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Optimum Design of Distributed Mass and Stiffness Structural Systems Under Constraints
OPTIMUM DESIGN OF DISTRIBUTED MASS AND STIFFNESS STRUCTURAL SYSTEMS UNDER CONSTRAINTS R. A. Rand* and C . N . Shen Rensselaer Polytechnic Institute, Troy, New York, USA ABSTRACT S h e l l s t r u c t u r e s a r e c l a s s i c examples o f d i s t r i b u t e d parameter systems. I n t h i s paper an approach i s developed t o o b t a i n t h e optimum (minimum w e i g h t ) c o n f i g u r a t i o n o f a g e n e r a l s h e l l s t r u c t u r e s u b j e c t e d t o simultaneous frequency and s t r e s s c o n s t r a i n t s . The c o n s t r a i n e d frequenc i e s a r e determined by d i s c r e t i z i n g t h e s h e l l u s i n g a f i n i t e - e l e m e n t t e c h n i q u e and t h e n m i n i m i z i n g s p e c i a l R a y l e i g h q u o t i e n t s d e f i n e d f o r t h e f i n i t e - e l e m e n t model. The optimum c o n f i g u r a t i o n i s determined by o p t i m a l d e c i s i o n o f a v e c t o r o f m a t e r i a l and geometric d e s i g n parameters u s i n g a m o d i f i e d Complex technique. The approach i s demonstrated by a p p l i c a t i o n t o a c o n i c a l s h e l l o f e l l i p t i c a l c r o s s - s e c t i o n s u b j e c t e d t o i n e r t i a l and e x t e r n a l l y a p p l i e d l o a d i n g s . An a n a l y t i c a l s o l u t i o n t o t h i s problem, o b t a i n e d u s i n g l e a s t squares techniques and Lagrange m u l t i p l i e r s , i s presented f o r comparison. INTRODUCTION Continuous s t r u c t u r a l systems a r e c l a s s i c examples o f d i s t r i b u t e d parameter systems. H i s t o r i c a l l y , t h e dynamic response o f such systems has been determined by f o r m u l a t i o n and s o l u t i o n o f a s e t o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s . I n g e n e r a l , t h i s s e t o f e q u a t i o n s i s so comp l e x t h a t numerical methods a r e r e q u i r e d t o o b t a i n t h e s o l u t i o n . For example, c o n s i d e r a s h e l l s t r u c t u r e o f general shape. The e q u a t i o n s governing t h e dynamic response o f t h e s h e l l a r e f o r m u l a t e d by f i r s t c o n s i d e r i n g t h e response o f a d i f f e r e n t i a l element, as shown i n F i g . 1.
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F i g . 1.
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F i g . 2. D e f i n i t i o n o f l a m i n a t e parameters a , tL, v
D i f f e r e n t i a l s h e l l element
The response o f t h i s element i s d e s c r i b e d by a s e t o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s c o n s i s t i n g of f i v e e q u i l i b r i u m e q u a t i o n s , s i x f o r c e - s t r a i n r e l a t i o n s , and s i x s t r a i n - d i s p l a c e m e n t r e l a t i o n s . A complete d e r i v a t i o n o f these e q u a t i o n s i s g i v e n by Kraus ( 1 ) . The response o f t h e complete s h e l l s t r u c t u r e i s o b t a i n e d by i n t e g r a t i n g t h i s s e t o f e q u a t i o n s o v e r t h e e n t i r e s h e l l surface and a p p l y i n g a p p r o p r i a t e i n i t i a l and boundary c o n d i t i o n s . The r e s u l t i n o e q u a t i o n s a r e solved, f o r a p a r t i c u l a r c o n f i g u r a t i o n and l o a d i n g , e i t h e r by f i n i t e - d i f f e r e n c e techniaues o r by d i r e c t numerical i n t e g r a t i o n .
*Now w i t h General E l e c t r i c N u c l e a r Energy D i v i s i o n , San Jose, C a l i f o r n i a , U.S.A. 511
R. A . Rand and C . N .
Shen
An a l t e r n a t i v e t o t h i s c l a s s i c a l approach has e v o l v e d o v e r t h e p a s t s e v e r a l y e a r s . Before f o r m u l a t i o n o f t h e g o v e r n i n g e q u a t i o n s t h e c o n t i n u o u s s t r u c t u r e , w h i c h i n t h e c l a s s i c a l app r o a c h has an i n f i n i t e number o f degrees o f freedom, i s i d e a l i z e d as an assembly o f d i s c r e t e elements, o f f i n i t e dimension, w i t h assumed response c h a r a c t e r i s t i c s . These elements a r e i n t e r c o n n e c t e d a t a f i n i t e number o f nodal p o i n t s . The d i s p l a c e m e n t components o f t h e nodal p o i n t s become t h e degrees o f freedom f o r t h e s t r u c t u r e . The response of t h e s t r u c t u r e i s obt a i n e d by combining t h e response o f t h e d i s c r e t e elements such t h a t e q u i l i b r i u m and compatib i l i t y c o n d i t i o n s a r e s a t i s f i e d a t a l l nodal p o i n t s . However, s i n c e t h e s i z e of t h e elements remains f i x e d and t h e number o f degrees o f freedom i s f i n i t e , i n t e g r a t i o n s o v e r t h e e n t i r e s t r u c t u r e a r e r e p l a c e d by f i n i t e summations and