef i c ti t io u s t i m e s ~ - 1 nd ~ - 2 i l l b e n e c e s s a r y in t he de sc r ip t i on of t he i n i t i a lcondi t ions .r e s p e c t t o t he s p a t i a l v a r i a b l e E, ( a t any t im e T . and posi t ion t i ) a r e

T h e d if f e re n c e f o r m u l a s u s ed t o r e p r e s e n t d e r i v a t i v e s w i t h

J

i = 1, 2 , . . . , N-1

In o r d e r to be cons i s t en t wi th Re fe rence 2, s i m p l e b a c k w a r d a n d f o r w a r dd i f f e r e n c e e x p r e s s i o n s a r e u s e d a bo ve i n t r e a t i n g b o u nd a ry p o in t s. M o r eac cu ra t e p roc edu re s i nvolving t he i n troduc tion of f i c t i t ious po int s a r ed i s c u s s e d i n R e f e r e n c e s 4 and 20.which y i e ld s t he num er i ca l r e su l t s of t h i s pap e r, t ake s advan tage of suchi m p r o v e d d i f f e r e n c e e x p r e s s i o n s a t b o u nd a ry p o i n t s.

A c o m p u t e r p r o g r a m d e v el o pe d a t S & ID ,

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The H oubo lt me th od f e a t u r e s t he u se o f an imp rove d backw ard d i f f e r e ncet e c h ni q u e f o r t h e d e t e r m i n a t i o n of t h e t i m e d e r i v a t i v e s .b a s e d o n f i t t i ng a t h i r d d e g r e e p o l y n o m ia l i n T t h r o u g h f o u r c o n s e c u t i v ed i s c r e t e t e m p o r a l p o i nt s.of Z i w i th r e s p e c t t o t i m e i s given by

i , J

T h i s m e t h o d i s

T h e g e n e r a l e x p r e s s i o n f o r t h e s e c o n d d e r i v a t i v e

= ( Z Z i , j - 5 z i , j - 1 4 z i , j - 2 - z i , J 3 ) / c 2..2 5)

j = 1 , 2 , . . .It i s i m p o r t a n t t o n ot e th a t t h e i n i t i a l i n c r e m e n t s of c a l c u l a t i o n ( a t j = 1 , 2 )i n t he a bove e x p r e s s io n i n vo lve t he eva lua t i on of f unc t i o ns a t nega t i v e va l u e sof t im e . Th es e f i c t i t i o us func t i ons z i , -1, z i , -2 c a n be d e t e r m i n e d f r o m t hep r e s c r i b e d i n i t i a l c o n d it i o ns of t h e p r o b l e m z i , 0, i i , 0, and i'i, 0.u t i li z i n g t h e d i f f e r e n c e f o r m u l a s s u g g e s t e d b y H o ub o lt 3 t o s t a r t t h e p r o c e s s ,

t h e fo ll owin g exp re s s io ns c an be ob t a ined :

By

2=

-H e r e , z i , o Y z i., a r e c o l u m n m a t r i c e s r e p r e s e n t i n g t h e c o ef fi c ie n tsof t h e a p p r o p r i a t e F o u r i e r e x p a n s i o n s f o r t h e d i m e n s i o n l e s s i n i t i a l d i s p l a c e -m e n t , v e l o c i t y, a n d a c c e l e r a t i o n c o n d i t i o n s , r e s p e c t i v e l y.

and Z i ,

By u si ng Eq ua t i on s (2 3 -26 ) t he d i f f e r en t i a l eq ua t i ons (18 ) an d bound a ry

c o nd i t i ons (20) c a n b e e x p r e s s e d by t h e s e t of a l g e b r a i c e q u a t i o n s

H e r e

-12 -


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- w h e r e t h e s u b s c r i p t 0 r e f e r s t o the end cond i ti ons a t s = 0 , and

w h e r e t he s u b s c r i p t N r e f e r s t o t he bounda ry cond i t i ons a t s = 5i = 1, 2 , , N-1, t he coe ff i c i en t s a r e g iven by

F o r

Ai = ( 2 E i / A ) F i

Bi = - ( 4 E i / A ) 2 A ( G i - f f j D i / c 2 )

C i = ( 2 E i / A ) - Fi

2= ]-E. zo T E z o

j2Aei, - j z i , j - 1 J z , j - 2 f q j z i , j - 3 j2ADi

2i , j

A s s u m i n g t h a t g i , j i s known, it i s ev iden t t ha t t he above equa t ions 27)a r e of t h e s a m e f o r m a s t h o se t r e a t e d i n t h e s t a t i c p r o b le m .t o s o l v e t h e s e e q u at io n s i s a d i r e c t m a t r i x e l im i n a t io n t e ch n iq u e ( P o t t e r s '

involves re la t ing z i , j to z i t i , j b y e x p r e s s i o n s of t h e f o r m

