Kamal 1986

7/26/2019 Kamal 1986

http://slidepdf.com/reader/full/kamal-1986 1/26

Composite Structures 5 (1986) 177-202

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K . K a m a l a n d S . D u r v a s u l a

Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012,

India

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a p p r o a c h . T h e a n a l y s is i s p r e s e n t e d i n a u n i f ie d f o r m s o a s t o i n c o r p o r a t e

a l l d i f f e r e n t c o m b i n a t i o n s o f l a m i na t e b o u n d a r y c o n d i ti o n s a n d w i t h f u l l

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

p l a t e . A p a r a m e t r i c s t u d y i s m a d e u s i n g a b e a m c h ar ac te ris ti c f u n c t i o n a s

t h e a d m i s s i b l e f u n c t i o n f o r t h e n u m e r i c a l c a lc ul ati on s.

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p u b l i s h e d r e su l ts . T h e e f f e c t o f p a r a m e t e r s s u c h a s t h e a s p e c t r a t io o f

p l a t e s p l y - a n g l e n u m b e r o f l ay er s a n d a ls o t h e t h i c kn e s s r a ti o s o f p l i e s i n

l a m i n a t e s o n t h e f r e q u e n c i e s o f th e l a m i n a t e i s s y s te m a t i c a l ly s t u d i e d . I t i s

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n u m e r i c a l v a l u e s o f t h e t h ic k n e s s r a ti o s e x i st w h i c h i m p r o v e t h e v i b r a ti o n

c h a r a c te r is t ic s o f s u c h l a m i n a t e s . N u m e r i c a l v a l u e s o f t h e n o n -

d i m e n s i o n a l f r e q u e n c i e s a n d n o d a l p a t t er n s u s i n g t h e t h i c k n e s s r a t io

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

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

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177

Composite Structures

0263-8223/86/ 03-50 © Elsevier App lied S cience Publishers L td,

England, 1986. Printed in Grea t Britain

7/26/2019 Kamal 1986


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C o m p o s i t e m a t e r i a l s h a v e f u l l y e s t a b l i s h e d t h e m s e l v e s a s w o r k a b l e

e n g i n e e r i n g m a t e r i a l s a n d a r e n o w q u i t e e x t e n s i v e l y u s e d i n v a r i o u s

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

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om e studies o n free vibration o f composite laminates 79

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

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

ou t s t a nd i ng s t r eng t h , s ti ff nes s, l ow s pec if ic g r av it y and un i qu e f l ex ib i li ty

i n d e s i g n c a p a b i li ty t o b e s t m a t c h t h e r e q u i r e m e n t s o f a g i v e n s t ru c t u ra l

a p p l i c a t i o n . A n y l a m i n a t e d s tr u c t u re o b t a i n e d b y s ta c k i n g t w o o r m o r e

s u c h c o m p o s i t e l a y e r s o f t e n p o s s e ss e s b e n d i n g - s t r e t c h i n g c o u p l in g . ~T h e

e f f e c t o f s u c h c o u p l i n g is fo u n d t o r e d u c e t h e o v e r a ll b e n d i n g s ti ff n es s o f

t he compos i t e l ami na t e . ~ - 3 H ow ever , i t i s pos s i b l e t o e l i mi na t e s uch

c o u p l i n g e f f e c t s4 a n d h e n c e m a x i m i s e th e f u n d a m e n t a l f r eq u e n c y o f t h e

l am i na t e i f, i n des i gn , ce r t a i n r equ i s i t e th i cknes s r a ti o s o f t he l aye rs i n t he

l a m i n a t e a r e u s ed .5

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

e n e r g y a p p r o a c h f o r t h e s o l u ti o n o f c la m p e d , l a m i n a t e d c o m p o s i t e p la t es

has be en ca r r i ed ou t by va r i ous au t ho r s . 6-9 A s h t on and W ad do up s t°

ana l y s ed t he an i s o t r o p i c p l a t e s u s i ng a dou b l e s e r i e s i n R i t z s s o l u t i on .

B e r t a n d M a y b e r r y H u s e d t h e R a y l e i g h - R i t z a p p r o a c h t o d e t e r m i n e t h e

n a t u r a l f r e q u e n c i e s o f l at e ra l v i b ra t io n o f l a m i n a t e d p la t e s w i t h c l a m p e d

e d g e s . T h e i r a n a ly s is , h o w e v e r , is c o n f i n e d t o t h e s q u a r e l a m i n a t e

ge om et r i e s . W hi t n ey , ~2 u s i ng the Fou r i e r s e r ie s app r o ach , s t ud i e d t he

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

u n s y m m e t r i c a l l y l a m i n a t e d r e c t a n g u l a r p l a t e . L i n a n d K i n g 13 a n a l y s e d

t h e r e c t a n g u l a r u n s y m m e t r ic a l l y l a m i n a t e d p la t e f o r f re e t r a n s v e r se

v i b r a t i o n u s i n g B o l o t i n s a s y m p t o t i c m e t h o d . T h e i r a n a ly s is p r o v i d e s a

h i g h e r e s t i m a t e o f t h e f r e q u e n c y v a l u e s s i n c e t h e e f f e c t o f s h e a r

d e f o r m a t i o n , i n - p l a n e a n d r o t a to r y i n e r t ia a re n o t i n c l u d e d i n t h e

ana lys i s .

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c o m p o s i t e p l a t e s i n d i c a t e d t h a t , t h o u g h n u m e r o u s c o m p o s i t e p l a t e

ana l y s es a r e ava i l ab l e , w h a t i s d i r ec t l y u s e f u l t o a d es i gne r t o s yn t hes i s e

op t i m a l des i gns o f l am i na t e s app ea r t o be ve r y m eag r e i nd eed . B e r t ~415

ca r r i e d ou t a p i o ne e r i n g e f f o r t in t h is d i rec t i on w h i ch , s ubs equ en t l y , i s

f o l l o w e d b y R a o a n d S i n gh . 6 R e c e n t l y , a n e f f o r t h a s b e e n m a d e a s

r ega r ds t he buc k l i ng beh av i o u r o f com pos i t e p l a t e s , ~7~8 pa r t icu l a r ly t o

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

v a l u e s o f l a m i n a t e s . A d a l i ~9 e m p l o y e d a m a t h e m a t i c a l a p p r o a c h i n t h e

o p t i m a l d e s i g n o f a n a n t i -s y m m e t r ic an g l e -p l y l a m i n a t e .

I n t h i s pap e r a s ma l l de f l ec t i on , t h i n , l aye r ed com pos i t e p l a t e ana l y s is

b a s e d o n t r a n s v e r s e s h e a r d e f o r m a t i o n t h e o r y is p r e s e n t e d . T h e

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180

K. Kamal S. Durvasula

R a y l e i g h - R i t z e n e r g y a p p r o a c h is e m p l o y e d t o o b ta i n t h e n a t u r a l

f r e q u e n c i e s o f t h e g e n e r a l r e c t a n g u l a r l a m i n a t e w i t h a n y g i v e n

c o m b i n a t i o n o f b o u n d a r y c o n d i t i o n s . T h i s a n a l y s i s i n c l u d e s m i x e d

t r i g o n o m e t r i c b e a m c h a r a c t er i s ti c f u n c ti o n s . T h e i n te g r a l v a lu e s o f s u c h

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

c o m b i n a t i o n o f i n - p la n e a s w e ll as o u t - o f - p l a n e b o u n d a r y c o n d i t i o n s , a re

e v a l u a t e d i n R e f . 2 0 . T h e p r e s e n t v i b r a t i o n a n a l y s i s , f i n a l l y , l e a d s t o a