t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n s o f t h e cont i n u o u s s t r u c t u r e a r e r e p l a c e d by a system o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s o f l a r g e dimens i o n , w h i c h a r e i d e a l l y s u i t e d f o r s o l u t i o n on l a r g e s c a l e d i g i t a l computers. The m a j o r d i f f e r e n c e between t h i s new approach, now commonly r e f e r r e d t o as t h e f i n i t e - e l e m e n t method, and t h e c l a s s i c a l approach i s t h a t a p p r o x i m a t i o n s a r e i n t r o d u c e d i n t h e f o r m u l a t i o n o f t h e g o v e r n i n g e q u a t i o n s , r a t h e r t h a n i n t h e i r s o l u t i o n . The f i n i t e - e l e m e n t method has been a p p l i e d t o v a r i o u s t y p e s o f s t r u c t u r e s w i t h e x c e l l e n t r e s u l t s ( R e f s . 2,3). There i s c o n s i d e r a b l e i n t e r e s t i n d e v e l o p i n g automated t o o l s f o r s e e k i n g optimum s t r u c t u r a l c o n f i g u r a t i o n s ( R e f . 4 ) . The s t r u c t u r a l o p t i m i z a t i o n problem i s g e n e r a l l y posed as t h a t o f d e t e r m i n i n g t h e v a l u e o f a s e t o f v a r i a b l e d e s i g n parameters such t h a t an o b j e c t i v e f u n c t i o n i s n ~ i n i m i z e d( o r maximized) s u b j e c t t o a p p r o p r i a t e d e s i o n c o n s t r a i n t s . The o b j e c t i v e f u n c t i o n i s d e f i n e d i n such a way t h a t i t i s a measure o f t h e "goodness" o f t h e s t r u c t u r e under c o n s i d e r a tion. Because o f t h e i r h i g h s t r u c t u r a l e f f i c i e n c y , d i s t r i b u t e d parameter s h e l l s t r u c t u r e s a r e used i n h i g h performance a p p l i c a t i o n s where w e i g h t i s most o f t e n t h e p r i m e measure o f t h e "goodness" o f t h e d e s i g n . O t h e r measures such as f a b r i c a t i o n c o s t s and r e f u r b i s h a b i l i t y c o u l d a l s o e n t e r i n t o t h e e v a l u a t i o n o f t h e d e s i g n , b u t i n t h i s s t u d y o n l y t h e s t r u c t u r a l w e i g h t w i l l be i n c l u d ed i n t h e o b j e c t i v e f u n c t i o n and t h e terms "minimum w e i g h t d e s i g n " and "optimum d e s i g n " w i l l be used i n t e r c h a n g e a b l y . To f u r t h e r i n c r e a s e t h e s t r u c t u r a l e f f i c i e n c y , s h e l l s t r u c t u r e s a r e o f t e n c o n s t r u c t e d o f a f i b e r - r e i n f o r c e d composite. Composite m a t e r i a l s have been employed w i t h e v e r - i n c r e a s i n g f r e quency i n r e c e n t y e a r s , e s p e c i a l l y i n h i g h performance s h e l l s t r u c t u r e s . A l t h o u g h a composite m a t e r i a l i s , i n g e n e r a l , d e f i n e d as a m a t e r i a l system composed of a m i x t u r e o r c o m b i n a t i o n of two o r more c o n s t i t u e n t s t h a t d i f f e r i n f o r m a n d / o r c o m p o s i t i o n and t h a t a r e e s s e n t i a l l y i n s o l u b l e i n one a n o t h e r , t h i s d i s c u s s i o n d e a l s e x c l u s i v e l y w i t h h i g h performance f i b e r comp o s i t e s , i . e . composite m a t e r i a l s i n w h i c h h i g h s t r e n g t h f i b e r s a r e embedded w i t h p r e f e r r e d o r i e n t a t i o n s i n a ( r e l a t i v e l y ) l o w s t r e n g t h m a t r i x . Such composites a r e f a b r i c a t e d by s t a c k i n g l a m i n a e c o n s i s t i n g o f s t i f f f i b e r s embedded i n a s o f t m a t r i x , as shown s c h e m a t i c a l l y i n F i g . 2. The f i b e r o r i e n t a t i o n a n g l e a , l a m i n a t h i c k n e s s tL, and f i b e r volume f r a c t i o n v F f o r each lami n a a r e ( m a t e r i a l ) d e s i g n parameters. However, i n h i g h performance a p p l i c a t i o n s t h e s h e l l s a r e l i k e l y t o be s u b j e c t e d t o e x t e r n a l e x c i t a t i o n s . To a v o i d h a r m f u l resonance e f f e c t s , t h e s h e l l s must be designed s u b j e c t t o cons t r a i n t s on t h e i r n a t u r a l f r e q u e n c i e s ( R e f s . 5, 6, 7 ) . These c o n s t r a i n t s a r e u s u a l l y o f t h e form
n a t u r a l frequency o f t h e s h e l l i n q u e s t i o n and w A , w a r e d e s i g n conwhere w i s t h e ith 'i s t r a i n t $ d e t e r m i n e d from t h e f r e q u e n c i e s o f t h e e x t e r n a l e x c i t a t i o n s . 7 C o n s t r a i n t s o f t h e f o r m ( I ) , ( 2 ) a r e imposed on t h e n a t u r a l f r e q u e n c i e s t o m a i n t a i n s t r u c t u r a l i n t e g r i t y under t h e time-dependent l o a d i n g s . To i n s u r e s t r u c t u r a l i n t e g r i t y under any t i m e independent l o a d i n g s , s t r e s s c o n s t r a i n t s o f t h e f o r m
where oi i s t h e a p p l i e d s t r e s s and si imposed.
