dynamic of structure

7/23/2019 Dynamic of structure

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Journal of Sound and Vibration (1992) 152(1), 27-37

VIB R TI ON N LYSIS OF THICK COMPOS ITE

CL MPED CONIC L SHELLS OF V RYI NG THICKNESS

K . R . S IVADAS AND N . GANESAN

Department of Applied Mechanics, Indian Institute of Technology Madras, Madras 600 036, India

Receioed 9 July

1990,

and in reoised orm

22

Nooember

1990)

Thick shells m ade o f composite materials have been analyzed by using a higher order

shell theory. In this th eory thickness no rm al strain an d tw o transverse strains are included.

A hig her order th ree-n ode d isoparametric axisymm etric finite element is used to solve the

problem. Nu me rical experiments with the p resent element indicate tha t this element yields

accurate vib ration results w ith very few elements. In the present stud y, the suitabili ty of

different theories used for vibration studies has been investigated. Three theories are com-

pared , v iz . Love ' s f irst approxim at ion she ll theory , an improved theory wi th shear deform a-

tion and rotatory inert ia , and a shell theory with thickness normal strain and shear

deformat ions. I t i s foun d tha t the shear deformat ions have an apprec iable e ffect on the

vib ratio n characteristics o f com para tively thick shells, especially composite shells. A p ara-

metric study has been conducted to study the effects of various geometric propert ies of

shell on th e free vibr ation characteristics of conical isotropic a nd com posite she lls. Th e

effect of mass distribution on the natural frequencies is also studied in the present work.

1 . I N T R O D U C T I O N

L a m i n a t e d s h e l l s tr u c t u r e s a r e i n c r e as i n g l y u s e d i n v a r i o u s f i e ld s s uc h a s c h e m i c a l , m e c h a n -

i c a l, m a r i n e a n d e s p e c i a l ly a e r o s p a c e a p p l i c a t i o n s , e t c. , a s s t r u c t u r a l e l e m e n t s b e c a u s e o f

t h e i r h i g h s p e c if ic p r o p e r t i e s . M a n y r e s e a r c h e r s h a v e a n a l y z e d t h i n s h e l l s t r u c t u r e s t o

o b t a i n t h e i r v a r i o u s c h a r a c t e r i s t ic s . B e r t a n d F r a n c i s [ 1 ] c o n d u c t e d a d e t a i l e d li t e r a t u r e

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

c o n t a i n i n g c o m p o s i t e m a t e r i a l s . I n t h e i r r e v i e w , t h e y d i s c u s s e d t h e v a r i o u s s h e l l t h e o r i e s

u s e d f o r c o m p o s i t e s h e l l s .

F o r c o m p o s i t e s h e ll s, th e e f f ec ts o f s h e a r d e f o r m a t i o n a n d r o t a t o r y i n e r t ia , i n t h e c a s e

o f v i b r a t io n , a r e c o m p a r a t i v e l y l a rg e c o m p a r e d w i t h t h o s e f o r i s o t ro p i c s h e ll s. I n r e c e n t

y e a r s , c o n s id e r a b l e a t t e n t i o n h a s b e e n p a i d t o t h e d e v e l o p m e n t o f a p p r o p r i a t e s h el l t h e o r ie s

t h a t c a n a c c u r a t e l y p r e d i c t t h e v a r i o u s c h a r a c t e r i s t i c s . M a n y r e s e a r c h e r s h a v e a n a l y z e d

t h i c k s h e l ls o f d i f f e r e n t c o n f i g u r a t i o n s a n d l e ve ls o f s o p h i s t i c a t i o n . K a p a n i a [2 ] h a s p r e -

s e n t e d a l i t e r a t u r e r e v ie w o f v a r i o u s r e s e a r c h w o r k i n t h e f i el d o f t h i c k l a m i n a t e d s h e ll s .

C i r c u l a r c y l i n d r ic a l s h el ls h a v e b e e n a n a l y z e d b y m a n y r e s e a rc h e r s, u s i n g h i g h e r o r d e r

t h e o ri e s . S o m e o f t h e w o r k r e p o r t e d i s l is t ed b el o w . H u t c h i n s o n a n d E 1 - A z ha r i [3 ] a n a l y z e d

t h e v i b r a t i o n c h a r a c t e r i s t i c s o f t h i c k i s o t r o p i c s h e l l u s i n g t h r e e - d i m e n s i o n a l s h e l l t h e o r y .

P a r k a n d S t a n l e y [4 ] a n d A h m a d [5 ] a n a l y z e d th i c k s h e ll s u s i n g th e f i n it e e l e m e n t m e t h o d .

K a n t

et al.

[6 ] u s e d a h i g h e r o r d e r t h e o r y o f c o m p o s i t e l a m i n a t e t o a n a l y z e th e t r a n s i e n t

b e h a v i o u r , u s i n g th e f in i te e l e m e n t m e t h o d . I t w a s f o u n d t h a t i n t h e c o m p o s i t e l a m i n a t e

t h e i r t h e o r y p r e d i c t e d m o r e a c c u r a t e r e s u lt s a s c o m p a r e d w i t h t h o s e o f f ir s t o r d e r s h e a r

d e f o r m a t i o n t h e o r i e s .