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

e i g e n v a l u e s a n d e i g e n v e c t o r s ar e m a d e o n a D E C 1090 c o m p u t e r u si ng a n

a v a i l a b l e e i g e n v a l u e s u b r o u t i n e . W h i l e t h e a n a l y s i s i s v a l i d f o r a g e n e r a l

r e c t a n g u l a r l a m i n a t e d p l a t e , w i th a n y g iv e n c o n d i t i o n s o f s i m p l y

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

l a m i n a t e d p l a t e s w i t h b o u n d a r y c o n d i t i o n o f C I t y p e . ~3 T h e n u m e r i c a l

r e s u lt s o b t a i n e d c o n s t i tu t e u p p e r b o u n d s t o t h e e x a ct e i g e n v a l u es a n d

p r o v i d e a n e w s o u r c e o f c o m p a r i s o n f o r a s se s s m e n t o f t h e r e s ul ts o b t a i n e d

u s i n g v a r i o u s o t h e r s o l u t i o n t e c h n i q u e s . T h e p r e s e n t a n a l ys i s is t h e n u s e d

t o s t u d y t h e e f f e c t o f th i c k n e s s ra t io s o f la m i n a e o n t h e n a t u r a l f r e q u e n c y

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

s i m p l y s u p p o r t e d r e c t a n g u la r l a m i n a t e w a s m a d e e a r li e r b y t h e a u t h o r s ~

u s i n g t r i g o n o m e t r i c s i n e s e r i e s a s t h e a d m i s s i b l e f u n c t i o n s . I n a l l t h e s e

s t u d i e s i t i s f o u n d t h a t 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 a n t i - s y m m e t r i c ,

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

r e d u c i n g t h e d e g r a d a t i o n in t h e o v e r a ll s ti ff n es s o f th e l a m i n a t e s i m p l y by

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

e l i m i n a t e b e n d i n g - s t r e t c h i n g c o u p l in g .

2 P R O B L E M F O R M U L A T I O N A N D A N A L Y S IS

L a m i n a t e d p l a t e s a r e b a s ic a ll y c o m p o s e d o f N l a y e rs ; t h e p l a n e s t re s s

c o n s t i t u t i v e e q u a t i o n f o r t h e k t h l a y e r is g i v e n as

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, j a r e t h e r e d u c e d e l a s t ic s ti f fn e s s c o e f f i c ie n t s o f t h e k t h l a y e r

t r a n s f o r m e d t o t h e x , y d i r e c t io n s . M o d i f i e d c o n s ti t u ti v e r e la t i o n s f o r t h e

s t r e ss a n d m o m e n t r e s u lt a n t s o f t h e l a m i n a t e , r e p o r t e d i n R e f . 21 , a r e

e m p l o y e d t o i n c l u d e t h e s h e a r d e f o r m a t i o n s in th e l a m i n a t e .

T h e s t r a in e n e r g y e x p r e s s i o n , f o r m u l a t e d a n d p r e s e n t e d i n R e f . 2 1 , is

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Som e studies on free vibration of comp osite laminates 181

e m p l o y e d in t h e p r e s e n t s tu d y . B y i n c l u d i n g t h e i n - p la n e a n d r o t a t o r y

i ne r t i a s , t he m ax i m um k i ne t i c ene r gy Tmax o f t he s y s t em, h ow eve r , i s

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o r t h o t r o p y , o r t h o t r o p y in b e n d i n g a n d w h i c h p o ss e ss e s t h e u n c o u p l e d

l a m i n a t e b e h a v i o u r is k n o w n t o h a v e t h e h i g h e s t s ti ff n es s, a n d t h u s , t h e

h i g h e s t f u n d a m e n t a l f r e q u e n c y o f v ib ra ti on .2 H o w e v e r , i n s o m e d e s ig n

s i t u a t io n s w h i c h d o n o t n e c e ss a ri ly r e q u i r e a b a l a n c e d l a m i n a t e ,

u n s y m m e t r i c o r a n t i - sy m m e t r ic l a m i n a t e s c h e m e s a r e a ls o u s e d . I t is f el t

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

pa r t i a l l y o r ev en t o t a l l y be e l i m i na t ed by ado p t i ng ce r t a i n s pec if ic ,

t h i cknes s d i s t r i bu t i on o f t he l aye r s . U s e o f s uch t h i cknes s r a t i o s o f t he

l a m i n a e r e s u lt s i n a c o n s e q u e n t i n c r e a se i n t h e o v e r a l l s ti ff n es s a n d h e n c e

a n i m p r o v e m e n t o f v i b r a ti o n c h a r ac t er is ti c s o f t h e l a m i n a t e . T h i s

i m p o r t a n t s t ru c t u r a l b e h a v i o u r o f a l a m i n a t e is c l ea r ly s e e n a n d v e r i f ie d

h e r e t h e o r e ti c a ll y , a n d is f o u n d t o b e i n d e p e n d e n t o f t h e b o u n d a r y

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

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82 K. Kamal S. Durvasula

I n w h a t f o l lo w s , t h e R a y l e i g h - R i t z m e t h o d is u s e d t o o b ta i n a n

a p p r o x i m a t e s o l u ti o n o f t h e v ib r a t io n p r o b l e m o f t h e l a m i n a t e d p l a te a n d

t h e d i s p l a c e m e n t f u n c t i o n s

U, V, W

a n d r o t a t io n s / 3 ~ , fi r a r e a s s u m e d a s

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m = l n = l

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m = l n = l

M N

W = ~ ~ G~,.cb~,.(sx)qJ~.(rl) (3c)

m = l n = l

M N

/3~ = ~ ~ X ,..tbx,.(()qJ~ .(rl) (3d)

m l n l

M N

~Y

= Z E Ymnl~ym(~)l~ly( l~)

3 e )

m = l n = l

w h e r e 4~um ((), v,. ((), xm ((), ym SO), (bw,. ~) a n d tou.0q), to~ .(~), tox.( 'q),

tO y. ~ /) , tO w. 7 ) a r e a d m i s s i b l e f u n c t i o n s s a t i s f y i n g t h e g e o m e t r i c b o u n d a r y

c o n d i t i o n s o f t h e l a m i n a t e u n d e r c o n s i d e r a t io n . T h e c o e f f ic i e n ts E , . ., F m . ,

G , . . in e q n ( 3 ) a r e t h e n d e t e r m i n e d f r o m t h e u s u a l c o n d i t io n s a s

0 O m a x - T m a x ) = 0 4 a )

8Emn

O Um~-

Tmax) 0 4b )

8Fm.

t~ V m a x - T m a ~ ) = 0 4 C )

0 G , . .

O Umax-T m a x ) = 0 4 d )

~Xm.

a ( U m . . - T m~) = 0 (4e )

aY, . .

7/26/2019 Kamal 1986


Som e studies on free vibration o f com posite laminates 183

w h i c h r e s u l t in a se t o f l in e a r , h o m o g e n e o u s , s i m u l t a n e o u s a l g e b r a ic

e q u a t i o n s , w h i c h c a n b e c o n c i s e ly p u t i n t h e m a t r i x f o r m a s

[ S ] k . [M ] ]

,~

Fr~

G,~

Y~

= 0 5)

w h e r e t h e s t i ff n e ss m a t r i x [ S] is c o m p o s e d o f f if t e en s u b m a t r ic e s . T h e

e l e m e n t s o f th e s e s u b m a t r i c e s a re g i ve n in A p p e n d i x A . T h e m a s s m a t r i x

[ M ] h a s e l e v e n s u b m a t r i c e s a n d i s s h o w n i n A p p e n d i x A . E a c h s u b m a t r i x

o f t h e [ S ] a n d [ M ] m a t r i c e s is o f t h e o r d e r o f ( ( M × PC) × ( M × IV )).