i s t h e l a m i n a t e s t r e n g t h i n t h e ith direction, are also
Distributed Mass and Stiffness Structural Systems
I n l i Y i h ~O T tile above d i s c u s s i o n , t h e o b j e c t i v e s of t h i s s t u d y were t o develop a t e c h n i o u e t o determine t h e n a t u r a l frequencies o f g e n e r a l composite s h e l l s ; develop a t e c h n i q u e t o o p t i m i z e such s h e l l s s u b j e c t t o n a t u r a l frequency and s t r e s s c o n s t r a i n t s ; and demonstrate t h e techniques by a p p l i c a t i o n t o a t y p i c a l problem. To accomplish these o b j e c t i v e s , a frequency c a l c u l a t i o n method c o u p l i n g t h e f i n i t e - e l e m e n t t e c h n i q u e f o r d i s c r e t i z a t i o n of c o n t i n u o u s s t r u c t u r a l systems w i t h a s p e c i a l R a y l e i g h q u o t i e n t t e c h n i q u e i s developed. A m i n i m i z a t i o n t e c h n i q u e der i v e d by m o d i f y i n g t h e "Complex Method" o f Box(8) i s a l s o developed. T h i s m i n i m i z a t i o n techn i q u e i s c o m p a t i b l e w i t h t h e frequency c a l c u l a t i o n techniaue. A s o l u t i o n i s p r e s e n t e d i n which t h e optimum c o n f i g u r a t i o n f o r a d i s t r i b u t e d parameter s h e l l s t r u c t u r e i s o b t a i n e d t h r o u g h o p t i m a l d e c i s i o n o f a v e c t o r o f d e s i g n parameters which j n c l u d e s b o t h m a t e r i a l and geometric v a r i a b l e s . The s h e l l i s s u b j e c t e d t o simultaneous frequency and s t r e s s c o n s t r a i n t s . T h i s paper d e a l s p r i m a r i l y w i t h t h e frequency c a l c u l a t i o n technique. The o p t i m i z a t i o n p o r t i o n o f the study i s described i n less d e t a i l . ANALYSIS Problem D e f i n i t i o n A d i s t r i b u t e d parameter system c o n s i s t i n g o f a s l e n d e r c o n i c a l s h e l l o f e l l i p t i c a l c r o s s s e c t i o n w i t h a m e t a l l i c end r i n g i s r i g i d l y a t t a c h e d t o a f o u n d a t i o n and s u b j e c t e d t o i n e r t i a l l o a d i n g a l o n g i t s a x i s and p e r i o d i c t r a n s v e r s e ( e x t e r n a l ) l o a d i n g normal t o i t s a x i s and p a r a l l e l t o t h e m i n o r e l l i p t i c a l a x i s , as shown s c h e m a t i c a l l y i n F i g . 3. Such l o a d i n g s would o c c u r on a s h e l l used as t h e p r i m a r y s t r u c t u r a l member o f a h e l i c o p t e r r o t o r blade, as shown schema t i c a l l y i n F i g . 4.
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F i g . 4. A p p l i c a t i o n o f t a p e r e d e l l i p t i c a l s h e l l as p r i m a r y s t r u c t u r a l member o f r o t o r b l a d e (schematic)
For t h e case shown i n F i g . 4 t h e i n e r t i a l l o a d i n g a r i s e s due t o t h e r o t a t i o n o f t h e s h e l l about an a x i s normal t o i t s c e n t e r l i n e and t h e t r a n s v e r s e l o a d i n g i s t h e l i f t generated by t h i s r o t a t i o n . The r i n g i s used so t h a t r e a c t i o n s a t t h e attachment p o i n t w i l l n o t r e s u l t i n h i g h l o c a l stresses i n the s h e l l wall. The she:l i s assumed t o be a f i b e r - r e i n f o r c e d l a m i n a t e c o n s t r u c t e d by s t a c k i n g f i b e r g l a s s - e p o x y laminae. The f i b e r o r i e n t a t i o n a n g l e a, t h i c k n e s s t , and f i b e r volume f r a c t i o n v f o r each lamina a r e ( m a t e r i a l ) d e s i g n parameters. F o r a n a l y s \ s purposes we assume t h e l a m i k a t e c o n s i s t s o f f o u r laminae. The v a l u e s o f vF, t and t h e magnitudes of a a r e equal f o r each laminae; t h e s i g n of a v a r i e s such t h a t t h e l a m ~ n a k ei s balanced. The l a m i n a t e geometry i s shown i n F i g . 5. M a t e r i a l p r o p e r t i e s o f t h e f i b e r g l a s s and epoxy c o n s t i t u e n t s o f t h e l a m i n a t e a r e shown i n Table 1 . The s i z e of t h e m e t a l l i c a t t a c h i n g r i n g i s c h a r a c t e r i z e d by a s i z e parameter b, where b i s a ( g e o m e t r i c ) d e s i g n parameter. The geometry o f t h e r i n g i s shown i n F i g . 6. TABLE 1 M a t e r i a l P r o p e r t i e s o f Composite C o n s t i t u e n t s
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e 2 ( E l a s t i c modulus, l b / i n )
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F i g . 6.
A t t a c h i n g r i n g geometry
For t h e h e l i c o p t e r b l a d e a p p l i c a t i o n c o n s i d e r e d i n t h i s problem t h e o v e r a l l s h e l l geometry and l o a d i n g s a r e as shown i n F i g . 7. The a x i a l l o a d i n g f v a r i e s a l o n g t h e l e n g t h o f t h e s h e l l b u t i s assumed t o be independent of t i m e T . The t r a n s v e r s e l o a d i n g n i s assumed t o be u n i f o r m a l o n g t h e l e n g t h of t h e s h e l l and t o c o n s i s t o f a l a r g e time-independent component no due t o t h e l i f t generated by b l a d e r o t a t i o n and a time-dependent component n ' a r i s i n g f r o m unsteady aerodynamics ( f l u t t e r ) . The e x c i t a t i o n f r e q u e n c y o f n ' i s assumed t 8 v a r y randomly between 0 and w ' To a v o i d t h e harmful resonance e f f e c t s o f laPge t i p d e f l e c t i o n s and l a r g e bending s t r e s e e s i n t h e s h e l l which would o c c u r i f t h e s h e l l was e x c i t e d a t c l o s e t o a n a t u r a l frequenc y by n o o ' , t h e fundamental t r a n s v e r s e (bending) frequency wT o f t h e s h e l l i s c o n s t r a i n e d such that
.
