nasa - fully-coupled fluid-structure vibration analysis using nastran

7/15/2019 Nasa - Fully-Coupled Fluid-Structure Vibration Analysis Using Nastran

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e v a l u a t e d . N A S T R A N i s u s e d t o p e r f o r m n o r m a l m o d e s ( S O L 1 0 3 ) a n d f o r c e d - r e s p o n s e a n a l y s e s

( S O L 1 0 8 , 1 1 1 ) o n c y l i n d r i c a l a n d c u b i c u i d / s t r u c t u r e m o d e l s . E a c h m o d e l i s d i s c r e t i z e d u s i n g

n i t e e l e m e n t m e t h o d s . B u l k d a t a l e c a r d s u n i q u e t o t h e s p e c i c a t i o n o f a u i d / s t r u c t u r e m o d e l

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

T h e s e s o l u t i o n s a r e u s e d t o e v a l u a t e N A S T R A N ' s s o l u t i o n s f o r a c c u r a c y . A p p e n d i c e s t o t h i s

w o r k i n c l u d e N A S T R A N d a t a p r e s e n t e d i n f r i n g e p l o t f o r m , F O R T R A N s o u r c e c o d e l i s t i n g s

w r i t t e n i n s u p p o r t o f t h i s w o r k , a n d N A S T R A N d a t a l e u s a g e r e q u i r e m e n t s f o r e a c h a n a l y s i s .

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!

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S e c t i o n 1 : I n t r o d u c t i o n

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

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

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

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

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

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

e l e m e n t a n a l y s i s p a c k a g e c a l l e d M S C / N A S T R A N ( v e r s i o n s 6 7 a n d 6 8 ) i n t h e a n a l y s i s o f f u l l y

c o u p l e d u i d / s t r u c t u r e p r o b l e m s . N o r m a l m o d e s a n a l y s i s a n d f o r c e d r e s p o n s e a n a l y s i s i s u s e d

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

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

a c c u r a c y o f a N A S T R A N - g e n e r a t e d s o l u t i o n i s d e t e r m i n e d t h r o u g h c o m p a r i s o n w i t h c l a s s i c a l

a n a l y t i c s o l u t i o n s .

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

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

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

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

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

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

2



p r o b l e m . F o r t h e n o r m a l m o d e s p r o b l e m s , b o t h q u a d r a t i c a n d l i n e a r e l e m e n t m e s h e s a r e u s e d .

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

M S C / P A T R A N w a s u s e d a s a g r a p h i c a l p r e - a n d p o s t - p r o c e s s o r . F E M m o d e l s a n d b u l k

d a t a l e s w e r e c o n s t r u c t e d w i t h i n P A T R A N . H o w e v e r , a t t h e t i m e o f t h i s w o r k , P A T R A N d i d

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

b u l k d a t a l e o u t s i d e o f P A T R A N b e f o r e i t w a s s u b m i t t e d t o N A S T R A N f o r a n a l y s i s . T h e s e

m o d i c a t i o n s a r e d i s c u s s e d i n t h i s p a p e r .

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

t h e s e p r o b l e m s . A p p e n d i x A t a b u l a t e s N A S T R A N ' s n u m e r i c r e s u l t s f o r e a c h p r o b l e m a n d

c o m p a r e s i t t o t h e a p p r o p r i a t e a n a l y t i c r e s u l t . M S C / P A T R A N w a s u s e d t o p r o d u c e t h e f r i n g e

p l o t s s h o w n . I t s h o u l d b e n o t e d t h a t P A T R A N u s e s a l i n e a r i n t e r p o l a t i o n b e t w e e n n o d e s i n

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

e l e m e n t m e s h e s h a v i n g a n e q u a l n u m b e r o f n o d e s w i l l a p p e a r i d e n t i c a l i n a P A T R A N - g e n e r a t e d

f r i n g e p l o t . A p p e n d i x B l i s t s t h e F O R T R A N c o d e s w r i t t e n i n s u p p o r t o f t h i s w o r k . S e v e r a l

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

N u m e r i c a l R e c i p e s : T h e A r t o f S c i e n t i c C o m p u t i n g ( F O R T R A N V e r s i o n )

1

. A p p e n d i x C

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

S e c t i o n 2 : T h e C u b i c P r o b l e m

T h e u i d / s t r u c t u r e i n t e r a c t i o n p r o b l e m w a s r s t c o n s i d e r e d f o r a c u b i c d o m a i n . T h i s

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

i s r e a d i l y s o l v a b l e i n C a r t e s i a n c o o r d i n a t e s . T o f u r t h e r f a c i l i t a t e s o l u t i o n o f t h e e q u a t i o n u s i n g

a n a l y t i c m e t h o d s , b o u n d a r y c o n d i t i o n s o f P = 0 w e r e e n f o r c e d o n a l l s u r f a c e s o f t h e u i d n o t i n

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

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

3



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

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

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

s t r a i g h t f o r w a r d . A s k e m a t i c r e p r e s e n t a t i o n o f t h e f o r c e d r e s p o n s e u i d / s t r u c t u r e m o d e l f o r

t h e c u b i c g e o m e t r y i s s h o w n i n F i g u r e 1 .

X

Y

Z X

Y

Z

Fluid

Thin Plates (2)

Point

Forces (2)

Fluid/Structure Cubic Geometry, Exploded View

F i g u r e 1 : E x p l o d e d v i e w o f t h e u i d / s t r u c t u r e m o d e l f o r t h e c u b i c g e o m e t r y .

T h e F r e e F l u i d P r o b l e m

F o r a n i d e a l , s t a t i o n a r y u i d , t h e a c o u s t i c p r e s s u r e e l d i s d e s c r i b e d b y t h e w a v e e q u a t i o n

r

2

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2

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w h e r e , i n C a r t e s i a n c o o r d i n a t e s ,

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4



U s i n g t h e s e p a r a t i o n o f v a r i a b l e s t e c h n i q u e

2

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

c o n d i t i o n s

P ( 0 ; y ; z ; t ) = P ( A ; y ; z ; t ) = 0

P ( x ; 0 ; z ; t ) = P ( x ; A ; z ; t ) = 0

P ( x ; y ; 0 ; t ) = P ( x ; y ; A ; t ) = 0

( 3 )

a n d a s s u m i n g a h a r m o n i c t i m e d e p e n d a n c e , t h e s o l u t i o n t o E q u a t i o n 1 i s

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m y

A

s i n

k z

A

( 4 )

w h e r e D

n m k

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

!

n m k

a r e g i v e n b y

!

n m k

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2

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2

( 5 )

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

M S C / N A S T R A N . O n e m o d e l w a s c o n s t r u c t e d u s i n g 1 0 0 0 l i n e a r H E X 8 e l e m e n t s a n d 1 3 3 1

n o d e s . T h e s e c o n d m o d e l u s e d 2 1 6 q u a d r a t i c e l e m e n t s a n d 1 2 2 5 n o d e s . T h e t h i r d m o d e l

w a s d i s c r e t i z e d w i t h 1 0 0 0 q u a d r a t i c H E X 2 0 e l e m e n t s a n d 4 9 6 1 n o d e s . T h e u i d i n e a c h o f

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

1 . B o u n d a r y c o n d i t i o n s i n a g r e e m e n t w i t h E q u a t i o n 3 w e r e a l s o a p p l i e d t o t h e e a c h o f t h e

m o d e l s .

S y m b o l P r o p e r t y N a m e M a t e r i a l P r o p e r t y V a l u e


1 0 : 3 2 1 0

6

p s i

h T h i c k n e s s o f s t r u c t u r e

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5 i n

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1 3 : 6 2 0 2 1 0

3

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0 . 3 3 4

s


2 : 5 3 8 3 2 1 0

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3

T a b l e 1 P r o p e r t i e s f o r t h e c u b i c m o d e l .

5



T o s p e c i f y a u i d e l e m e n t w i t h i n N A S T R A N , s e v e r a l b u l k d a t a c a r d s m u s t b e s p e c i e d o r

c h a n g e d . T h e b r i e f o u t l i n e o f t h e s e c a r d s w h i c h f o l l o w s i s a s s h o w n i n t h e M S C / N A S T R A N

Q u i c k R e f e r e n c e G u i d e , V e r s i o n 6 8 . T h e r e a d e r i s r e f e r r e d t o t h i s s o u r c e f o r m o r e d e t a i l

r e g a r d i n g t h e s e c a r d s . T h e r s t c a r d t o b e m o d i e d i s t h e G R I D c a r d . T h e g e n e r a l f o r m a t

o f t h i s c a r d i s a s s h o w n i n F i g u r e 2

3

. A \ - 1 " i n t h e C D e l d o f t h e G R I D c a r d i s u s e d t o

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

b e d e n e d f o r v o l u m e e l e m e n t s .

G R I D I D C P X 1 X 2 X 3 C D P S S E I D

I D

G r i d p o i n t i d e n t i c a t i o n n u m b e r .

C P I d e n t i c a t i o n n u m b e r o f c o o r d i n a t e s y s t e m i n w h i c h t h e l o c a t i o n o f t h e g r i d p o i n t i s

d e n e d .

X 1 , X 2 , X 3 L o c a t i o n o f t h e g r i d p o i n t i n c o o r d i n a t e s y s t e m C P .

C D

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

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

P S

P e r m a n e n t s i n g l e - p o i n t c o n s t r a i n t s a s s o c i a t e d w i t h t h e g r i d p o i n t .

S E I D S u p e r e l e m e n t i d e n t i c a t i o n n u m b e r .

F i g u r e 2 : N A S T R A N G R I D b u l k d a t a c a r d f o r m a t .

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

m a t e r i a l p r o p e r t i e s c a r d i s s h o w n i n F i g u r e 3

3

. N o t e t h a t t h e M A T 1 0 c a r d s p e c i e s m a t e r i a l

p r o p e r t i e s f o r u i d e l e m e n t s o n l y .

M A T 1 0 M I D B U L K R H O C

M I D

M a t e r i a l i d e n t i c a t i o n n u m b e r .

B U L K

B u l k m o d u l u s .

R H O M a s s d e n s i t y .

C

S p e e d o f s o u n d

F i g u r e 3 : N A S T R A N M A T 1 0 b u l k d a t a c a r d f o r m a t .

6



T h e t h i r d b u l k d a t a c a r d t h a t m u s t b e s p e c i e d i s t h e P S O L I D c a r d , s h o w n i n F i g u r e

4

3

. F o r u i d e l e m e n t s i n t h e m o d e l , t h e F C T N e l d o f t h e P S O L I D c a r d m u s t b e s p e c i e d

a s \ P F L U I D " .

P S O L I D P I D M I D C O R D M I N S T R E S S I S O P F C T N

P I D P r o p e r t y i d e n t i c a t i o n n u m b e r .

M I D I d e n t i c a t i o n n u m b e r o f a M A T 1 , M A T 4 , M A T 5 , M A T 9 , o r M A T 1 0 e n t r y .

C O R D M

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

I N I n t e g r a t i o n n e t w o r k .

S T R E S S

L o c a t i o n s e l e c t i o n f o r s t r e s s o u t p u t .

I S O P I n t e g r a t i o n s c h e m e .

F C T N

F l u i d e l e m e n t a g . ( C h a r a c t e r : } P F L U I D } i n d i c a t e s a u i d e l e m e n t , } S M E C H }

i n d i c a t e s a s t r u c t u r a l e l e m e n t ; d e f a l u t = } S M E C H }

F i g u r e 4 : N A S T R A N P S O L I D b u l k d a t a c a r d f o r m a t .

M S C / N A S T R A N p e r f o r m e d a n o r m a l m o d e s a n a l y s i s ( S O L 1 0 3 ) o n t h e l i n e a r a n d

q u a d r a t i c m o d e l s . T h e N A S T R A N - c a l c u l a t e d f r e q u e n c y f o r t h e l i n e a r H E X 8 m o d e l m o d e

1 , 1 , 1 w a s 2 3 6 8 . 7 7 H z . F o r t h e 1 2 2 5 { n o d e q u a d r a t i c H E X 2 0 m o d e l , t h e e i g e n f r e q u e n c y f o r t h i s

m o d e w a s 2 3 5 9 . 1 8 H z . T h e 4 9 6 1 { n o d e q u a d r a t i c H E X 2 0 m o d e l y i e l d e d a f r e q u e n c y o f 2 3 5 9 . 3 2

H z . F o r c o m p a r i s o n w i t h N A S T R A N ' s r e s u l t s , t h e n a t u r a l f r e q u e n c i e s f o r t h e s e m o d e l s c a n

b e d e t e r m i n e d a n a l y t i c a l l y u s i n g E q u a t i o n 5 . F r o m t h i s r e l a t i o n , t h e r s t n a t u r a l f r e q u e n c y

f o r e a c h m o d e l i s 2 3 5 9 . 0 5 H z , m o d e 1 , 1 , 1 . A f r i n g e p l o t o f t h i s m o d e s h a p e f o r e a c h o f t h e

m o d e l s i s s h o w n i n F i g u r e s A 1 , A 2 , a n d A 3 . I t s h o u l d b e n o t e d t h a t i n t h e s e ( a n d s u b s e q u e n t )

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

T h e N A S T R A N s o l u t i o n w a s c o m p a r e d f o r a c c u r a c y a g a i n s t t h e a n a l y t i c s o l u t i o n , E q u a -

t i o n 4 . A f r i n g e p l o t d i s p l a y i n g t h e d i e r e n c e b e t w e e n a n a l y t i c a n d n u m e r i c s o l u t i o n s a t e a c h

n o d e i n t h e m o d e l w a s c r e a t e d f o r e a c h c a s e . T h e e r r o r p l o t f o r t h e l i n e a r H E X 8 c u b e , m o d e

1 , 1 , 1 i s s h o w n i n F i g u r e A 4 . T h e e r r o r p l o t f o r t h e 1 2 2 5 { n o d e q u a d r a t i c H E X 2 0 c u b e , m o d e

7



1 , 1 , 1 i s s h o w n i n F i g u r e A 5 . F i g u r e A 6 s h o w s t h e e r r o r p l o t f o r t h e 4 9 6 1 { n o d e q u a d r a t i c m o d e l .

T h i s a n a l y s i s w a s r e p e a t e d f o r m o d e 1 , 3 , 1 o f t h e l i n e a r m o d e l a n d m o d e 1 , 1 , 3 o f t h e

q u a d r a t i c m o d e l s . T h e s e m o d e s h a p e s a r e s h o w n i n F i g u r e s A 7 , A 8 , a n d A 9 . T h e a s s o c i a t e d

e r r o r p l o t s f o r e a c h o f t h e s e m o d e s h a p e s a r e s h o w n i n F i g u r e s A 1 0 , A 1 1 , a n d A 1 2 . A s u m m a r y

o f t h e n o r m a l m o d e s a n a l y s i s f o r t h e u i d c u b e i s s h o w n i n T a b l e 2 . H e r e , t h e m a x i m u m

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

a n d n u m e r i c s o l u t i o n s .

M o d e l

T y p e

E l e m e n t s N o d e s

M o d e

S h a p e

E i g e n v a l u e

E r r o r

M a x i m u m

E r r o r i n

M o d e l

1 0 0 0 L i n e a r H E X 8

1 3 3 1

0 . 4 1 1 7 %

6 : 8 3 2 1 0

0 6

2 1 5 Q u a d r a t i c H E X 2 0

1 2 2 5

0 . 0 0 5 5 %

0 . 0 0 1 5

1 0 0 0 Q u a d r a t i c H E X 2 0

4 9 6 1

1 , 1 , 1

0 . 0 0 0 7 %

0 . 0 0 0 2

1 0 0 0 L i n e a r H E X 8

1 3 3 1

3 . 1 3 4 %

0 . 5 1 0

2 1 5 Q u a d r a t i c H E X 2 0

1 2 2 5

0 . 3 1 3 7 %

0 . 4 5 9 3

F l u i d o n l y

1 0 0 0 Q u a d r a t i c H E X 2 0

4 9 6 1

1 , 3 , 1

0 . 0 4 3 2 %

0 . 3 8 6

T a b l e 2 R e s u l t s f o r c u b i c g e o m e t r y , n o r m a l m o d e s a n a l y s i s .

T h e F r e e P l a t e P r o b l e m f o r t h e F l u i d / S t r u c t u r e M o d e l

N e x t , t h e m o d e l w a s m o d i e d b y b o u n d i n g t w o o p p o s i n g f a c e s o f t h e u i d c u b e w i t h t h i n ,

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

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

o f m o t i o n

4

:

D r

4

w +

s

h

@

2

w

@ t

2

= 0 ( 6 )

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

e x u r a l r i g i d i t y , i s g i v e n b y

D =

E h

3

1 2 ( 1 0

2

)

( 7 )

8



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

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

w ( x ; y ; t ) = e

i ! t

1

X

n = 1

1

X

m = 1

C

m n

s i n

m x

A

s i n

n y

A

( 8 )

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

!

m n

=

m

A

2

+

n

A

2

s

D

s

h

( 9 )

T h e F r e e F l u i d P r o b l e m f o r t h e F l u i d / S t r u c t u r e M o d e l

F o r t h e i d e a l u i d b e t w e e n t h e p l a t e s , t h e w a v e e q u a t i o n ( E q u a t i o n 1 ) s t i l l a p p l i e s , b u t

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

P ( 0 ; y ; z ; t ) = P ( A ; y ; z ; t ) = 0

P ( x ; 0 ; z ; t ) = P ( x ; A ; z ; t ) = 0

@ P

0

x ; y ;

0 A

2

; t

1

@ z

=

f

@

2

w ( x ; y ; t )

@ t

2

@ P

0

x ; y ;

A

2

; t

1

@ z

= 0

f

@

2

w ( x ; y ; t )

@ t

2

( 1 0 )

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

a r e i n c o n t a c t . A p p l y i n g t h e s e b o u n d a r y c o n d i t i o n s y i e l d s t h e f o l l o w i n g e x p r e s s i o n f o r P :

P ( x ; y ; z ; t ) = 0 e

i ! t

1

X

n = 1

1

X

m = 1

D

m n

s i n

m x

A

s i n

n y

A

c o s (

m n

z ) ( 1 1 )

w h e r e

2

m n

=

2

A

2

0

m

2

+ n

2

1

0

!

2

c

2

o

( 1 2 )

a n d

D

m n

=

f

!

2

C

m n

m n

s i n

0

A

2

m n

1

( !

2

m n

0 !

2

)

( 1 3 )

T h r e e c u b i c u i d / s t r u c t u r e m o d e l s w e r e c o n s t r u c t e d f o r a N A S T R A N n o r m a l m o d e s

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

p r o p e r t i e s s h o w n i n T a b l e 1 . T h e n u m b e r a n d t y p e o f e l e m e n t s a n d n o d e s w e r e a s s h o w n i n

9



T a b l e 3 . T h e b o u n d a r y c o n d i t i o n s o f E q u a t i o n 1 0 w e r e a p p l i e d t o t h e u i d s u r f a c e s i n t h e

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

M o d e l M a t e r i a l E l e m e n t T y p e N u m b e r o f N o d e s

T o t a l N o d e s i n

M o d e l

S t r u c t u r e 2 0 0 L i n e a r Q U A D 4

2 4 2

1

F l u i d 1 0 0 0 L i n e a r H E X 8

1 3 3 1

1 5 7 3

S t r u c t u r e 7 2 Q u a d r a t i c Q U A D 8

2 6 6

2

F l u i d 2 1 6 Q u a d r a t i c H E X 2 0

1 2 2 5

1 4 9 1

S t r u c t u r e 2 0 0 Q u a d r a t i c Q U A D 8

6 8 2

3

F l u i d 1 0 0 0 Q u a d r a t i c H E X 2 0

4 9 6 1

5 6 4 3

T a b l e 3 N A S T R A N m o d e l s f o r t h e c u b i c g e o m e t r y u i d / s t r u c t u r e p r o b l e m

F o r m o d e l s c o n t a i n i n g b o t h u i d a n d s t r u c t u r e e l e m e n t s , i t i s n e c e s s a r y i n t h e N A S T R A N

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

s t r u c t u r e p o r t i o n s o f t h e m o d e l . T h i s i s a c c o m p l i s h e d b y u s i n g t h e N A S T R A N A C M O D L

c a r d . T h e g e n e r a l f o r m a t o f A C M O D L c a r d i s s h o w n i n F i g u r e 5

3

. F o r t h i s w o r k , e a c h

s t r u c t u r e g r i d i n t h e S S E T c a r d h a d a c o r r e s p o n d i n g u i d g r i d i n t h e F S E T c a r d .

A C M O D L

I N T E R I N F O R F S E T S S E T F S T O L

I N T E R

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

I N F O R I n d i c a t e s w e t h e r F S E T a n d S S E T a r e t o b e u s e d t o d e n e t h e u i d - s t r u c t u r e i n t e r f a c e

F S E T

I d e n t i c a t i o n n u m b e r o f a S E T 1 e n t r y t h a t c o n t a i n s a l i s t o f u i d g r i d p o i n t s o n t h e

i n t e r f a c e .

S S E T

I d e n t i c a t i o n n u m b e r o f a S E T 1 e n t r y t h a t c o n t a i n s a l i s t o f s t r u c t u r a l g r i d p o i n t s o n t h e

i n t e r f a c e .

F S T O L

T o l e r a n c e , i n u n i t s o f l e n g t h , u s e d i n d e t e r m i n i n g t h e u i d - s t r u c t u r e i n t e r f a c e .

F i g u r e 5 : M S C / N A S T R A N A C M O D L b u l k d a t a c a r d f o r m a t .

E n t r y \ b y h a n d " o f t h e n o d e s r e q u i r e d f o r t h e A C M O D L c a r d w o u l d b e a n e x t r a o r d i n a r i l y

t e d i o u s t a s k , p a r t i c u l a r l y f o r m o r e c o m p l i c a t e d o r l a r g e r m o d e l s . S e v e r a l F O R T R A N c o d e s

10



w e r e c r e a t e d t o w r i t e t h i s d a t a f o r t h e A C M O D L c a r d f r o m d a t a a l r e a d y i n t h e b u l k d a t a l e .