The me thod used

m e t h o d 7 ) which i s d e s c r i b e d i n d e t a i l in R e f e r e n c e 2. T h e b a s i c p r o c e d u r e

-13 -


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w h e r e

( i = 2 , 3 , . . . , N-1)

- l B ~ C ; ~ A . ~

3 3 )

F r o m E q ua ti on s ( 3 2 ) an d t he l a s t of E qua t i o n 27), t h e c o r r e c t v a l u e of z N , jca n be ca l cu l a t ed a s f o l l ows :

A l l t he r ema in ing z i,s t a r t i n g f r o m t he c a l c u l a t e d v a l u e ZN, j .i n ve r s ions o f 4 x 4 m a t r i c e s .

a r e c a l c u l a te d in t h e r e v e r s e o r d e r u si ng E q u a ti o n 32)The compu ta t i on s i n vo lve on ly

F i n a l l y, z o , i s g i v e n b y

r 1

It c a n be s e e n t h a t g i , j i n E q ua t io n 30), a n d i n t u r n i n t h e s o l u t i onZ. To s t a r t t h eepen ds on t h e p r ev io us so lu t i ons z i , j - l , z i , j - 2y v3*p r o c e s s a t j = 1 , gi, 1 i s d e t e r m i n e d f r o m t h e k no wn i n i t i a c o n d i t i o n s , a n d

t he r e s u l t i n g e q ua t io n s a r e s o l v ed by P o t t e r s ' m e t h o d t o g iv e z i , 1.

z i ,a r e s o lv e d f o r zi , 2.

1, j '

By u s i ngand t he i n i t i a l co nd i t i on s , g i , 2 i s c o m p u t e d , a n d t h e r e s u l t i n g e q u a t i o n s

T h e c y c l e i s r e p e a t e d t o d e t e r m i n e s u b s eq u e n t v a l u e sof z i , j .

It sho u ld be no t e d t h a t t he so lu t i on s z i , o b t a in e d ab o v e r e p r e s e n t t h es h e l l r e s p o n s e f o r a p a r t i c u l a r F o u r i e r i n d e x n.u n s y m m e t r i c lo a d s , t h e c o m p l e t e s o l u t i o n is o b t a in e d b y p e r f o r m i n g t h ea p p r o p r i a t e s u m m a t i o n of t he F o u r i e r c o e f f i c ie n t s u s in g t he F o u r i e re x p ans ions de sc r i bed b y Equ a t i ons (15).G r e i f 4 , t h e H ou bo lt m e t h o d of s o l u t i o n f o r t h e d y n a m i c r e s p o n s e r e d u c e s t ot h e s t a t i c s o lu t io n a s c b ec o m es unbounded .

F o r t he g e n e r a l c a s e of

A s poin ted ou t by Jo hnso n and

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T he b a s i c n u m e r i c a l m e th o d d i s c u s s e d a b ov e c a n e a s i l y b e e x t e n d edto she l l s w i th d i s co n t i nu i t i e s an d b r a n ch cond i ti ons by fo ll ow ing t he pa r a l l e ldeve lo p men t o f Re f e r e nce 2 f o r t h e c a s e of t h e s t a t i c s of s h e l l s .of t h i s t ype ca n a l so cons id e r p rob l em s in w hi ch t h e m a t e r i a l a n d g e o m e t r i cp r o p e r t i e s of t h e s h e l l a r e ti m e -d e pe n de n t.B i a nd C i i n Equa t i ons (28), (29) and (30) m u s t be r e c o m p u t e d a t e a c h t i m ei n t e r v a l .

A n a n a l y s i s

In su c h c a s e s , t h e m a t r i c e s A i ,

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NUMERICAL RESULTS

T he d y n a m i c s h e l l a n a l y s i s d e s c r i b e d i n t h i s r e p o r t w a s p r o g r a m m e dfor so lu t ion on a d ig i ta l co mp uter ( IBM 7094).c o m p u t e r p r o g r a m i s d e s c r i b e d i n Volume 5 of th i s repor t .r e s u l t s a r e p r e s e n t e d f o r a s a m p l e p ro b le m s i m ul a ti n g t he v e r t i c a l i m p a c tof a t y p i c a l r e - e n t r y v e h i c le d u r i n g w a t e r i m p a c t .