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The f a c t o r k i s chosen such t h a t a p p l i c a t i o n o f t h e c o n t r a i n t ( 4 ) w i l l i n s u r e t h a t t h e response o f t h e s h e l l t o n ' w i l l be s m a l l r e l a t i v e t o t h e response t o no. I n o r d e r t o i n s u r e s t r u c t u r a l i n t e g r i t y undep t h e ( t i m e - i n d e p e n d e n t ) l o a d i n g s f, n and t o i n s u r e t h a t t h e r i n g serves i t s purpose o f p r e v e n t i n g h i g h l o c a l s t r e s s e s a t t h e a t t a c h 8 e n t p o i n t , s t r e s s c o n s t r a i n t s o f t h e form
and a s t i f f n e s s c o n s t r a i n t o f t h e form
a r e a l s o imposed. I n e q u a t i o n ( 6 ) uL i s t h e a p p l i e d s t r e s s i n t h e l a m i n a t e i n t h e f i b e r d i r e c t i o n and sL !s t h e l a m i n a t e s t r e n g t h i n t h e f i b e r d i r e c t i o n . S i m i l a r l y , i n e q u a t i o n ( 7 ) o N i s t h e a p p l i e d s t r e s s i n t h e l a m i n a t e i n t h e d i r e c t i o n t r a n s v e r s e t o t h e f i b e r s and s i s t h e l a m i n a t e s t r e n g t h i n t h a t d i r e c t i o n . C o n s t r a i n t s ( 6 ) , ( 7 ) a r e a p p l i e d a t t h e poin! o f maximum s t r e s s i n t h e s h e l l . I n e q u a t i o n (8) kR i s t h e r i n g a x i a l s t i f f n e s s p e r u n i t l e n g t h o f circumference and k i s a l o w e r l i m i t f o r kR. R0
Since t h e s h e l i i s used i n an a i r b o r n e a p p l i c a t i o n i t must have minimum weight. The d e s i g n problem becomes t h a t o f d e t e r m i n i n g t h e m a t e r i a l d e s i g n parameters a , t L , v F and t h e geometric d e s i g n parameter b such t h a t t h e s h e l l has minimum w e i g h t and s a t i s f i e s t h e frequency cons t r a i n t ( 4 ) and t h e s t r e s s and s t i f f n e s s c o n t r a i n t s ( 6 ) , ( 7 ) , ( 8 ) . I n a d d i t i o n t o these d e s i g n c o n s t r a i n t s i t i s a l s o r e q u i r e d t h a t t be g r e a t e r t h a n .0325" (a m a n u f a c t u r i n g c o n s t r a i n t ) , t h a t v be g r e a t e r t h a n . 4 b u t l e s s t h k n . 8 ( a m a n u f a c t u r i n g c o n s t r a i n t ) , and t h a t a be g r e a t e r t h a n l & o b u t l e s s t h a n 800 (a m a n u f a c t u r i n g c o n s t r a i n t ) .
Distributed Mass and Stiffness Structural Systems
f = axla1 load a t : n = n o m l load at z a , = 6 a , = ? b , = d b2=2.66' i = 60
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F i g . 7.
S h e l l geometry and l o a d i n g
Problem F o r m u l a t i o n The d e s i g n parameters a r e assembled t o form a v e c t o r u o f problem v a r i a b l e s . The v e c t o r u can i n c l u d e b o t h g e o m e t r i c v a r i a b l e s which d e s c r i b e t h e s i z e o r shape o f t h e s h e l l o r a s s o c i a t e d components and m a t e r i a l v a r i a b l e s which d e s c r i b e t h e e l a s t i c p r o p e r t i e s o r g e o m e t r i c c o n f i g u r a t i o n o f t h e l a m i n a t e c o m p r i s i n g t h e s h e l l w a l l . As i n d i c a t e d above, t h e r e a r e f o u r d e s i g n parameters f o r t h e problem c o n s i d e r e d i n t h i $ paper: t h e f i b e r o r i e n t a t i o n a n g l e a (measured w i t h r e s p e c t t o t h e s h e l l a x i a l c o o r d i n a t e s ) , t h e lamina t h i c k n e s s tL ( n o t e t h a t t h e t o t a l s h e l l t h i c k n e s s can be o b t a i n e d once tL i s determined), t h e f i b e r volume f r a c t i o n vF, and t h e r i n g s i z e parameter b. Thus u becomes
where
ul
= f i b e r o r i e n t a t i o n a n g l e a (degrees)
u2 = lamina t h i c k n e s s tL ( i n c h e s ) u3 = f i b e r volume f r a c t i o n vF
u4 = r i n g s i z e parameter b ( i n c h e s )
1
The w e i g h t w o f t h e s h e l l and i t s a s s o c i a t e d components and t h e d e s i g n (and m a n u f a c t u r i n g ) c o n s t r a i n t s g r e t h e n expressed i n terms o f u. The s p e c i f i c c o n s t r a i n t s depend upon t h e s p e c i f i c problem u n d e r r c o n s i d e r a \ i o n b u t w i l j i n c l u d e i n e q u a l i t y c o n s t r a i n t s o f t h e form
2 where wi, 'I , 'I a r e as d e f i n e d p r e v i o u s l y . The magnitude o f wi(u) must be determined numer*i Bi u s i n g t h e s p e c i a l R a y l e i g h q u o t i e n t t e c h n i q u e t o be d e s c r i b e d below. Thus ically g . ( u ) and i t s d e r i v a t e s , i f r e q u i r e d , must be e v a l u a t e d n u m e r i c a l l y . T h i s c o u l d c r e a t e comput a t i o n a l d i f f i c u l t i e s during the minimization o f the objective function. I n l i g h t o f t h i s d i s c u s s i o n , t h e s p e c i f i c problem under c o n s i d e r a t i o n becomes t h a t o f m i n i m i z i n g t h e o b j e c t i v e f u n c t i o n f o ( u ) , where fo(u) = wo(u) s u b j e c t t o t h e frequency c o n s t r a i n t g,(u)
=
a 0
'I2-'?