T h e s e c o d e s a r e l i s t e d i n A p p e n d i x B o f t h i s w o r k .

F r o m E q u a t i o n 9 , t h e r s t n a t u r a l f r e q u e n c y o f t h e p l a t e i s 4 8 4 . 5 4 H z , m o d e 1 , 1 . F o r

t h e l i n e a r Q U A D 4 m o d e l , t h e N A S T R A N - c a l c u l a t e d e i g e n f r e q u e n c y f o r m o d e 1 , 1 o f t h e p l a t e

w a s 4 8 6 . 8 6 H z . F o r t h e 2 6 6 { n o d e q u a d r a t i c Q U A D 8 m o d e l , i t w a s 4 8 1 . 0 2 H z . T h e 6 8 2 { n o d e

Q U A D 8 m o d e l y i e l d e d a r e s u l t o f 4 8 2 . 3 7 H z . T h i s m o d e s h a p e i s s h o w n f o r e a c h o f t h e Q U A D 4

a n d Q U A D 8 m o d e l s i n F i g u r e s A 1 3 , A 1 4 , a n d A 1 5 . F r o m t h e s e g u r e s , i t i s a p p a r e n t t h a t a

s t r o n g c r o s s - c o u p l i n g b e t w e e n t h e f r o n t ( z =

A

2

) a n d r e a r ( z = 0

A

2

) p l a t e s i s o c c u r r i n g . U n l i k e

t h e p a r t i a l l y - c o u p l e d a n a l y t i c s o l u t i o n d e r i v e d a b o v e , N A S T R A N a s s u m e s a f u l l y c o u p l e d

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

r e a r p l a t e i n t h e N A S T R A N s o l u t i o n . I n t h e u n c o u p l e d a n a l y t i c s o l u t i o n , t h e r s t m o d e s h a p e

o f t h e s y s t e m i s a 1 , 1 m o d e s h a p e f o r o n e p l a t e a n d a s t a t i o n a r y ( 0 , 0 ) s h a p e f o r t h e o t h e r .

T h e u n c o u p l e d a n a l y t i c s o l u t i o n w a s u s e d t o a n a l y z e t h e m o d e s h a p e s p r e s e n t o n t h e f r o n t

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

f o r a l l t h r e e m o d e l s i s s h o w n i n F i g u r e s A 1 6 , A 1 7 , a n d A 1 8 .

T h i s a n a l y s i s w a s r e p e a t e d f o r p l a t e m o d e 2 , 2 . T h i s m o d e s h a p e i s s h o w n i n F i g u r e s

A 1 9 , A 2 0 , a n d A 2 1 . T h e a s s o c i a t e d e r r o r p l o t s a r e s h o w n i n F i g u r e s A 2 2 , A 2 3 , a n d A 2 4 . A

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

l i s t e d i n T a b l e 4 .

M o d e l T y p e E l e m e n t s N o d e s

M o d e

S h a p e

E i g e n v a l u e

E r r o r

M a x i m u m

E r r o r i n

M o d e l

T a b l e 4 N o r m a l m o d e s a n a l y s i s f o r s t r u c t u r e p o r t i o n o n l y o f t h e

u i d / s t r u c t u r e m o d e l , c u b i c g e o m e t r y . ( C o n t i n u e d ) . . .

11



2 0 0 L i n e a r Q U A D 4

2 4 3

0 . 4 7 7 8 %

0 1 : 2 9 2 1 0

0 3

7 2 Q u a d r a t i c Q U A D 8

2 6 6

0 . 7 2 6 5 %

0 1 : 8 2 2 1 0

0 3

2 0 0 Q u a d r a t i c Q U A D 8

6 8 2

1 , 1

0 . 4 4 8 1 %

0 1 : 1 7 2 1 0

0 3

2 0 0 L i n e a r Q U A D 4 2 4 3 2 . 8 9 8 %

9 5 : 5 2 1 0

0 3

7 2 Q u a d r a t i c Q U A D 8

2 6 6

0 . 7 2 7 5 %

1 3 3 : 4 2 1 0

0 3

F l u i d / s t r u c t

2 0 0 Q u a d r a t i c Q U A D 8

6 8 2

2 , 2

0 . 5 6 2 8 %

4 8 : 9 2 1 0

0 3

T a b l e 4 N o r m a l m o d e s a n a l y s i s f o r s t r u c t u r e p o r t i o n

o n l y o f t h e u i d / s t r u c t u r e m o d e l , c u b i c g e o m e t r y .

F o r t h e u i d p o r t i o n s o f t h e s y s t e m , f r i n g e p l o t s o f N A S T R A N c a l c u l a t e d m o d e 1 , 1 , 0 a r e

s h o w n i n F i g u r e s A 2 5 ( l i n e a r H E X 8 e l e m e n t s ) , A 2 6 ( 1 2 2 5 { n o d e q u a d r a t i c H E X 2 0 m o d e l ) ,

a n d A 2 7 ( 4 9 6 1 { n o d e q u a d r a t i c H E X 2 0 m o d e l ) . F o r t h e l i n e a r m o d e l , N A S T R A N c a l c u l a t e d

a f r e q u e n c y f o r t h i s m o d e o f 1 9 3 4 . 0 9 H z . F o r t h e 1 2 2 5 { n o d e q u a d r a t i c m o d e l , i t w a s 1 9 2 6 . 2 6

H z . T h e 4 9 6 1 { n o d e H E X 2 0 m o d e l y i e l d e d 1 9 2 6 . 1 7 H z . A f r i n g e p l o t s h o w i n g t h e d i e r e n c e

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

A 2 8 , A 2 9 , a n d A 3 0

T h i s a n a l y s i s w a s r e p e a t e d f o r m o d e 1 , 1 , 1 o f t h e u i d f o r b o t h m o d e l s . T h i s m o d e s h a p e

i s s h o w n i n F i g u r e s A 3 1 , A 3 2 , a n d A 3 3 . T h e a s s o c i a t e d e r r o r p l o t s a r e s h o w n i n F i g u r e s A 3 4 ,

A 3 5 , a n d A 3 6 . A s u m m a r y o f t h e n o r m a l m o d e s o f v i b r a t i o n f o r t h e u i d i s l i s t e d i n T a b l e 5 .


M o d e

S h a p e

E i g e n v a l u e

E r r o r

M a x i m u m

E r r o r i n

M o d e l

1 0 0 0 L i n e a r H E X 8

1 3 3 1

0 . 4 6 3 7 %

5 : 6 2 1 0

0 6

2 1 6 Q u a d r a t i c H E X 2 0

1 2 2 5

0 . 0 0 5 2 %

7 2 8

: 0

2 1 0

0 6

1 0 0 0 Q u a d r a t i c H E X 2 0

4 9 6 1

1 , 1 , 0

0 . 0 0 0 7 %

9 6 : 1 2 1 0

0 6

1 0 0 0 L i n e a r H E X 8

1 3 3 1

0 . 4 1 1 7 %

6 : 6 2 1 0

0 6

2 1 6 Q u a d r a t i c H E X 2 0

1 2 2 5

0 . 0 0 5 4 %

1 : 5 3 2 2 1 0

0 3

F l u i d / s t r u c t

1 0 0 0 Q u a d r a t i c H E X 2 0

4 9 6 1

1 , 1 , 1

0 . 0 0 0 7 %

1 9 8 : 6 2 1 0

0 6

T a b l e 5 N o r m a l m o d e s a n a l y s i s f o r u i d p o r t i o n

o n l y o f u i d / s t r u c t u r e m o d e l , c u b i c g e o m e t r y .

12



T h e F o r c e d F l u i d / S t r u c t u r e P r o b l e m

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

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

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

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

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

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

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

D r

4

w +

@ w

@ t

+

s

h

@

2

w

@ t

2

= F

(

x ; y ; t ) ( 1 4 )

w h e r e i s t h e v i s c o u s d a m p i n g c o e c i e n t a n d F ( x , y , t ) i s a h a r m o n i c p o i n t f o r c e a p p l i e d t o t h e

p l a t e . F o r t h e c a s e w h e r e t h i s f o r c e i s a p p l i e d t o t h e c e n t e r o f t h e p l a t e , i t c a n b e w r i t t e n a s :

F ( x ; y ; t ) = F

o

e

i ! t

x 0

A

2

y 0

A

2

( 1 5 )

A p p l y i n g s i m p l y - s u p p o r t e d b o u n d a r y c o n d i t i o n s t o t h e p l a t e , t h e s o l u t i o n t o E q u a t i o n 1 4 i s

g i v e n b y

w ( x ; y ; t ) = e

i ! t

1

X

m = 1

1

X

n = 1

C

m n

s i n

m x

A

s i n

n y

A

( 1 6 )

E x p a n d i n g F ( x , y , t ) i n t e r m s o f t h e e i g e n f u n c t i o n s o f t h e p l a t e , w e h a v e

F ( x ; y ; t ) = e

i ! t

1

X

m = 0

1

X

n = 0

F

m n

s i n

m

2

s i n

n

2

( 1 7 )

w h e r e

F

m n

=

4 F

o

A

2

( 1 8 )

a n d t h u s

C

m n

=

4 F

o

A

2

s

h

s i n

0

m

2

1

s i n

0

n

2

1

( !

2

m n

0 !

2

+ 2 i ! !

m n

)

( 1 9 )

13



N o t e t h a t i n E q u a t i o n 1 6 , t h e v i s c o u s d a m p i n g c o e c i e n t h a s b e e n r e p l a c e d b y a f r e q u e n c y -

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

s

h

= 2 !

m n

( 2 0 )

F o r t h e u i d i n t h e r e g i o n b e t w e e n t h e p l a t e s , E q u a t i o n 1 a n d b o u n d a r y c o n d i t i o n s 1 0 s t i l l

a p p l y . U s i n g t h e a b o v e r e s u l t s f o r t h e p l a t e s , t h e e q u a t i o n f o r a c o u s t i c p r e s s u r e a t a p o i n t

i n t h e u i d i s n o w w r i t t e n

P ( x ; y ; z ; t ) = 0 e

i ! t

1

X

n = 1

1

X

m = 1

D

m n

s i n

m x

A

s i n

n y

A

c o s (

m n

z ) ( 2 1 )

w h e r e a g a i n

2

m n

=

2

A

2

0

m

2

+ n

2

1

0

!

2

c

2

o

( 2 2 )

b u t

D

m n

=

4 F

o

A

2

s

h

f

!

2

s i n

0

n

2

1

s i n

0

m

2

1

m n

s i n

0

A

2

1

( !

2

m n

0 !

2

+ 2 i ! !

m n

)

( 2 3 )

F o r t h e n u m e r i c a l a n a l y s i s b y N A S T R A N o f t h i s f o r c e d - r e s p o n s e p r o b l e m p o i n t f o r c e s

o f { 5 s i n ( ! t ) ^n l b s . w e r e a p p l i e d t o t h e c e n t e r o f e a c h p l a t e i n t h e l i n e a r m o d e l . B o u n d a r y

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

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

A d i r e c t f r e q u e n c y r e s p o n s e a n a l y s i s ( S O L 1 0 8 ) w a s p e r f o r m e d b y N A S T R A N o v e r a

f r e q u e n c y r a n g e f r o m z e r o t o 5 0 0 0 H z . S t r u c t u r a l d a m p i n g i n t h e m o d e l w a s s p e c i e d b y u s i n g

t h e P A R A M G c a r d i n t h e N A S T R A N b u l k d a t a l e . F i g u r e A 3 7 s h o w s t h e d i s p l a c e m e n t o f a

n o d e l o c a t e d a t t h e c e n t e r o f t h e f r o n t p l a t e f o r t h e l i n e a r Q U A D 4 m o d e l . F i g u r e A 3 8 s h o w s

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

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

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

14



A m o d a l f r e q u e n c y r e s p o n s e a n a l y s i s ( S O L 1 1 1 ) w a s n e x t p e r f o r m e d b y N A S T R A N o n

t h e s a m e c u b i c - g e o m e t r y m o d e l . T h e r s t t w e n t y N A S T R A N - c a l c u l a t e d u i d a n d s t r u c t u r a l

m o d e s w e r e u s e d b y N A S T R A N t o c a l c u l a t e t h e o v e r a l l r e s p o n s e o f t h e m o d e l . I t s h o u l d

b e n o t e d t h a t t h e t w e n t i e t h n a t u r a l f r e q u e n c y o f t h e s t r u c t u r e , a s c a l c u l a t e d b y N A S T R A N ,

o c c u r s a t a p p r o x i m a t e l y 3 0 0 0 H Z , w h i l e t h e t w e n t i e t h n a t u r a l f r e q u e n c y o f t h e u i d o c c u r s

a b o v e 5 0 0 0 H z . T h e r e s u l t s o f t h i s a n a l y s i s f o r a f r e q u e n c y r a n g e o f z e r o t o 5 0 0 0 H z a r e s h o w n

i n F i g u r e s A 3 9 f o r a p o i n t a t t h e c e n t e r o f t h e f r o n t p l a t e , a n d A 4 0 f o r a p o i n t m i d w a y b e t w e e n

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

S e c t i o n 3 : T h e C y l i n d r i c a l P r o b l e m

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

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

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

o f t h e w a v e e q u a t i o n u s i n g t h e s e p a r a t i o n o f v a r i a b l e s t e c h n i q u e , b o u n d a r y c o n d i t i o n s o f P = 0

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

F o r m o d e l s i n w h i c h a s t r u c t u r e w a s p r e s e n t , a t h i n c y l i n d r i c a l s h e l l w a s u s e d . B o u n d a r y

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

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

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

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

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

15



X

Y

ZX

Y

Z

Fluid/Structure Cylindrical Geometry, Exploded View

Point Forces (2)

Thin, Cylindrical Shell

Fluid Interior to Shell

F i g u r e 6 : E x p l o d e d v i e w o f u i d / s t r u c t u r e m o d e l f o r t h e c y l i n d r i c a l g e o m e t r y .


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

r

2

P 0

1

c

2

o

@

2

P

@ t

2

= 0 ( 2 4 )

w h e r e r

2

P i s g i v e n b y

r

2

P =

@

2

P

@ r

2

+

1

r

@ P

@ r

+

1

r

2

@

2

P

@

2

+

@

2

P

@ z

2

( 2 5 )

A s s u m i n g a h a r m o n i c t i m e d e p e n d a n c e a n d h o m o g e n o u s b o u n d a r y c o n d i t i o n s :

P ( r ; ; 0 ) = P ( r ; ; l ) = 0

P ( a ; ; z ) = 0

j P ( 0 ; ; z ) j < 1 ( b o u n d e d n e s s )

P ( r ; ; z ) = P ( r ; + 2 n ; z ) ( p e r i o d i c i t y )

( 2 6 )

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

P ( r ; ; z ; t ) = e

i ! t

1

X

j = 1

1

X

m = 1

1

X

n = 0

J

n

r

j

r

a

s i n

m z

l

[ B

m n j

c o s ( n ) + C

m n j

s i n ( n ) ] ( 2 7 )

16



w h e r e J

n

i s t h e n t h o r d e r B e s s e l f u n c t i o n , r

j

i s i t s j t h r o o t , a n d a a n d l a r e t h e r a d i u s a n d

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

j m n

a r e g i v e n b y

!

j n m

= c

o

r

r

j

a

2

+

m

l

2

( 2 8 )

T h r e e c y l i n d r i c a l u i d v o l u m e s w e r e d i s c r e t i z e d f o r a n a l y s i s b y M S C / N A S T R A N . O n e

m o d e l w a s c o n s t r u c t e d u s i n g 2 2 4 0 l i n e a r W E D G E 6 e l e m e n t s a n d 1 4 4 9 n o d e s , o n e u s e d 3 4 8

q u a d r a t i c W E D G E 1 5 e l e m e n t s a n d 1 2 0 0 n o d e s , a n d t h e t h i r d u s e d 2 2 4 0 q u a d r a t i c W E D G E 1 5

e l e m e n t s a n d 6 6 0 9 n o d e s . T h e u i d i n e a c h o f t h e m o d e l s w a s g i v e n t h e m a t e r i a l p r o p e r t i e s

f o r d e n s i t y a n d s p e e d o f s o u n d s h o w n i n T a b l e 6 . G e o m e t r i c d i m e n s i o n s a r e a l s o s h o w n i n t h i s

t a b l e . B o u n d a r y c o n d i t i o n s i n a g r e e m e n t w i t h E q u a t i o n 2 6 w e r e a p p l i e d t o e a c h m o d e l .

S y m b o l P r o p e r t y N a m e M a t e r i a l P r o p e r t y V a l u e


1 0 : 3 2 1 0

6

p s i

a

R a d i u s o f c y l i n d e r

1 i n

c

o

A c o u s t i c s p e e d o f s o u n d

1 3 : 6 2 0 2 1 0

3

i n / s e c

h T h i c k n e s s o f s t r u c t u r e

0 . 0 6 2 5 i n

l L e n g t h o f t h e c y l i n d e r

5 i n

P o i s s i o n ' s R a t i o

0 . 3 3 4

s


2 : 5 3 8 3 2 1 0

0 4

s l u g s = i n

3

f


1 : 1 7 0 2 1 0

0 7

s l u g s = i n

3

T a b l e 6 P r o p e r t i e s f o r c y l i n d r i c a l m o d e l .

F o r e a c h m o d e l , N A S T R A N p e r f o r m e d a n o r m a l m o d e s a n a l y s i s ( S O L 1 0 3 ) . T h e n a t u r a l

f r e q u e n c i e s f o r t h i s g e o m e t r y c a n b e d e t e r m i n e d a n a l y t i c a l l y f r o m E q u a t i o n 2 8 f o r c o m p a r i s o n

t o N A S T R A N ' s r e s u l t s . A n a l y t i c a l l y , t h e r s t n a t u r a l f r e q u e n c y f o r t h i s m o d e l i s 5 3 8 7 . 9 1

H z , m o d e 1 , 0 , 1 . T h e N A S T R A N l i n e a r W E D G E 6 m o d e l c a l c u l a t e d a n e i g e n f r e q u e n c y o f

5 4 4 5 . 6 7 H z f o r t h i s m o d e . T h e q u a d r a t i c 3 4 8 { e l e m e n t m o d e l y i e l d e d a f r e q u e n c y 5 3 9 2 . 3 8 H z .

T h e m o d e 1 , 0 , 1 e i g e n f r e q u e n c y f o r t h e 2 2 4 0 { e l e m e n t W E D G E 1 5 m o d e l w a s 5 3 8 8 . 0 6 H z . A

n o r m a l i z e d f r i n g e p l o t o f t h i s m o d e s h a p e i s s h o w n f o r e a c h m o d e l i n F i g u r e s A 4 1 , A 4 2 , a n d

17



A 4 3 . T h e N A S T R A N n u m e r i c s o l u t i o n w a s c o m p a r e d t o t h e a n a l y t i c s o l u t i o n , E q u a t i o n 2 7 .

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

a t e a c h n o d e i s s h o w n i n F i g u r e s A 4 4 , A 4 5 , a n d A 4 6 .

T h i s a n a l y s i s w a s r e p e a t e d f o r m o d e 1 , 1 , 3 f o r e a c h m o d e l . T h i s m o d e s h a p e i s s h o w n i n

F i g u r e s A 4 7 , A 4 8 , a n d A 4 9 . H o w e v e r , a t t h e t i m e o f t h i s w r i t i n g , i t w a s n o t p o s s i b l e t o c r e a t e

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

r o t a t i o n a l s y m m e t r y i s p r e s e n t , N A S T R A N i n t r o d u c e s a n a r b i t r a r y p h a s e a n g l e w i t h r e s p e c t

t o t h e a n a l y t i c s o l u t i o n i n t o i t s n u m e r i c s o l u t i o n . T h a t i s , t h e n u m e r i c s o l u t i o n i s a c c u r a t e ,

b u t i t s m o d e s h a p e i s r o t a t e d b y s o m e a n g l e a b o u t i t s a x i s o f s y m m e t r y . I t w a s n o t p o s s i b l e t o

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

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

T a b l e 7 . T h e m a x i m u m m o d e l e r r o r i s a g a i n t h e l a r g e s t d i e r e n c e a n y w h e r e i n t h e m o d e l

b e t w e e n t h e n o r m a l i z e d a n a l y t i c a n d n u m e r i c s o l u t i o n s .


M o d e

S h a p e

E i g e n v a l u e

E r r o r

M a x i m u m

E r r o r i n

M o d e l

2 2 4 0 L i n e a r W E D G E 6

1 4 4 9

1 . 0 7 %

0 . 0 1 7

3 4 8 Q u a d r a t i c W E D G E 1 5

1 2 0 0

8 2 : 8 7 2 1 0

0 3

%

0 . 0 0 5

2 2 4 0 Q u a d r a t i c W E D G E 1 5

6 6 0 9

1 , 0 , 1

6 : 5 0 2 1 0

0 3

% 7 7 4 : 2 2 1 0

0 6

2 2 4 0 L i n e a r W E D G E 6

1 1 4 9

3 . 0 4 % N / A

3 4 8 Q u a d r a t i c W E D G E 1 5

1 2 0 0

0 . 3 5 6 % N / A

F l u i d o n l y

2 2 4 0 Q u a d r a t i c W E D G E 1 5

6 6 0 9

1 , 1 , 3

0 . 0 1 5 % N / A

T a b l e 7 R e s u l t s f o r c y l i n d r i c a l g e o m e t r y , n o r m a l m o d e s a n a l y s i s .