3 A P P L I C A T I O N T O F U L L Y C L A M P E D L A M I N A T E

A n a l y s i s o f t h e v i b r a t i o n p r o b l e m r e p r e s e n t e d b y e q n ( 5) is s p e c i al is e d

h e r e t o a l a m i n a t e d p la te w i t h c la m p e d b o u n d a r y c o n d i t io n s o f t y p e C I ,

w h e r e i n t h e b o u n d a r y c o n d it io n s t a k e t h e f o r m

0W

U = V = W - - - - j ~ x = B y = 0 (6)

0n

T h e a d m i s s i b le f u n c t i o n s f o r U V W B x ~ y s a t i s f y i n g t h e a b o v e

b o u n d a r y c o n d i t io n s a r e g i ve n as

(bum(~:) = ~bvm(S) = ~b x,.(() = ~bym(~:) = x/2-s inm ~r~ :;

(7a)

tO..(~) = qJv.( O) = **.( 0) = O r.(n) = X /~s inm ro

t o )

~bw,. ~) = cos hflm ~:- cosf lm ~:- a , . (s inh /3m ~- s inf lm~)

(7c)

Sw.( 0 ) = cosh /3 . 0 - cos /3 ,n - a . ( s in hB ,r l - s in B .n) (7d)

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184 K . Kam al S . Durva sula

w h e r e i n , n u m e r i c a l v a l u e s o f o r ,,, a , , / 3 , , , 13, c o r r e s p o n d i n g t o

c o n d i t i o n s a r e o b t a i n e d f r o m t h e t a b le s a s g i v e n i n R e f . 2 2.

t h e e n d

4 N U M E R I C A L C A L C U L A T I O N S

N u m e r i c a l c a l c u l a ti o n s ar e c a rr i e d o u t o n a D E C 1090 c o m p u t e r a n d a

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

a s s u m e d s e ri e s . T h e c o m b i n a t i o n M = 4 , N = 4 , a t w h i c h r e li a b le v a l u e s

a r e o b t a i n e d , c o r r e s p o n d s t o 16 t e r m s e a c h i n t h e a s s u m e d s e ri e s l e a d i n g

t o o v e r a l l s t if f n e s s a n d m a s s m a t r i c e s I S ] a n d [ M ] o f t h e o r d e r o f (8 0 x 8 0 ) .

T A B L E

Composite Lamina Properties

C om pos i t e m a t e r i a l

P r op e rt ie s G F R P ~ G F R P B F R P O F R P G r a p h it e- F R P

E d E 2

1.053 2.%3 11.481 5 13-749 6 40.0

v 12 0-242 0-25 0.28 0.34 0-25

G 12/ E 2 0.023 4 0 453 2 0 275 9 0.371 65 0 5

aFabric lamina proper ties as reported in Ref. 11.

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

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

c o n d i t i o n s a r e f i rs t c o m p a r e d w i t h t h e r e su l ts o b t a i n e d e a r l i e r u s in g o t h e r

s o l u t i o n t e c h n i q u e s . T h e p r e s e n t a na l ys i s is t h e n u s e d t o s t u d y th e e f f e c t

o f v a r i o u s d e s i g n p a r a m e t e r s , s u c h a s th e a s p e c t r a t io o f th e p l a t e a /

b = 1/3 , 1/2 , 2/ 3 an d 1), pl y an gl es (0 = 0 _+20 , ___45 , ___60 a n d 90°),

p l y t h i c k n e s s o n n a t u r a l f r e q u e n c y o f v i b r a t io n o f f o u r a n d s ix l a y e r e d ,

a n t i - s y m m e t r i c l a m i n a t e s o f c o n s t a n t s i de to t h i c k n e s s r a t io a / t = 60) to

m i n i m i s e t h e s h e a r d e f o r m a t i o n ef fe c t. F o u r c o m m o n l y u se d c o m p o s i t e

m a t e r i a l s , n a m e l y , b o r o n / e p o x y ( B F R P ) , g la s s /e p o x y ( G F R P ) , o r g a n i c

f ib re r e i n f o r c e d p l a st ic ( O F R P ) a n d g r a p h i t e / e p o x y ( G r - F R P ) a r e

c o n s i d e r e d w i t h t h e i r m a t e r i a l p r o p e r t i e s a s g i v e n i n T a b l e 1.

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S o m e s t u d ie s o n f r e e v i b r a ti o n o f c o m p o s i t e l a m i n a te s

185

5 RESULTS AND DISCUSSIONS

The numerical results obtained from a set of laminates with various

lamination schemes and with clamped boundaries of the C1 type are

found to compare well with available results using the present analysis

Table 2). A numerical study of convergence in the example case of a four

layered cross-ply, rectangular, laminated GFRP) plate with boundaries

of type C1 is presented in Table 3. Convergence for frequency parameter

T A B L E

Comparison of the Fundamental Frequency Parameter of a Two-layered Cross-ply

Clamped Laminate (C1- C1-C1- C1) a

k* = o J b 2 ~

( b / t

= 50)

b / a P r e s e n t s o l u t i o n A s y m p t o t i c s o l u t i o n F o u r i e r s o l u t i o n

Re f . 13 Re f . 12

1 23.413 23.638 24.527

2 16.925 17-111 21.171

3 16.506 16-640 16.950

4 16.463 16-543 17.200

5 16.130 16.509 16.974

a E I / E ~

= 40,

G12/E2

= 0.5, v12 = 0.25.

T A B L E 3

Conver gence Study: Rectangular Laminated, C1-C1--C1-C1 Plate a

k n = ~ / p t / E 2 t 3 to na 2

Numb er of layers

= 4 0°10°190°190° al b

= 2/3

M N O r d e r o f m a t r i x k t k 2 k 3 k 4 k s

1 1 (5 ×5) 7 1630 . . . .

2 2 (20 x 20) 7 1620 11 60 0 17 561 22 274 - -

2 3 (30 × 30) 7 1184 11 598 16 885 18 165 22 280

3 3 (45 × 45) 7 1064 1 1 5 1 6 16 855 17 991 22 270

4 4 (80 × 80) 7 1063 1 1 4 7 9 1 6 8 4 9 17 990 22 060

a E 1 / E 2

= 1 053, v12 = 0 242,

GI2/E2 = 0 0234 .

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1 86 K . K a m a l , S . D u r v a s u l a

T A B L E 4

N a tu ra l F req u en c ies in H z o f a T w o - lay e red A n g le -p ly L am in a ted R ec tan g u la r P la te

a / b

= 3 /2 ) w ith A l l E d g es C lamp e d ; (C 1 -C I -C I -C 1 ) c

M o d e n o . P r e s e n t c a l c u la t i o n Q u o t e d i n R e f . 1 1

0 ° / 0 ° l a m i n a t e

l 1 , 1 )

2 2 ,1)

3 1 ,2 )

4 3 , l )

5 2 ,2)

6 3 ,2 )

7 1 ,3 )

8 (2 ,3 )

9 (3 ,3 )

4 5 0 / 4 5 ° l a m i n a t e

l 1 , 1 )

2 2, 1)

3 1 , 2 )

4 3 , 1 )

5 2 , 2 )

6 3 , 2 )

7 1 , 3 )

8 2 , 3 )

9 3 , 3 )

79 101

121 ~ 112 b

1 1 9 156

194 230

200 244

223 288

281 372

303 425

461 456 b

373 480

385 563

75 95

109 a 104 h

121 145

178 223

190 225

233 267

280 344

313 415

445 a 427 b

330 460

391 527

aT heo ret ica l values ( fo r equ ivalen t f ib re lamina) .

bE x p er im en ta l v a lu es ( f ab r i c lamin a) .