C o m p o s i t e c i r c u la r c y l in d r i c a l s h el ls h a v e b e e n a n a l y z e d b y m a n y r e s e a r c h e rs [ 7 -1 4 ],

b u t v e r y f e w re s u l ts a r e a v a i l a b le f o r c o n i c a l t h ic k s h e ll s. M o s t o f t h e w o r k r e p o r t e d i n

27

0022-460x/92/010027 + 11 03.00/0 1992 Acad emicPress Limited


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28

K R SIVADA S AND N GANESAN

the f ield i s for i so t ropic conical shel l s . Sr in ivasan and Hosur [15] and Takahashi e t a l . [16]

ana lyzed e xac t ly the ax i sym me t r i c v ib ra t ion charac t e r i s ti cs o f i so t rop ic con ica l shell s o f

vary ing th i ckness us ing a f i r s t o rder shear defo rmat ion theory . Sun and Whi tney [17]

ana lyzed ax i symm et r i c v ib ra t ion o f she lls wi th a qua dra t i c shear defo rm at ion shel l t heory .

A hm ed [18] ana lyzed a x i symm et r i c p l ane- s t r a in v ib ra t ions o f a t h i ck she ll. Tak aha sh i e t

a l . [19 , 20 ] ana lyzed the asym m et r i c v ib ra t ion beh av io ur o f an i so t rop ic con ica l she ll wi th

axial ly varying th ickness .

In t he p resen t s tudy the f r ee v ib ra t ion beh av iou r o f t h ick con ica l she lls o f i so t rop ic and

o r t h o t r o p i c ma t e r i a ls h a s b een an a l y zed . A co m p ar i s o n o f th r ee t h eo ri e s u s ed f o r th e

ana lys i s o f com pos i t e shells has be en ma de to f i nd to wh at ex t en t a par t i cu l a r t he ory can

be used . The theor i es co m par ed a re t h in shell t heory (Lo ve s f ir s t app rox im at ion she ll

t heory (Th in ) ) , f i r s t o rder shear defo rmat ion she l l t heory (H5) and a quadra t i c shear

d e f o r ma t i o n s h e l l t h eo r y w i t h t h i ck n es s n o r ma l s t r a i n ( H 7 ) . A q u ad r a t i c ax i s y mmet r i c

i s o p a r ame t r i c f i n i t e e l emen t h a s b een u s ed i n t h e s o l u t i o n p r o ced u r e . Th e 3 - D p r o b l em

becomes 2 -D af t e r express ing the c i r cumferen t i a l var i ab l e dependence in a Four i e r se r i es .

N u m er i ca l i n t eg r a ti o n h as b een d o n e t o o b t a i n t h e s t if fn e s s an d mas s m a t ri ce s. T o s t u d y

the e f fec t o f var iou s para m eter s on the f r equencies o f the she ll, a pa ram et r i c s tudy ha s

been m ade . T he e f fec t o f th i ckness var i a t i on a long the ax i a l d i r ec t ion wi th a cons t ra in t on

the to t a l mass o f t he shel l fo r a par t i cu l a r l eng th to smal l end rad ius r a t i o (p ) on the

na tu ra l f r equencies i s a l so made .

To ch eck t h e co mp u t a t i o n a l i mp l emen t a t i o n , t h e p r e s en t r e s u l t s a r e co mp ar ed w i t h

pub l i shed va lues .

2 . F O R M U LA TI O N

There a re m any she ll t heor i es o f d i f fe ren t levels o f soph i s t i ca t ion . M os t o f the t heo r i es

r e ta i n o n e o r m o r e o f th e a s s u mp t i o n s o f Lo v e s fi rs t ap p r o x i m a t i o n s he ll t h eo r y . Th e r e

are shell t heor i es i n wh ich the assum pt ions o f the van i sh ing o f t he t r ansver se norm al s t ress

an d p r e s e r v a t i o n o f t h e n o r ma l e l emen t a r e ab an d o n ed . M ar q u e z [2 1] , D o n n e l l [ 22 ] an d

K rau s [23] have d i scussed the genera l shell t heor ies . Th e the ory wh ich is used in t he p resen t

w o r k ( H 7 ) i s o n e p r o p o s ed b y N ag h d i an d d i s cu s sed b y K r a u s [2 3].

To accommodate t he t h i ckness normal s t r a in , t he d i sp l acement s i n t he s l an t l eng th ( s ) ,

c i r cumferen t i a l (0 ) and perpend icu lar t o t he r e fe rence su r face ( z ) co -o rd ina t e d i r ec t ions

a r e r ep r e s en t ed b y

u ( s , O , z ) = U o ( S, 0 ) + z u ' ( s , 0 ) , v ( s , O , z ) = V o (S , O ) + z v ' ( s , 0 ) ,

w ( s , O , z ) = w o ( s , 0 ) + z w ' ( s , 0 ) + ( z 2 / 2 ) w ( s , 0 ) . ( 1 )

The genera l s t r a in -d i sp l ac em ent r e l a t ions i n li near e las t i c it y i n an o r thog ona l she ll co -

o rd ina t e sys t em have been g iven by Kra us [23]. Par t icu l a r i z ing the s t r a in -d i sp l ac em ent

re l a t ions fo r t he con ica l co -o rd ina t es , i .e ., t he s l an t leng th ( s ) , the c i r cum feren t i a l d i r ec t ion