T h e F r e e S h e l l P r o b l e m

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

i n t e r a c t i o n . W e b e g i n b y d e n i n g a t h i n , n i t e l y - l o n g c y l i n d r i c a l e l a s t i c s h e l l l l e d w i t h a

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

18



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

t h e D o n n e l l - M u s h t a r i e q u a t i o n s o f m o t i o n a r e :

5

1

C

2

L

@

2

u

@ t

2

0

@

2

u

@ z

2

0

1 +

2 a

2

@

2

u

@

2

0

1 +

2 a

2

@

2

v

@ z @

0

a

@ w

@ z

= 0

1

C

2

L

@

2

v

@ t

2

0

1 +

2 a

@

2

u

@ z @

0

1 0

2

@

2

v

@ z

2

0

1

a

2

@

2

v

@

2

0

1

a

2

@ w

@

= 0

1

C

2

L

@

2

w

@ t

2

+

a

@ u

@ z

+

1

a

2

@ v

@

+

1

a

2

w +

h

2

1 2

r

4

w = 0

( 2 9 )

w h e r e

C

L

=

E

s

( 1 0

2

)

1

2

( 3 0 )

O n t h e L H S o f E q u a t i o n 2 9 , u ( z ; ) ; v ( z ; ) ; w ( z ; ) r e p r e s e n t d i s p l a c e m e n t s i n t h e a x i a l ,

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

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

5

:

v ( 0 ; ) = v ( l ; ) = 0

w ( 0 ; ) = w ( l ; ) = 0

( 3 1 )

t h e c o m p l e t e s o l u t i o n t o E q u a t i o n 2 9 c a n b e w r i t t e n a s :

u

(

; z ) =

e

i ! t

1

X

n = 0

1

X

m = 0

c o s

m z

l

[

A

m n

c o s

(

n ) + A

?

m n

s i n

(

n ) ]

v ( ; z ) = e

i ! t

1

X

n = 0

1

X

n = 0

s i n

m z

l

[ B

m n

s i n ( n ) + B

?

m n

c o s ( n ) ]

w ( ; z ) = e

i ! t

1

X

n = 0

1

X

m = 0

s i n

m z

l

[ C

m n

c o s ( n ) + C

?

m n

s i n ( n ) ]

( 3 2 )

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

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

6

. T h i s w o r k w i l l u s e t h e u n - s t a r r e d s o l u t i o n .

T h a t i s , w e w i l l t a k e

u ( z ; ) = e

i ! t

1

X

n = 0

1

X

m = 0

A

m n

c o s

m z

l

c o s ( n )

v ( z ; ) = e

i ! t

1

X

n = 0

1

X

m = 0

B

m n

s i n

m z

l

s i n ( n )

w ( z ; ) = e

i ! t

1

X

n = 0

1

X

m = 0

C

m n

s i n

m z

l

c o s ( n )

( 3 3 )

19



T h e e n d c o n d i t i o n s s h o w n i n E q u a t i o n 3 1 r e p r e s e n t \ s h e a r d i a p h r a g m s }

7

. N o n | a x i a l d i s -

p l a c e m e n t s a t t h e e n d s o f t h e c y l i n d r i c a l s h e l l ( a t z = 0 a n d z = l ) a r e z e r o . S u b s t i t u t i n g

E q u a t i o n 3 3 i n t o E q u a t i o n 2 9 y i e l d s :

2

6

4

0

0

2

0

1 0

2

n

2

+

2

m n

1

1 +

2

n

1 +

2

n

0

0

1 0

2

2

0 n

2

+

2

m n

1

0 n

0 n

1 +

h

2

1 2 a

2

0

2

+ n

2

1

2

0

2

m n

3

7

5

2

6

4

A m n

B

m n

C

m n

3

7

5

=

2

6

4

0

0

0

3

7

5

( 3 4 )

w h e r e

2

m n

=

s

0

1 0

2

1

a

2

!

2

m n

E

( 3 5 )

a n d

=

m a

l

( 3 6 )

F o r n o n - t r i v i a l s o l u t i o n s t o E q u a t i o n 3 4 , t h e d e t e r m i n a n t o f t h e c o e c i e n t m a t r i x i s s e t e q u a l

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

2

m n

7

:

6

m n

0 K

2

4

m n

+ K

1

2

m n

0 K

0

= 0 ( 3 7 )

w h e r e

K

2

= 1 +

3 0

2

0

n

2

+

2

1

+

h

2

1 2 a

2

0

n

2

+

2

1

2

K

1

=

1 0

2

( 3 + 2 )

2

+ n

2

+

0

n

2

+

2

1

2

+

3 0

1

0

h

2

1 2 a

2

0

n

2

+

2

1

3

K

0

=

1 0

2

0

1 0

2

1

4

+

h

2

1 2 a

2

0

n

2

+

2

1

4

( 3 8 )

T h e r o o t s o f t h i s e q u a t i o n i n c o m b i n a t i o n w i t h E q u a t i o n 3 5 c a n b e u s e d t o c a l c u l a t e t h e

n a t u r a l f r e q u e n c i e s !

m n

o f t h e c y l i n d r i c a l s h e l l .

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

E p s t e i n - K e n n a r d e q u a t i o n s o f m o t i o n . U n l i k e t h e D o n n e l l - M u s h t a r i f o r m u l a t i o n , t h e E p s t e i n -

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

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

t h e l i t e r a t u r e [ 7 ] . H e r e , i t i s n o t e d t h a t , w i t h r e s p e c t t o t h e n a t u r a l f r e q u e n c i e s o f v i b r a t i o n ,

E q u a t i o n 3 7 b e c o m e s

7

6

m n

0 ( K

2

+ k 1 K

2

)

4

m n

+ ( K

1

+ k 1 K

1

)

2

m n

0 ( K

0

+ k 1 K

0

) = 0 ( 3 9 )

20



w h e r e , f o r t h e E p s t e i n - K e n n a r d t h e o r y ,

1 K

2

=

+ 3

0

0

0

2 0 8

2

+ 3

3

1

2

2 ( 0 )

2

0

0

9 0 3 7 + 9

2

+

3

1

2 ( 0 )

2

0

2

0

n

2

+

2

1

( 0 )

2

1 K

1

=

0

3 + 8 0 5

2

0

3

1

2

2 ( 0

)

+

( 2 + ) n

2

2

0

0

6 + 4 0 8

2

+ 3

3

0 8

4

1

4

4 ( 0

)

0

2

0

n

2

+

2

1

3

2 ( 0

)

0

0

2 6 0 6 0 + 4 0

2

0 3

3

0 8

4

1

2

n

2

2 ( 0 )

0

0

3 0 2 2 + 0

2

1

n

4

2 ( 0 )

1 K

0

=

2

( 0 )

"

0

2 + 6 0 2

2

0 3

3

1

4

2 ( 0 )

+ 4

2

n

2

+ n

4

0

+

0

6

0

7 0 5

0

4

n

2

0 8

2

n

4

0 2 n

6

#

( 4 0 )

a n d

k =

h

2

2 a

2

( 4 )

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


F o r t h e u i d - l l e d r e g i o n i n s i d e t h e c y l i n d r i c a l s h e l l , t h e w a v e e q u a t i o n i n c y l i n d r i c a l

c o o r d i n a t e s ( E q u a t i o n 2 4 ) c a n b e u s e d . T h e b o u n d a r y c o n d i t i o n s n o w b e c o m e

P ( r ; ; 0 ) = P ( r ; ; l ) = 0

@ P ( a ; ; z )

@ r

=

f

@

2

w ( ; z )

@ t

2

j P ( 0 ; ; z ) j < 1 ( b o u n d e d n e s s )

P ( r ; ; z ) = P ( r ; + 2 n ; z ) ( p e r i o d i c i t y )

( 4 2 )

w h e r e w ( ; z ) r e p r e s e n t s t h e r a d i a l d i s p l a c e m e n t o f t h e c y l i n d r i c a l s h e l l . T h a t i s , j u s t a s i n t h e

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

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

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

5

:

P ( r ; ; z ) = e

i ! t

1

X

n = 0

1

X

m = 1

D

m n

J

n

(

m n

r ) s i n

m z

l

c o s ( n ) ( 4 3 )

w h e r e

D

m n

=

C

m n

f

!

2

m n

2

n

a

J

n

(

m n

a ) 0

m n

J

n + 1

(

m n

a )

3

( 4 4 )

21



a n d

2

m n

=

!

m n

c

o

2

0

m

l

2

( 4 5 )

T h r e e c y l i n d r i c a l u i d / s t r u c t u r e m o d e l s w e r e c o n s t r u c t e d f o r N A S T R A N n o r m a l m o d e s

a n a l y s i s . T h e t h r e e m o d e l s w e r e d i s c r e t i z e d a s s h o w n i n T a b l e 8 . T h e u i d a n d s t r u c t u r e

e l e m e n t s w i t h i n e a c h o f t h e m o d e l s w e r e g i v e n t h e m a t e r i a l p r o p e r t i e s s h o w n i n T a b l e 6 .

B o u n d a r y c o n d i t i o n s a s s h o w n i n E q u a t i o n s 3 1 a n d 4 2 w e r e a p p l i e d o n e a c h m o d e l .

M o d e l M a t e r i a l E l e m e n t T y p e

N u m b e r o f

N o d e s

T o t a l N o d e s i n

M o d e l

S t r u c t u r e 4 8 0 L i n e a r Q U A D 4

5 0 4

1

F l u i d

2 2 4 0 L i n e a r W E D G E 6 1 4 4 9

1 9 5 3

S t r u c t u r a l 1 9 2 Q u a d r a t i c Q U A D 8

6 0 8

2

F l u i d 6 7 2 Q u a d r a t i c W E D G E 1 5

2 1 2 1

2 7 2 9

S t r u c t u r e

4 8 0 Q u a d r a t i c Q U A D 8

1 4 8 8

3

F l u i d 2 2 4 0 Q u a d r a t i c W E D G E 1 5

6 6 0 9

8 0 9 7

T a b l e 8 N A S T R A N m o d e l s f o r u i d / s t r u c t u r e p r o b l e m , c y l i n d r i c a l g e o m e t r y .

F r o m E q u a t i o n 3 5 , u s i n g E p s t e i n - K e n n a r d t h e o r y , t h e n a t u r a l f r e q u e n c y o f t h e c y l i n d r i c a l

s h e l l f o r m o d e 1 , 1 i s 6 2 4 8 . 9 9 H z . U s i n g t h e l i n e a r Q U A D 4 m o d e l , N A S T R A N c a l c u l a t e d a n

e i g e n f r e q u e n c y o f 6 2 5 1 . 1 4 H z . T h e 1 9 2 { e l e m e n t Q U A D 8 m o d e l y i e l d e d a f r e q u e n c y f o r t h i s

m o d e o f 6 2 5 6 . 3 4 H z . A f r e q u e n c y o f 6 2 4 5 . 5 5 H z w a s c a l c u l a t e d u s i n g t h e 4 8 0 { e l e m e n t Q U A D 8

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

A 5 0 , A 5 1 , A 5 2 , A 5 3 , A 5 4 , A 5 5 , A 5 6 , A 5 7 , a n d A 5 8 . T h e \ b r e a t h i n g m o d e s " o f t h e s h e l l ,

i . e . m o d e s w h e r e m = 1 a n d n = 0 i n E q u a t i o n 3 3 o c c u r ( u s i n g E p s t e i n - K e n n a r d t h e o r y ) f o r t h i s

m o d e l a t 1 2 , 3 1 8 . 1 9 H z , 1 9 , 5 8 5 . 9 1 H z a n d 3 5 , 0 3 7 . 3 9 H z . T h e s e c o n d f r e q u e n c y , 1 9 , 5 8 5 . 9 1 H z ,

i s s h o w n f o r e a c h m o d e l i n F i g u r e s A 5 9 , A 6 0 , A 6 1 , A 6 2 , A 6 3 , A 6 4 , A 6 5 , A 6 6 , a n d A 6 7 , w i t h

e r r o r p l o t s f o r t h e r a d i a l d i s p l a c e m e n t s i n F i g u r e s A 6 8 , A 6 9 , a n d A 7 0 .

22



F o r t h e u i d i n s i d e t h e c y l i n d r i c a l s h e l l , f r i n g e p l o t s o f N A S T R A N - c a l c u l a t e d m o d e 1 , 0 , 1

i s s h o w n i n F i g u r e s A 7 1 ( l i n e a r W E D G E 6 e l e m e n t s ) , A 7 2 ( 6 7 2 { e l e m e n t W E D G E 1 5 m o d e l )

a n d A 7 3 ( 2 2 4 0 { e l e m e n t W E D G E 1 5 m o d e l ) . A f r i n g e p l o t s h o w i n g t h e d i e r e n c e b e t w e e n

t h e a n a l y t i c a n d n u m e r i c s o l u t i o n s o f t h i s m o d e i s s h o w n i n F i g u r e s A 7 4 , A 7 5 , a n d A 7 6 .

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

c y l i n d e r i s s h o w n i n T a b l e 9 .


M o d e

S h a p e

E i g e n v a l u e

E r r o r

M a x i m u m

E r r o r i n

M o d e l

2 2 4 0 L i n e a r W E D G E 6 1 4 4 9

0 . 1 0 2 %

9 8

:

4 2 1 0

0 6

6 7 2 Q u a d r a t i c W E D G E 1 5

2 1 2 1

2 : 9 3 7 2 1 0

0 4

%

0 . 0 8 2 4

2 2 4 0 Q u a d r a t i c

W E D G E 1 5

6 6 0 9

1 , 0 , 1

7 : 3 4 2 1 0

0 5

% 1 0 1 : 3 2 1 0

0 6

4 8 0 L i n e a r Q U A D 4

5 0 4

0 . 1 4 3 % N / A

1 9 2 Q u a d r a t i c Q U A D 8

6 0 8

0 . 1 1 8 % N / A

4 8 0 Q u a d r a t i c Q U A D 8

1 4 8 8

1 , 1

0 . 0 5 5 % N / A

4 8 0 L i n e a r Q U A D 4

5 0 4

0 . 3 9 7 %

0 : 0 8 7 2 1 0

0 6

1 9 2 Q u a d r a t i c Q U A D 8

6 0 8

0 . 2 9 4 %

- 0 . 0 2 7 0

F l u i d / s t r u c t .

4 8 0 Q u a d r a t i c Q U A D 8

1 4 8 8

1 , 0

0 . 0 4 0 %

1 3 : 2 7 9 2 1 0

0 3

T a b l e 9 R e s u l t s f o r c y l i n d r i c a l u i d / s t r u c t u r e g e o m e t r y , n o r m a l m o d e s a n a l y s i s .

T h e F o r c e d F l u i d / S t r u c t u r e P r o b l e m

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

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

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

g i v e n b y

5

:

2

6

4

L

1 1

L

1 2

L

1 3

L

2 1

L

2 2

L

2 3

L

3 1

L

3 2

L

3 3

3

7

5

D M

2

6

4

u

v

w

3

7

5

=

1 0

2

E h

2

6

4

f

z

f

0 f

r

3

7

5

( 4 6 )

23



w h e r e

L

1 1

= 0 !

2

+ 2 i

L

1 2

= 0

1 +

2

a

@

2

@ z @

L

1 3

= 0

a

@

@ z

L

2 1

= 0

1 +

2 a

@

2

@ z @

L

2 2

=

1

C

2

L

@

2

@ t

2

+

0

1 0

2

1

E h

@

@ t

0

1 0

2

@

2

@ z

2

0

1

a

2

@

2

@

2

L

2 3

= 0

1

a

2

@

@

L

3 1

=

a

@

@ z

L

3 2

=

1

a

2

@

@

L

3 3

=

1

C

2

L

@

2

@ t

2

+

0

1 0

2

1

E h

@

@ t

+

1

a

2

+

h

2

1 2

r

4

( 4 7 )

a n d f

z

; f

; a n d f

r

r e p r e s e n t f o r c e s i n t h e a x i a l , c i r c u m f e r e n t i a l , a n d r a d i a l d i r e c t i o n s . H e r e

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

5

. A p p l y i n g f o r c e s

t o t h e c y l i n d e r o f

f

z

= 0

f

= 0

f

r

= F

o

s i n ( ! t )

z 0

l

2

[ ( ) + ( 0 ) ] ;

( 4 8 )

e x p a n d i n g t h e s e f o r c e s i n t e r m s o f t h e e i g e n f u n c t i o n s o f t h e s h e l l a n d o n c e a g a i n u s i n g E q u a t i o n

3 3 w i t h t h e b o u n d a r y c o n d i t i o n s o f E q u a t i o n 3 1 , E q u a t i o n 4 6 y i e l d s

2

6

4

L

1 1

L

1 2

L

1 3

L

2 1

L

2 2

L

2 3

L

3 1

L

3 2

L

3 3

3

7

5

2

6

4

A

m n

B

m n

C

m n

3

7

5

= 0

1

s

h

2

6

4

0

0

F

m n

3

7

5

( 4 9 )

24



w h e r e n o w

L

1 1

= 0 !

2

+ 2 i ! !

m n

+

2

C

2

L

+

1 0

2 a

2

n

2

C

2

L

L

1 2

= L

2 1

= 0

1 +

2 a

n C

2

L

L

1 3

= L

3 1

= 0

a

C

2

L

L

2 2

= 0 !

2

+ 2 i ! !

m n

+

1 0

2

2

C

2

L

+

n

2

a

2

C

2

L

L

2 3

= L

3 2

=

n

a

2

C

2

L

L

3 3

= 0 !

2

+ 2 i ! !

m n

+

1

a

2

C

2

L

+

h

2

C

2

L

1 2

0

2

a

2

+ n

2

1

2

( 5 0 )

=

m

l

( 5 1 )

a n d F

m n

i s g i v e n b y

F

m n

=

4

l

F

o

s i n

m

2

[ c o s ( n ) + 1 ] ( 5 2 )

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

a f r e q u e n c y - d e p e n d e n t d a m p i n g t e r m .

F o r t h e u i d i n s i d e t h e s h e l l , w e p r o c e e d a s w e d i d f o r t h e u n d a m p e d , f r e e v i b r a t i o n c a s e .

T h a t i s , a p p l y i n g t h e b o u n d a r y c o n d i t i o n s o f E q u a t i o n 4 2 t o t h e c y l i n d r i c a l w a v e e q u a t i o n

( E q u a t i o n 2 4 ) , t h e s o l u t i o n f o r t h e a c o u s t i c p r e s s u r e e l d b e c o m e s :

5

P ( r ; ; z ) = e

i ! t

1

X

n = 0

1

X

m = 1

D

m n

J

n

(

m n

r ) s i n

m z

l

c o s ( n ) ( 5 3 )

w i t h

D

m n

=

C

m n

f

!

2

m n

2

n

a

J

n

(

m n

a

)

0

m n

J

n

+ 1

(

m n

a

)

3

( 5 4 )

a n d

2

m n

=

!

m n

c

o

2

0

m

l

2

( 5 5 )

F o r t h e N A S T R A N a n a l y s i s o f t h i s p r o b l e m , a c y l i n d r i c a l n i t e e l e m e n t m o d e l w a s

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

25



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

w e r e m o d i e d a s s h o w n i n T a b l e 1 0 . T h e s t r u c t u r a l a n d u i d e l e m e n t s w e r e g i v e n t h e m a t e r i a l

p r o p e r t i e s s h o w n i n T a b l e 6 .

R a d i u s ( a )

1 0 . 0 i n

L e n g t h ( l )

5 0 . 0 i n

S h e l l T h i c k n e s s ( h )

0 . 0 6 2 5 i n

T a b l e 1 0 G e o m e t r i c d i m e n s i o n s f o r t h e c y l i n d r i c a l

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

B o u n d a r y c o n d i t i o n s i n a g r e e m e n t w i t h E q u a t i o n s 3 1 a n d 4 2 w e r e a p p l i e d t o t h e a p p r o -

p r i a t e s t r u c t u r a l a n d u i d e l e m e n t s r e s p e c t i v e l y . T w o p o i n t f o r c e s o f 5 0 s i n ( ! t )

^

k l b s . , i n p h a s e

a n d b o t h p o i n t i n g r a d i a l l y i n w a r d w e r e l o c a t e d a t = 0 ; z =

l

2

a n d = ; z =

l

2

. N A S T R A N

p e r f o r m e d a d i r e c t f r e q u e n c y r e s p o n s e ( S O L 1 0 8 ) o v e r t h e f r e q u e n c y r a n g e z e r o t o 5 0 0 0 H z .

A n a l y t i c a n d N A S T R A N - g e n e r a t e d n u m e r i c s o l u t i o n s f o r t h e f r e q u e n c y r a n g e z e r o t o 1 0 0 0

H z a r e s h o w n i n F i g u r e s A 7 8 a n d A 7 7 . F i g u r e A 7 8 s h o w s t h e d i s p l a c e m e n t v s . f r e q u e n c y o f

a s t r u c t u r a l n o d e l o c a t e d a t r = a ; = 0 ; z =

l

2

. F i g u r e A 7 7 s h o w s t h e a c o u s t i c p r e s s u r e o f

a u i d n o d e a t l o c a t i o n r =

a

2

; = 0 ; z =

l

2

.

S e c t i o n 4 : C o n c l u s i o n s

M S C / N A S T R A N h a s b e e n u s e d t o n u m e r i c a l l y c a l c u l a t e a n u m b e r o f d i e r e n t m o d e

s h a p e s f o r a c u b i c a n d a c y l i n d r i c a l a c o u s t i c c a v i t y . T h e a c c u r a c y o f t h e N A S T R A N f u l l y -

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

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

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

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

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

t o a m o d e l c o n t a i n i n g b o t h u i d a n d s t r u c t u r e e l e m e n t s . N A S T R A N w a s a b l e t o c o m p u t e t h e

26



n o r m a l m o d e s f o r e a c h m o d e l q u i c k l y a n d a c c u r a t e l y . I n s u m , N A S T R A N ' s c a l c u l a t i o n o f t h e

n o r m a l m o d e s ( S O L 1 0 3 ) f o r a m o d e l t e n d e d t o b e s i m p l e , d i r e c t , q u i c k , a n d a c c u r a t e .