T
0
R. A. Rand and C. N. Shen the stress constraints g2(u) = sL(u)-oL(u)
0
the r i n g s t i f f n e s s constraint
the f i b e r o r i e n t a t i o n constraint
t h e lamina t h i c k n e s s c o n s t r a i n t
t h e f i b e r volume f r a c t i o n c o n s t r a i n t
and t h e r i n g s i z e c o n s t r a i n t
Equations (12) through (20) a r e d e r i v e d i n d e t a i l elsewhere (Ref. 11). We n o t e t h a t t h e problem i s a g e n e r a l methematical programming problem w i t h o n l y i n e q u a l i t y c o n s t r a i n t s .
...
The o b j e c t i v e f u n c t i o n fo(u) and t h e c o n s t r a i n t s g . ( u ) , i = 2, , 8 a r e c l o s e d form f u n c t i o n s of u, b u t s i n c e w ( u ) must be determined n u m e r i c a l l y , t h e c o n s t r a i n t g ( u ) i s n o t a c l o s e d form f u n c t i o n o f u. ~ 6 ter a n s v e r s e frequency w T ( u ) i s determined by f i r s t b i s c r e t i z i n g t h e c o n t i n u ous s h e l l u s i n g t h e f i n i t e - e l e m e n t t e c h n i q u e and t h e n m i n i m i z i n g a s p e c i a l R a y l e i g h t q u o t i e n t d e f i n e d f o r t h e d i s c r e t e model. The techniques a r e d e s c r i b e d below. Thus t h e s o l u t i o n t o t h e problem formulated i n e q u a t i o n s (12) t h r o u g h (20) i n v o l v e s a m i n i m i z a t i o n w i t h i n a m i n i m i z a t i o n . D i s c r e t i z a t i o n of D i s t r i b u t e d Parameter S h e l l For a s h e l l s t r u c t u r e , d i s c r e t i z a t i o n i s performed i n b o t h t h e a x i a l and c i r c u m f e r e n t i a l d i r e c t i o n s , as shown s c h e m a t i c a l l y i n F i g . 8. The number o f elements r e q u i r e d t o model t h e s h e l l i s a f u n c t i o n o f t h e response c h a r a c t e r i s t i c s o f t h e element b e i n g used and t h e accuracy d e s i r e d . I n g e n e r a l , accuracy i n c r e a s e s w i t h i n c r e a s i n g number o f elements, as shown s c h e m a t i c a l l y i n F i g . 9. However, no f i r m r u l e s f o r t h e number o f elements r e q u i r e d f o r a p a r t i c u l a r problem can be f o r m u l a t e d . The response o f any element i i n a d i s c r e t i z e d s h e l l model i s d e s c r i b e d by t h e r e l a t i o n
where Ki = s t i f f n e s s m a t r i x f o r element i Mi = mass m a t r i x f o r element i
..
q . = v e c t o r o f nodal p o i n t displacement components f o r element i 1
A fi
=
v e c t o r o f a p p l i e d nodal p c i n t f o r c e components f o r element i
2 denotes d / d l Z , where 7 i s time. The f o r m u l a t i o n s o f Ki, M. i n e q u a t i o n and t h e s u p e r s c r i p t doubly(21) depend upon t h e t y p e o f element used i n t h e d i s c r e t i z a t i o n . For t h i s s t u d y curved q u a d r i l a t e r a l s h e l l element i s developed u t i l i z i n o t h e smooth s u r f a c e i n t e r p o l a t i o n
a
Distributed Mass and Stiffness Structural Systems
ERROR
\
F i g . 8.
Discretization o f shell structure u s i n g f i n i t e elements
F i g . 9. Accuracy o f f i n i t e - e l e m e n t c a l c u l a t i o n as a f u n c t i o n o f number o f elements
f u n c t i o n s o f P i a n ( 9 ) . A d e s c r i p t i o n o f t h e element and d e t a i l s o f t h e d e r i v a t i o n s o f Ki, a r e presented elsewhere (Refs. 10, 1 1 ) .