(0 ) and the d i rec t ion perpend icu lar t o t he r e fe rence su r face (ou tward) ( z ) , and subs t i t u t i ng

i n t o t h em t h e d i s p l acemen t f u n c t i o n s ( 1 ) , o n e o b t a i n s

e , = O U o /O S + z ~ u ' / ~ s ,

1 [ ( ~ 0 U o + U 0 s i n c p + ~ _ ] + z ( ~ 3 v ' + u ' s i n - - | z 2 w ]

E O - - - -

l + z / R 2 c~O r R 2 : c~O r ~ + R 2 / + Z R 2 A '

E z = W ' -1- ZW ,


3/11

F R E E V I B R A T I O N O F T H I C K C O N I C A L S H E L L S 29

- -

sin rp

,

Os Os 1 + 2/R 2 O0 r O0 r

y ~ = OWo/OS+ z Ow ' /Os+ (z2/2) Ow /Os+ u' ,

1 ( ~ O w + z O w ' - t z Z t ~ w v I -v ') , (2)

YO Z-l+z /R ~2 O0 r O0 2r O0 R2

w here r i s the re fe re nce su rface rad ius a t an y po in t , q~ i s the semi-ver tex ang le an d R2 i s

t h e r ad iu s o f cu rv a tu re .

T h e s t r a in an d k in e t ic en e rg i e s a re r ep re s en t ed b y t h e s t r a in e n e rg y ,

S.E . = [ ~ ' rD~ d V (3)

J ~

an d k in e ti c en e rg y ,

w h e r e

. E . = ~ p g a - g d V , 4 )

V

~T= { e , , e0 , e~ , r ,0 , r= ,

r O z }

5)

D a re t h e t h ree -d im en s io n a l el a st ic co n s t an t s ,

g r = {fi , b , ~} , (6 )

an d ( ' ) r ep re s en t s d if f e ren t ia t i o n wi th r e s p ec t t o t im e .

T o d e r iv e t h e s h e ll t h eo ry w i th o u t t h e t h ick n es s n o rm a l s t r a i n ( im p ro v ed ) , t h e d i s p lace -

m en t fu n c t i o n (1 ) i n t h e r ad i a l d i r ec t i o n i s m o d i f i ed t o b eco m e w(x , O , z )= Wo(X, 0) ,

an d b y m o d i fy in g eq u a t i o n (2 ) acco rd in g ly , o n e o b t a in s t h e s t r a i n -d i s p l acem en t r e la t i o n s

co r re s p o n d in g t o t h is t h eo ry (H5 ) .

3 . NUMERICAL ANALYSIS

A th ree -n o d ed i s o p a ram e t r i c s em i -an a ly ti ca l f in i te e l em en t w i th 21 d eg rees o f f r eed o m

can b e u s ed t o s o lv e t h e p ro b l em . T h i s e l em en t h a s b een d i s cu s s ed b y W eav e r an d J o h n s to n

[24] . The co-o rd ina te in the s lan t l eng th d i rec t ion i s rep resen ted by

3

s= E N~s , , (7)

i ~ l

wh ere t h e Ni a re t h e s h ap e fu n c t i o n s , g iv en b y

N , = ( ~ z - ~ ) / 2 , N 2 = 1 - ~ 2 N 3 = ( ~ 2 + ~ ) /2 ,

i n wh ich ~ is t h e i s o p a ram e t r i c ax i al co -o rd in a t e an d s / i s t h e n o d a l co -o rd in a t e .

S imi la r ly , one can rep rese n t the seven depe nde n t var iab les uo , vo , Wo, u ' , v ', w ' and w

in t e rm s o f s h ap e fu n c t i o n s an d n o d a l q u an t i t i e s . Fo u r i e r ex p an s io n s a re u s ed i n t h e

c i rcumferen t ia l d i rec t ion .

T h e s t r a in m a t r i x { t } can b e r ep re s en t ed b y

{ I~ }T = ~ l e , 8 )

wh ere q e is t h e d i s p l acem en t v ec to r (n o d a l q u an t i ti e s ) .


4/11

3 0 K R S I V A D A S A N D N G A N E S A N

S u b s t i t u t i n g e q u a t i o n ( 8 ) i n e q u a t i o n ( 3 ) y ie l d s t h e s t r a i n e n e r g y o f a n e l e m e n t a s

1 T

S .E . = -'qeKe,qe,

w h e r e K~n is t h e s t if f ne s s m a t r i x o f t h e j t h e l e m e n t , g i v e n b y

Ken = f BX D B d V.

S i m i l a r l y , u p o n s u b s t i t u t i n g t h e d i s p l a c e m e n t f u n c t i o n i n e q u a t i o n ( 6 ) , t h e v e l o c i ty v e c t o r

b e c o m e s

g = 0 9 B q e . ( 9 )

U p o n s u b s t i t u t i n g e q u a t i o n ( 9 ) i n t h e k i n e t i c e n e r g y e x p r e s s i o n ( 5 ) , t h e k i n e t i c e n e r g y

b e c o m e s

I 2 T

K.E . = ~o~ qeMenqe, Men pB 'TB dV. (10 )

l

B y c o m b i n i n g a l l t h e e l e m e n t m a t r i c e s o n e o b t a i n s t h e s ti ff n es s m a t r i x K n a n d t h e m a s s

m a t r i x M , o f t h e s h e ll fo r a p a r t i c u l a r c i r c u m f e r e n ti a l m o d e ( n ). F i n a l ly , s o l v in g t h e

e i g e n v al u e p r o b l e m K , q - o ~ 2 M , q = 0 y i e ld s t h e n a t u r a l f r eq u e n c ie s .