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

t h e f o r c e d r e s p o n s e a n a l y s i s e s , p a r t i c u l a r l y w h e n t h e d i r e c t m e t h o d ( S O L 1 0 8 ) w a s u s e d . I n

g e n e r a l , r a t h e r t h a n u s e s y s t e m m e m o r y , N A S T R A N w r i t e s d a t a t o l e s d u r i n g t h e s o l u t i o n

o f a n i t e e l e m e n t p r o b l e m . A l t h o u g h m o s t o f t h e s e l e s a r e d e l e t e d w h e n t h e N A S T R A N

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

I f e n o u g h f r e e s p a c e i s n o t a v a i l a b l e f o r u s e b y N A S T R A N , t h e p r o b l e m c a n n o t b e s o l v e d .

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

s u b m i t t e d t o N A S T R A N a s f o u r s e p a r a t e p r o b l e m s , e a c h c o v e r i n g a r a n g e o f 2 5 0 f r e q u e n c i e s .

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

s o l u t i o n f o r t h e e n t i r e f r e q u e n c y r a n g e o f i n t e r e s t . A p p e n d i x C o f t h i s w o r k c o n t a i n s a l i s t i n g

o f t h e s i g n i c a n t l e s p r o d u c e d b y N A S T R A N d u r i n g e a c h a n a l y s i s a n d t h e s i z e o f e a c h l e .

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

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

I n s h o r t , N A S T R A N ' s n o r m a l m o d e s a n a l y s i s w a s v e r y r o b u s t . F o r a r e a s o n a b l e m o d e l

d i s c r e t i z a t i o n , i t s r e s u l t s c a n b e c o n s i d e r e d a c c u r a t e . H o w e v e r , f o r t h e c a s e s w h e r e a f o r c e d

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

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

A c k n o w l e d g m e n t s

T h e a u t h o r s g r a t e f u l l y a c k n o w l e d g e t h e a s s i s t a n c e o f T r a v i s T u r n e r a n d R i c h a r d S .

M a t t h e w s , w i t h o u t w h o m t h i s w o r k w o u l d n o t h a v e b e e n p o s s i b l e

27



R e f e r e n c e s

[ 1 ] W i l l i a m H . P r e s s . N u m e r i c a l R e c i p e s : T h e A r t o f S c i e n t i c C o m p u t i n g . C a m b r i d g e

U n i v e r s i t y P r e s s , r s t e d i t i o n , 1 9 9 0 .

[ 2 ] T y n M y i n t - U w i t h L o k e n a t h D e b n a t h . P a r t i a l D i e r e n t i a l E q u a t i o n s f o r S c i e n t i s t s a n d

E n g i n e e r s . P T R P r e n t i c e - H a l l , t h i r d e d i t i o n , 1 9 8 7 .

[ 3 ] M i c h a e l R e y m o n d a n d M a r k M i l l e r E d i t o r s . M S C / N A S T R A N Q u i c k R e f e r e n c e G u i d e ,

V e r s i o n 6 8 . T h e M a c N e a l - S c h w i n d l e r C o r p o r a t i o n , 1 9 9 4 .

[ 4 ] A r t h u r W . L e i s s a . V i b r a t i o n o f P l a t e s . U S G P O , N A S A S P - 1 6 0 , 1 9 6 8 .

[ 5 ] H . C . L e s t e r a n d S . L e f e b v r e . P i e z o e l e c t r i c A c t u a t o r M o d e l s f o r A c t i v e S o u n d a n d V i b r a t i o n

C o n t r o l o f C y l i n d e r s . P r o c e e d i n g s o f t h e C o n v e r e n c e o n R e c e n t A d v a n c e s i n A c t i v e C o n t r o l

o f S o u n d a n d V i b r a t i o n , V i r g i n i a P o l y t e c h n i c I n s t i t u t e a n d S t a t e U n i v e r s i t y , B l a c k s b u r g ,

V i r g i n i a , A p r i l 1 5 { 1 7 , 1 9 9 1 , T e c n o m i c P u b l i s h i n g C o m p a n y , 1 9 9 1 .

[ 6 ] R i c h a r d H . M a c N e a l . N A S T R A N T h e o r e t i c a l M a n u a l . T h e M a c N e a l - S c h w i n d l e r C o r p o -

r a t i o n , 1 9 7 2 .

[ 7 ] A r t h u r W . L e i s s a . V i b r a t i o n o f S h e l l s . U S G P O , N A S A S P - 2 8 8 , 1 9 7 3 .

28



A p p e n d i x A F i g u r e s

0.

-.06667

-.1333

-.2000

-.2667

-.3333

-.4000

-.4667

-.5333

-.6000

-.6667

-.7333

-.8000

-.8667

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-1.000

X

Y

Z

F i g u r e A 1 : M o d e 1 , 1 , 1 f o r c u b i c g e o m e t r y , 1 0 0 0 l i n e a r H E X 8 e l e m e n t s , 1 3 3 1 n o d e s .

X

Y

Z

1.000

.9333

.8667

.8000

.7333

.6667

.6000

.5333

.4667

.4000

.3333

.2667

.2000

.1333

.06667

.00000004500

X

Y

Z

F i g u r e A 2 : M o d e 1 , 1 , 1 f o r c u b i c g e o m e t r y ,

2 1 5 q u a d r a t i c H E X 2 0 e l e m e n t s , 1 2 2 5 n o d e s .

29



1.000

.9333

.8667

.8000

.7333

.6667

.6000

.5333

.4667

.4000

.3333

.2667

.2000

.1333

.06667

.00000004500

X

Y

Z


1 0 0 0 q u a d r a t i c H E X 2 0 e l e m e n t s , 4 9 6 1 n o d e s .

.000006831

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X

Y

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F i g u r e A 4 : M o d e 1 , 1 , 1 e r r o r f o r c u b i c g e o m e t r y ,

1 0 0 0 l i n e a r H E X 8 e l e m e n t s , 1 3 3 1 n o d e s .

30



X

Y

Z

.000003375

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X

Y

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.000006602

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Y

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31



.4341

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X

Y

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F i g u r e A 7 : M o d e 1 , 3 , 1 f o r c u b i c g e o m e t r y , 1 0 0 0 l i n e a r H E X 8 e l e m e n t s , 1 3 3 1 n o d e s .

X

Y

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.6045

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32



1.000

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Y

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.5222

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X

Y

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F i g u r e A 1 0 : M o d e 1 , 3 , 1 e r r o r f o r c u b i c g e o m e t r y ,

1 0 0 0 l i n e a r H E X 8 e l e m e n t s , 1 3 3 1 n o d e s .

33



X

Y

Z

.4593

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X

Y

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34



1.000

.9333

.8667

.8000

.7333

.6667

.6000

.5333

.4667

.4000

.3333

.2667

.2000

.1333

.06667

.00000004500

X

Y

Z

F i g u r e A 1 3 : M o d e 1 , 1 f o r s t r u c t u r e p o r t i o n o f u i d / s t r u c t u r e

c u b e . 2 0 0 l i n e a r Q U A D 4 e l e m e n t s , 2 4 2 n o d e s .

X

Y

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.1314

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X

Y

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F i g u r e A 1 4 : M o d e 1 , 1 f o r s t r u c t u r e p o r t i o n o f

u i d / s t r u c t u r e c u b e . 7 2 q u a d r a t i c Q U A D 8 e l e m e n t s , 2 6 6 n o d e s .

35



0.

-.06667

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c u b e . 2 0 0 q u a d r a t i c Q U A D 8 e l e m e n t s , 6 8 2 n o d e s .

X

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.0000000000000

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X

Y

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F i g u r e A 1 6 : M o d e 1 , 1 e r r o r f o r c u b i c u i d / s t r u c t u r e g e o m e t r y ( f r o n t

p l a t e o n l y ) . 2 0 0 q u a d r a t i c Q U A D 4 e l e m e n t s , 2 4 2 n o d e s .

36



X

Y

Z

.0000000000000

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p l a t e o n l y ) . 7 2 q u a d r a t i c Q U A D 8 e l e m e n t s , 2 6 6 n o d e s .

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Y

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F i g u r e A 1 8 : M o d e 1 , 1 e r r o r f o r c u b i c u i d / s t r u c t u r e g e o m e t r y

( s t r u c t u r e o n l y ) . 2 0 0 q u a d r a t i c Q U A D 8 e l e m e n t s , 6 8 2 n o d e s .

37



1.000

.8667

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c u b e . 2 0 0 l i n e a r Q U A D 4 e l e m e n t s , 2 4 2 n o d e s .

X

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1.000

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X

Y

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u i d / s t r u c t u r e c u b e . 7 2 q u a d r a t i c Q U A D 8 e l e m e n t s , 2 6 6 n o d e s .

38



1.000

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u i d / s t r u c t u r e c u b e . 2 0 0 Q U A D 8 e l e m e n t s , 6 8 2 n o d e s .

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Y

Z

F i g u r e A 2 2 : M o d e 2 , 2 e r r o r f o r c u b i c u i d / s t r u c t u r e g e o m e t r y

( f r o n t p l a t e o n l y ) . 2 0 0 l i n e a r Q U A D 4 e l e m e n t s , 2 4 2 n o d e s .

39



X

Y

Z

.1340

.1161

.09825

.08038

.06252

.04466

.02679

.008931

-.008932

-.02680

-.04466

-.06252

-.08038

-.09825

-.1161

-.1340

X

Y

Z


p l a t e o n l y ) . 7 2 q u a d r a t i c Q U A D 8 e l e m e n t s , 2 6 6 n o d e s .

.04894

.04242

.03589

.02937

.02284

.01631

.009789

.003263

-.003263

-.009789

-.01631

-.02284

-.02937

-.03589

-.04242

-.04894

X

Y

Z


p l a t e o n l y ) . 2 0 0 q u a d r a t i c Q U A D 8 e l e m e n t s , 6 8 2 n o d e s .

40



0.

-.06667

-.1333

-.2000

-.2667

-.3333

-.4000

-.4667

-.5333

-.6000

-.6667

-.7333

-.8000

-.8667

-.9333

-1.000

X

Y

Z

F i g u r e A 2 5 : M o d e 1 , 1 , 0 f o r u i d p o r t i o n o f u i d / s t r u c t u r e

c u b e . 1 0 0 0 l i n e a r H E X 8 e l e m e n t s , 1 3 3 1 n o d e s .

X

Y

Z

0.

-.06667

-.1333

-.2000

-.2667

-.3333

-.4000

-.4667

-.5333

-.6000

-.6667

-.7333

-.8000

-.8667

-.9333

-1.000

X

Y

Z


c u b e . 2 1 6 q u a d r a t i c H E X 2 0 e l e m e n t s , 1 2 2 5 n o d e s .

41



1.000

.9333

.8667

.8000

.7333

.6667

.6000

.5333

.4667

.4000

.3333

.2667

.2000

.1333

.06667

.00000004500

X

Y

Z


c u b e . 1 0 0 0 q u a d r a t i c H E X 2 0 e l e m e n t s , 4 9 6 2 n o d e s .

.000005216

.000004495

.000003773

.000003052

.000002331

.000001610

.0000008880

.0000001670

-.0000005540

-.000001276

-.000001997

-.000002718

-.000003440

-.000004161

-.000004882

-.000005603

X

Y

Z

F i g u r e A 2 8 : M o d e 1 , 1 , 0 e r r o r f o r u i d / s t r u c t u r e g e o m e t r y

( u i d o n l y ) . 1 0 0 0 l i n e a r H E X 8 e l e m e n t s , 1 3 3 1 n o d e s .

42



X

Y

Z

.0007277

.0006779

.0006281

.0005783

.0005285

.0004787

.0004290

.0003792

.0003294

.0002796

.0002298

.0001801

.0001303

.00008049

.00003071

-.00001907

X

Y

Z

F i g u r e A 2 9 : M o d e 1 , 1 , 0 e r r o r f o r u i d / s t r u c t u r e g e o m e t r y

( u i d o n l y ) . 2 1 6 q u a d r a t i c H E X 2 0 e l e m e n t s , 1 2 2 5 n o d e s .

.0000007150

-.000005742

-.00001220

-.00001866

-.00002511

-.00003157

-.00003803

-.00004449

-.00005094

-.00005740

-.00006386

-.00007031

-.00007677

-.00008323

-.00008969

-.00009614

X

Y

Z

F i g u r e A 3 0 : M o d e 1 , 1 , 0 e r r o r f o r c u b i c u i d / s t r u c t u r e g e o m e t r y

( u i d o n l y ) . 1 0 0 0 q u a d r a t i c H E X 2 0 e l e m e n t s , 4 9 6 2 n o d e s .

43



1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

X

Y

Z


c u b e . 1 0 0 0 l i n e a r H E X 8 e l e m e n t s , 1 3 3 1 n o d e s .

X

Y

Z

1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

X

Y

Z


c u b e . 2 1 6 q u a d r a t i c H E X 2 0 e l e m e n t s , 1 2 2 5 n o d e s .

44



1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

X

Y

Z


c u b e . 1 0 0 0 q u a d r a t i c H E X 2 0 e l e m e n t s , 4 9 6 2 n o d e s .

.000006617

.000005802

.000004987

.000004173

.000003358

.000002544

.000001729

.0000009140

.00000009900

-.0000007150

-.000001530

-.000002345

-.000003160

-.000003974

-.000004789

-.000005603

X

Y

Z

F i g u r e A 3 4 : M o d e 1 , 1 , 1 e r r o r f o r c u b i c u i d / s t r u c t u r e

g e o m e t r y ( u i d o n l y ) . 1 0 0 0 l i n e a r H E X 8 e l e m e n t s , 1 3 3 1 n o d e s .

45



X

Y

Z

.001532

.001328

.001124

.0009195

.0007151

.0005108

.0003065

.0001022

-.0001022

-.0003065

-.0005108

-.0007151

-.0009195

-.001124

-.001328

-.001532

X

Y

Z


( u i d o n l y ) . 2 1 6 q u a d r a t i c H E X 2 0 e l e m e n t s , 1 2 2 5 n o d e s .

.0001986

.0001721

.0001456

.0001192

.00009268

.00006620

.00003972

.00001324

-.00001324

-.00003972

-.00006620

-.00009268

-.0001192

-.0001456

-.0001721

-.0001986

X

Y

Z


( u i d o n l y ) . 1 0 0 0 q u a d r a t i c H E X 2 0 e l e m e n t s , 4 9 6 2 n o d e s .

46



0 1000 2000 3000 4000 500010

-5

10-4

10-3

10-2

10-1

W-Displacement vs. Frequency

Frequency (Hz)

D i s p l a c e m e n t ( l o g )

Fluid/structure cube, 1200 Elements, 1573 Nodes, NASTRAN direct frequency response analysis

Numeric So

Analytic S

F i g u r e A 3 7 : D i s p l a c e m e n t a t t h e c e n t e r o f t h e z = 5 p l a t e o n t h e

u i d / s t r u c t u r e c u b e . N A S T R A N d i r e c t f r e q u e n c y r e s p o n s e a n a l y s i s . 1 2 0 0

l i n e a r e l e m e n t s , 1 5 7 3 n o d e s . A n a l y t i c a n d n u m e r i c s o l u t i o n s s h o w n .

1000 2000 3000 4000 5000

10-6

10-5

10-4

10-3

10-2

10-1

100

Acoustic Pressure vs. Frequency

Fluid/structure cube, 1200 Elements, 1573 Nodes, NASTRAN direct frequency response analysis

Frequency (Hz)

A c o u s t i c P r e s s u r e ( l o g )

Numeric So

Analytic S

F i g u r e A 3 8 : A c o u s t i c p r e s s u r e a t t h e p o i n t ( 2 . 5 , 2 . 5 , 3 . 5 ) i n t h e u i d / s t r u c t u r e

c u b e . N A S T R A N d i r e c t f r e q u e n c y r e s p o n s e a n a l y s i s . 1 2 0 0 l i n e a r

e l e m e n t s , 1 5 7 3 n o d e s . A n a l y t i c a n d n u m e r i c s o l u t i o n s s h o w n .

47



0 1000 2000 3000 4000 5000

10-5

10-4

10-3

10-2

10-1

W-Displacement vs. Frequency

Fluid/structure cube, 1200 Elements, 1573 Nodes, NASTRAN modal frequency response analysis

Frequency (Hz)


Numeric So

Analytic S

F i g u r e A 3 9 : D i s p l a c e m e n t a t t h e c e n t e r o f t h e z = 5 p l a t e o n t h e

u i d / s t r u c t u r e c u b e . N A S T R A N m o d a l f r e q u e n c y r e s p o n s e a n a l y s i s . 1 2 0 0


1000 2000 3000 4000 5000

10-6

10-5

10-4

10-3

10-2

10-1

100

Frequency (Hz)


Acoustic Pressure vs. Frequency

Fluid/structure cube, 1200 Elements, 1573 Nodes, NASTRAN modal frequency response analysis

Numeric So

Analytic S

F i g u r e A 4 0 : A c o u s t i c p r e s s u r e a t t h e p o i n t ( 2 . 5 , 2 . 5 , 3 . 5 ) i n t h e u i d / s t r u c t u r e

c u b e . N A S T R A N m o d a l f r e q u e n c y r e s p o n s e a n a l y s i s . 1 2 0 0 l i n e a r

e l e m e n t s , 1 5 7 3 n o d e s . A n a l y t i c a n d n u m e r i c s o l u t i o n s s h o w n .

48



1.000

.9333

.8667

.8000

.7333

.6667

.6000

.5333

.4667

.4000

.3333

.2667

.2000

.1333

.06667

.00000004500

X

Y

Z

F i g u r e A 4 1 : F l u i d m o d e 1 , 0 , 1 f o r c y l i n d r i c a l g e o m e t r y ,

2 2 4 0 l i n e a r W E D G E 6 e l e m e n t s , 1 4 4 9 n o d e s .

X

Y

Z

1.000

.9333

.8667

.8000

.7333

.6667

.6000

.5333

.4667

.4000

.3333

.2667

.2000

.1333

.06667

.00000004500

X

Y

Z


3 4 8 q u a d r a t i c W E D G E 1 5 e l e m e n t s , 1 2 0 0 n o d e s .

49



0.

-.06667

-.1333

-.2000

-.2667

-.3333

-.4000

-.4667

-.5333

-.6000

-.6667

-.7333

-.8000

-.8667

-.9333

-1.000

X

Y

Z


2 2 4 0 q u a d r a t i c W E D G E 1 5 e l e m e n t s , 6 6 0 9 n o d e s .

.01687

.01575

.01462

.01350

.01237

.01125

.01012

.008998

.007873

.006748

.005623

.004499

.003374

.002249

.001124

-.0000004860

X

Y

Z

F i g u r e A 4 4 : F l u i d m o d e 1 , 0 , 1 e r r o r f o r c y l i n d r i c a l

g e o m e t r y , 2 2 4 0 l i n e a r W E D G E 6 e l e m e n t s , 1 4 4 9 n o d e s .

50



X

Y

Z

.004925

.004488

.004051

.003614

.003177

.002740

.002303

.001866

.001429

.0009924

.0005555

.0001185

-.0003184

-.0007554

-.001192

-.001629

X

Y

Z


g e o m e t r y , 3 4 8 q u a d r a t i c W E D G E 1 5 e l e m e n t s , 1 2 0 0 n o d e s .

.0007742

.0007226

.0006709

.0006193

.0005677

.0005161

.0004645

.0004129

.0003613

.0003097

.0002581

.0002064

.0001548

.0001032

.00005161

.0000000000000

X

Y

Z


g e o m e t r y , 2 4 4 0 q u a d r a t i c W E D G E 1 5 e l e m e n t s , 6 6 0 9 n o d e s .

51



1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

X

Y

Z


2 2 4 0 l i n e a r W E D G E 6 e l e m e n t s , 1 4 4 9 n o d e s .

X

Y

Z

1.000

.8668

.7334

.6001

.4667

.3334

.2000

.06667

-.06667

-.2000

-.3334

-.4667

-.6001

-.7334

-.8668

-1.000

X

Y

Z


3 4 8 q u a d r a t i c W E D G E 1 5 e l e m e n t s , 1 2 0 0 n o d e s .

52



1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

X

Y

Z


2 2 4 0 q u a d r a t i c W E D G E 1 5 e l e m e n t s , 6 6 0 9 n o d e s .

1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

1R

T

Z

X

Y

Z

F i g u r e A 5 0 : U - d i s p l a c e m e n t ( a x i a l ) o f c y l i n d r i c a l s h e l l ,

m o d e 1 , 1 . 4 8 0 l i n e a r Q U A D 4 e l e m e n t s , 5 0 4 n o d e s .

53



1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

1 R

T

Z

X

Y

Z

F i g u r e A 5 1 : V - d i s p l a c e m e n t ( c i r c u m f e r e n t i a l ) o f c y l i n d r i c a l

s h e l l , m o d e 1 , 1 . 4 8 0 l i n e a r Q U A D 4 e l e m e n t s , 5 0 4 n o d e s .

1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

1R

T

Z

X

Y

Z

F i g u r e A 5 2 : W - d i s p l a c e m e n t ( r a d i a l ) o f c y l i n d r i c a l s h e l l ,

m o d e 1 , 1 . 4 8 0 l i n e a r Q U A D 4 e l e m e n t s , 5 0 4 n o d e s .

54



X

Y

Z

1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

X

Y

Z


m o d e 1 , 1 . 1 9 2 q u a d r a t i c Q U A D 8 e l e m e n t s , 6 0 8 n o d e s .