Mi
The response o f t h e ( c o m p l e t e ) she1 1 s t r u c t u r e i s d e s c r i b e d by assembling t h e response equat i c n s o f t h e i n d i v i d u a l elements. T h i s assembly y i e l d s a l a r g e s e t of e q u a t i o n s of t h e form
where
#
~ e n o t e sa m a t r i x c o m p o s i t i o n r a t h e r t h a n a d i r e c t summation. An example o f m a t r i x c o m p o s i t i o n f o r a s i m p l e two element mcdel i s p r e s e n t e d elsewhere (Ref. 1 1 ) . We s i m p l i f y e q u a t i o n (22) b y w r i t i n g 1
where K = (assembled) s t i f f n e s s m a t r i x f o r t h e s h e l l M = (assembled) mass m a t r i x f o r t h e s h e l l
q = v e c t o r o f nodal p o i n t displacement components f o r e n t i r e s h e l l f
A
= v e c t o r o f a p p l i e d nodal p o i n t f o r c e components f o r e n t i r e s h e l l
T y p i c a l l y , t h e s e t (23) c o n t a i n s s e v e r a l hundred e q u a t i o n s and t h e s o l u t i o n i s o b t a i n e d u s i n g m a t r i x techniques developed f o r l a r g e s e t s o f e q u a t i o n s . Frequency C a l c u l a t i o n Using R a y l e i g h Q u o t i e n t The o r d i n a r y d i f f e r e n t i a l e q u a t i o n s (23) o b t a i n e d by d i s c r e t i z i n g t h e s h e l l can be d i r e c t l y i n t e g r a t e d w i t h r e s p e c t t o t i m e t o o b t a i n t h e complete s h e l l response h i s t o r y . Imposing cons t r a i n t s on t h i s response r a t h e r t h a n on t h e n a t u r a l f r e q u e n c i e s would r e s u l t i n a t r u e o p t i m a l c o n t r o l problem, which c o u l d be t r e a t e d w i t h i n t h e framework o f t h e techniques developed. However, t h e computation c o s t s f o r a l l b u t t h e s i m p l e s t c o n f i g u r a t i o n s and l o a d i n g s would be p r o h i b i t i v e . We make t h e problem one o f o p t i m a l d e c i s i o n r a t h e r t h a n o p t i m a l c o n t r o l by imposi n g c o n s t r a i n t s o n l y 'on t h e n a t u r a l f r e q u e n c i e s and n o t on t h e a c t u a l response. The n a t u r a l frequencies o f t h e s h e l l d e s c r i b e d m a t h e m a t i c a l l y by e q u a t i o n (23) a r e o b t a i n e d by setting fA = 0
i n e q u a t i o n (23) and s o l v i n g t h e r e s u l t i n g f r e e v i b r a t i o n e q u a t i o n
R . A . Rand a n d C. N . Shen
The c l a s s i c a l method f o r s o l v i n g e q u a t i o n ( 2 4 ) t o o b t a i n t h e s h e l l n a t u r a l f r e q u e n c i e s i s t o assume a s o l u t i o n o f t h e form j w-r q = ye
(25)
where
and s u b s t i t u t e t h i s s o l u t i o n i n t o e q u a t i o n ( 2 4 ) .
l21
The r e s u l t i n g frequency e q u a t i o n becomes
-wM+K y = 0
(26)
E q u a t i o n (26) i s s o l v e d by numerical techniques f o r n 3 values o f n a t u r a l frequency wi and c o r r e s p o n d i n g nodal v e c t o r yl, where n i s t h e number o f degrees o f freedom i n t h e f i n i t e element system. However, i n such syst&rrs t h e o r d e r o f t h e m a t r i c e s M and K i s so l a r g e t h a t c o m p u t a t i o n a l c o s t s and s t o r a g e requirements become excessive. One way t o a v o i d these problems i s t o s e l e c t "key" degrees o f freedom and reduce t h e o r d e r o f system o f e q u a t i o n s u s i n g a r e d u c t i o n o f t h e t y p e proposed by Guyan ( 1 2 ) . T h i s t e c h n i q u e i s a r b i t r a r y i n t h e method of s e l e c t i n g t h e "key" nodes. I n t h i s s t u d y t h e f r e q u e n c i e s a r e o b t a i n e d f r o m K, M u s i n g a R a y l e i g h - t y p e technique, which r e s u l t s i n s i g n i f i c a n t l y l o w e r computational c o s t s . The t e c h n i q u e i s based upon R a y l e i g h ' s p r i n c i p l e , which s t a t e s t h a t i n t h e fundamental mode o f v i b r a t i o n o f an e l a s t i c system t h e d i s t r i b u t i o n o f k i n e t i c and p o t e n t i a l e n e r g i e s i s such as t o make t h e frequency a minimum (Ref. 1 3 ) . By d e f i n i t i o n , t h e R a y l e i g h q u o t i e n t r ( q ) a s s o c i a t e d w i t h q i s g i v e n by 7
T T I n e q u a t i o n (27) q Kq i s p r o p o r t i o n a l t o t h e p o t e n t i a l ( s t r a i n ) energy i n t h e model and q Mq i s p r o p o r t i o n a l t o t h e k i n e t i c energy i n t h e model. Thus, as a consequence o f R a y l e i g h ' s p r i n c i p l e , i f r ( q ) i s m i n i m i z e d w i t h r e s p e c t t o a, t h e n as t h e minimum i s approached
when w i s t h e fundamental f r e q u e n c y o f t h e she1 1, y1 i s t h e c o r r e s p o n d i n g model v e c t o r , and c l i s a sEale f a c t o r which a r i s e s due t o t h e f a c t t h a t i f y1 i s a s o l u t i o n t o e q u a t i o n (24) f o r can o n l y be determined t o w i t h i n an a r b i t r a r y conw12, t h e n clyl i s a l s o a s o l u t i o n . Thus yl stant. T h i s method can be extended t o f i n d successive v a l u e s o f wi2 and yi f o r t h e h i g h e r modes b y i n t r o d u c i n g o r t h o g o n a l i t y c o n s t r a i n t s . Thus, i f r ( q ) i s m i n i m i z e d s u b j e c t t o t h e c o n s t r a i n t t h a t q be o r t h o g o n a l t o y l , so
then
I n general, i f r ( q ) i s minimized subject t o t h e cciistraint; ni modal v e c t o r s , so
then
t h a t q be o r t h o g o n a l t o t h e f i r s t
D i s t r i b u t e d Mass a n d S t i f f n e s s S t r u c t u r a l S y s t e m s
M a t h e m a t i c a l l y , these r e s u l t s a r e expressed as
w
ni+l
=
min r ( q ) 9
I
,2
qT~yi = 0, i = 1.2
G n.+l G n 1
,..., ni
o
\
1
Bv t h e use o f R a v l e i q h ' s ~ r i n c i ~ tl hee frequencv c a l c u l a t i o n i s t r a n s f o r m e d from a d i r e c t ( n u m e r i c a l ) c a l c u l a t i o n t o an lrnconstrained ( f o r w 1 2 ) o r a c o n s t r a i n e d ( f o r wi2, 2 < i < n ) A princip?e m i n i m i z a t i o n . A s i m i l a r scheme has been p r e s e n t e d b Fox and Kapoor (Ref. 14) reason f o r t h e t r a n s f o r r i a t l o n o f t h e frequency c a l c u f a t i o n from a d i r e c t t o an i n d i r e c t one i s t h a t t h e assembly o f t h e ( l a r g e ) m a t r i c e s K,M can be r e p l a c e d by t h e assembly o f v e c t o r s Ka and Mq. T h i s assembly proceeds a n a l o g o u s l y t o t h e assembly o f K, M.