L i n e a r v a r i a t i o n o f t h e t h i c k n e s s a l o n g t h e a x i a l d i r e c t i o n , s y m m e t r i c a b o u t t h e m i d -

l e n g t h o f t h e s h e ll , is a s s u m e d a n d i s r e p r e s e n t e d b y h = h o( 1 + k J 2 s - 1 1 ) . W i t h t h e t o t a l

m a s s o f t h e s h e l l c o n s i d e r e d c o n s t a n t , t h e s h e l l t h i c k n e s s a t s = 1 /2 , w h e r e t h e m i n i m u m

th ickness occu rs , i s g iven by h0 and i s

ho=hav( l

+ k / 2 ) ; h e r e

s = s / l , k

i s t he th i ckness

v a r i a t i o n p a r a m e t e r a n d h a~ i s t h e a v e r a g e t h i c k n e s s .

F o r t h e t h i n s h e ll t h e o r y , t h e s h e ll s a r e a n a l y z e d b y u s i n g a t w o - n o d e d r i n g e le m e n t w i t h

e i g h t d e g r ee s o f fr e e d o m p e r e l e m e n t . T h e a u t h o r s m a d e a c o n v e r g e n c e s tu d y f o r t h e t w o -

n o d e d e l e m e n t a n d f o u n d t h a t 1 5 e l e m e n t s w i l l g i v e a g o o d c o n v e r g e n c e ( w h i c h i s n o t

p r e s e n t ed ) , a n d h e n c e 15 e l e m e n t s w e r e u s e d f o r t h e c o m p u t a t i o n .

C l a m p e d - c l a m p e d b o u n d a r y c o n d i t i o n s a r e u s e d i n t h e p r e s e n t a n a l y s i s a n d a r e g i v e n

by u0 = v0 = w0 = u ' = v ' = w ' = w = 0 at s = 0 an d 1.

4 . C O N V E R G E N C E S T U D Y A N D C O M P A R I S O N

R e s u l t s o f a c o n v e r g e n c e s t u d y f o r t h e th r e e - n o d e d i s o p a r a m e t r i c r i n g e le m e n t a r e s h o w n

i n F i g u r e 1. T h e v a l u e s o f t h e t h ic k n e s s p a r a m e t e r

h* (=a/h , ,v )

and semi -ve r t ex ang le ~0

use d fo r th i s f igu re we re 5 and 30 , r e spec t ive ly . A l so he re a i s t he re fe ren ce su r face r ad iu s

a t t h e s m a l l e n d o f t h e s h el l. I t c a n b e s e e n t h a t c o n v e r g e n c e is o b t a i n e d w i t h v e r y f e w

e l em e n t s . I n t h e p r e s e n t s t u d y f iv e e l e m e n ts w e r e u s ed f o r c o m p u t a t i o n . A c o m p a r i s o n o f

t h e f r e q u e n c y p a r a m e t e r s , ~ , =~/p/E2

ko) ,

i s s h o w n i n T a b l e 1 . I t i s s e e n f r o m t h e t a b l e

t h a t t h e p r e s e n t r e s u l t s t a l l y w i t h r e f e re n c e v a l u e s [ 1 5 ]. T h e f i r s t tw o f r e q u e n c i e s c o n v e r g e

w i t h f o u r e l e m e n t s , w i th e r r o r s o f 1 . 2 % a n d 3 -5 % , r e s p ec ti v el y . F o r h i g h e r m o d e s m o r e

e l e m e n t s a re r e q u i r e d t o o b t a i n c o r r e s p o n d i n g p e r c e n t a g e e rr o r s.

5 . R E S U L T S A N D D I S C U S S I O N

T h e d i m e n s i o n s a n d p r o p e r t i e s u s e d a r e a s f o ll o w s : s m a l l e n d r a d i u s o f t h e s h el l, 0 . 1 m ;

i s o t r o p i c m a t e r i a l p r o p e r t i e s ( s t e e l ) : Y o u n g ' s m o d u l u s E , 2 101~ N / m 2 ; P o i s s o n r a t i o v

0 . 3 ; m a s s d e n s i t y p 7 8 0 0 k g / m 3; o r t h o t r o p i c m a t e r i a l p r o p e r t i e s: Y o u n g s ' m o d u l u s i n f ib r e

d i r e c t i o n E ~ , 1 .4 1 0 ~ N / m 2 ; Y o u n g s ' m o d u l u s i n t ra n s v e r s e d i r e c t i o n E 2 , 15 x 1 0~ N / m 2 ;


5/11

F R E E V I B R A T I O N O F T H I C K C O N I C A L S H E L L S 31

g a

u_ I

I

._ . I

2 4 6

N u m b e r o f e le m e n t s

F i g u r e 1 . C o n v e r g e n c e s t u d y . / J = 2 - 9 2 4 , h = 1 0, ~ o = 2 0 . , m = 1 ; 17 , m = 2 .