X

Y

Z

62.94

54.54

46.15

37.76

29.37

20.98

12.59

4.196

-4.196

-12.59

-20.98

-29.37

-37.76

-46.15

-54.54

-62.94

X

Y

Z

F i g u r e A 5 4 : V - d i s p l a c e m e n t ( c i r c u m f e r e n t i a l ) o f c y l i n d r i c a l s h e l l ,


55



X

Y

Z

61.08

52.94

44.79

36.65

28.50

20.36

12.22

4.072

-4.072

-12.22

-20.36

-28.50

-36.65

-44.79

-52.94

-61.08

X

Y

Z



1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

1R

T

Z

X

Y

Z


m o d e 1 , 1 . 4 8 0 q u a d r a t i c Q U A D 8 e l e m e n t s , 1 4 8 8 n o d e s .

56



1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

1R

T

Z

X

Y

Z

F i g u r e A 5 7 : V - d i s p l a c e m e n t ( c i r c u m f e r e n t i a l ) o f c y l i n d r i c a l s h e l l ,


1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

1R

T

Z

X

Y

Z



57



60.82

52.71

44.60

36.49

28.38

20.27

12.16

4.055

-4.055

-12.16

-20.27

-28.38

-36.49

-44.60

-52.71

-60.82

1 R

T

Z

X

Y

Z

F i g u r e A 5 9 : U - d i s p l a c e m e n t ( a x i a l ) o f c y l i n d r i c a l s h e l l , m o d e 1 , 0

( b r e a t h i n g m o d e ) . 4 8 0 l i n e a r Q U A D 4 e l e m e n t s , 5 0 4 n o d e s

.00005914

.00005131

.00004348

.00003565

.00002782

.00001999

.00001216

.000004335

-.000003495

-.00001132

-.00001915

-.00002698

-.00003481

-.00004264

-.00005047

-.00005830

1R

T

Z

X

Y

Z


s h e l l , m o d e 1 , 0 ( b r e a t h i n g m o d e ) . 4 8 0 l i n e a r Q U A D 4

e l e m e n t s , 5 0 4 n o d e s . E i g e n v e c t o r n o t n o r m a l i z e d t o o n e .

58



0.

-1.263

-2.527

-3.790

-5.054

-6.317

-7.580

-8.844

-10.11

-11.37

-12.63

-13.90

-15.16

-16.42

-17.69

-18.95

1 R

T

Z

X

Y

Z

F i g u r e A 6 1 : W - d i s p l a c e m e n t ( r a d i a l ) o f c y l i n d r i c a l s h e l l , m o d e 1 , 0

( b r e a t h i n g m o d e ) . 4 8 0 l i n e a r Q U A D 4 e l e m e n t s , 5 0 4 n o d e s .

X

Y

Z

1.000

.8667

.7333

.6000

.4667

.3333

.2000

.06667

-.06667

-.2000

-.3333

-.4667

-.6000

-.7333

-.8667

-1.000

X

Y

Z


( b r e a t h i n g m o d e ) . 1 9 2 q u a d r a t i c Q U A D 8 e l e m e n t s , 6 0 8 n o d e s .

59



X

Y

Z

1.000

.8674

.7347

.6021

.4695

.3368

.2042

.07156

-.06107

-.1937

-.3263

-.4590

-.5916

-.7242

-.8569

-.9895

X

Y

Z

F i g u r e A 6 3 : V - d i s p l a c e m e n t ( c i r c u m f e r e n t i a l ) o f c y l i n d r i c a l s h e l l , m o d e 1 , 0


X

Y

Z

0.

-.06667

-.1333

-.2000

-.2667

-.3333

-.4000

-.4667

-.5333

-.6000

-.6667

-.7333

-.8000

-.8667

-.9333

-1.000

X

Y

Z

F i g u r e A 6 4 : W - d i s p l a c e m e n t ( a x i a l ) o f c y l i n d r i c a l s h e l l , m o d e 1 , 0


60



60.44

52.38

44.32

36.26

28.21

20.15

12.09

4.029

-4.029

-12.09

-20.15

-28.21

-36.26

-44.32

-52.38

-60.44

1R

T

Z

X

Y

Z


( b r e a t h i n g m o d e ) . 4 8 0 q u a d r a t i c Q U A D 8 e l e m e n t s , 1 4 8 8 n o d e s .

.002703

.002343

.001982

.001622

.001262

.0009011

.0005406

.0001802

-.0001803

-.0005408

-.0009012

-.001262

-.001622

-.001983

-.002343

-.002704

1R

T

Z

X

Y

Z


s h e l l , m o d e 1 , 0 ( b r e a t h i n g m o d e ) . 4 8 0 q u a d r a t i c Q U A D 8

e l e m e n t s , 1 4 8 8 n o d e s . E i g e n v e c t o r n o t n o r m a l i z e d t o o n e .

61



18.91

17.65

16.39

15.13

13.87

12.61

11.35

10.09

8.825

7.565

6.304

5.043

3.782

2.522

1.261

.000004292

1R

T

Z

X

Y

Z

F i g u r e A 6 7 : W - d i s p l a c e m e n t ( r a d i a l ) o f c y l i n d r i c a l s h e l l , m o d e 1 , 0


X

Y

Z

.0000005070

-.00002158

-.00004366

-.00006574

-.00008783

-.0001099

-.0001320

-.0001541

-.0001762

-.0001982

-.0002203

-.0002424

-.0002645

-.0002866

-.0003087

-.0003307

X

Y

Z

F i g u r e A 6 8 : W - d i s p l a c e m e n t ( r a d i a l ) e r r o r f o r c y l i n d r i c a l s h e l l ,

m o d e 1 , 0 ( b r e a t h i n g m o d e ) . 4 8 0 l i n e a r Q U A D 4 e l e m e n t s , 5 0 4 n o d e s .

62



X

Y

Z

.00001526

-.001782

-.003579

-.005377

-.007174

-.008972

-.01077

-.01257

-.01436

-.01616

-.01796

-.01976

-.02155

-.02335

-.02515

-.02695

X

Y

Z

F i g u r e A 6 9 : W - d i s p l a c e m e n t ( r a d i a l ) e r r o r f o r c y l i n d r i c a l s h e l l , m o d e 1 , 0


X

Y

Z

.001328

.001238

.001147

.001057

.0009670

.0008767

.0007865

.0006963

.0006061

.0005158

.0004256

.0003354

.0002452

.0001549

.00006472

-.00002551

X

Y

Z

F i g u r e A 7 0 : W - d i s p l a c e m e n t ( r a d i a l ) e r r o r f o r c y l i n d r i c a l s h e l l , m o d e 1 , 0


63



1.000

.9333

.8667

.8000

.7333

.6667

.6000

.5333

.4667

.4000

.3333

.2667

.2000

.1333

.06667

.00000004500

1 R

T

Z

X

Y

Z

F i g u r e A 7 1 : A c o u s t i c p r e s s u r e i n s i d e c y l i n d r i c a l s h e l l , u i d

m o d e 1 , 0 , 1 . 2 2 4 1 l i n e a r W E D G E 6 e l e m e n t s , 1 4 4 9 n o d e s .

X

Y

Z

0.

-.06667

-.1333

-.2000

-.2667

-.3333

-.4000

-.4667

-.5333

-.6000

-.6667

-.7333

-.8000

-.8667

-.9333

-1.000

X

Y

Z

F i g u r e A 7 2 : A c o u s t i c p r e s s u r e i n s i d e c y l i n d r i c a l s h e l l , u i d m o d e

1 , 0 , 1 . 6 7 2 q u a d r a t i c W E D G E 1 5 e l e m e n t s , 2 1 2 1 n o d e s .

64



1.000

.9333

.8667

.8000

.7333

.6667

.6000

.5333

.4667

.4000

.3333

.2667

.2000

.1333

.06667

.00000004500

1R

T

Z

X

Y

Z

F i g u r e A 7 3 : A c o u s t i c p r e s s u r e i n s i d e c y l i n d r i c a l s h e l l , m o d e

1 , 0 , 1 . 2 2 4 1 q u a d r a t i c W E D G E 1 5 e l e m e n t s , 6 6 0 9 n o d e s .

.00009835

.00009179

.00008524

.00007868

.00007212

.00006557

.00005901

.00005245

.00004590

.00003934

.00003278

.00002623

.00001967

.00001311

.000006557

.0000000000000

1R

T

Z

X

Y

Z

F i g u r e A 7 4 : M o d e 1 , 0 , 1 e r r o r f o r c y l i n d r i c a l u i d / s t r u c t u r e g e o m e t r y

( u i d o n l y ) . 2 2 4 1 l i n e a r W E D G E 6 e l e m e n t s 1 4 4 9 n o d e s .

65



X

Y

Z

.08239

.07690

.07141

.06592

.06042

.05493

.04944

.04394

.03845

.03296

.02746

.02197

.01648

.01099

.005493

.000000003000

X

Y

Z

F i g u r e A 7 5 : M o d e 1 , 0 , 1 e r r o r f o r c y l i n d r i c a l u i d / s t r u c t u r e g e o m e t r y ( u i d

o n l y ) . 6 7 2 q u a d r a t i c W E D G E 1 5 e l e m e n t s , 2 1 2 1 n o d e s .

.0001013

.00009457

.00008782

.00008106

.00007431

.00006755

.00006080

.00005404

.00004729

.00004053

.00003378

.00002702

.00002027

.00001351

.000006756

.0000000000000

1R

T

Z

X

Y

Z

F i g u r e A 7 6 : M o d e 1 , 0 , 1 e r r o r f o r c y l i n d r i c a l u i d / s t r u c t u r e g e o m e t r y ( u i d

p o r t i o n o n l y ) . 2 2 4 1 q u a d r a t i c W E D G E 1 5 e l e m e n t s , 6 6 0 9 n o d e s .

66



200 400 600 800 1000

10-3

10-2

10-1

Numeric and Analytic Displacements vs. Frequency, Cylindrical Geometry

Radial Displacement at location (x,y,z) a,0,L/2 (node 283)

Frequency (Hz)

R a d i a l D i s p l a c e m e n t ( l o g )

W Analytic

W Numeric

F i g u r e A 7 7 : R a d i a l d i s p l a c e m e n t a t t h e c o o r d i n a t e s r = a ; = 0 ; z =

l

2

f o r t h e

u i d / s t r u c t u r e c y l i n d e r . N A S T R A N d i r e c t f r e q u e n c y r e s p o n s e a n a l y s i s . 2 7 2 0


67



200 400 600 800 1000

10-6

10-5

10-4

10-3

10-2

10-1

100

Numeric and Analytic Acoustic Pressure Fields vs. Frequency, Cylindrical Geometry

Acoustic Pressure at location (x,y,z) a/2,0,L/2 (node 1218)

Frequency (Hz)

A c o u s t i c P r e s s u r e ( l o g )

P Analytic

P Numeric

F i g u r e A 7 8 : A c o u s t i c p r e s s u r e a t t h e c o o r d i n a t e s r =

a

2

; = 0 ; z =

l

2

f o r t h e

u i d / s t r u c t u r e c y l i n d e r . N A S T R A N d i r e c t f r e q u e n c y r e s p o n s e . 2 7 2 0

l i n e a r e l e m e n t s , 1 9 5 3 n o d e s . A n a l y t i c a n d n u m e r i c s o l u t i o n s s h o w n

68



A p p e n d i x B : F O R T R A N C o d e s

T h i s A p p e n d i x c o n t a i n s a s o u r c e c o d e l i s t i n g o f t h e F O R T R A N p r o g r a m s w r i t t e n i n

s u p p o r t o f t h i s w o r k . A b r i e f d e s c r i p t i o n o f e a c h o f t h e s e c o d e s i s s h o w n i n T a b l e 1 1 .

G e n e r a l D a t a M a n i p u l a t i o n

p r e u i d . f N A S T R A N b u l k d a t a l e i s m o d i e d s u c h t h a t i t c o n t a i n s u i d e l e m e n t s

o n l y .

p c h 2 r e s . f C o n v e r t s N A S T R A N . p c h l e t o a P A T R A N - r e a d a b l e . r e s l e . F l u i d

p r e s s u r e s a t e a c h n o d e a r e w r i t t e n a s } d i s p l a c e m e n t s } i n t h e x - d i r e c t i o n .

c o n v e r t . f C o n v e r t s N A S T R A N . p c h l e t o a T E C P L O T d a t a l e . F o r u s e w i t h

f o r c e d - r e s p o n s e a n a l y s i s

W r i t t e n f o r C u b i c G e o m e t r y M o d e l s

r e a d e r . f C o m p a r e s a n a l y t i c a n d n u m e r i c n o r m a l m o d e s s o l u t i o n s f o r u i d - o n l y c u b e

a t e a c h n o d e .

s e t m a k e r . f C r e a t e s A C M O D L c a r d f o r a c u b i c g e o m e t r y .

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

u i d / s t r u c t u r e c u b e a t e a c h n o d e .

f o r r e s . f

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

c u b i c g e o m e t r y .

W r i t t e n f o r C y l i n d r i c a l G e o m e t r y M o d e l s

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

u i d - o n l y c y l i n d e r .

c a r d . f C r e a t e s A C M O D L c a r d f o r a c y l i n d r i c a l g e o m e t r y .

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

E p s t i e n - K e n n a r d t h e o r y .

f o r c e d 2 . f

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

c y l i n d r i c a l g e o m e t r y .

T a b l e 1 1 D e s c r i p t i o n o f F O R T R A N c o d e s u s e d .

T h e c o d e s l i s t e d i n t h i s A p p e n d i x h a v e b e e n w r i t t e n s p e c i c a l l y f o r t h i s s t u d y a n d s h o u l d

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

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

69



P r o g r a m p r e u i d . f

c1234&123456789012

program pre_fluid

character*8 minus1,fst8,snd8,trd8,frt8,fth8,six8,

& sth8,eth8,nth8,tth8

character*20 zfile,ofile

minus1=’-1 ’

write(*,*)’ENTER BDF FILE NAME’

read(*,1)zfile

1 fo rm at (a )ofile=’P’//zfile

open(unit=10,file=zfile,status=’old’)

open(unit=11,file=ofile,status=’unknown’)

100 read(10,1121,end=200)fst8,snd8,trd8,frt8,fth8,six8

& , st h8, et h8 ,nt h8 ,t th 8

if(fst8.eq.’GRID ’)then

write(11,1121)fst8,snd8,trd8,frt8,fth8,six8,minus1,

& eth8,nth8,tth8

elseif(fst8.eq.’MAT1 ’)then

fst8=’MAT10 ’

trd8=’ ’

frt8=’1.170E-7’

fth8=’13620.0 ’

write(11,1121)fst8,snd8,trd8,frt8,fth8,six8

elseif(fst8.eq.’PSOLID ’)then

frt8=’ ’

fth8=’ ’

six8=’ ’

sth8=’ ’

eth8=’PFLUID ’

nth8=’ ’

tth8=’ ’

write(11,1121)fst8,snd8,trd8,frt8,fth8,six8,sth8,

& eth8,nth8,tth8

elseif(fst8(1:6).eq.’ASSIGN’)then

elseif(fst8(1:4).eq.’TIME’)then

write(11,*)’TIME 600’

do 110 ii=1,600

read(10,1121,end=200)fst8,snd8,trd8,frt8,fth8,six8

& , st h8, et h8 ,nt h8 ,t th 8

if(fst8(1:4).eq.’CEND’)goto 120

110 continue

120 write(11,*)’CEND’

elseif(fst8.eq.’ METHO’)then

write(11,*)fst8,’D(STRUCT)’,snd8(2:8)

write(11,*)fst8,’D(FLUID)’,snd8(2:8)

elseif(fst8.eq.’ VECTO’)then

write(11,*)’ DISPLACEMENT(SORT1,PUNCH) = ALL’

else

write(11,1121)fst8,snd8,trd8,frt8,fth8,six8,sth8,

& eth8,nth8,tth8

endif

goto 100

200 close(10)

close(11)

1121 format(10a8)

stop

end

P r o g r a m p c h 2 r e s . f

C1234&123456789012

program pch2res

character*72 title,subtitle,label,analysistype,datatype

character*30 ifile,ofile

character*15 subcase,eigen

character*8 icase,inumber,sn,mn

character*6 mode,cont

write(*,*)’ENTER PUNCH FILE’

read(*,1)ifile

write(*,*)’ENTER NUMBER OF DATA POINTS’

read(*,*)numdata

open(unit=11,file=ifile,status=’old’)

1101 read(11,2,end=205)title,iline

read(11,2)subtitle,iline

read(11,2)label,iline

read(11,2)analysistype,iline

read(11,2)datatype,iline

1 fo rm at (a )

2 format(a72,i8)

c

read(11,3)subcase,icase,iline

read(11,4)eigen,freq,mode,inumber,iline

3 format(a15,5x,a8,44x,i8)

4 format(a15,E13.4,2x,a6,a8,28x,i8)

if(analysistype(2:11).eq.’EIGENVECTO’)then

c extract job name

i=0

105 i=i+1

if(ifile(i:i).ne.’.’.and.ilen.lt.20)then

ofile(1:i)=ifile(1:i)

ilen=i

goto 105

else

endif

c extract mode number

i=0

im=0

106 i=i+1

if(inumber(i:i).ne.’ ’.and.im.le.8)then

im=im+1mn(im:im)=inumber(i:i)

goto 106

elseif(i.lt.8)then

goto 106

else

endif

c extract subcase number

i=0

is=0

107 i=i+1

if(icase(i:i).ne.’ ’.and.is.le.8)then

is=is+1

sn(is:is)=icase(i:i)

goto 107

elseif(i.lt.8)then

goto 107

else

endif

c create output file name

write(*,*)inumber,mn

write(*,*)icase,sn

ofile=ifile(1:ilen)//’_mode’//mn(1:im)//’.dis.’//sn(1:is)else

write(*,*)analysistype(1:11)

ofile=’error.dis.1’

endif

c

defmax=1

nwidth=6

ndmax=int(numdata/2)

open(unit=12,file=ofile,status=’unknown’)

write(12,1003)eigen,freq

write(12,1111)numdata,numdata,defmax,ndmax,nwidth

write(12,5)title

write(12,5)subtitle

5 f or ma t( a7 2)

1003 format(a15,E13.4)

do 100 i=1,numdata

read(11,1001)cont,node,typevar,a1,a2,a3,iline

read(11,1002)cont,a4,a5,a6,iline

write(12,1112)node,a1,a2,a3,a4,a5,a6

1111 format(2i9,e15.9,2i9)

1112 format(i8,(5e13.7))1001 format(a6,i8,a4,3(5x,e13.4),i8)

1002 format(a6,12x,3(5x,e13.4),i8)

100 continue

close(12)

goto 1101

205 close(11)

stop

end

P r o g r a m c o n v e r t . f

program convert

c

c Program to convert xxx.pch file from NASTRAN forced-response

c analysis to PATRAN-readable XY-Plot files or TECPLOT

c formatted data files.

c

c Input consists of punch file from NASTRAN forced-response

c analysis. The total number of nodes (maxn) in the model, the

c number of nodes calculated (iwant), the number of frequencies

c (maxfreq) and the output format must be programed prior to

c compiling.

c

c Output consists of XY-Plot or TECPLOT data for displacement

c magnitudes at a point vs. frequency. XYPlot files contain

c magnitude or phase information for all nodes. Tecplot files

c contain infromation for phase and magnitude for only one node.

c

c Written by C.M.Fernholz 48221

c

c Declarations

c

implicit real*8(a-h,o-y)

70



implicit complex(z)

c

parameter (maxn=5643,maxfreq=126,iwant=3,pi=3.141592654)

c

character*72 title,subtitle,label,analysistype,datatype

character*30 ifile,ofileRe,ofileIm,ofileu,ofilev,ofilew,

& ofilex,ofiley,ofilez

character*15 subcase, point,displace

character*8 icase,inumber,sn,mn


character*4 typevarc

dimension disr(6),disi(6),frequency(maxfreq),node(maxn),

& freqRe(maxfreq,maxn,6),freqIm(maxfreq,maxn,6),

& udata(maxn,maxfreq),utheta(maxn,maxfreq),

& vdata(maxn,maxfreq),vtheta(maxn,maxfreq),

& wdata(maxn,maxfreq),wtheta(maxn,maxfreq),

& uphase(maxn,maxfreq),vphase(maxn,maxfreq),

& w ph as e( ma xn ,m ax fr eq ), si ze (3 )

c

logical patran,tecplot

c

c Begin program

c

c Choose output type

c

patran=.FALSE.

tecplot=.TRUE.

c

write(*,*)’Enter punch file name’

read(*,1)ifile

1 fo rm at (a )

cc Open punch file

open(unit=10,file=ifile,status=’old’)

c

do 50 i=1,iwant

c

c Read headers from punch file

c

read(10,1000)title,iline




read(10,1000)datatype,iline

read(10,1005)subcase,icase,iline

read(10,1010)point,node(i)

c

c Read in frequencies, displacements

c

do 100 j=1,maxfreq

c

read(10,1100)frequency(j),typevar,

& (disr(k),k=1,3),iline

read(10,1105)cont,(disr(k),k=4,6),ilineread(10,1105)cont,(disi(k),k=1,3),iline

read(10,1105)cont,(disi(k),k=4,6),iline

c

do 9 k=1,3

zdisp=cmplx(disr(k),disi(k))

partone=real(zdisp)

parttwo=imag(zdisp)

size(k)=SQRT(partone*partone+parttwo*parttwo)

9 continue

c

c Store real and imaginary displacements for each frequency in an

c array. Note: used if PATRAN fringe plots of displacements for

c a given frequency are desired.

c

do 160 k=1,6

freqRe(j,i,k)=disr(k)

freqIm(j,i,k)=disi(k)