C E D E 2

= 1.053, v12 = 0.242,

G I 2 / E 2 = 0 . 0 2 3 4 ,

E2 = 2 560 000 lb /in 2, t = 0.04 2 .

a = 9 ,

4

p = 0 .000 197 lb-s -/m ,

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

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

q u i t e s a t i s f a c t o r y i n d e e d . I n T a b l e s 4 a n d 5 , n u m e r i c a l r e s u l t s f o r t h e f ir st

n i n e f r e q u e n c i e s i n H z , o f r e c t a n g u l a r , t w o - p l y a n d f o u r - p l y l a m i n a t e s

w i t h d i f f e r e n t f ib r e o r i e n t a t i o n s w i th m a t e r i a l a n d g e o m e t r y , a s u s e d b y

L i n

3

a n d B e r t a n d M a y b e r r y , ~t a r e p r e s e n t e d . I t a p p e a r s f r o m t h e c o m -

p a r i s o n ( T a b l e s 4 a n d 5 ) t h a t t h e r e s u l t s o f t h e t h e o r e t i c a l a n a l y s i s o f R e f .

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So m e s t ud i e s on f r ee v i b ra ti on o f com pos i t e l am i na te s

187

T A B L E 5

N a t u r a l F r e q u e n c i e s in H z o f a F o u r - l a y e r e d A n g l e - p l y L a m i n a t e d R e c t a n g u l a r P l a t e

( a / b

= 3 /2 ) w i t h A l l E d g e s C l a m p e d ( C 1 - C 1 - C 1 - C 1 ) c

M od e no . Present ca lcu la tion Qu oted in Ref . 13 Q uo ted in Ref . I1

4 5 ° / 4 5 ° / - 4 5 ° / - 4 5 0 l a m i n a t e

I

(1, 1) 150 159 180

2 (2, 1) 242 252 280

3 (1 ,2 ) 356 391 425

4 (3, 1) 380 401 430

5 (2 ,2) 466 486 512

6 (3, 2) 561 638 660

7 (1 , 3) 627 737 795

8 (2, 3) 660 836 875

9 (3 ,3) 969 993 1005

0° / / 90~ /90 ° lamina te

1 1 , 1 )

160 190

217 a 193 b

2 (2, 1) 237 300

3 (1 ,2 ) 386 442

4 (3 , 1 ) 404 460

5 (2 , 2) 453 550

6 ( 3 , 2 ) 5 6 2 7 1 0

7 ( 1 , 3 ) 6 0 0 8 2 0

908 b 860 b

8 (2, 3) 743 925

9 (3, 3) 779 1080

a T h e o r e t i c a l v a l u e s ( f o r e q u i v a l e n t f i b r e l a m i n a ) .

bE x p e r i m e n t a l v a l u e s ( fa b r i c l a m i n a ) .

eE l ~ E 2

= 1-053 , v i2 = 0 .242 ,

GI2/E2 = 0 .0234,

E2 = 2 560 00 0 lb / in 2 , t = 0 .042 .

a = 9 , p = 0 .000 197 lb- s2 / in 4 ,

11 a r e o f do ub t f u l a c c u r a c y . C l o s e r e xa m i na t i on o f t he a na l y s is o f R e f . 11

i nd i c a t e s t ha t t he a na l y s i s i s a pp l i c a b l e on l y t o t he c a s e o f s qua r e

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

a p p e a r i n t h e p r e v i o u s a n a ly s is o f R e f . 1 1, w h e r e a s t h e r e su l ts p r e s e n t e d

a r e f o r h 4= 1 . I n T a b l e 5 n a t u r a l f r e q u e n c i e s o f v ib r a t io n o f a c l a m p e d

l a m i n a t e d p l a te o b t a i n e d u s in g t h e p r e s e n t a n a ly s is , e m p l o y i n g R a y l e i g h -

R i t z s m e t h o d , a r e a ls o c o m p a r e d w i th t h e re s u lt s o b t a i n e d b y L i n a n d

K i n g , 13 u s i n g B o l o t i n s a s y m p t o t ic m e t h o d . T h e s li g h tl y h i g h e r e s t i m a t e

7/26/2019 Kamal 1986


1 8 8

K. Kamal, S. Durvasula

o f R e f . 13 is e x p e c t e d a s t h e e f f e c t s o f s h e a r d e f l e c t i o n a n d i n - p l a n e a n d

r o t a t o r y i n e r t i a s h a v e n o t b e e n c o n s i d e r e d i n t h e i r a n a ly s is .

T h e v a r i a t io n o f t h e n o n - d i m e n s i o n a l f r e q u e n c y p a r a m e t e r k , w i th t h e

p l a t e a s p e c t r a t i o f o r a c la m p e d , a n t i - s y m m e t r ic a n g le - p ly l a m i n a t e

r e v e a l s t h a t a s q u a r e l a m i n a t e

a/b

= 1 ) , a s i t s h o u l d b e , h a s t h e

m a x i m u m f r e q u e n c i e s o f v i b r a t i o n F ig . 1 ). I n F ig . 2 t h e f u n d a m e n t a l

f r e q u e n c y p a r a m e t e r is p l o t t e d a s a f u n c t io n o f th e t h i c k n e s s ra t io

h/t2)

f o r a n e x a m p l e c a s e o f a f o u r - l a y e r e d , a n t i - s y m m e t r i c , a n g l e - p l y l a m i n a t e

s h o w n i n F ig . 3 a ) f o r d i f f e r e n t f ib r e o r i e n t a t i o n s 0 = 0 ° , - +2 0° . -+30° ,

+ 6 0 ° , + 4 5 ° a n d 9 0 °) . I t is o b s e r v e d t h a t f o r a l l o r i e n t a t i o n s , e x c e p t f o r

l a m i n a t e s w i t h a l a y - u p s c h e m e o f 0 = 0 ° , 9 0 ° o r + 4 5 ° , f o r w h i c h c a s e

i n d i v i d u a l p l y t h i c k n e s s e s h a v e n o i n f l u e n c e o n t h e s t if f n e ss o f t h e

l a m i n a t e , t h e f r e q u e n c y p a r a m e t e r s r e a c h th e i r m a x i m u m a t t h e th i c k n e s s

r a t i o o f 0 .4 1 4 2 . T h i s r e v e a l s t h a t a t th i s t h i c k n e s s r a t io t h e c o u p l i n g

6 0

5 0

C

GFRP } - o - - - - I

N=4B45 ° C'l /

4 0

kn

3 0

n = 3

(3 /1 )

2

= 2 ( 2 / 1 )

1 0 -

i'1--I

O

0 - 1 0 . 3

112 )

I / I )

0 ' 5 0 .7 0 9 1 .0

o b

F i g , 1 . F r e q u e n c y p a r a m e t e r o f f o u r - p ly l a m i n a t e El~E2 = 1 . 0 5 3 , GLT/ET = 0 5

ULT = 0 25 .

7/26/2019 Kamal 1986


om e stud ies on r ee v ibra t ion of composi te lamin ates 1 8 9

1 o . o [ e -_ _ _ Z ~ 9 o_ ~

9 . 5 V / G F R P ~ I

~ _

I

9

2 5 | I I I I

0 . 1 0 . 3 0 . 5 0 . 7 0 . 9

t ~ t z

F i R 2 . V a r i a t i o n o f f r e q u e n c y p a r a m e t e r w i t h t h ic k n e s s r a t io f o r a n a n ti -s y m m e t r ic ,

s q u a r e , f o u r - p l y l a m i n a t e n o t e : fa l s e o r i g i n ) .

' v-'//,s ' / ' 9 ~ / / / / / / / A ~,

~

o \ \ \ \ \ \ \ \ q ,.

; / / ; ~ . W / / / / / / A ~,

b )

~ . \ \ \ \ \ \ ' . ( - ( , ~

H / . / / / / / ; ~ ; ~

J_ ~ . - ' / / / / / ~ 4 ~ .

.\ \ \ \ \ ~ - .~

Y / / / / . d ~ ,

y / . / . / / / J , ~

F i g . 3 . a ) F o u r - p l y l a m i n a t e s c h e m e ; b ) s ix - p ly l a m i n a t e s c h e m e .