TABLE

Com par i son o f requ enc y param eters , ~ n = O)

M od e Present Reference [ 15] Erro r (%)

1 1 853 1.845 0 43

2 2 107 2 088 0.91

3 2 697 2 594 3 97

4 3.409 3 274 4.12

5 3 794 3 565 6 42

Y o u n g s ' m o d u l u s i n t h i c k n e s s d i r e c t i o n E 3 , 1 .5 x 1 01 N / m 2 ; s h e a r m o d u l i G i 2 = G i 3 = G 2 3,

6 x 109 N /m 2; Po i s son ra t io s v l2 = Via =

v23,

0 .2 1 ; m a s s d e n s i t y 1 60 0 k g / m 3. H e r e 1 , 2 a n d

3 r e p r e s e n t t h e p r o p e r t i e s i n t h e d i r e c t i o n s o f t h e f i b re , t ra n s v e r s e t o t h e f i b re a n d p e r p e n d i c -

u l a r t o t h e s u r f a c e o f t h e s h e ll . T h e r e s u l t s f o r c o n s t a n t t h i c k n e s s s h el ls a r e p r e s e n t e d i n

F i g u r e s 2 - 5 .

I n F i g u r e 2 is s h o w n a c o m p a r i s o n o f d i f f e re n t s h el l t h e o r ie s u s e d f o r t h e p r e d i c t i o n o f

n a t u r a l f r e q u e n c i e s o f i s o t r o p i c a n d o r t h o t r o p i c s h e ll s w h e n p= 3 0 a n d / ~ = 1. I n F i g u r e

2 ( a ) a r e s h o w n t h e v a r i a t i o n s o f t h e f r e q u e n c y p a r a m e t e r s , ~ ., o f i s o t r o p i c s h e l ls w i t h r e s p e c t

t o t h e c i r c u m f e r e n t i a l m o d e n u m b e r , n , f o r d i f fe r e n t v a l u e s o f th e t h i c k n e s s p a r a m e t e r , h *

=a/h, ,v) w h e n m = 1 ( f ir s t a x i a l w a v e ) . I n t h i s f i g u re L o v e ' s fi r st a p p r o x i m a t i o n s h e l l

t h e o r y ( T h i n ) , f i rs t o r d e r s h e a r d e f o r m a t i o n t h e o r y ( H S ) a n d t h e t h e o r y w i t h th i c k n e s s

n o r m a l s t r a i n ( H 7 ) a r e c o m p a r e d . I t c a n b e s e e n t h a t t h e e f fe c t o f t h e th i c k n e ss n o r m a l

s t r a i n i s v e r y s m a l l i n t h e r a n g e o f g e o m e t r i c p a r a m e t e r s c o n s i d e r e d . I n f a c t , t h e e f f ec t o f

t h e t h i c k n e s s n o r m a l s t r a i n i s t o i n c r e a s e s l i g h t l y t h e n a t u r a l f r e q u e n c i e s . B u t i t is s e en

t h a t t h e d i f f e re n c e s i n t h e v a l u e s o f t h e n a t u r a l f r e q u e n c i e s o f sh e l ls o b t a i n e d b y u s i n g t h i n

s h e ll t h e o r y a n d t h i c k s h e l l t h e o r i e s , r e s p e ct i v e ly , a r e l a r g e f o r l o w e r v a l u e s o f h * . W h e n

h * = 5 a n d n = 9 t h e d i f fe r e n c e i n n a t u r a l f r e q u e n c i e s o b s e r v e d i s a b o u t 3 0 % f o r t h e i s o t r o p i c

s h e ll . W h e n h * i n c r e a s e s, i .e . , w h e n t h e s h e ll b e c o m e s t h i n n e r a n d t h i n n e r , t h e n , a s e x p e c t e d ,

a l l t h e t h r e e s h e l l t h e o r i e s p r e d i c t m o r e o r l es s t h e s a m e r e s u l ts .


6/11

3

}

ut

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4

a)

n

3

13

2

o ~ o

o l

0 5 6

5 I ( b )

I

13

o

o

3 - o

5 ~

o h * =

o / o o I

0 3 6 9

C i r c u m f e r e n t i a lm o d en u m b e r

F i g u r e 2 . E f f e c t o f h * o n f r e q u e n c y p a r a m e t e r s o f c o n i c a l s h e l l s ; m = 1 . ( a ) I s o t r o p i c ; ( b ) o r t h o t r o p i c .

I- 7, T h i n ; O , H 5 , H 7 .

I n t h e c a s e o f o r t h o t r o p i c s h e ll s th e n a t u r e o f t h e v a r i a t io n s o f n a t u r a l f r e q u e n c i e s is

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

F i g u r e 2 ( b ) . I t c a n b e s e e n t h a t t h e e f fe c t o f s h e a r d e f o r m a t i o n is h i g h e r f o r c o m p o s i t e

s h el ls c o m p a r e d w i t h t h a t f o r i s o t r o p i c sh e ll s. W h e n n = 8 a n d h * = 5 , t h e n a t u r a l f r e q u e n c y

p r e d i c t e d b y t h in s h e ll t h e o r y is a l m o s t 1 0 0 h i g h e r t h a n t h a t p r e d i c t e d b y t h i c k sh e ll

t h e o r y . I n t h e c a s e o f o r t h o t r o p i c s h e ll s, t h e r e is s o m e d i f fe r e n c e d u e t o s h e a r d e f o r m a t i o n

w h e n h * = 2 5 ; t h i s i n d i c a t e s t h a t t h e f ir s t a p p r o x i m a t i o n s h e ll t h e o r y i s n o t s u i t a b l e f o r

c a l c u l a t i n g t h e v a r i o u s c h a r a c t e r i s t i c s o f s h e ll s h a v i n g l o w e r v a l u e s o f h * ( h * < 2 5 ) . F o r

o r t h o t r o p i c s h e l l s, a l s o , t h e e f f e c t o f t h i c k n e s s n o r m a l s t r a i n i s v e r y s m a l l.