160 continue

c

c Store displacement magnitudes for each node

c

if (patran) thenudata(i,j)=size(1)

vdata(i,j)=size(2)

wdata(i,j)=size(3)

endif

c

if (tecplot) then

udata(node(i),j)=size(1)

vdata(node(i),j)=size(2)

wdata(node(i),j)=size(3)

endif

c

c Store displacement phase angles for each node

c

if (disr(1).EQ.0.0) then

if (disi(1).GT.0.0) then

utheta(i,j)=90.0

elseif (disi(1).LT.0.0) then

utheta(i,j)=-90.0

elseif (disi(1).EQ.0.0) then

utheta(i,j)=0.0

endif

else

utheta(i,j)=(180.0/pi)*ATAN(disi(1)/disr(1))

endif

c

if (disr(2).EQ.0.0) thenif (disi(2).GT.0.0) then

vtheta(i,j)=90.0


vtheta(i,j)=-90.0

else

vtheta(i,j)=0.0

endif

else

vtheta(i,j)=(180.0/pi)*ATAN(disi(2)/disr(2))

endif

c

if (disr(3).EQ.0.0) then

if (disi(3).GT.0.0) then

wtheta(i,j)=90.0


wtheta(i,j)=-90.0

else

wtheta(i,j)=0.0

endif

else

wtheta(i,j)=(180.0/pi)*ATAN(disi(3)/disr(3))

endifc

if (tecplot) then

uphase(node(i),j)=utheta(i,j)

vphase(node(i),j)=vtheta(i,j)

wphase(node(i),j)=wtheta(i,j)

endif

c

100 continue

50 continue

c

close(10)

c

c Dummy print

write(*,*)title

c

c Magnitude output files for u,v, and w displacements

c

if (patran) then

ofileu=’magu.xyd’

ofilev=’magv.xyd’

ofilew=’magw.xyd’

elseif (tecplot) thenofileu=’magu.plt’

ofilev=’magv.plt’

ofilew=’magw.plt’

endif

c

c Phase angle ouput files for u,v, and w displacements

c

if (patran) then

ofilex=’phau.xyd’

ofiley=’phav.xyd’

ofilez=’phaw.xyd’

elseif (tecplot) then

ofilex=’phau.plt’

ofiley=’phav.plt’

ofilez=’phaw.plt’

endif

c

open(unit=40,file=ofileu,status=’unknown’)

open(unit=50,file=ofilev,status=’unknown’)

open(unit=60,file=ofilew,status=’unknown’)

open(unit=70,file=ofilex,status=’unknown’)

open(unit=80,file=ofiley,status=’unknown’)open(unit=90,file=ofilez,status=’unknown’)

c

if (patran) then

do 300 l=1,iwant

c

write(40,1300)’XYDATA,U-DISP, NODE’,node(l)

write(50,1300)’XYDATA,V-DISP, NODE’,node(l)

write(60,1300)’XYDATA,W-DISP, NODE’,node(l)

write(70,1300)’XYDATA,U-PHASE NODE’,node(l)

write(80,1300)’XYDATA,V-PHASE NODE’,node(l)

write(90,1300)’XYDATA,W-PHASE NODE’,node(l)

c

do 350 m=1,maxfreq

write(40,1305)frequency(m),udata(l,m)

write(50,1305)frequency(m),vdata(l,m)

71



write(60,1305)frequency(m),wdata(l,m)

write(70,1305)frequency(m),utheta(l,m)

write(80,1305)frequency(m),vtheta(l,m)

write(90,1305)frequency(m),wtheta(l,m)

350 continue

c

300 continue

c

do 14 n=40,90,10

write(n,1310)’END’

14 continuec

elseif (tecplot) then

c

write(40,1400)’TITLE = "Magnitude for u-displacements"’

write(50,1400)’TITLE = "Magnitude for v-displacements"’

write(60,1400)’TITLE = "Magnitude for w-displacements"’

write(70,1401)’TITLE = "Phase Angle for u-displacements"’

write(80,1401)’TITLE = "Phase Angle for v-displacements"’

write(90,1401)’TITLE = "Phase Angle for w-displacements"’

c

do 11 n=40,60,10

write(n,1405)’VARIABLES = "Frequency","Displacement"’

write(n,1410)’ZONE T="Numeric Solution", I =’,maxfreq

11 continue

c

do 12 n=70,90,10

write(n,1406)’VARIABLES = "Frequency","Phase"’

write(n,1411)’ZONE T="Phases", I =’,maxfreq

12 continue

c

write(*,*)’Enter desired node number for analysis’

read(*,2)l2 format(i)

c

do 450 m=1,maxfreq

write(40,1415)frequency(m),udata(l,m)

write(50,1415)frequency(m),vdata(l,m)

write(60,1415)frequency(m),wdata(l,m)

write(70,1415)frequency(m),uphase(l,m)

write(80,1415)frequency(m),vphase(l,m)

write(90,1415)frequency(m),wphase(l,m)

450 continue

c

endif

c

do 13 n=40,90,10

close(n)

13 continue

c

1000 format(a72,i8)

1005 format(a15,5x,a8,44x,i8)

1010 format(a13,5x,i8,46x,i8)

1100 format(4x,e13.4,a1,3(5x,e13.4),i8)

1105 format(a6,12x,3(5x,e13.4),i8)1200 format(a15,e13.4)

1205 format(2i9,e15.9,2i9)

1210 format(a72)

1215 format(i8,(5e13.7)/e13.7)

1300 format(a19,i4)

1305 format(f10.3,2x,f10.6)

1310 format(a3)

1400 format(a39)

1401 format(a41)

1405 format(a38)

1406 format(a31)

1410 format(a30,i5)

1411 format(a20,i5)

1415 format(f10.3,2x,e13.5)

1420 format(a25,i5)

c

9999 stop

end

P r o g r a m r e a d e r . f

program reader

c

c Program to compare analytical and numerical displacement

c calculations for cubic geometry, fluid only.

c

c Input files include:

c NASTRAN-generated neutral file: used to determine

c geometric location of each node in the model. Neutral

c file should contain only node location information.

c NASTRAN-generated punch file: used to determine the

c displacement of each node, as calculated by NASTRAN.

c

c Output files:

c read.out: includes information about the maximum

c displacement error for each direction and the maximum

c displacement error magnitude for the system.

c error.dis.1: PATRAN-readable displacement file that

c can be used to display the displacement errors for the

c m od el in fr in ge p lo t f or m.

c

c It is necessary to specify the number of nodes (maxn) and the

c number of elements (maxe) in the model prior to compiling.

c

c Declarations

c

implicit real*8(a-h,o-z)c

parameter (maxn=4961, maxe=1000)

c

character*15 zfile,pchfile,eigen

character*8 count,typevar

character za(maxn)*1

c

dimension x(maxn),y(maxn),z(maxn),

& aprod(maxn),

& p x( ma xn ), py (m ax n) ,p z( ma xn ),

& nod e( ma xn) ,e rr or( ma xn ),

& ndf(maxn),junk(6)

c

c Begin program

c

write(*,*)’Enter neutral file name’

read(*,1)zfile

write(*,*)’Enter punch file name’

read(*,1)pchfile

1 f or mat (a )

c

c Define geometry (cube)c

side=5.0

pi=3.141592654

idn=1.0

c

c Mode shape of interest (nnn=x-dir,mmm=y-dir,kkk=z-dir)

c

nnn=1

mmm=1

kkk=-3

c

c Open neutral file


c Open punch file

open(unit=20,file=pchfile,status=’old’)

c Open output file

open(unit=30,file=’read.out’,status=’unknown’)

c

c Sort loop to determine maximum displacement in punch file

c Punch file should contain data only for mode of interest

c

bigx=0bigy=0

bigz=0

c

do 50 j=1,(maxn-1)

c

read(20,1004) cont,node(j),typevar,

& px(j),py(j),pz(j),iline

read(20,1005) cont,a4,a5,a6,iline

c

if (ABS(px(j)).GT.ABS(bigx)) bigx=px(j)

if (ABS(py(j)).GT.ABS(bigy)) bigy=py(j)

if (ABS(pz(j)).GT.ABS(bigz)) bigz=pz(j)

c

50 continue

c

if (bigx.EQ.0.0) bigx=1.0

if (bigy.EQ.0.0) bigy=1.0

if (bigz.EQ.0.0) bigz=1.0

c

write(30,1)’Error: Analytic vs. Numeric Solutions’

write(30,1015)’Cubic geometry, mode ’,nnn,mmm,kkk

write(30,1010)’bigx = ’,bigxwrite(30,1010)’bigy = ’,bigy

write(30,1010)’bigz = ’,bigz

c

rewind(unit=20)

c

100 read(10,1000)idpacket,idn,iv,kc,n1,n2,n3,n4,n5

c

if (idpacket.eq.99) then

close(10)

close(20)

goto 500

c

elseif (idpacket.eq.1) then

c

c Determine location of grid point

72



read(10,1001) x(idn),y(idn),z(idn)

read(10,1002) icf,za(idn),ndf(idn),ncnfig,ncid,

& (junk(i),i=1,6)

c

c Determine analytic displacement of grid point, mode (1,1,1)

ax=sin((nnn*pi*x(idn))/side)

ay=sin((mmm*pi*y(idn))/side)

az=sin((kkk*pi*z(idn))/side)

aprod(idn)=ax*ay*az

c

c Determne grid point displacement from punch fileread(20,1004) cont,node(idn),typevar,

& px(idn),py(idn),pz(idn),iline

read(20,1005) cont,a4,a5,a6,iline

px(idn)=px(idn)/bigx

py(idn)=py(idn)/bigy

pz(idn)=pz(idn)/bigz

c

else

write(*,*)’Invalid packet ID’

stop

endif

c

goto 100

c

c Dummy print

500 write(*,*)’write to read.out completed’

c

c Calculate displacement error at each node, max error for system

c

bigerror=0

c

do 250 m=1,maxnc

error(m)=aprod(m)-px(m)

c

if (ABS(error(m)).GT.bigerror) then

bigerror=error(m)

endif

c

250 continue

c

write(30,1010)’Max error ’,bigerror

c

close(30)

c

c Write PATRAN input file

c

eigen=’$EIGENVALUE =’

freq=bigerror

defmax=1

nwidth=6

ndmax=int(maxn/2)

c

open(unit=40,file=’error.dis.1’,status=’unknown’)write(40,1100)eigen,freq

write(40,1101)maxn,maxn,defmax,ndmax,nwidth

write(40,1102)’$TITLE GOES HERE’

write(40,1102)’$SUBTITLE=LOAD_CASE_ONE’

c

do 200 l=1,maxn

write(40,1103)node(l),error(l),0.0,0.0,0.0,0.0,0.0

200 continue

c

close(40)

c

1000 format(i2,8i8)

1001 format(3e16.9)

1002 format(i1,1a1,3i8,2x,6i1)

1003 format(4e16.9)

1004 format(a6,i8,a4,3(5x,e13.4),i8)

1005 format(a6,12x,3(5x,e13.4),i8)

1010 format(a10,e18.9)

1015 format(a21,i2,ix,i2,1x,i2)

1100 format(a15,e13.4)

1101 format(2i9,e15.9,2i9)

1102 format(a72)1103 format(i8,(5e13.7))

c

stop

end

P r o g r a m s e t m a k e r . f

program setmaker

c

c Program to produce NASTRAN ACMODL set cards for a cubic geometry

c as well as fluid/solid grid set. If grid point is determined to

c be a fluid point, grid co-ordinate remains "-1". If grid point

c is a solid, co-ordinate ID is changed to "0." (solid).

c

c Input consists of original GRID data only from the NASTRAN

c bulk data file. It is necessary to specifiy the total number

c of grids in the model (maxn) prior to compiling.

c

c Output consists of two files:

c grids.out: contains fluid/solid grid points for .bdf file

c s et 55 5: c on ta in s A CM OD L s et c ar ds .

c In the ACMODL output file, SET1=555 contains the solid grid

c points and SET1=666 contains the fluid grid points. Line

c continuation markers start at "AAAAAAA".

cc Assumptions: Structure nodes are assumed to lie in two planes

c only; one at z=0 and the other at z=5. If a fluid node as a z

c coordinate of either 0 or 5, it is assumed to lie on the fluid/

c structure interface and will be included in the ACMODL card

c set. It is assumed that every structure node is on the fluid/

c structure interface. The structure nodes are assumed to be

c sequential (ie xxx THRU xxx).

c

c Program written by C.M.Fernholz (48221)

c

c Declarations

c

implicit real*8(a-h,o-z)

c

parameter (maxn=5643)

c

character*8 grid,set,thru,cont,con2

character*15 gridfile,yfile,zfile

c

logical test,check

c

dimension inode(maxn),m(8)c

c Begin program

c

write(*,*)’Enter GRID file name’

read(*,2)gridfile

write(*,*)’Enter minimum structure grid point ID’

read(*,1)nodemin

write(*,*)’Enter maximum structure grid point ID’

read(*,1)nodemax

1 f or mat (i )

2 f or mat (a )

c

yfile=’grids.out’

zfile=’set555’

c

c Open file containing grid information

open(unit=10,file=gridfile,status=’old’)

c Open file for ACMODL gridset

open(unit=20,file=zfile,status=’unknown’)

c Open file for grids output

open(unit=30,file=yfile,status=’unknown’)

cc Read grid file, determine fluid and solid grids, write new gridset

c

do 20 i=1,maxn

read(10,1000)grid,node,xx,yy,zz,fs

if (node.LT.nodemin.OR.node.GT.nodemax) then

fs=-1

write(30,1000)grid,node,xx,yy,zz,fs

elseif (node.GE.nodemin.OR.node.LE.nodemax) then

fs=0

write(30,1000)grid,node,xx,yy,zz,fs

else

write(*,*)’WARNING: Grid type indeterminate’

endif

20 continue

c

close(10)

rewind(unit=30)

c

c Write structure SET card

c

isset=555

set=’SET1’t hr u= ’ T HR U’

write(20,1005)set,isset,nodemin,thru,nodemax

c

c Read grid file, determine which fluid points lie on interface

c

jj=0

do 50 i=1,maxn

c

read(30,1000)grid,node,xx,yy,zz,fs

c

if (node.LT.nodemin.OR.node.GT.nodemax) then

c

if (zz.EQ.(0.0).OR.zz.EQ.(5.0)) then

jj=jj+1

inode(jj)=node

73



endif

c

endif

c

50 continue

c

close(30)

c

c Write fluid SET card

c

55 ifset=666c

c Logic for line continuation markers

c Note: "A" = char(65), "+" = char(43)

c

m(1)=43

do 60 i=2,8

m(i)=65

60 continue

c

cont=char(m(1))//char(m(2))//char(m(3))//char(m(4))//

& char(m(5))//char(m(6))//char(m(7))//char(m(8))

con2=cont

c

c Write first seven fluid points to SET card (first line)

write(20,1010)set,ifset,(inode(k),k=1,7),cont

c

c Write groups of remaining fluid grid nodes eight at a time

c

imax=INT((jj-7)/8)

ithing=8

icount=1

check=.TRUE.c

100 if (check) then

cont=con2

test=.TRUE.

80 if (test) then

i=8

if (m(i).GE.90) then

m(i)=65

m(i-1)=m(i-1)+1

if (m(i-1).LT.90) then

test=.FALSE.

else

i=i-1

if (i.LT.2) then

write(*,*)’Too many grid points’

goto 999

endif

endif

else

m(i)=m(i)+1

test=.FALSE.

endif85 goto 80

endif

c

con2=char(m(1))//char(m(2))//char(m(3))//char(m(4))//


c

write(20,1015)cont,(inode(k),k=ithing,ithing+7),con2

c

ithing=ithing+8

icount=icount+1

if (icount.GT.imax) then

check=.FALSE.

endif

goto 100

endif

c

c Write last line of file

c

cont=con2

write(20,1020)cont,(inode(k),k=ithing,jj)

c

close(20)c

1000 format(a8,i8,8x,3F8.4,i8)

1005 format(a8,2i8,a8,i8)

1010 format(a8,i8,7i8,a8)

1015 format(a8,8i8,a8)

1020 format(a8,8i8)

c

999 stop

end

P r o g r a m p l a t e . f

program plate

c

c Program to compare analytic and numeric displacement

c calculations for the cubic fluid/structure geometry.

c Errors for w-displacement and pressure are determined.

c

c Input files include:

c PATRAN-generated punch (.pch) file. Numeric dis-

c placements for each node are read from this file.

c PATRAN-generated neutral (.out) file containing

c node information only. Geometric node locations

c are read from this file.

cc Output files include:

c error.out: contains information about the maximum

c error occuring anywhere in the model.

c error.dis.1: PATRAN-readable displacement file that

c can be used to display the error results for the

c model in fringe plot form.

c

c It is necessary to specify the number of nodes (maxn),

c the number of elements (maxe), and the mode shape of

c interest prior to compiling.

c

c Assumptions:

c Analytic model uses an uncoupled solution. NASTRAN

c uses a coupled one. For the first structural mode,

c coupling will occur between the two plates of the

c model. The mode shape of the plate that is moving

c will be superimposed (with some attenuation of the

c amplitude) upon the other plate. To account for this,

c displacement amplitudes in the two plates are handled

c seperately in this code. The rear plate (at z=0) is

c assumed to be stationary. The error.dis file can be

c modified to delete the nodes corresponding to thec rear plate.

c

c Written by C.M.Fernholz (48221)

c

c Declarations

c

implicit real (a-h,o-z)

c

parameter (maxn=1573,maxe=1200,pi=3.141592654)

c

character*72 title,subtitle,label,analysistype,

& datatype,header

character*30 pchfile,neufile,header2

character*15 subcase,eigen

character*8 icase,an,mn


character*5 solidtype,fluidtype

character*1 flustr

c

dimension px(maxn),py(maxn),pz(maxn),node(maxn),

& w(maxn),p(maxn),serror(maxn),ferror(maxn),

& nodefluid(maxn),nodesolid(maxn),& x(m ax n) ,y( ma xn ),z (m ax n),

& junk(6),itrash(9)

c

logical fluid

c

c Begin program

c

c Choose mode shape of interest (m=x-dir,n=y-dir,k=z-dir)

c

m=-2

n=2

k=0

c

c Define cubic geometry

c

side=5.0

thick=0.0625

c

c FEM geometry

c

ii=0

jj=0fluidtype=’HEX8’

solidtype=’QUAD4’

c

write(*,*)’Fluid or structure mode? (f/s)’

read (*,1)flustr

write(*,*)’Enter desired mode (eigenvalue) number’

read (*,2)modenumber

write(*,*)’Enter punch (.pch) file name’

read (*,1)pchfile

write(*,*)’Enter neutral (.out) file name’

read (*,1)neufile

1 f or mat (a )

c

if (flustr.EQ.’f’) then

fluid=.TRUE.

74



elseif (flustr.EQ.’s’) then

fluid=.FALSE.

endif

c

c Open punch file

open (unit=10,file=pchfile,status=’old’)

c Open neutral file

open (unit=20,file=neufile,status=’old’)

c Open error output data file

open (unit=30,file=’error.out’,status=’unknown’)

cwrite(*,*)’Enter min structure grid ID for front plate’

read (*,2)nodeminf

write(*,*)’Enter max structure grid ID for front plate’

read (*,2)nodemaxf

write(*,*)’Enter min structure grid ID for back plate’

read (*,2)nodeminb

write(*,*)’Enter max structure grid ID for back plate’

read (*,2)nodemaxb

2 fo rm at (i )

c

c Read displacements from punch file

c

100 bigx=0.0

bigy=0.0

bigzf=0.0

bigzb=0.0

c

read(10,1000)title,iline




read(10,1000)datatype,ilineread(10,1005)subcase,icase,iline

read(10,1010)eigen,freq,mode,inumber,iline

c

do 150 j=1,maxn

c

read(10,1015)cont,node(j),typevar,px(j),py(j),pz(j),

& iline

read(10,1020)cont,a4,a5,a6,iline

c

if (node(j).LT.nodeminb.OR.node(j).LT.nodeminf.OR.

& node(j).GT.nodemaxb.OR.node(j).GT.nodemaxf) then


endif

c


c

if (node(j).LE.nodemaxb.AND.node(j).GE.nodeminb) then

if (ABS(pz(j)).GT.ABS(bigzb)) bigzb=pz(j)

elseif (node(j).LE.nodemaxf.AND.node(j).GE.nodeminf) then

if (ABS(pz(j)).GT.ABS(bigzf)) bigzf=pz(j)

endif

c150 continue

c

if (inumber.NE.modenumber) goto 100

c

close(10)

c



if (bigzf.EQ.0.0) bigzf=1.0

if (bigzb.EQ.0.0) bigzb=1.0

c

c Write to error.out file

c

write(30,1100)’Analytic vs. Numeric Solutions’

write(30,1100)’Cubic Fluid/Structure Geometry’

write(30,1105)maxe,solidtype,’,’,fluidtype,’elements’

write(30,1110)maxn,’nodes’

c

if (fluid) then

write(30,1130)’Fluid analysis’

elseif (.NOT.fluid) then

write(30,1130)’Solid analysis’endif

c

write(30,1115)’bigx = ’,bigx

write(30,1115)’bigy = ’,bigy

write(30,1115)’bizgf= ’,bigzf

write(30,1115)’bigzb= ’,bigzb

c

c Normalize displacements

c

do 200 j=1,maxn

px(j)=px(j)/ABS(bigx)

py(j)=py(j)/ABS(bigy)

if (node(j).LE.nodemaxb.AND.node(j).GE.nodeminb) then

pz(j)=pz(j)/bigzb

elseif(node(j).LE.nodemaxf.AND.node(j).GE.nodeminf)then

pz(j)=pz(j)/bigzf

endif

200 continue

c

c Read grid point locations from neutral file

c

read(20,1200)(itrash(i),i=1,9)

read(20,1205)header

read(20,1200)(itrash(i),i=1,9)

read(20,1210)header2

c300 read(20,1200)idpacket,idn,iv,kc,n1,n2,n3,n4,n5

c

if (idpacket.EQ.99) then

close(20)

goto 350

c

elseif (idpacket.EQ.1) then

c

read(20,1215)x(idn),y(idn),z(idn)

read(20,1220)icf,za,ndf,ncnfig,ncid,(junk(i),i=1,6)

c

if (idn.LE.nodemaxb.AND.idn.GE.nodeminb) then

c

ii=ii+1

nodesolid(ii)=idn

w(ii)=0.0

c

serror(ii)=w(ii)-pz(idn)

c

elseif (idn.LE.nodemaxf.AND.idn.GE.nodeminf) then

c

ii=ii+1nodesolid(ii)=idn

ax=sin(m*pi*x(idn)/side)

ay=sin(n*pi*y(idn)/side)

w(ii)=ax*ay

c

serror(ii)=w(ii)-pz(idn)

c

else

c

jj=jj+1

nodefluid(jj)=idn

ax=sin(m*pi*x(idn)/side)

ay=sin(n*pi*y(idn)/side)

az=cos(k*pi*z(idn)/side)

p(jj)=ax*ay*az

c

ferror(jj)=p(jj)-px(idn)

c

endif

c

else

write(*,*)’WARNING: Invalid packet ID’c

endif

c

goto 300

c

c Determine maximum errors in model

c

350 bigerrorb=0.0

bigerrorf=0.0

c

do 400 i=1,ii

if (nodesolid(i).LE.nodemaxb.AND.