T he u s e r ' s m a n u a l f o r t heN u m e r i c a l

T h e m a t h e m a t i c a l m o d e l c o n s i d e r e d i s i l l u s t r a te d i n F i g u r e 1 with theshe l l s t r uc tu re be ing a sha ll ow sphe r i ca l she l l of r ad iu s of cu rv a tu reR = 175. 6 ins . and having an opening angle of 19 .53 ' . F o r conve nience , a na x i s y m m e t r i c p r o b l e m i s c o n s i d e r e d (i. e. , the impact po in t co inc ides wi ththe apex of t he she l l ) and on ly the Fou r i e r componen t co r r e spond in g t o theindex n = 0 i s r e q u i r e d .cha rac t e r i s t i c of s andwich o r l aye r she l l con f igu ra t i on wh ich i s t yp i ca l l yu s ed i n r e - e n t r y s h e l l s t r u c t u r e s .s t i ff ness (d) of the she l l a r e both se t equal to 3 . 33 x l o 6 l b s . / i n . , w h ic hco r r e sp ond s t o a s andwich she l l having 0. 05 in . s t ee l fac ings and 1 . 95 in.h o n ey c om b c o r e .f o r i l l u s t r a t i v e p u r p o s e s a nd t h a t t r a n s v e r s e s h e a r d i s t o r t i o n e f f e c t s , a l th o u ghneg lec t ed i n t h i s ana ly s i s , m ay be s ign if i cant f o r such a s andwich conf igu -r a t i on .a r e a ( m oh ) = 9. 7 x l b s s e c 2 / i n . 3, P o i s s o n ' s r a t i o v = 0. 33, and modulusof e las t ic i ty E = 2 9 . 7 x 1 0 p s i .r i g id ly c l amped a t i t s bounda ry t o a h e a v i er m a s s ( s i m u la t e d r e - e n t r ycap su l e ) so t ha t t he i r combined we igh t i s 10 , 000 l b s .

T h e s t i f fn e s s p a r a m e t e r s w e r e s e l e c t e d t o b e

The ex t ens iona l s t i f fn e s s (b ) and f l exu ra l

It should be r ecognized t ha t t h i s con f igu ra t i on was s e l e c t ed

O t h e r p r o p e r t i e s of t h e s h e l l a r e a s f ol lo w s: M a s s p e r u ni t s u r f a c e

R e f e r r i n g t o F i g u r e 1 , t he s h e l l i s c o n s i d e r e d

T h e p r e s s u r e a p p li e d t o t h e s h e l l is g i ve n in F i g u r e 3 f o r a v e r t i c a limpac t i n to w a te r a t an i n i t i a l ve loc i t y of 30 fp s .the f i r s t t e r m of Equa tion ( 1 2 ) i s s e e n t o conta in a s ing ular i ty a t the edge ofthe we t t ed su r f ace . Th i s s i ngu l a r i t y i s i n t eg rab l e , howeve r, so t h a t t h ef o r c e a c t i ng o n t h e s h e l l i s f i n it e , a n d t he a p p l i e d p r e s s u r e a t e a c h s t a ti onpoint i s a v e r a g e d o v e r a c o m p l e t e s p a t i a l i n c r e m e n t .

T he p r e s s u r e l oa di ng f r o m

T h e r e s p o n s e of t h e s h e l l d ue t o t h e s e a p p li e d r ig i d- b od y p r e s s u r e s i si l l u s t r a t e d i n F i g u r e 5 -1 6.s h e l l , c a l c ul a t io n s w e r e m a d e up t o 1 5 m i l l is e c o n d s f r o m t h e t i m e of i m p a c t( s e e F i g u r e s 5 -8 ).ma x im um acce l e r a t i on o f t he ce n t e r of g r av i t y of t he veh i c le (9 .5 m s . ).r e s p o n s e q u a n ti t i es s ho w n i n F i g u r e 5 - 8 r e p r e s e n t t h e d i s p l a c e m e n t W ,m e r i d i o n a l be nd in g m o m e n t M i , m e r i d io n a l m e m b r a n e f o r c e N 6 , and ve loc i t y