2 . 0

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=

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o o s

C i r c u m f e r e n t i a lm o d e n u m b e r

F i g u r e 3 . E f f e c t o f p o n f r e q u e n c y p a r a m e t e r s o f c o n i c a l s h e l ls : m = 1. ( a ) I s o t r o p i c ; ( b ) o r t h o t r o p i c .

I n F i g u r e 3 is s h o w n t h e e f f ec t o f t h e l e n g t h p a r a m e t e r , p

=l/a)

o n t h e f r e q u e n c y

p a r a m e t e r , Z , o f a C C c o n i c a l s h el l o f c o n s t a n t t h ic k n e s s w h e n ( p= 3 0 o a n d h * = 1 0. T h e

v a r i a t i o n s o f t h e f r e q u e n c y p a r a m e t e r s o f i s o t r o p i c sh e ll s a r e s h o w n i n F i g u r e 3 ( a ) . I t c a n


7/11

F R E E V I B R A T I O N O F T H I C K C O N I C A L S H E L L S

33

b e s e e n t h a t w h e n p = 0 . 5 t h e f r e q u e n c y p a r a m e t e r i n c re a s e s a s n i n c re a s e s. W h e n p > 0- 5

t h e f r e q u e n c y p a r a m e t e r s d e c r e a s e i n it ia l ly a n d l a t e r in c r e a s e a s n is in c r e a s e d . I t c a n b e

s e e n t h a t t h e r a t e o f v a r i a t i o n o f ; t i n c re a s e s a s ~ i n c re a s e s. I n F i g u r e 3 ( b ) a r e s h o w n t h e

v a r i a t i o n s o f ;t o f a n o r t h o t r o p i c s h el l w i t h r e s p e c t t o n f o r d i ff e r e n t v a l u e s o f g w h e n t h e

w i n d i n g a n g l e a = 0 . T h e v a r i a t i o n o f 2 w i t h r e s p e c t t o n i s m o r e o r l e ss t h e s a m e a s

o b s e r v e d i n t h e c a s e o f i s o t r o p i c s h e ll s, e x c e p t t h a t t h e l o w e s t v a l u e o f ; t is h i g h e r f o r

o r t h o t r o p i c s h el ls .

2 4

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2 0

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o

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~ 1 ' 6

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1 . 2

0 3 0 6 0 9 0

S e m i - v e r t e x angle ~o

( d e g r e e s )

F i g u r e 4 . E f f e c t o f ~0 o n f r e q u e n c y p a r a m e t e r s o f C C o r t h o t r o p i c c o n i c a l s h e l l s ; r n = 1 , h * = 1 0 , ~0= 30 .

F i g u r e 4 i s s h o w n t h e e f f e ct o f t h e s e m i - v e r te x a n g le , cp, o n t h e f r e q u e n c y p a r a m e t e r s o f

C C c o n i c a l sh e ll s w h e n h * = 1 0. F o r a ll th e v a l u e s o f n t h e f r e q u e n c y p a r a m e t e r , Z , d e c r e a s e s

a s i n c r e a s e s . I t is s e e n t h a t t h e n a t w h i c h t h e l o w e s t n a t u r a l f r e q u e n c y o c c u r s s h i f ts a s

cp i n c r e a s e s . I t i s a s s u m e d t h a t t h e w i n d i n g a n g l e a is z e r o . A s i m i l a r t r e n d h a s b e e n

o b s e r v e d i n t h e c a s e o f i s o t r o p i c s h e ll s ( n o t p r e s e n t e d ) .

I n F i g u r e 5 i s s h o w n t h e e ff e c t o f t h e w i n d i n g a n g le , c t, o n t h e f r e q u e n c y p a r a m e t e r s o f

s in g le l a y e r ( a ) C C c o n i c a l s he ll s w h e n h * = 1 0. F o r t h is c a s e a s e m i - v e r t e x a n g l e o f ~ =

3 0 h a s b e e n a s s u m e d . W h e n n > 6 a n d n = 0 , th e f r e q u e n c y p a r a m e t e r , Z , a l w a y s i n c r e a se s

a s a i s i n c r e a s e d . W h e n n = 1 a n d 5 t h e r e i s a n i n c r e a s e a n d l a t e r a d e c r e a s e a s a i s

i n c r ea s e d . F o r a ll o t h e r v a l u e s o f n t h e f r e q u e n c y p a r a m e t e r s d e c r e a s e . A s i m i la r t r e n d h a s

b e e n o b s e r v e d f o r a t h r e e - l a y e r sy m m e t r i c a l l a m i n a t e a/O/a) shel l .