& nodesolid(i).GE.nodeminb) then

if (ABS(serror(i)).GT.ABS(bigerrorb)) then

bigerrorb=serror(i)

endif

elseif (nodesolid(i).LE.nodemaxf.AND.

& n ode so li d(i ). GE .no de mi nf) t he n

if (ABS(serror(i)).GT.ABS(bigerrorf)) then

bigerrorf=serror(i)

endif

endif400 continue

c

write(30,1120)’Maximum error in front plate’,bigerrorf

write(30,1120)’Maximum error in rear plate ’,bigerrorb

c

bigerror=0.0

c

do 450 i=1,jj

if (ABS(ferror(i)).GT.ABS(bigerror)) then

bigerror=ferror(i)

endif

450 continue

c

write(30,1125)’Maximum error in fluid’,bigerror

c

75



close(30)

c

c Open error.dis file

open (unit=40,file=’error.dis.1’,status=’unknown’)

c


defmax=1

nwidth=6

ndmax=INT(maxn/2)

c

if (.NOT.fluid) thensubtitle=’$SUBTITLE = STRUCTURE ERROR’

elseif (fluid) then

subtitle=’$SUBTITLE = FLUID ERROR’

endif

c



write(40,1310)title

write(40,1310)subtitle

c

if (.NOT.fluid) then

c

do 500 i=1,ii

write(40,1315)nodesolid(i),0.0,0.0,

& serror(i),0.0,0.0,0.0

500 continue

c

elseif (fluid) then

c

do 550 i=1,jj

write(40,1315)nodefluid(i),ferror(i),

& 0.0,0.0,0.0,0.0,0.0550 continue

c

endif

c

close(40)

c

1000 format(a72,i8)

1005 format(a15,5x,a8,44x,i8)

1010 format(a15,e13.4,2x,a6,i8,28x,i8)

1015 format(a6,i8,a4,3(5x,e13.4),i8)

1020 format(a6,12x,3(5x,e13.4),i8)

1100 format(a30)

1105 format(i4,1x,a5,a1,1x,a5,1x,a8)

1110 format(i4,1x,a5)

1115 format(a7,f)

1120 format(a28,1x,f)

1125 format(a22,1x,f)

1130 format(a14)

1200 format(i2,8i8)

1205 format(a72)

1210 format(a30)

1215 format(3e16.9)1220 format(i1,1a1,3i8,2x,6i1)

1300 format(a15,e13.4)

1305 format(2i9,e15.9,2i9)

1310 format(a72)

1315 format(i8,(5e13.7))

c

stop

end

P r o g r a m f o r r e s . f

program forres

c

c Program to determine the analytic solution for the forced

c frequency response of a cubic fluid/structure geometry.

c Model consists of fluid cube with sides of length A having

c structrual plates located at z = -A/2,A/2.

c

c No input files required. Fluid and structure material

c properties must be programmed prior to compiling.

c

c Output consists of five files:

c struc.eig: Eigen values of the structure

c sfreq.plt: Disp. vs. freq. of the structure at a point

c sphas.plt: Phase angle vs. freq. of the struct at a point

c pfreq.plt: Pressure vs. freq. of a point in the fluid

c pphas.plt: Phase angle vs. freq. of the fluid at a point

c Output files are written in TECPLOT format.

c

c A non-coupled model is assumed. Structure obeys non-homogeneous

c plate equation of motion. Fluid obeys homogeneous wave equation.

c


c

c Declarations

c

implicit real (a-h,o-y)

implicit complex (z)

c

parameter (nfreq=1000,step=5.0,pi=3.141592654)

c

dimension zqmn(30,30), zdisp(nfreq),

& z qp re ss (3 0, 30 ), zp re ss (n fr eq )

c

logical once,print

c

c Begin programc

once=.TRUE.

print=.TRUE.

c

c Fluid and structure material properties

c

c Two pi

twopi=2.0*pi

c Length of a side (inches)

A=5.0

c Thickness of plates (inches)

h=0.0625

c Young’s Modulus for the plates (psi)

E=10.3e6

c Density of the plate (slugs/in**3)

rhos=2.5383e-4

c Damping coefficient

eta=0.005

c Amplitude of input force (lbs)

F=5.0

c Poission ratio for plate

uu=0.334c Speed of sound in fluid (in/sec)

co=13620.0

c Density of the fluid (slugs/in**3)

rhof=1.17e-7

c

c Structure stiffness

d=(E*h**3)/(12*(1-uu*uu))

c

c Point of interest

c

write(*,2)’Enter x co-ordinate’

read(*,1)x

write(*,*)’Enter y co-ordinate’

read(*,1)y

write(*,2)’Enter z co-ordinate’

read(*,1)ez

write(*,*)’Entered: ’,x,y,ez

1 f or ma t( f4 .2 )

2 f or mat (a )

c

freq=0.0

skip=step*twopizo=(0.0,0.0)

zi=(0.0,1.0)

c

open(unit=10,file=’struct.eig’,status=’unknown’)

open(unit=20,file=’sfreq.plt’,status=’unknown’)

open(unit=25,file=’sphas.plt’,status=’unknown’)

open(unit=30,file=’pfreq.plt’,status=’unknown’)

open(unit=35,file=’pphas.plt’,status=’unknown’)

c

do 100 i=1,nfreq

c

freq=i*skip

zdisp(i)=zo

zpress(i)=zo

c

do 200 m=1,20

do 250 n=1,20

c

wo=((pi*pi*(n*n+m*m))/(A*A))*SQRT(d/(rhos*h))

c

if (print) then

write(10,1000)’Mode’,m,’,’,n,’and& Frequency’,wo/twopi

endif

c

zdenom=cmplx(wo*wo-freq*freq,2.0*eta*wo*freq)

zqmn(m,n)=4.0*F*sin(m*pi/2.0)*sin(n*pi/2.0)/

& (A*A*rhos*h*zdenom)

zdisp(i)=zdisp(i)+zqmn(m,n)*

& sin(m*pi*x/A)*sin(n*pi*y/A)

c

alf=-(freq/co)*(freq/co)+

& (n*pi/A)*(n*pi/A)+

& (m*pi/A)*(m*pi/A)

if (alf.LT.0.0) then

alf=SQRT(ABS(alf))

76



zalpha=cmplx(0.0,alf)

else

alf=SQRT(alf)

zalpha=cmplx(alf,0.0)

endif

c

zqpress(m,n)=-(rhof*freq*freq*zqmn(m,n))/zalpha

zpress(i)=zpress(i)+zqpress(m,n)*

& sin(m*pi*x/A)*

& sin(n*pi*y/A)*

& (csin(zalpha*ez)+& (1 +cc os (z alp ha *A ))* cc os (za lp ha *e z)/

& csin(zalpha*A))

c

250 continue

200 continue

c

call tecplot(freq,zpress(i),zdisp(i),

& frequency,pmag,ppha,dmag,dpha)

c

if (once) then

write(20,2000)’TITLE = "Analytic W-Displacements"’

write(25,2000)’TITLE = "Analytic W-Phase Angles "’

write(30,2000)’TITLE = "Analytic U-Displacements"’

write(35,2000)’TITLE = "Analytic U-Phase Angles "’

c

write(20,2005)’VARIABLES = "Frequency","Magnitude"’

write(25,2010)’VARIABLES = "Frequency","Phase"’

write(30,2005)’VARIABLES = "Frequency","Magnitude"’

write(35,2010)’VARIABLES = "Frequency","Phase"’

c

do 300 j=20,35,5

write(j,2015)’ZONE T="Analytic Solution"’300 continue

c

once=.FALSE.

endif

c

write(20,2020)frequency,dmag

write(25,2020)frequency,dpha

write(30,2020)frequency,pmag

write(35,2020)frequency,ppha

c

print=.FALSE.

100 continue

c

close(10)

do 305 j=20,35,5

close(j)

305 continue

c

1000 format(a4,1x,i2,a1,i2,1x,a13,1x,f10.2)

2000 format(a34)

2005 format(a35)

2010 format(a31)2015 format(a26)

2020 format(f10.2,1x,e13.5)

c

stop

end

c

c *****************************************************************

subroutine tecplot(freq,zpres,zdisp,fre,pmag,ppha,dmag,dpha)

c

c Subroutine to format data for use by TECPLOT

c

c Declarations

c



c

parameter (pi=3.141592654)

c

c Begin subroutine

c

c Frequency output

cfre=freq/(2*pi)

c

c Pressure output

c

preal=real(zpres)

pimag=imag(zpres)

pmag=SQRT(preal*preal+pimag*pimag)

c

if (preal.EQ.0.0) then

if (pimag.GT.0.0) then

ppha=(pi/2.0)

elseif (pimag.LT.0.0) then

ppha=-(pi/2.0)

else

ppha=0.0

endif

else

ppha=ATAN(pimag/preal)

endif

c

c Displacement output

c

dreal=real(zdisp)

dimag=imag(zdisp)

dmag=SQRT(dreal*dreal+dimag*dimag)

cif (dreal.EQ.0.0) then

if (dimag.GT.0.0) then

dpha=(pi/2.0)

elseif (dimag.LT.0.0) then

dpha=-(pi/2.0)

else

dpha=0.0

endif

else

dpha=ATAN(dimag/dreal)

endif

c

return

end

c

c *****************************************************************

complex function ccos(z)

c

c Function to return the cosine of a complex argument

c

c Declarations

cimplicit real (a-h,o-y)


c

c Begin function

c

argr=real(z)

argi=imag(z)

c

partone=cos(argr)*cosh(argi)

parttwo=-1*sin(argr)*sinh(argi)

c

ccos=cmplx(partone,parttwo)

c

return

end

c

c *****************************************************************

complex function csin(z)

c

c Function to return the sine of a complex argument

c

c Declarationsc

real argr,argi,partone,parttwo

complex z

c

c Begin function

c

argr=real(z)

argi=imag(z)

c

partone=sin(argr)*cosh(argi)

parttwo=cos(argr)*sinh(argi)

c

csin=cmplx(partone,parttwo)

c

return

end

P r o g r a m c o m p a r e . f

program compare

c

c Program to compare analytical and numerical displacement

c calculations of first normal mode for a cylinder-shaped

c geometry.

c

c Input files:

c PATRAN-generated neutral file: used to determine geometric

c location of each node in the model. Neutral file should

c contain only node location information.

c NASTRAN-generated punch file: used to determine the

c displacement of each node, as calculated by NASTRAN.

c It is necessary to specify the total number of nodes (maxn)

c and the number of elements (maxe) in the model prior to

c compiling.

c Output files:

c error.out: includes information about the maximum

77



c di sp la ce men t

c e rr or f or t he s ys te m.

c error.dis.1: PATRAN-readable displacement file that can be

c used to display the displacement errors for the model in

c fr in ge pl ot for m.

c


c

c Declarations


cparameter (maxn=6609, maxe=2440, pi=3.141592654)

c

character*15 zfile,pchfile,eigen

character*8 count,typevar

character za(maxn)*1

c

dimension x(maxn),y(maxn),z(maxn),

& a x( ma xn ), ay (m ax n) ,a z( ma xn ),

& p x( ma xn ), py (m ax n) ,p z( ma xn ),

& node(maxn),error(maxn),aprod(maxn),

& ndf(maxn), junk(6)

c

c Begin Program

c

write(*,*)’Enter Neutral File Name’

read(*,1)zfile

write(*,*)’Enter Punch File Name’

read(*,1)pchfile

1 fo rm at (a )

c

c Define Geometry

axis=5.0radius=1.0

c

c Zero-order Bessel function, first root

root=2.40482556

c

c Open neutral file


c Open punch file

open(unit=20,file=pchfile,status=’old’)

c Open error data output file

open(unit=30,file=’error.out’,status=’unknown’)

c

c Sort loop to determine maximum displacement in punch file

c

bigx=0.0

bigy=0.0

bigz=0.0

c

do 50 j=1,maxn

c

read(20,1004) cont,node(j),typevar,

& px(j),py(j),pz(j),ilineread(20,1005) cont,a4,a5,a6,iline

1004 format(a6,i8,a4,3(5x,e13.4),i8)

1005 format(a6,12x,3(5x,e13.4),i8)

c



if (ABS(pz(j)).GT.ABS(bigz)) bigz=pz(j)

c

50 continue

c



if (bigz.EQ.0.0) bigz=1.0

c

c Dummy print

write(*,*)’Displacements read’

c

c Read in data from neutral file

c

100 read(10,1000)idpacket,idn,iv,kc,n1,n2,n3,n4,n5

1000 format(i2,8i8)

cif (idpacket.EQ.99) then

close(10)

close(20)

goto 500

c

elseif (idpacket.EQ.1) then

c

c Determine location of grid point

read(10,1001)x(idn),y(idn),z(idn)

read(10,1002)icf,za(idn),ndf(idn),ncnfig,ncid,

& (junk(i),i=1,6)

1001 format(3e16.9)

1002 format(i1,1a1,3i8,2x,6i1)

c

c Determine analytical displacement of grid point

rad=SQRT(x(idn)*x(idn)+y(idn)*y(idn))

axial=sin((pi*z(idn))/axis)

radial=BESSJ0((rad*root)/radius)

aprod(idn)=axial*radial

c

c Normalize grid point displacements from punch file

px(idn)=px(idn)/bigx

py(idn)=py(idn)/bigy

pz(idn)=pz(idn)/bigz

c

elsewrite(*,*)’Invalid packet ID’

stop

endif

c

goto 100

c

c Calculate displacement error at each node, max error for system

c

500 bigerror=0.0

c

do 250 m=1,maxn

error(m)=aprod(m)-px(m)

if (ABS(error(m)).GT.ABS(bigerror)) bigerror=error(m)

250 continue

c

write(30,1)’Error: Analytic vs. Numeric Solutions’

write(30,1)’Cylinder, fluid only, mode 1,0,1’

write(30,*)’bigx = ’,bigx

write(30,*)’bigy = ’,bigy

write(30,*)’bigz = ’,bigz

write(30,*)’bigerror= ’,bigerror

cc Dummy print

write(*,*)’Errors determined’

c

c Write PATRAN input file

c


freq=bigerror

defmax=1

nwidth=6

ndmax=INT(maxn/2)

c

open(unit=40,file=’error.dis.1’,status=’unknown’)



write(40,1102)’$TITLE GOES HERE’

write(40,1102)’$SUBTITLE=LOAD_CASE_ONE’

c

do 200 l=1,maxn

write(40,1103)node(l),error(l),0.0,0.0,0.0,0.0,0.0

200 continue

c

close(40)close(30)

c

1100 format(a15,e13.4)

1101 format(2i9,e15.9,2i9)

1102 format(a72)

1103 format(i8,(5e13.7))

c

999 stop

end

P r o g r a m c a r d . f

program card

c

c Program to producte NASTRAN ACMODL set cards for a cylindrical

c geometry as well as fluid/solid grid set. If grid point is

c determined to be a fluid point, grid co-ordinate ID remains

c "-1". If grid point is a solid, co-ordinate ID is changed to

c "0" (solid).

c

c Input data consists of original GRID data only from the

c NASTRAN bulk data file.

c

c Output files:

c "grids.out" contains fluid/solid grid points

c "set555" contains ACMODL set cards. SET1 555 = solid grids,

c SET1 666 = fluid grids on fluid/solid interface. Line

c continuation markers start with "AAAAAAA"

c

c Assumptions: program matches each structure grid point with its

c closest fluid grid point. Model can only represent a fluid

c volume surrounded by a structural shell. The shell only touches

c the fluid on one side. 2D elements must have nodal configura-

c tions compatible with 3D elements (eg QUAD4 2D elements with HEX8

c 3D elements).

c Program verifies that no fluid grid appears twice in the ACMODL

78



c fluid set card. Structure grid points are assumed to be

c sequential (ie xxx THRU xxx).

c

c Written by: Christian M. Fernholz 4-8221

c

c Declarations

c Note: necessary to assign maxn=number of grids in model,

c


c

parameter (maxn=868)c

character*8 grid,set,thru,cont,con2

character*15 gridfile,yfile,zfile

character*1 type(maxn)

c

logical repeat,test,check

c

dimension node(maxn),m(8),

& yf(maxn),xf(maxn),zf(maxn),nodf(maxn),

& ys(maxn),xs(maxn),zs(maxn),nods(maxn),

& ytemp1(maxn),xtemp1(maxn),ntemp1(maxn),

& x te mp 2(m ax n) ,nt em p2 (ma xn )

c

c Begin program

c

write(*,*)’Enter grid file name’

read(*,2)gridfile

write(*,*)’Enter minimum structure grid point ID’

read(*,1)nodemin

write(*,*)’Enter maximum structure grid point ID’

read(*,1)nodemax

1 fo rm at (i )2 fo rm at (a )

c

zfile=’set555’

yfile=’grids.out’

c

pi=3.141592654

rad=1.0

c

c Error tolerances

toly=0.001

tolz=0.001

tolr=0.001

c

c Initialize node array

do 10 i=1,maxn

node(i)=9999

10 continue

c

c Open file containing grid information

open (unit=10,file=gridfile,status=’old’)

c Open file for ACMODL set cards

open (unit=20,file=zfile,status=’unknown’)c Open file for solid/fluid grid output

open (unit=30,file=yfile,status=’unknown’)

c

c Read grid file, determine fluid/solid grids, write new gridset

c

do 20 i=1,maxn

c

read(10,1000)grid,nod,xx,yy,zz,fs

c

if (nod.LT.nodemin.OR.nod.GT.nodemax) then

fs=-1

write(30,1000)grid,nod,xx,yy,zz,fs

elseif (nod.GE.nodemin.OR.nod.LE.nodemax) then

fs=0

write(30,1000)grid,nod,xx,yy,zz,fs

else

write(*,*)’WARNING: Grid type indeterminate’

endif

20 continue

c

close(10)

rewind(unit=30)c

c Write structure grid point ACMODL set card

c

isset=555

set=’SET1’

t hr u= ’ T HR U’

write(20,1005)set,isset,nodemin,thru,nodemax

c

c Read grid co-ordinates (x,y,z) from new grid file

c

j=0

k=0

irad=0

c

do 50 i=1,maxn

c

read(30,1000)grid,nod,xx,yy,zz,fs

c

if (nod.LT.nodemin.OR.nod.GT.nodemax) then

j=j+1

xf(j)=xx

yf(j)=yy

zf(j)=zz

nodf(j)=nod

radius=SQRT(xx*xx+yy*yy)

if (radius.LT.(rad+tolr).AND.radius.GT.(rad-tolr)) thenirad=irad+1

endif

elseif (nod.GE.nodemin.OR.nod.LE.nodemax) then

k=k+1

xs(k)=xx

ys(k)=yy

zs(k)=zz

nods(k)=nod

else

write(*,*)’ERROR: Grid type indeterminate’

goto 999

endif

50 continue

c

c Dummy print

write(*,*)’Grids successfully read’

write(*,*)’Fluid grids total: ’,j

write(*,*)’Structure grids total: ’,k

c Dummy print

write(*,*)’Number of surface structure nodes: ’,k

write(*,*)’Number of surface fluid nodes: ’,irad

cc Verify that two fluid grid points do not occupy same location

c

repeat=.FALSE.

n=0

c

do 700 i=1,(j-1)

do 710 ii=(i+1),j

c

if (xf(i).EQ.xf(ii).AND.

& yf(i).EQ.yf(ii).AND.

& zf(i).EQ.zf(ii)) then

repeat=.TRUE.

n=n+1

endif

c

710 continue

700 continue

c

if (repeat) then

write(*,*)’WARNING: ’,n,’ Fluid grids identical’

endif

cc Match structure grid points to closest fluid grid point

c

do 500 i=1,k

c

c Find all fluid grids with same z coordinate

c

ll=1

do 505 l=1,j

if (zf(l).GT.(zs(i)-tolz).AND.