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190 K . K a m a l S . Du r v a s u l a

1 0 .0 - - G F R P * _ ~ _ _ ~ C 1

o /b = I 0 '[ [ l ~ r + e

t l / t 3 : l O c ~

N = 6

C~

e : 0 ° 9 0 °

8 °

9 8

k 1 J 30 ° 8 t 6 O° ~ 70o

9 6 I

_ _ I

I t i

0 2 0 3 ' 0 4 0 5 . 0

t 2 / t 3

F i g . 4 . V a r i a t i o n o f f r e q u e n c y p a r a m e t e r w i t h t h ic k n e s s ra t i o f o r a n a n t i - s y m m e t r i c ,

s q u a r e , s i x - p ly l a m i n a t e n o t e : f a l s e o r i g i n ).

s t i f f n e s s t e n d s t o v a n i s h . T h e l e s s e r is t h e c o u p l i n g e f f e c t , t h e s t i f fe r t h e

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

s e n s i t i v e t o t h e t h i c k n e s s r a t i o f o r 0 = + 2 0 ° o r 7 0° in c o m p a r i s o n w i t h t h e

o t h e r o r i e n t a t i o n s . T h i s is a t t r i b u t e d t o t h e fa c t t h a t t h e e x i s te n c e o f th e

c o u p l i n g e f f e c t , i n a c l a m p e d s q u a r e p l a t e , i n t h e r e g i o n o f 0 = -+ 20 ° o r

7 0 ° is m a x i m u m a n d h e n c e t h is c r it ic a l t h i c k n e s s r a t i o o f 0 .4 1 4 2 p l a y s a

m a j o r r o l e i n i n cr e a s in g t h e f r e q u e n c y p a r a m e t e r o r b u c k l i n g v a l u e s 4 f o r

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

l a m i n a t e a s i n F i g. 3 b ) i s m a d e a n d t h e v a r ia t io n o f f r e q u e n c y p a r a m e t e r

k~ w i t h t h e t h i c k n e s s r a t i o

t 2 / t 3 )

i s p l o t t e d i n F ig . 4 f o r a s e t o f g i v e n

t ~ / t ~ .

T h e f r e q u e n c y p a r a m e t e r , o n c e a g a in , i s f o u n d t o po s s es s a m a x i m u m a t a

t h i c k n e s s r a t i o

t 2 / t 3

= 2 -0 f o r a n e x a m p l e c a s e o f l t / t 3 = 1 , i r r e s p e c t i v e o f

t h e f i b r e o r i e n t a t i o n in th e l a m i n a t e . F o r o t h e r r a t io s

o f t l / t 3

s u c h v a l u e s o f

t 2 / t 3 c a n b e o b t a i n e d f r o m F i g. 5 . F u r t h e r , it is s e e n t h a t t h e e f f e c t o f t h is

t h i c k n e s s r a t io is m o r e p r o n o u n c e d f o r t h e la m i n a t e w i t h fi b re o r ie n t -

a t i o n s o f - +2 0° o r -+70 ° .

N u m e r i c a l v a l u e s o f t h e n o n - d i m e n s io n a l f u n d a m e n t a l f r e q u e n c y

p a r a m e t e r k~ o f f o u r - a n d s i x - la y e r e d a n ti - s y m m e t r i c s q u a r e l a m i n a t e s ,

7/26/2019 Kamal 1986


om e stud ies on r ee v ibra t ion o f composi te lamin ates

191

14

1 0

le

~ 6

2

O. 414 2 I I

0 2 . 0 4 . 0 6 . 0

t i l t 3

Fig. 5. Locus of d esign thicknessratios in a six-ply laminate.

4 0

3 5

3 0

k~

2 5

- - G r - F R P

C~ C i C r Cf

. . . . . O F R P . . . .

f

B F R P

2

1 5

0 o

G F R P

I L 1 I

4 0 o 8 0 °

e

F i g . 6 . D e s i g n f r e q u e n c y p a r a m e t e r v a l u e s f o r f o u r a n d s ix la y e r e d l a m i n a t e s .

u s i n g t h e r e s p e c t i v e o p t i m a l t h i c k n e s s ra t io s fo r v a r i o u s c o m p o s i t e

m a t e r ia l s , a r e t h e n p r e s e n t e d i n t h e f o r m o f g r a p h s F i g . 6 ) w h i c h s h o u l d

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

f u n d a m e n t a l f r e q u e n c y i s f o u n d to b e h i g h e s t f o r a c l a m p e d , s q u a r e ,

7/26/2019 Kamal 1986


192

% ,

=

2 2 6 5


/ 7 ] ': : : - '

4 5 5 6 4 5 5 6 7 2

? 9

7 8 5 3 ? 8 5 3 I t ( )

0

I I C I O

4 5 ° / ' - 4 5 ° / 4 5 4 5 ]

F •

i

L ...........

k

:

3 3 5

m ]

L

k n = 2 0 7 4

T l r ~

i

4 8 2 9 4 8 2 9 6 5 3 9 8 9 8 4 8 9 8 4 1 0 1 3 7

[ 0 ° / 9 < 3 ° / 0 ° 1 9 0 ° ]

~ L Z _

3 5 . 9 2 4 8 3 8 5 3 0 5 7 3 2 9 7 4 5 6 8 7 2 4

~ 3 0 ° 1 6 0 ° 6 0 ° 3 0 ° ]

Fig 7 Nodal patterns of BFRP composite laminates

I O I 3 7

1 0 6

3 7

laminate with fibre orientat ions in each layer at either 0 ° or 90 ° i.e. a case

of a unidirect ional laminate) and is least for 0 = -+45 . This is eviden t as

the former laminates fall into the symmetric lay-up schemes with no

bending-twist ing couplings and thus have a higher stiffness, whereas , the

latter laminate cases possess less stiffness. The dip in the value of kl is

more fo r a highly anisotropic laminate such as Gr-FRP. Nodal patte rns of

the represen tative case of an anti-symmetric and symmetric laminates

mad e up of boro n/ep oxy are reported in Fig. 7.

6 CONCLUSION

A free vibration analysis for the calculation of natural frequencies of

rectan gula r laminates employing modified shear deformat ion theory and

using the Rayleigh-Ritz energy method is carried out. The analysis is

applicable to any possible type of combinations of in-plane and out-of-

plane boundary conditions by using the corresponding displacement

func tions, as approp riate. Results thus obtained for an example case of a

clamped laminate provide a source of comparison for assessment of such

results obtained using other solution techniques. Numerical results

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Som e studies on free vibration o f com posite laminates

193

o b t a i n e d u s i n g th i s a n al y si s f o r a n a n t i - s y m m e t r i c l a m i n a t e o f a g i v e n

m a s s s h o w e d t h a t s u c h a l a m i n a t e c a n b e m a d e t o p o s s e s s a h i g h e r

f r e q u e n c y o f v i b r a t io n b y u s in g c e r t a in o p t i m a l t h i c k n e s s r a ti o s i n f o u r -

p l y a s w e l l a s s i x -p l y l a m i n a t e s . T h e e f f e c t o f s u c h t h i c k n e s s r a t i o s o n 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 f o r a n a n t i - s y m m e t r i c s q u a r e l a m i n a t e w i t h + -20

a n d +-70 l a y - u p is f o u n d t o b e m o r e a d v a n t a g e o u s i n d e s ig n s o f s u c h

l a m i n a t e s .

R E F E R E N C E S

I . P i s t e r , K . S . , F lexura l v ib ra tions o f th in l am ina ted p la tes , J. Acoust . Soc.

A m .

31 (1959) 233.

2 . Jone s , R . M . , Buck l ing and v ib ra t ion o f unsym me t r ica l ly am ina ted c ross-p ly

rec t an g u l a r p l a t e s , A I A A J o u r n a l 11(12) (1973) 1626-32.