M a n y r e s e a r c h e r s h a v e a n a l y z e d c o n i c a l s h el ls , b u t i n m o s t o f t h e s e a n a l y se s t h i n s h e ll

t h e o r y h a s b e e n u s e d . S o m e r e s e a r c h w o r k h a s b e e n p u b l i s h e d w h i c h d e a l s w i th t h i c k

c o n i c a l s h el ls b y u s i n g f ir st o r d e r s h e a r d e f o r m a t i o n t h e o r y , a n d w i t h s o m e t y p i c a l c a s e s

o f v a r i a b l e t h ic k n e s s s h el l. A c c o r d i n g t o t h e a u t h o r s k n o w l e d g e , n o p u b l i s h e d r e su l t s a r e

a v a i l a b le o n t h e e f f ec t o f t h e m a s s d i s t r i b u t io n i n th e a x i al d i r e c t io n w i t h a c o n s t r a i n t o n

t h e t o t a l m a s s f o r a p a r t i c u l a r v a l u e o f / a . I n t h e p r e s e n t p a p e r , t h e a u t h o r s h a v e m a d e a n

a t t e m p t t o s t u d y t h is e f f ec t f o r i s o t r o p i c a n d o r t h o t r o p i c c o n i c a l s he lls .

I n F i g u r e s 6 ( a ) a n d ( b ) a r e s h o w n t h e n o r m a l i z e d f r e q u e n c i e s ( o / c 0 0 ) , w h e r e co is t h e

f r e q u e n c y o f t h e t a p e r e d s h el l a n d coo is th e c o r r e s p o n d i n g n a t u r a l f r e q u e n c y o f th e a v e r a g e

t h i c k n e s s s h e ll , o f a C C c o n i c a l s h e l l f o r d i f f e r e n t v a l u e s o f g . I n t h i s , / 3 i s t h e r a t i o o f t h e

m a x i m u m t h i ck n e s s t o t h e m i n i m u m t h ic k n e s s . T h e r e is a l in e a r s y m m e t r i c v a r i a t i o n o f

t h i c k n e ss a l o n g t h e a x i a l d i r e c t i o n w i th m a x i m u m t h i ck n e s s a t t h e e n d s. A s e m i - v e r te x

a n g l e o f 3 0 is c o n s i d e r e d i n t h i s c a s e .


8/11

4 K R S I V A D A S A N D N ( I A N E S A N

5 2

n = 9 ~

~ Z 4

g

0 8 1 , I ~ J , I

0 3 0 6 0 9 0

Winding angle, a degrees)

F i g u r e 5 . E f f e c t o f a o n f r e q u e n c y p a r a m e t e r s o f s i n g l e l a y e r C C c o n i c a l s h e l l s ;

m = 1 . h * =

1 0 ,

q ~= 3 0 .

T h e n o r m a l i z e d f r e q u e n c i e s o f t h e s h el l w h e n p = 0 . 5 a n d m = 1 a r e s h o w n i n F i g u r e

6 ( a ) . T h e f i g u re s h o w s a d e c r e a s i n g t r e n d i n th e n o r m a l i z e d f r e q u e n c i e s f o r a ll th e v a l u e s

o f f l a s n i s i n c r e a se d . W h e n n < 4 t h e n o r m a l i z e d f r e q u e n c i e s a r e m a x i m u m : i .e . , t h e

m a x i m u m i n c r e a s e i n n a t u r a l f r e q u e n c i e s d u e t o m a s s d i s t r ib u t i o n i s o b s e r v e d i n th i s

r e g i o n . T h e n o r m a l i z e d f r e q u e n c y m a y f a l l b e l o w u n i t y : i .e ., t h e n a t u r a l f r e q u e n c y o f t h e

t a p e r e d s h e l l b e c o m e s l e s s t h a n t h a t o f t h e a v e r a g e t h i c k n e s s s h e l l, f o r h i g h e r c i r c u m f e r e n -

t ia l m o d e s . I t i s s e e n t h a t t h e n o r m a l i z e d f r e q u e n c i e s a r e h i g h e r f o r h i g h e r v a l u e s o f f t.

F o r o t h e r v a l u e s o f , t h e n o r m a l i z e d f r e q u e n c i e s i n c r e a s e i n it ia l ly a n d d e c r e a s e la te r , a n d

e v e n t u a l l y fa l l b e l o w u n i t y a s n i s i n c r e a se d . I n t h e s e c a s e s t h e m a x i m u m v a l u e s o f t h e

n o r m a l i z e d f r e q u e n c i e s o c c u r a t d i f fe r e n t v a l u e s o f n . W h e n = 1 a n d 2 t h e y o c c u r a t n =

3 , a n d w h e n = 5 t h e y o c cu r a t n = 2. F r o m F i g u r e 6 o n e c a n s ee th a t t h e m a x i m u m v a l u e s

o f t h e n o r m a l i z e d f r e q u e n c i e s a r e l o w e r f o r h i g h e r v a l u e s o f p . I t i s a l s o o b s e r v e d t h a t t h e

o

3

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z

1 - 5

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3 6 9

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1 - 2

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0 . 8

O

b )

, I , I ~

3 6 9

C i rc um fe r en ti a l m o d e n u m b e r

F i g u r e 6 . E f f e c t o f f l o n n o r m a l i z e d f r e q u e n c i e s i s o t r o p i c ) ; n z

= 1 , tp = 3 0 . ( a ) # = 0 .5 ; ( b ) # = 2 .


9/11

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a )

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1 . 0

0 . 9

0 . 8 , I

0 3

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0 3 6 9 6 9

C i r c u m f e r e n t i a l m o d e n u m b e r

F i g u r e

7 . E f f e c t o f / ~

o n n o r m a l i z e d f r e q u e n c i e s i s o t r o p i c ) ;

m = 2 , = 3 0 . ( a ) I t = 0 - 5 ; ( b ) I t = 2 .