In o r de r t o de t e rmine t he peak r e spo nse of t he

Th i s t im e i s we l l beyond the t im e of o ccu r r e nce of t heT h e

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W a t t he apex of t h e she l l , ob t a ined by u s ing a t i m e i nc r em e n t A t o f . 0 5 m s .I n s p e c t i o n of th e r e s u l t s i n d i c a t e s th a t p e a k s h e l l r e s p o n s e o c c u r s a ta p p r o x i m a t e l y 2. 2 m s . a f t e r t he i m p a ct .a p p r e c i a b l y be fo r e t h e t i m e w he n t h e m a x i m u m f o r c e a c t s o n th e s h e l l a t9. 5 m s . T h r e e - d i m e n s i o n a l p lo t s o f t h e d i s p l a c e m e n t W a n d t h e m e r i d i o n a ls t r e s s e s ?E, at th e e x t r e m e f i b e r o f t h e s h e l l a r e p l ot te d i n F i g u r e s 9 an d 10v e r s u s t i m e and m e r i d i o n a l d i s t a nc e o n t h e s h e l l s u r f a c e m e a s u r e d f r o m t h eimp ac t po in t .a f t e r i m p a c t . H o w e v e r, m a x i m u m s t r e s s e s a t s u b s eq u e n t t i m e s te n d t ofo l low the mo veme nt of the edge of the wet ted su r f ac e .m a x i m u m s t r e s s e s m a y o c c u r a t po in ts o t h e r t h a n t h e a p ex .

T h i s p e a k i s e x p e r i e n c e d

P e a k s h e l l s t r e s s e s o c c u r a t th e a p e x of t h e s h e l l a t 2 . 2 m s .

T h u s, a t l a t e r t i m e s ,

D o tt e d c u r v e s o n F i g u r e s 5 a n d 6 r e p r e s e n t r e s u l t s o b ta i ne d wh en t h ep r e s s u r e s a r e a p p li ed s t a t ic a l ly , i. e . , w h en t h e i n e r t i a e f f e c t s of th e s h e l la r e n e g le c t ed .r e s p o n s e , a n d , a s o n e m i g h t e x p e c t , t h e d y n a m i c r e s p o n s e o s c i l l a t e s a b o u tt h e s t a t i c c r a w l c u r v e s .

T h e s e r e su l t s a r e s o m e t i m e s r e f e r r e d t o a s th e s t a t ic c r a w l

I n o r d e r t o a s c e r t a i n t h e e f f e c t of t h e s h e l l f l e x i b i li t y on t h e r e s p o n s e ,c a l cu l a ti o n s w e r e a l s o m a d e f o r t h e c a s e w h e r e t h e f l e x u r a l s t i f f n es s w a sred uc ed by one-ha l f .40 l a r g e r , and t he p e a k s t r e s s a t th e e x t r e m e f i b e r s w a s 3 5 . 770 l a r g e r .T he p e a k re s p o n s e o c c u r r e d a t 2. 8 m s , s ho wi ng t h a t f o r a m o r e f l e x ib l e s h e l lt h e p e a k r e s p o n s e i s de l ay e d .

It w a s o b s e r v e d t h a t t h e p e a k d e f l e c ti o n W w a s a b o u t

A s d i s c u s s e d i n R e f e r e n c e 4 h e a c c u r a c y of t h e H ou bo lt m e t h o d v a r i e sw i th th e s i z e of t h e t i m e i n c r e m e n t ( E ) used .t i m e i n c r e m e n t E s h ou l d b e s m a l l e r th a n 1 / 5 0 t h of t h e p e r i o d of t h e p a r t i c u l a rm o d e o f v i br at i on i n o r d e r t h a t i m p o r t a n t o u t p ut q u a n t i t ie s i n t h a t m o d e a r enot s ign i f ican t ly dam ped .a p e x d i s p l a c e m e n t a n d b en di ng m o m e n t , o b t a i n e d b y v a r y i n g t h e t i m e i n c r e -m e n t A t be tw een 0 . 0 1 m s . and 0 . 3 ms. T h e r e s u l t s of F i g u r e 11 r e v e a l th a texce l l en t co nve rgenc e i n t h e d e f l e c t i on W o c c u r s w h e n a t i m e i n c r e m e n t of0 . 0 2 5 m s . i s s e l e c t e d . A s m i g h t b e e x p e c t e d , c o n v e rg e n c e i s s l o w e r f o rthe bending moment M E s i n c e t h i s q u a n ti t y i s c o m p u te d n u m e r i c a l l y f r o md e r iv a t i ve s of t h e d i s p l ace me n t s .d a m p e d f o r a t i m e i n c r e m e n t o f 0 . 0 5 m s . , i n t h e i n t e r e s t s of e c on o m y, t h i st i m e i n c r e m e n t w a s s e l e c t e d f o r th e l o ng e r r e s p o n s e c u r v e s of F i g u r e s 5 to8.s t a t i o n s ( N ) w as v a r i e d b e tw e e n 60 and 120.