3 . Re i s s n e r , E . an d S t av sk y , Y . Ben d i n g an d s t re t ch in g o f ce r t a i n t y p es o f

h e t e ro g en e o u s ae l o t ro p i c e l a st ic p l a te s ,

J . A p p l . M ech . 2 8

(1961) 402--8.

4 . S h a r m a , S . , I y e n g a r , N . G . R . a n d M u r t h y , P . N . , S o m e c o m m e n t s o n

coupl ing e f fec t s o f ang le-p ly lam ina tes , J. Struct. Mech. 7(4 ) (1979) 473--82.

5 . K am a l , K . an d D u rv as u l a , S . , Pa ram e t r i c s t u d y o f v i b ra ti o n ch a rac t e r is t ic s

o f s i mp l y s u p p o r t ed l ami n a t ed r ec t an g u l a r p l a t e s , Pap e r p re s en t ed a t t h e

techn ica l sess ion o f 3 4 th A G M o f A er o . S o c i . o f l n d ia 11 Dec . , 1982 , l iT

Mad ras , I n d i a .

6 . Be r t , C . W . , Re cen t r e s ea rch in co mp o s i te an d s an d w i ch p l a te d y n ami cs ,

S h o c k V i b . D i g .

11(10) (Oct. 1979) 13-23.

7 . Le i s sa , A . W . , Ad van ces in v ib ra t ion , buck l ing and pos t buck l ing s tud ies

o n c o m p o s i t e p l a te s , Com pos i te s t ruc tures ed . by I . H . Marsha l l , E l sev ier

A pp l ied S c ience Pub l i shers, Lo ndo n , 1981 , pp . 312-34 .

8 . Be r t , C . W . , Res e a rch o n d y n ami cs o f co m p o s i te an d s an d w i ch p l a te s ,

S h o c k V i b . D i g .

14(10) (1982) 17-34.

9 . G i b s o n , R . F . , R ecen t r e sea rch o n d y n am i c mech an i ca l p ro p e r t i e s o f f ib r e

r e i n fo rced co mp o s i t e ma t e r i a l s an d s t ru c t u re s , S h o ck V ib . D ig . 15(2)

(1983) 3-15.

10. A sh to n , J . E . an d Wa ddou ps , M . E . , Analys i s o f an i so t rop ic p la tes , J .

C o mp o s i t e M a ter ia l s 3 (1969) 148--65.

1 1. B e r t , C . W . an d M ay b e r ry , B . L . , F ree v i b ra ti o n s o f u n s y m me t r i ca l ly

l am i n a t ed an i s o t ro p i c p l a te s w i t h c l amp ed ed g es ,

J . Co mp os i te Mater ia ls 3

(1969) 282-93.

1 2. W h i t n ey , J . M . , T h e e f f ec t o f b o u n d a ry co n d i ti o n s o n t h e r e s p o n s e o f

l a m i n a t e d c o m p o s i t e s,

J . C om pos i te Mater ia l s

4 (1970) 192-203.

1 3. L i n , C . C . an d K i n g , W . W . , F re e t ran s v e r s e v i b ra ti o n s o f rec t an g u l a r

u n s y mmet r i ca l l y l ami n a t ed p l a t e s , J . S o u n d a n d V ib r a t i o n 36(1) (1974)

91-103 .

1 4. Be r t , C . W . , O p t i ma l d e s ig n o f a co mp o s i te ma t e r i a l p l a te t o max i mi s e t h e

f u n d a m e n t a l f r e q u e n c y ,

J . S o u n d a n d V ib r a ti o n

50(2) (1977) 22 37.

7/26/2019 Kamal 1986


194 K. Kamal S. Durvasula

15. B er t , C. W . , Design o f c lamped compos i te mater ia l p la tes to maximise

f u n d a m e n t a l f r eq u e n c y , J. Mechanical Design Trans. A SM E 1 (1978)

274-8.

16. R ao , S . S . and Singh , K. , O pt im um des ign o f lamina tes wi th na tu ra l

f r eq u en cy co n s t r a in t s , J. S ou nd and Vibration 67 09 79 ) 101-12.

17. Sh arm a, S . , Iyengar , N. G. R. a nd Mur thy , P . N. , The buck l ing o f

an t i sym m etr ic lamin a ted ang le-p ly and c ross-p ly p la tes ' , Fibre Science and

Technology 13 (1980) 29-48.

18. Son i , P . J . an d Iyengar , N. G . R . , Op t imal des ign o f c lam ped lam ina ted

co m p o s i t e p l a t e s , Fibre Science a nd Technology 19 (1983) 281-96.

19. Ad al i , S . , M ul t iob jec t ive des ign o f an an t i symm etr ic ang le-p ly lamina te by

n o n l i n e a r p r o g r a m m i n g , J. Mechanism Transmissions and Auto ma tion in

Design Trans . A SM E 105 09 83 ) 214-19.

20 . Kamal , K. , Numerical values of in tegrals of trigono-beam characterist ic

fu n c t io n s Dep ar tm en t o f Ae r o sp ace En g . I n d ian I n s t i t u t e o f Sc i en ce ,

Ban g a lo r e , Rep o r t No . A E .3 7 1 S , 1 9 82 .

2 1. Kam a l , K . an d Du r v asu la , S ., On m ac r o m ech an ica l b eh av io u r o f co m p o s i t e

l am in a te s , Composite Structures in press.

2 2. Ch an g , T i sh - Ch u m an d Cra ig, R . R . , O n n o r m a l m o d e s o f u n if o rm b e am s

En g in ee r in g M ech an ic s Resea r ch L ab o r a to ry , T h e Un iv e r si ty o f Tex as a t

Au s t in , R ep o r t EM R L 1 06 8, 1 9 .

23. D urv asu la, S. , Bh atia , P. and Nair , P. S. ,

Num erical values o f in tegrals o f

beam characteristic functions Dep ar tm en t o f Ae r o sp ace En g in ee r in g ,

Ind ia n Ins t i tu te o f Sc ience , Bang alore , Re po r t No . A E 220S, 1969 .

A P P E N D I X A

T h e s t i ff n e s s m a t r i x [ S ] i n e q n ( 5) is c o m p o s e d o f f if t ee n s u b m a t r i c e s a n d

i s g i v e n a s

[ S ] =

- f S l ] E S 2 ] I -S 3 ] I S 4 ] I S 5 ]

I S 2 ] I S 6 ] I S 7 ] [ S s ] [ S 9 ]

[ S 3 ] I S 7 ] [ S l o ] [ S I l l [ S I 2 ]

[ s . ] [ s d [ s ] [ s . ] [ s ~ ]

[ sd [ sd [ s d [ s ] [ s d

7/26/2019 Kamal 1986


om e studies on free vibration o f com posite laminates 9 5

a n d t h e m a s s m a t r i x [ M ] i s g i v e n as

[ M ] =

- p [ l ] 0 [ m , ] [ m : ] 0

o p [ l ] [m3] o [m4]

[m ] In3] In s ] [m~] [aT]

[m ~ ] o [m ~ ] ~ [ t ] o

I

o [ m 4 ] [m T ] o ~ [ ]

w h e r e

[ m ,] m ..~ - 2 q 2 1 4 ~ . J 4 ~

[ m - ] . . . .