5

c i rc u m f e r e n t ia l m o d e n u m b e r n a t w h i c h t h e n o r m a l i z e d f r e q u e n c y f a ll s b e l o w u n i t y sh i ft s

t o l o w e r v a l u e s o f n a s / ~ i n c r e a se s .

I n F i gu r e s 7 a ) a n d b ) a r e s h o w n t h e n o r m a l i ze d f r e q u e n c ie s o f C C i s o t r o p i c c o n i c a l

s h e l l f o r d i ff e r en t v a l u e s o f / ~ w h e n m = 2 a n d = 3 0 . I n t h i s c a s e , t h e n o r m a l i z e d f r e q u e n -

c i e s d e c r e a s e a s n i n c r e a se s . F o r a s h o r t sh e l l p = 0 . 5 ) m o s t o f t h e n o r m a l i z e d f r e q u e n c i e s

a r e a l w a y s h i g h e r t h a n u n i t y i n t h e r a n g e o f n c o n s i d e re d . W h e n /~ = 1 a n d n > 4 t h e

n o r m a l i z e d f r e q u e n c i es a r e le s s t h a n u n i t y . W h e n p = 2 a n d 5 , a ll th e n o r m a l i ze d f r e q u e n -

c i e s a r e le s s t h a n u n i t y e x c e p t w h e n p = 5 a n d n = 0 .

1 . 5

3 1 4

g

, ~ 1 . 3

o

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( a )

( b )

1 . 5

1 . 3

1 . 1

, I , 0 9 , I

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1 - 1 , I , I ,

0 3 6 9

F i g u r e 8 . E f f ec t o f ~ o n n o r m a l iz e d f re q u e n c i es o r t h o t r o p i c ) ; m 1 , a = 0 , q ~ = 3 0 . ( a ) I t = 0 . 5 ; ( b ) I t = 2 .

I n F ig u r e s 8 a ) a n d b ) a r e s h o w n t h e n o r m a l i z e d fr e q u e n c ie s o f a C C o r t h o t r o p i c

c o n i c a l s h e ll f o r d if fe r e n t v a lu e s o f ~ a n d ~ w h e n a = 0 a n d ~ = 3 0 . T h e t r e n d s o f t h e

v a r i a t io n s o f t h e n o r m a l i z e d f r e q u e n c i e s a r e m o r e o r le s s t h e s a m e a s o b s e r v e d i n t h e c a s e

o f i s o t r o p i c s h e l ls , b u t t h e p e r c e n t a g e i n c r e a s e i n n a t u r a l f r e q u e n c i e s is s l ig h t l y h i g h e r t h a n

f o r i s o t r o p i c s h e l ls . W h e n r n = 2 , t h e n o r m a l i z e d f r e q u e n c i e s a r e h i g h e r t h a n i n t h e is o t r o p i c

c a se . T h e c o r r e s p o n d i n g v a r i a t io n i s s h o w n i n F i g u r e 9 . In a l l t h e c a se s t h e p e r c e n t a g e

v a r i a t io n o f n a t u r a l f re q u e n c i e s a r e h i g h e r fo r h i g h e r v a lu e s o f 8 .


10/11

36

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3

=

N-

g

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1 1

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1 0

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0 3 6 9

i r c u m f e r e n t i a l

m o d e n u m b e r

Figure 9. Eff ect of fl on no rmalized frequencies (orthotropic): m= 2 a = 0 ~, (0=30 . (a) p =0.5; (b) p =2.

6 . C O N C L U S I O N S

A h i g h e r o r d e r s h el l t h e o r y f o r c o n i c a l c o m p o s i t e s h el ls h a s b e e n d e v e l o p e d t o a n a l y z e

t h e f r e e v i b r a t i o n c h a r a c t e r i s t i c s . T h e e f f ec t o f t h i c k n e s s s h e a r i s l a r g e f o r l o w e r v a l u e s o f

t h e s m a l l e n d r a d i u s t o t h i c k n e s s r a t i o , a n d is g r e a t e r f o r o r t h o t r o p i c s h el ls t h a n f o r

i s o t r o p i c s h el ls . T h e l o w e s t f r e q u e n c y o f t h e s h el l c a n b e i n c r e a s e d b y d i s t r i b u t i n g t h e m a s s

a l o n g t h e a x i a l d i r e c t i o n .

R E F E R E N C E S

1. C. W . BER T and P . H. FRANCIS 1974 American Institute o f Aeronautics an d Astronautics Journal

1 2 ( 9 ), 1 1 7 3 -1 1 8 6 . Co m p o s i t e m a te r i a l m e c h a n i c s : s t r u c tu r a l m e c h a n i c s .

2 . R. K . KAPANIA 1989 Transactions o f the American Society o f Me chanical Engineers Journal of

Pressure Vessel Technology

111, 88-96 . A rev iew on the ana lys is o f lamin a ted shel ls .

3 . J , R. HUC H]NSON an d S. A. EL-AZHA RI 1986 Transactions of the American Society of M echanical

Engineers Journal o f App lied Mechanics 53 , 641-646 . Vib ra t ions o f f ree sha llow c i rcu la r

cylinders .

4 . K . C. PARK and G. M . STANLEY 1986 Transactions q/ the American Society of Mechanical

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