It h a s b e e n e s t i m a t e d 6 t h a t t he

In F i g u r e s 1 1 a n d 12 a r e p r e s e n t e d r e s u l t s f o r

S i n c e t h e r e s u l t s a r e n o t s i g ni f ic a n tl y

No a p p r e c i a b l e v a r i a t i o n w a s n o t i c e d i n t h e r e s u l t s w h e n t h e n u m b e r of

E x p e r i m e n t a l o b s e r v at i o n s s ho w t h a t a lt h ou g h p e a k p r e s s u r e s o c c u r a tthe edge of the wet ted su r f ac e , t h ey a r e by n o m e a n s i n fi n i te , a s p r e d i c t e dby t h e p r e s e n t l i n e a r h y d r od y n a m i c a n a l y s i s .of t h e s e i n fi ni te p r e s s u r e s o n t h e s h e l l r e s p o n s e , a p r o f i l e w h o s e r a d i u s w a s

I n o r d e r t o a s c e r t a i n t h e e f f ec t

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9970 of t he p r ed i c t ed r ad iu s was app l ied to the she l l .w a s excluded f rom the forc ing func t ion . Res u l t s of F ig u re 13 , us ing a t imeincretnent of 0. 1 m s . , show an 8. 5 30 r educ tion i n t he peak bending mom en t .The imp l i ca t i on i s t ha t t he she l l r e sponse i s s e n s it i v e t o t h e p r e s s u r e p e a kat the edge of the wet ted s ur fa ce .du r ing exp e r im en t s , t he r e su l t s of t h i s s tudy would app ea r t o be conse rva t i ve .

Thus , the inf in i te peak

S ince i n f in i t e peaks a r e no t ob t a ined

F i g u r e 8 i l l u s t r a t e s t he no rm a l ve loci t y of t he su r f a ce of t he sh e l lr e l a t i ve t o the cen t e r of g r av i t y of the veh i cl e . These r e s u l t s i nd i ca t e t ha tt he ve loc it y of t he sh e l l a ch i eves va lues i n t he o rd e r of t he impac t ve loc i t y.Becau se t he p r e s s u re s a r e compu ted on t he ba s i s of t he ve loc i t y of a r i g idmov ing su r f ace , one wou ld susp ec t t ha t s t ru c t u r a l de fo rm a t ions m ay havea n a p p r e c i a b l e e f f e c t o n t h e p r e s s u r e d i s tr i b u ti o n s .of t h i s hyd roe l a s t i c i n t e r ac t i on i s g iven i n Vo lume 3 of t h i s r e p o r t .

A c o m p l e t e d i s c u s s i o n


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PRECEDING PAGE BLANK NOT F I L 7 X 3 .

CONCLUSION

In conc lu s ion , f o r t h e s a m p l e p r o b l em c o n s i d e r e d , it h a s b e e n o b s e r v e dt h a t d u ri n g w a t e r i m p a c t t h e p ea k s h e l l r e s p o n s e s b a s e d o n r i gi d -b o dy p r e s -s u r e s o c c u r a t a t i m e a p p r e c ia b l y b e fo r e th e m a x i m u m f o r c e s a r e a p p li e dt o th e s h e l l s u r f a c e . I n a d d i ti o n , t h e p r e s e n t l i n e a r h y d r o d y n a m i c t h e o r yg i v e s a c o n s e r v a t i v e r e s u l t b e c au s e it p r e d i c t s a n i nf i ni te p e ak p r e s s u r e a tt he e dge o f t he we t t e d su r f a ce .r e l a t i v e t o t h e c e n t e r of g r a v i t y of t h e v e h i c l e i n d i c a t e s t h a t t he e f f e c t s oft h e h y d r o e l a s t i c i n t e r a c t i o n b e t w e e n t h e s h e l l a n d t h e w a t e r m a y b e ofs i g n i f i c a n c e in t h e d e t e r m i n a t i o n of t h e p r e s s u r e s a n d t h e s h e l l r e s p o n s e s .

F i n a l l y, t h e hi gh v e l o c i t y of t h e s h e l l s u r f a c e

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J V

M

F i g u r e 1. Model o f Re-Entry Vehic le D u r i n g Impac t .

- 2 3 -


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LOCUS OF M A X I M U M

1 ACCELERATION

I N I T I A LI M PA C T

VELOCITY\ FPS

(0 ioT I M E IN M I L L I S E C O N D S

Figure 2 . Acceleration o f Center o f Gravity o f a SphereDuring Impact into Water

35

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