_

2 q l [ I ]

2 q 2 . - o o . . o r

[11114]

ql

- b I l l

I1

2 ( l , . , J ~ + x 2 ~ I ' l ~

I n . , , = p [ t] ~ .. ,, + a - r . . . . . . . ,

3 O l o o

[m 6 ]~ .,~ - ~ 1 4 ~ d 4 ~

r oo ol

[ aT ]m. = - - U t s ., , JS . ~

[ i ] . . = ~ ~

7/26/2019 Kamal 1986


196 K . K a m a l , S . D u r v a s u l a

T h e e l e m e n t s o f t h e s u b m a t r i c e s o f t h e s t i f f n e s s m a t r ix [ S ] a r e g i v e n a s

[ S . ] . . . . A l . a 2 1 1 t ~ , J l ~ ,~ + 2 A ~ 6 a 2 X l l ~ ) ,J l ~ ] 2 2 ~} 11

= A ~6a ) , I1 , .~J1 ,~

[ S 2 ] . . ~ =

[ S ] m.r~ =

[ s ° ~ =

[ S s ] m n r ~ i =

I S ] . . . =

[ s ~ ] ~ o ~ =

[ S d . . . =

[ s 9 ] . o , , =

[ S l O ] m n r ~ :

9 a 2 ~ 1 3 1 0 . 2 1 1 o o 2 2 o o 1 1

2 A L m ,+ 2 A l 6 a

1 3 , . , J 3 , ~ + 2 A 2 6 a h I 3 ,. ~ J3 , ~

~ , * 2 r ~ O l ~ I 0

- 1 - L A . f . 1 6 6 a 1 2 * m r J ~ n a

2 B l l t a l l l J l l ~ ) q _ 4 h a B l t j l m l l m 4_ 2 oo II

. . . . . - . . . . ,~ - 2 B 1 66 ah l l . , , J l , .

2 1 0 0 1 I I 0 0 3 0 0 1 1

2 B 1 1 2 a h 1 3 m , J 3 ,~ + 2 R B 1

16al3 m rJ3 ns +

2 B 1 2 6 a h 1 3 , .r J 3 ,~ +

2 I ) l 10

+ 2 B 1 6 6 a h 1 3 , .r J 3 ,~

1 2 0 0 2 1 0 0 2 1 1 0 1

2 B 2 1 1 a l 4 = , J 4 ~ ,

+ 2 / 3 2 ) 2 a h

1 4 ,, ,r J4 ,~ + 4 h B 2 1 6 a 1 4 , . , J 4 n ~

0 2 I 0 3 O 0 ) 2 2 0 1 I I

+ 2 B 2 1 6 a h 1 4 , . , J 4 , ~ 2B226ah 1 4 , .r J 4 ,~ + 4 B 2 6 6 h 1 4 , , , J 4 . s

2 2 ~ ) I I 2 I I 0 0 2 0 1 1 0

A z 2 a h / 2 , . , J 2 , = + A 6 6 a / 2 , . , J 2 , ~ + 2 A 2 6 a h 1 2 . J 2 =

10 OI 11 00 2 KI 11

2 B 1 12ah13, .~J3,~ + 2 B 1

+ 2 B l z 6 a X

I 3 , . , J 3 , ~

6a13, . ,J3,~

_]_ O l I0

2 B 1 6 6 a h 1 3 ~ , J 3 ,~

2 01 11) 3 IKI 11 t 1 00

2 B 1 2 6 a 2 ~ / 2 , , ,J 2 , ~ +

2 B 1 2 2 a h I 2 m , J 2 , ~ + 2 B 1 6 6 a h 1 2 , .r J 2 , ~

0 2 1 0 1 2 0 0 2 O l 1 1

2 B 2 1 2 a h 1 5 , , , , J 5 , ~ , + 2 B 2 1 6 a 1 5 m ~ J 5 , s + 4 B 2 2 6 a h (15,, , ,J5, ,~

I1) 02

+ 1 5 , . , J 5 . ~ ) + 3 oo 12 n m

B h I 5 , ., J 5 , ~ + 4 B 2 6 6 h I 5 , . ,J 5 , ~

2 I1

A ~ a l , . r J ~ , + 2 10 ) l + A s s a h l~mrJ,~

A 4 5 a

) ~ I , ~ , f ~

2 2 I1

+ D 2 1 1 - ~ h - ~ . . . . . .

( )

D 2 , , a : - 7 3 - V e ~ 2 , , a ~ r 2 , f ( 2

( 2 0 - ) F 1 °

+ D 2 1 6 h - -5 ~ - -~ D 2 1 6 h . . . . .

2 0 - 3 \

+ D 22 6X 3 - ~ O 2 2 6 X )

: ~ r J 2 g

7/26/2019 Kamal 1986


So m e s tud ies on free v ibra t ion o f compos i te l aminates

4

D 22 x 4 - - ~ D 2 2 2 h / ~ o j ~

20 - 2 \

+ D 2 66 X 2 - - ~ D 2 2 6 h ) 112.J12

197

[ S . ] . ~ 2 o ~ o o 2 o o o ~

=

A45a

hlamrJ4 ~

A 4 4 a I4 , . ~ J 4 , ~ +

1 1 2 O 0 1 0 - - ~ 2 O 0

+ ~ D l l l l 4 m r J 4 n s - - ~ D l l l l g m r J 4 n s

~ D 1 2 l o o 2 1 0 - 2 41o 402

12X 1 4 , . , J 4 , ~ - - f ~ D l l 2 X I

mr ns

+ D l 1 6 h

I 4 ~ , J 4 ~ + l 1 4 ~ m ,J 4 ~ ]

2 ]

- - ] 3 0 0 1 2

_ 2 0 D ~ X ¢ i4 H j 4 o ~ + i4 ,,o 2 j 4 ~ ) 2 D 1 2 6 h I4mrJ4ns

3h 2 -15 x . m . , , ,~

+

l 0 - - 3 0 0 1 2

3 - - ~ D 1 2 6h I4,,,~J4,,~

1 2 o l

I I 2 0

- 2

4 o l 4 l l

~ D 1 6 6h 1 4 , , , ~ J 4 ~ - - ~ D 1 6 6 h I ,,,~ J,

Sl2]mnr 2 1 0 0 0 2 2 0 0 0 1

= 2 A a s a

h I 5 , , , r J 5 , ~ + 2 A 5 5 a h l S m r J 5 , ~

2 1 0 - 2 o2 lo

+ D l1 2 2 o2 1o

5 , , , . J 5 , ~ - ~ h - r D l t 2 h I 5 , , , . J 5 ~

h 12 00 1 0 - 12 50O

+ D 1 1 6 T

I S m r J S m - - ~ D l l 6 X I 5 , , , . J ,,~

3 01 11 +

1 5 m r J 5 n s

D 1 2 6 h

I 5 m r J 5 n s

O l 2 0

7/26/2019 Kamal 1986


198 K . K a m a l , S. D u r v a s u l a

2 0 -

- 3 h 2 D 1 ~ ol t l

h ( 1 5 m J 5 , u + l ~ l I q 2 %

+ D 1 , , - - l ~ m l g 1 2 - 1 , ,X 4 1 5 ~ J 5 ~

2

, - - II ll

+ O l ~ X 215~ J 50 ~ - . T ~ X - D I , ~ I 5 . , ~ j 5= ,

- - 2 . . ( ~ . . m - - { ~ I1 11 (~ ) }

8 1 3 ]

. .. .. = . aa n a t L , . . s l . . ~ t D I , I I1 . . . . . .