& zf( l) .L T.( zs (i )+t ol z) ) t hen

ytemp1(ll)=yf(l)

xtemp1(ll)=xf(l)

ntemp1(ll)=nodf(l)

ll=ll+1

endif

505 continue

c

c Find fluid grids with same y coordinate

c

mm=1

do 510 mn=1,ll

if (ytemp1(mn).LT.(ys(i)+toly).AND.& y te mp 1( mn ). GT .( ys (i )- to ly )) t he n

xtemp2(mm)=xtemp1(mn)

ntemp2(mm)=ntemp1(mn)

mm=mm+1

endif

510 continue

c

c Find fluid grid with closest x coordinate

c

err=1000000.0

do 515 n=1,mm

error=ABS(xtemp2(n)-xs(i))

if (xtemp2(n).EQ.xs(i)) then

node(i)=ntemp2(n)

goto 500

79



elseif (error.LT.err) then

node(i)=ntemp2(n)

err=error

endif

515 continue

c

500 continue

c

c Dummy print

write(*,*)’Structure/fluid grids matched’

cc Check to see if any fluid points are repeated

c

repeat=.FALSE.

n=0

c

do 600 i=1,(k-1)

do 610 ii=(i+1),k

if (node(i).EQ.node(ii)) then

repeat=.TRUE.

n=n+1

endif

610 continue

600 continue

c

if (repeat) then

write(*,*)’WARNING:’,n,’ Nodes repeated in ACMODL card’

endif

c

c Write fluid grid point ACMODL set card

c

55 ifset=666

m(1)=43do 60 i=2,8

m(i)=65

60 continue

c

cont=char(m(1))//char(m(2))//char(m(3))//char(m(4))//


con2=cont

c

c Write first line of fluid grid points

write(20,1010)set,ifset,(node(n),n=1,7),cont

c

c Write remaining fluid grid points, eight at a time

c

imax=INT((k-7)/8)

c

ithing=8

icount=1

check=.TRUE.

c

100 if (check) then

cont=con2

test=.TRUE.80 if (test) then

i=8

if (m(i).GE.90) then

m(i)=65

m(i-1)=m(i-1)+1

if (m(i-1).LT.90) then

test=.FALSE.

else

i=i-1

if (i.LT.2) then

write(*,*)’Too many grid points’

goto 999

endif

endif

else

m(i)=m(i)+1

test=.FALSE.

endif

85 goto 80

endif

c

con2=char(m(1))//char(m(2))//char(m(3))//char(m(4))//& char(m(5))//char(m(6))//char(m(7))//char(m(8))

c

write(20,1015)cont,(node(n),n=ithing,ithing+7),con2

c

ithing=ithing+8

icount=icount+1

if (icount.GT.imax) then

check=.FALSE.

endif

goto 100

endif

c

c Write last line of file

cont=con2

write(20,1020)cont,(node(n),n=ithing,k)

c

close(30)

close(20)

c

1000 format(a8,i8,8x,3f8.4,i8)

1001 format(a8,i8,8x,3f8.4)

1005 format(a8,2i8,a8,i8)

1010 format(a8,i8,7i8,a8)

1015 format(a8,8i8,a8)

1020 format(a8,8i8)

c999 stop

end

P r o g r a m m o d e s m . f

program modesm

c

c Program to calculate natural frequencies of vibration for a

c cylindrical shell.

c

c No input files required. Shell material properties and

c geometric dimensions must be programmed prior to compiling.

c

c Output file freq.out contains mode numbers and natural

c frequency.

c

c Uses Epstein-Kennard theory and solution outlined in Leissa:

c Vibration of Shells, NASA SP-288

c

c Declarations



real l,nu,lam,k1,k2,k0

c

parameter (pi=3.141592654)

c

c Begin program

c

c Cylindrical shell, dimensions, material properties (Al)

c

c Length (in)

l=5.0

c Radius (in)

a=1.0

c Thickness (in)

h=0.0625

c Poission’s ratio

nu=0.334

c Young’s Modulus (psi)

E=10.3E6

c Density (slugs/in**3)

rho=2.5383E-4

c Square root of -1

zi=cmplx(0.0,1.0)

c

open(unit=10,file=’freq.out’,status=’unknown’)

write(10,1005)’Cylindrical Shell Natural Frequencies’

write(10,1010)’Epstein-Kennard Theory’

write(10,1015)’Mode’,’m’,’n’,’Roots (Re,Im):’,

& ’Root 1 (Hz)’,’Root 2 (Hz)’,’Root 3 (Hz)’

c

do 200 m=0,10

do 300 n=0,10

c

if (m.EQ.0.AND.n.EQ.0) goto 300

c

lam=(m*pi*a/l)

c

c Define Donnel-Mustari constants

c

k2=1.+.5*(3.-nu)*(n*n+lam*lam)+(h*h/(12.*a*a))*

& (n*n+lam*lam)*(n*n+lam*lam)

c

k1=.5*(1-nu)*((3.+2.*nu)*lam*lam+n*n+(n*n+lam*lam)*

& (n*n+lam*lam)+((3.0-nu)/(1.0-nu))*(h*h/(12.*a*a))*

& (n*n+lam*lam)*(n*n+lam*lam)*(n*n+lam*lam))

c

k0=.5*(1.-nu)*((1.-nu*nu)*(lam*lam*lam*lam)+

& (h*h/(12.*a*a))*(n*n+lam*lam)*(n*n+lam*lam)*

& (n*n+lam*lam)*(n*n+lam*lam))

c

c Modifying constants for Epstein-Kennard theory

c

delk2=(1+3*nu)/(1-nu)-(2-8*nu*nu+3*nu**3)*lam*lam/

& (2*(1-nu)**2)-(19-37*nu+19*nu*nu+nu**3)/

& (2*(1-nu)**2)-nu*nu*(n*n+lam*lam)/((1-nu)**2)

c

delk1=(3+8*nu-5*nu*nu-nu**3)*lam*lam/(2*(1-nu))+

& (2+nu)*n*n/2-(6+4*nu-8*nu*nu+3*nu**3)*lam**4/

80



& (4*(1-nu))-nu*nu*(n*n+lam*lam)**3/(2*(1-nu))-

& (26-60*nu+40*nu*nu-3*nu**3-8*nu**4)*lam*lam*n*n/

& (2*(1-nu))-(13-22*nu+10*nu*nu)*n**4/(2*(1-nu))

c

delk0=0.5*(1-nu)*((2+6*nu-2*nu*nu-3*nu**3)*lam**4/

& (2*(1-nu))+4*lam*lam*n*n+n**4-(1+nu)*lam**6/(1-nu)-

& (7-5*nu)*lam**4*n*n/(1-nu)-8*lam*lam*n**4-

& 2*n**6)

c

c Define cubic equation constants

cc2=k2+(h*h/(12*a*a))*delk2

c1=k1+(h*h/(12*a*a))*delk1

c0=k0+(h*h/(12*a*a))*delk0

c

c Solve cubic equation for omega squared. Use method outlined

c in CRC Standard Mathematical Tables, 23rd ed. page 105

c

c=(1.0/3.0)*(3.0*c1-c2*c2)

d=(1.0/27.0)*(-2.0*c2*c2*c2+9.0*c2*c1-27.0*c0)

c

delta=(d*d/4.0)+(c*c*c/27.0)

c

if (delta.LT.0.0) then

partone=-d/2.0

parttwo=SQRT(-1.0*delta)

zP=cmplx(partone,parttwo)

zQ=cmplx(partone,-parttwo)

else

partone=-d/2.0

parttwo=SQRT(delta)

zP=cmplx((partone+parttwo),0.0)

zQ=cmplx((partone-parttwo),0.0)endif

c

zP=zP**(1.0/3.0)

zQ=zQ**(1.0/3.0)

c

zfirst=-.5*(zP+zQ)

zsecnd=-.5*(zP-zQ)*SQRT(3.0)*zi

c

partone=real(zfirst)+real(zsecnd)

parttwo=imag(zfirst)+imag(zsecnd)

parttre=real(zfirst)-real(zsecnd)

partfor=imag(zfirst)-imag(zsecnd)

c

zroot1=zP+zQ+c2/3.0

zroot2=cmplx((partone+c2/3.0),parttwo)

zroot3=cmplx((parttre+c2/3.0),partfor)

c

zfreq1=SQRT(zroot1*E/(rho*(1-nu*nu)*a*a))/(2.0*pi)



c

write(10,1000)’Mode’,m,’,’,n,’and frequencies’,& zfreq1,zfreq2,zfreq3

c

300 continue

200 continue

c

close(10)

c

1000 format(a4,1x,i2,a1,i2,1x,a15,1x,3(f12.4,1x,f3.1,1x))

1005 format(a37)

1010 format(a22)

1015 format(a4,2x,a1,2x,a1,1x,a14,3(5x,a11))

c

stop

end

P r o g r a m f o r c e d 2 . f

program forced2

c

c Program to calculate natural frequencies of vibration for a

c cylindrical shell and the forced response over a range of

c frequencies.

c

c No input files required. NASTRAN-generated punch file is optional

c if comparison to a numceric theory is desired. Shell material

c properties and geometric dimensions must be programmed prior to

c compiling.

c

c Output files: freq.out contains mode numbers and natural

c frequencies.

c di spl ac e. out con ta in s dis pl ac eme nt i nf or mat io n

c written in TECPLOT form.

c

c Uses Donnell-Mustari theory and solution outlined in Leissa:

c Vibration of Shells, NASA SP-288

c

c Declarations

c



real l,nu,lam

c

parameter (pi=3.141592654,maxf=1000,modes=10)

c

dimension u(maxf),v(maxf),w(maxf),p(maxf),freq(maxf)

c

logical printc

c Begin program

c

c Cylindrical shell, dimensions, material properties (Al)

c

c Length (in)

l=50.0

c Radius (in)

a=10.0

c Thickness (in)

h=0.0625

c Poission’s ratio

nu=0.334

c Young’s Modulus (psi)

E=10.3E6

c Density of the structure (slugs/in**3)

rho=2.5383E-4

c Density of the fluid (slugs/in**3)

rhof=1.170e-7

c Speed of sound in fluid (in/sec)

co=13620.0

c Damping coefficienteta=0.005

c Applied force (lbs)

Fo=-50.0

c Square root of -1

zi=cmplx(0.0,1.0)

c Common coefficient (wave speed in structure)

cl2=E/(a*a*rho*(1-nu*nu))

c

c Frequency increment (Hz)

step=5.0

c Starting frequency

frequency=5.0*(2.0*pi)

c Print natural frequencies once

print=.FALSE.

c

c Determine location of interest

c

rr=a/2

tt=0.0

yz=l/2

c

1 f or mat (a )2 f or mat (f )

3 format(a13,f4.1,1x,f4.1,1x,f4.1)

c

tt=tt*(pi/180.0)

c

if (print) then

open(unit=10,file=’freq.out’,status=’unknown’)

write(10,1005)’Cylindrical Shell Natural Frequencies’

write(10,1010)’Donnell-Mustari Theory’

write(10,1015)’Mode’,’m’,’n’,’Roots (Re,Im):’,

& ’Root 1 (Hz)’,’Root 2 (Hz)’,’Root 3 (Hz)’

endif

c

c "Prep" tecplot output file

c

open(unit=20,file=’displace.out’,status=’unknown’)

write(20,1)’TITLE = "FORCED FLUID/STRUCTURE CYLINDER RESP."’

write(20,1)’VARIABLES = "FREQ","U","V","W","P"’

write(20,1)’ZONE T="Analytic"’

c

do 100 k=1,maxf

zu=cmplx(0.0,0.0)zv=cmplx(0.0,0.0)

zw=cmplx(0.0,0.0)

zp=cmplx(0.0,0.0)

do 200 m=1,modes

do 300 n=0,modes

c

if (m.EQ.0.AND.n.EQ.0) goto 300

c

c Calculate natural frequency for mode

c

lam=(m*pi*a/l)

c

c Define cubic equation constants

c

c2=1.+.5*(3.-nu)*(n*n+lam*lam)+(h*h/(12.*a*a))*

81



& (n*n+lam*lam)*(n*n+lam*lam)

c

c1=.5*(1-nu)*((3.+2.*nu)*lam*lam+n*n+(n*n+lam*lam)*

& (n*n+lam*lam)+((3.0-nu)/(1.0-nu))*(h*h/(12.*a*a))*

& (n*n+lam*lam)*(n*n+lam*lam)*(n*n+lam*lam))

c

c0=.5*(1.-nu)*((1.-nu*nu)*(lam*lam*lam*lam)+

& (h*h/(12.*a*a))*(n*n+lam*lam)*(n*n+lam*lam)*

& (n*n+lam*lam)*(n*n+lam*lam))

c

c Solve cubic equation for omega squared. Use method outlinedc in CRC Standard Mathematical Tables, 23rd ed. page 105

c

c=(1.0/3.0)*(3.0*c1-c2*c2)

d=(1.0/27.0)*(-2.0*c2*c2*c2+9.0*c2*c1-27.0*c0)

c

delta=(d*d/4.0)+(c*c*c/27.0)

c

if (delta.LT.0.0) then

partone=-d/2.0

parttwo=SQRT(-1.0*delta)

zPP=cmplx(partone,parttwo)

zQQ=cmplx(partone,-parttwo)

else

partone=-d/2.0

parttwo=SQRT(delta)

zPP=cmplx((partone+parttwo),0.0)

zQQ=cmplx((partone-parttwo),0.0)

endif

c

zPP=zPP**(1.0/3.0)

zQQ=zQQ**(1.0/3.0)

czfirst=-.5*(zPP+zQQ)

zsecnd=-.5*(zPP-zQQ)*SQRT(3.0)*zi

c

partone=real(zfirst)+real(zsecnd)

parttwo=imag(zfirst)+imag(zsecnd)

parttre=real(zfirst)-real(zsecnd)

partfor=imag(zfirst)-imag(zsecnd)

c

zroot1=zPP+zQQ+c2/3.0

zroot2=cmplx((partone+c2/3.0),parttwo)

zroot3=cmplx((parttre+c2/3.0),partfor)

c

zfreq1=SQRT(zroot1*E/(rho*(1-nu*nu)*a*a))



c

if (print) then

write(10,1000)’Mode’,m,’,’,n,’and frequencies’,

& zfreq1/(2.*pi),zfreq2/(2.*pi),zfreq3/(2.*pi)

if (m.EQ.modes) close(10)

endif

cif (imag(zfreq1).NE.0.0.OR.

& imag(zfreq2).NE.0.0.OR.

& i ma g( zf re q3 ). NE .0 .0 ) t he n

write(*,*)’Fatal error: natural frequency is ’,

& ’imaginary’

goto 9999

endif

c

call magnitude(zfreq1,freq1)



c

c Calculate force coefficient

c

lam=(m*pi/l)

c

Fmn=a*(2.0/(pi*l))*Fo*sin(m*pi/2.0)*(1.0+cos(n*pi))

c

freqA=-SQRT(-(Cl2/(2.0*a*a))*(nu-1.0)*(nu*lam*lam*

& a*a-n*n)/nu)

freqB=-SQRT(-(Cl2/(2.0*a*a))*(nu-1.0)*(nu*lam*lam*

& a*a+2.0*lam*lam*a*a+n*n))freqC1=SQRT((Cl2/a*a)*(lam*lam*a*a+n*n))

freqC2=SQRT(-(Cl2/(2.0*a*a))*(lam*lam*a*a+n*n)*(nu-1.0))

c

c Calculate cofactors for zAmn, zBmn, zCmn

c

zdet=cmplx((frequency*frequency-freq3*freq3),

& 2 .0 *e ta *f re qu en cy *f re q3 )*

& cmplx((frequency*frequency-freq2*freq2),

& 2 .0 *e ta *f re qu en cy *f re q2 )*

& cmplx((frequency*frequency-freq1*freq1),

& 2 .0 *e ta *f re qu en cy *f re q1 )

c

zcofA=cmplx((frequency*frequency-freqA*freqA),

& 2.0*eta*frequency*freqA)

zcofB=cmplx((frequency*frequency-freqB*freqB),

& 2.0*eta*frequency*freqB)

zcofC=cmplx((frequency*frequency-freqC1*freqC1),

& 2.0*eta*frequency*freqC1)*

& cmplx((frequency*frequency-freqC2*freqC2),

& 2.0*eta*frequency*freqC2)

c

zAmn=-(Fmn/(rho*h))*zcofA/(zdet)

zBmn= (Fmn/(rho*h))*zcofB/(zdet)

zCmn=-(Fmn/(rho*h))*zcofC/(zdet)

c

c Sum displacementsc

zu=zu+zAmn*cos(m*pi*yz/l)*cos(n*tt)

zv=zv+zBmn*sin(m*pi*yz/l)*sin(n*tt)

zw=zw+zCmn*sin(m*pi*yz/l)*cos(n*tt)

c

c Solve for pressure coefficient

c

alphsqr=(frequency*frequency)/(co*co)-(m*pi/l)*(m*pi/l)

c

if (alphsqr.LT.0.0) then

alph=SQRT(-1.0*alphsqr)

call cofimag(a,alph,n,bottom)

zDmn=(zCmn*rhof*frequency*frequency)/bottom

call bessimag(rr,alph,n,press)

c

elseif (alphsqr.GT.0.0) then

alph=SQRT(alphsqr)

call realcof(a,alph,n,bottom)

zDmn=(zCmn*rhof*frequency*frequency)/bottom

call bessreal(rr,alph,n,press)

c

elseif (alphsqr.EQ.0.0) thenwrite(*,*)’Alpha = 0.0’

c

endif

c

c Sum pressure

c

zp=zp+zDmn*press*sin(m*pi*yz/l)*cos(n*tt)

c

300 continue

200 continue

c

c Dummy print

write(*,4)’Complex data calculated for ’,k

4 format(a28,i4)

c

c Convert complex displacements, pressure to magnitudes

c

call magnitude(zu,u(k))

call magnitude(zv,v(k))

call magnitude(zw,w(k))

call magnitude(zp,p(k))

cc Dummy print

write(*,1105)zp

c

c Write to TECPLOT file

c

freq(k)=frequency/(2.0*pi)

write(20,1100)freq(k),u(k),v(k),w(k),p(k)

c

c Increment frequency

frequency=frequency+step*(2.0*pi)

c

100 continue

c

close(20)

c

1000 format(a4,1x,i2,a1,i2,1x,a15,1x,3(f12.4,1x,f3.1,1x))

1005 format(a37)

1010 format(a22)

1015 format(a4,2x,a1,2x,a1,1x,a14,3(5x,a11))

1100 format(f7.2,4(1x,e16.9))

1105 format(2(e16.9,1x))

c9999 stop

end

c

c *****************************************************************

subroutine magnitude(z,x)

c

c Subroutine to return the magnitude x of a complex argument z

c



c

partone=real(z)

parttwo=imag(z)

c

x=SQRT(partone*partone+parttwo*parttwo)

82



c

return

end

c

c *****************************************************************

subroutine realcof(a,alph,n,out)

c

c Subroutine to return the denominator of the pressure coefficient

c term. Used when the argument of the n-th order bessel function

c i s a real .



c

c Square root of -1

zi=cmplx(0.0,1.0)

c

x=alph*a

c

if (n.EQ.0) then

term1=BESSJ0(x)

term2=BESSJ1(x)

elseif (n.eq.1) then

term1=BESSJ1(x)

term2=BESSJ(2,x)

else

term1=BESSJ(n,x)

term2=BESSJ((n+1),x)

endif

c

c Construct denominator

c

out=(n/a)*term1-alph*term2c

return

end

c

c *****************************************************************

subroutine cofimag(a,alpha,n,out)

c

c Subroutine to return the denominator of the pressure coefficient

c term. Used when the argument of the n-th order bessel function is

c complex.

c



c

x=alpha*a

c

if (n.EQ.0) then

term1=BESSI0(x)

term2=BESSI1(x)

elseif (n.EQ.1) then

term1=BESSI0(x)

term2=BESSI(2,x)else

term1=BESSI(n,x)

term2=BESSI((n+1),x)

endif

c

c Construct denominator

c

out=alpha*term2+(n/a)*term1

c

return

end

c

c *****************************************************************subroutine bessreal(r,alpha,n,out)

c

c Subroutine to return the n-th order bessel function of a real

c argument.

c



c

zi=cmplx(0.0,1.0)

x=r*alpha

c

if (n.EQ.0) then

out=BESSJ0(x)


out=BESSJ1(x)

else

out=BESSJ(n,x)

endif

c

return

end

cc *****************************************************************

subroutine bessimag(r,alpha,n,out)

c

c Subroutine to return the n-th order modified bessel function of

c a real argument

c



c

x=r*alpha

c

if (n.EQ.0) then

out=BESSI0(x)


out=BESSI1(x)

else

out=BESSI(n,x)

endif

c

return

end

83



A p p e n d i x C : N A S T R A N F i l e U s e a g e S t a t i s t i c s

F i l e S i z e

( M B )

D B A L L S C R A T C H

A n a l y s i s T y p e

C a l c u l a t e d

E i g e n v a l u e s

R e q u e s t e d

F r e q u e n c i e s

N o d e s

4 . 6 8 6 0 . 4 5 9 1 0 - 1 3 3 1

1 9 . 7 5 9 5 . 9 4 7 1 0 - 4 9 6 1

6 . 6 6 0 0 . 9 7 5 1 4 - 1 5 7 3

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

5 . 8 9 0 0 . 5 0 0 1 0 - 1 4 4 9

2 4 . 4 7 8 6 . 5 5 4 1 0 - 6 6 0 9

1 4 . 0 2 5 3 . 2 2 8 4 7 - 1 9 5 3

6 1 . 5 3 8 1 7 . 5 6 4

S O L 1 0 3

5 1 - 8 0 9 7

2 3 . 4 0 0 6 . 5 5 4

S O L 1 0 8

- 2 5 0 5 6 4 4

2 4 . 3 9 6 7 . 4 0 6

S O L 1 1 1

2 0 2 5 0 5 6 4 4

4 8 . 9 3 1 1 6 . 6 8 7

S O L 1 0 8

- 2 5 0 1 9 5 3

F i g u r e 3 1 : M S C / N A S T R A N l e u s e a g e d a t a f o r D B A L L a n d S C R A T C H l e s .

nasa - fully-coupled fluid-structure vibration analysis using nastran

Documents

normal modes analysis

forced response analysis

normal modes problems

nastran data

normal modes of vibration

normal modes sol

accuracy of nastran

cubic uidstructure models