.
D e t e r m i n a t i o n o f S h e l l Transverse Frequency
WT
I n g e n e r a l , u n c o n s t r a i n e d m i n i m i z a t i o n o f r ( q ) y i e l d s t h e l o w e s t n a t u r a l frequency o f t h e d i s c r e t i z e d model. I f t h e c o n s t r a i n e d frequency i s n o t t h e l o w e s t frequency o f t h e s h e l l , t h e c a l c u l a t i o n o f t h e c o n s t r a i n e d frequency r e q u i r e s a c o n s t r a i n e d m i n i m i z a t i o n o f r ( q ) . The o p t i m a l d e c i s i o n o f u i n such problems t h u s r e q u i r e s a c o n s t r a i n e d m i n i m i z a t i o n ( t o o b t a i n t h e f r e q u e n c y ) w i t h i n a c o n s t r a i n e d m i n i m i z a t i o n ( t o o b t a i n u*) and t h e c o m p u t a t i o n a l c o s t s c o u l d become 1a r g e . I n t h i s s t u d y we develop a t e c h n i q u e by which t h e c a l c u l a t i o n o f t h e s h e l l t r a n s v e r s e b e n d i n r f r e q u e n c y w , i n c l u d i n g i n e r t i a l s t i f f e n i n g r e s u l t i n g from t h e a x i a l l o a d f due t o r o t a t i o n w R o f t h e shelT, i s t r a n s f o r m e d t o an u n c o n s t r a i n e d m i n i m i z a t i o n r e g a r d l e s s o f i t s o r d e r i n t h e ( o r d e r e d ) v e c t o r o f modal f r e q u e n c i e s . The t r a n s f o r m a t i o n i s performed by d e f i n i n g a s p e c i a l R a y l e i g h q u o t i e n t r ' ( q ' ) , where -r
.-
I n e q u a t i o n (33) q ' i s o b t a i n e d f r o m q as d e s c r i b e d below and uf i s t w i c e t h e p o t e n t i a l energy c o n t r i b u t e d t o t h e t r a n s v e r s e ) v i b r a t i o n mode c o r r e s p o n d i n g t o w T by t h e ( a x i a l ) i n e r t i a l l o a d f. I n t h i s s t u d y u i s g i v e n by (Ref. 11)
i
where ~ ( z i) s t h e t r a n s v e r s e displacement o f t h e s h e l l c e n t r o i d a t any c r o s s - s e c t i o n , as shown s c h e m a t i c a l l y i n F i g . 10. I n e q u a t i o n (33), t h e m o d i f i e d nodal p o i n t displacement v e c t o r q ' i s o b t a i n e d from q by s e t t i n g t o 0 those components i n q which, due t o symmetry o r r i g i d body m o t i o n c o n s i d e r a t i o n s , do n o t c o n t r i b u t e t o t h e e n e r g i e s i n t h e t r a n s v e r s e mode F o r example, a t node n o f element i shown and a r e s e t t o z e r o t o s a t i s f v symmetry i n F i g . 10, t h e ( r o t a t i o n a l ) degrees o f freedom requirements. For t h e s e f i v e degrees o f freedom, t h e t r a n s f o r m a t i o n from q t o q ' i s as f o l l o w s :
@
2
The corresponding components of t h e g r a d i e n t v r ' ( q l ) of r ' a r e a l s o s e t t o 0. The v a l u e o f w T 2 i s now o b t a i n e d by m i n i m i z a t i o n o f r ' ( q ' ) as g i v e n by e q u a t i o n ( 3 3 ) . The m i n i m i z a t i o n i s u n c o n s t r a i n e d r e g a r d l e s s of t h e a c t u a l o r d e r of wT, and t h e g r a d i e n t V r ' ( q t ) o f r ' ( q l ) can be o b t a i n e d b y d i f f e r e n t i a t i o n o f e q u a t i o n ( 3 3 ) , so any o f s e v e r a l g r a d i e n t techn i q u e s can be used. I n t h i s s t u d y t h e c o n j u g a t e g r a d i e n t method i s used (Ref. 8). T h i s s p e c i a l R a y l e i g h q u o t i e n t t e c h n i q u e can be a p p l i e d t o any c o n s t r a i n e d f r e q u e n c y f o r which t h e modal v e c t o r can be r e a s o n a b l y e s t i m a t e d . Optimum D e c i s i o n The d e c i s i o n ( m i n i n ~ i z a t i o n ) nroblem f o r m u l a t e d i n e q u a t i o n s (12) through (20) i s a n o n l i n e a r mathematical programming problem w i t h on1y inequal it y c o n s t r a i n t s . The optimum d e c i s i o n v e c t o r u* i s o b t a i n e d u s i n g two techniques: an " a n a l y t i c a l " s o l u t i o n i s o b t a i n e d u s i n g t h e c l a s s i c a l
R. A. Rand and C. N. Shen method o f Lagrange m u l t i p l i e r s and conpared w i t h a numerical s o l u t i o n o b t a i n e d u s i n g a m o d i f i e d Complex method. I n o r d e r t o e l i m i n a t e dimensional i n c o n s i s t e n c i e s i n t h e a n a l y t i c a l s o l u t i o n and t o e l i m i n a t e s c a l i n g ( n u m e r i c a l ) problems i n t h e numerical s o l u t i o n , t h e problem i s f i r s t transformed t o nondimensional c o o r d i n z t e space by d e f i n i n g a new d e c i s i o n v e c t o r v w i t h nondimensional components
F i g . 10.