5 o t

5 ~ l . l l l l l l ~ + D l t 6 h - ~ I I . , r J l , ~ [

3 h 2 - - , . , ~ - ~ -

1 0 5 m *~

- - - 3 h2 D 1 1 6 h l l , . ~ J l , ~ + D 1 6 6 h 2 ~ l l m ~ J l

5 2 O0 11

- 3 h 2 D l ~ h I I ,, ,~ J l r~

q

[ S ,4 ] . .. .. = Z , a a sa A l Z m ¢ J Z , ~ + D l = , k ~ I 2 ~ r J 2 ° 2 ~

-)

- - 5 i i ( ~ J

1 0 ~ 2 t o m z o rn ± D l l 6 h l 2 , .~ j 2 , ~

- 3 h -- -- TD l t2 . . . . . . . . . .

l O - - 4 5 ~) 11

- ~ D 1 ,6 12 'm~.J2'Xd+ D I 2 o A

~ / 2 j 2 s

1 0

2 5

Ol 10

- - ] 2 6~ k. / 2 m r J 2 n s + o o 11

3 h 2 D s (R) I1 / 2 m , J 2 . ~ ) + D l l 6 X ~ / 2 m ~ J 2 . ,

1 0 - - ~ O l 1 0

- 3 h 2 D 1 6 6 X ~ 1 2 , . ~ J 2 , ~

7/26/2019 Kamal 1986


Som e s tud i e s on r ee v i b r a t i on o f com pos i t e l am i na t es 199

[Sls]m~,~ 2 2 ~1 oo 3 ol Io

= A ~ s a h I 2 , .~ J 2 , ~ +

D 1 2 6 ~ / 2 ~ , J 2 ~

4 5 oo II 5 - 4 oo 11

+ D 1 2 2 h

- ~ [ 2 m r J 2 . s- - ~ D 1 2 2 X [ 2m r J2n s

D 1 2 6 X 3 5 / 2 ~ , . j 2 ~ _ 2 1 0 - 3 o l lo

- ~ D 1 2 6 h 12mrJ2~s

2 I 1 0 0 5 - - 2 I 1 0 0

+ 1 66 h I 2 , . , J 2 , ~ - - ~ - ~ D 1 6 6 h 1 2m ,J 2,~

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

d e r i v a t i v e s a r e d e f i n e d a s f o l l o w s :

( a ) I n t e g r a l s o f t r i g o n o - t r i g o n o f u n c t i o n s a n d t h e ir d e r i v a t i v e s

I I , . , , = ~ b L s o ) 4 ) q , s ¢ ) d s

= 4 , x ~ ( ~ ) 4 , x , ( O d ~

P q f P q

n , , , . r = , ¢ ,, , ,. ,, ( ~ : ) ' t ' , , r ( ~ : ) d ~ :

d

1

P q - - t k ~ ( ' O ) ' , b q ( ' O ) d ' o

l , , ,s -

0

1

= f o ~ ( n ) ~ b q s ( n ) d n

P,q - - S 1

J2 , , ,~ - ~bP x, , ( 'O ) q , ( 'O) d-o

f p q

= ~ y m ~ : ) ~ y r ~ ) d s ¢

o

P q f P q

1 3 m . = ~ b y , . s O ) 6 v , s ¢ ) d s

. 1

0

- - ~ , ~ m ( ~ ) 4 , Y , (~ ) d ~

= f ~ P . ( ~ ) ~ q ( ~ ) d ~

P , q - - ~ 1

J3n.s

- ~bP~. ' 0 ) o q ( ' 0 ) d ~

1

= f ~ b~ ( 'O )qJY ~ ( ' o )d 'o

( b ) I n t e g r a l s o f t r i g o n o - b e a r n c h a r a c t e r i s t i c f u n c t i o n s a n d t h e ir

d e r i v a t i v e s

f o

4P, q P q

b ,,m ~: ) 6 . , , (~ : ) d ~: P q -

m,,- = J4~ ,s -

qd u,,( O)~bqs( O) o

P q ~ 0 P q S

J s n , , - t 0 ~ ( , 7 ) , / , ( ~ ) d ' o

5 , . , .

6 v , , , (~) 6 w, (~ : ) d~: P 'q -

7/26/2019 Kamal 1986


2 0 0 K . K a m a l , S . D u r v a s u l a

c) Integrals of beam-beam characteristic function and their

derivatives

f 1

p q P ch q d

I

f I P q

J~P.'q = t~ ~. ,, ( O) $ ~., ('0) d'o

Numerical values of these integrals, obviously, depend upon the

combination of boundary conditions under consideration. These

numerical values of the integrals of the more frequently occurring

products have been calculated for ready use in Refs 20 and 23. The tables

of integral values used in the present analysis for an example problem of a

rectangular laminate with clamped boundary of C1 type, are reproduced

from Ref. 20 for ready reference Tables A1-A6).

T A B L E A

Integra ls I4~ , 15~r, J4~ and J5~m~

r

m 1 2 3 4 5

1 0-986 24 0.000 00 0 147 98 0-000 26 0-062 34

2 0.000 00 0.976 41 0.000 00 0.184 87 -0 .0 01 15

3 -0 .1 61 41 0.000 00 0.958 15 0.000 45 0.196 13

4 0-000 00 -0 .2 07 44 0-000 02 0 947 10 -0 .0 04 82

5 -0 032 92 0.000 00 -0 .232 22 0.000 48 0.936 95

T A B L E A 2

Integrals 14°~, ot

5mr, J4~ and J5'~lr

r

m 1 2 3 4 5

1 0-000 00 -2. 324 25 -0-00 0 01 -1.26 1 43 -0-0 00 75,

2 3.493 05 0-000 00 -3 618 27 -0-001 57 -2 .0 06 83

3 0-000 00 6-323 56 0.000 18 -4-691 21 -0 .1 25 82

4 0.302 46 0.000 00 9.157 36 -0 .0 06 82 -5 -718 88

5 0.000 00 0-754 57 0.000 00 11.991 17 -0 .0 71 55

7/26/2019 Kamal 1986


So m e s tudies on f ree v ibrat ion o f com posi t e laminates

T A B L E A

14m,, 15~r, ~r2 j5o~r

nteg rals ~u J4,,r and

201

r

m 1 2 3 4 5

1 -9 .7 33 80 0.000 03 -1 .45 9 54 0-033 30 -1- 059 24

2 0.000 O0 -38. 547 30 0-001 57 -7 .316 54 0.340 29

3 14.338 33 0.000 O1 -8 5.110 94 0.126 75 -1 7. 55 9 50

4 0.000 O0 32-759 17 0.004 09 -149.464 96 -1 .8 72 45

5 8.124 O1 -0 .0 00 03 57.294 76 0-048 70 -232.649 13

T A B L E A 4

14mr 15 . . . . ,~, and a- m r

In tegrals ~o ~o taro i~ ~o

m 1 2 3 4 5

1 -2. 79 6 56 1.379 83 -0 .565 74 0.854 61 -0 .315 51

2 2.028 77 -5 .6 59 66 1-732 79 - l . 132 56 1.198 71

3 2.838 61 2-170 70 -8 .4 84 93 1.862 78 -1. 74 8 O0

4 1-886 35 3.770 58 2.155 82 -11-301 36 1.859 25

5 2.023 23 2.077 96 4.714 07 2.145 46 -14. 247 36

T A B L E A 5

Integrals 14~, I5~, J4~ and J5~

r

m 1 2 3 4 5

1 -3 .7 29 62 -7.4 11 02 -4.7 62 16 -5.75 6 44 -3. 890 50

2 26.583 59 -5 .8 64 56 -20-731 40 -8- 593 35 - 15.896 64

3 -0 -6 72 42 66.655 41 -8-191 24 -39 .989 89 -12- 138 97

4 10-537 62 -3 -170 74 -124. 399 74 -10-429 56 -66 .009 80

5 1.616 42 22 227 33 -5. 46 6 04 199.797 35 -7. 767 17

7/26/2019 Kamal 1986


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T A B L E A

Integra ls / 4~ , /5~ , J4~ and J5~

m 1 2 3 4 5

1 62.568 24 35-307 25 68.397 14 67-715 85 93-630 87

2 14 35881 349-044 42 85 126 43 226.551 64 109 593 0

3 -63 509 23 37.834 66 1025.886 20 146.919 87 472.979 57

4 -1 .2 07 36 -232. 546 27 78-353 45 2259.082 90 20 2 54 3 4

5 -4 5. 26 6 48 11.497 62 -569-934 30 13 0- 25 7 5 4228-064 30

Kamal 1986

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Transcript of Kamal 1986