S h e l l m o t i o n i n t r a n s v e r s e mode (schematic)
Then t h e o p t i m a l d e c i s i o n problem becomes t h a t o f m i n i m i z i n a f o ( v ) = w0(v) s u b j e c t t o t h e nondimensional c o n s t r a i n t s yl(v)
=
2
nT ( v ) - 1 > 0
>0
y2(v) = sL(v)-uL(v)
y3(v) = sN(v)-aN(v) > O Y ~ ( v )= kR(V)-kR ( v )
>
0
0
y 5 ( v ) = (1-vl)(vl-.125) y ( v ) = v2-1 6
>0
>0
y 7( v ) = (1-v3)(v3-. 5) 2 0 y 8 ( v ) = v4-1
0
w h e r e n T i s a nondimension frequency g i v e n by
The a n a l y t i c a l s o l u t i o n t o t h i s problem i s o b t a i n e d u s i n g Lagrange m u l t i p l i e r s (Ref. 8 ) . we seek t h e s t a t i o n a r y p o i n t s o f a new o b j e c t i v e f u n c t i o n f ' ( v ) d e f i n e d as
Thus,
3
f l ( v ) = fo(v)+
=y
( v )
i= 1
where A . a r e t h e Lagrange m u l t i p l i e r s . posing the conditions
and s e l e c t i n g hi such t h a t
The s t a t i o n a r y p o i n t s f o r f ' ( v ) a r e determined by im-
Distributed Mass and Stiffness Structural Systems
Equations (40) and (41) comprise a s e t o f 12 e q u a t i o n s i n 12 unknowns. T h i s s e t i s s o l v e d t o d e t e r m i n e t h e s t a t i o n a r y p o i n t s o f f ' ( v ) . The s t a t i o n a r y p o i n t r e s u l t i n g i n t h e minimum v a l u e of f o ( v ) corresponds t o t h e optimum d e c i s i o n v e c t o r v*. Before t h e a n a l y t i c a l s o l u t i o n can proceed, t h e c o n s t r a i n t y ( v ) must be expressed as a f u n c t i o n o f v. T h i s i n t u r n r e q u i r e s t h a t o~ be expressed a4 an a n a l y t i c a l f u n c t i o n o f u. g e n e r a l f u n c t i o n a l form o f w T 2 ( u ) i s
The
where G r e p r e s e n t s t h e s h e l l e x t e r n a l geometry. I n t h i s problem wR,C a r e f i x e d and we c o u l d f i t a g e n e r a l power s e r i e s i n u., i = 1,2,3,4 t o an a r r a y of ( n u m e r i c a l l y c a l c u l a t e d ) values o f wT2. However, f o r t h i n s h e l l s we can s i m p l i f y t h e problem by assuming t h a t we can s e p a r a t e i n e r t i a l and bending e f f e c t s . Then e q u a t i o n (42) becomes WTL;
1
fl “l ( w R > ~ ) t f 2W '( ~ 9 ~ ) For f i x e d wR, C t h i s reduces t o
where a. i s a c o n s t a n t . The bending c o n t r i b u t i o n fZw(u) i n e q u a t i o n (44) i s now assumed t o be p r o p o r t i o n a l t o k /ms, where ks, ms a r e measures of s t i f f n e s s and mass, r e s p e c t i v e l y , f o r t h e s h e l l . Thus equa%ion ( 4 4 ) becomes
where a i s a c o n s t a n t . F i n a l l y , f o r t h i n l a m i n a t e d s h e l l s w i t h f i x e d c o n s t i t u e n t p r o p e r t i e s , k s which c o n t r o l s w T i s ( a p p r o x i m a t e l y ) g i v e n by
and
and e q u a t i o n ( 4 5 ) becomes w T 2 ( u ) = ao +a 1
U3COS
U1
l+u3
2 The c o n s t a n t s a a a r e determined by c a l c u l a t i n g w ( u ) f o r d i s c r e t e values of u!. u2, u3, and f i t t i n g equgiioA ( 4 6 ) t o t h e r e s u l t s u s i n g a l e a i t - s q u a r e s technique. The ana y t i c a l s o l u t i o n thus u t i l i z e s an approximate e x p r e s s i o n f o r LJ ( u ) . T The numerical s o l u t i o n t h e problem posed by e q u a t i o n s (36) and ( 3 7 ) i s o b t a i n e d u s i n g a modif i e d Complex method (Ref. 8 ) . T h i s method i s a t t r a c t i v e f o r numerical o p t i m i z a t i o n problems of t h e t y p e c o n s i d e r e d i n t h i s s t u d y because i t does n o t r e q u i r e e v a l u a t i o n o f t h e g r a d i e n t s of e i t h e r f ( v ) o r y . ( v ) , i = 1, . . . , 8. The b a s i c Complex method i s s i m i l a r t o t h e Simplex method eRcept t h a t t h e shape, as w e l l as t h e s i z e , o f t h e Complex v a r i e s . The i n i t i a l Complex i s e s t a b l i s h e d by t a k i n g any a c c e s s i b l e p o i n t as t h e s t a r t i n g v e r t e x . Each o f t h e r e m a i n i n g v e r t i c e s i s found by randomly d e f i n i n g a v e r t e x , t e s t i n g i t s a c c e s s i b i l i t y , and moving i t , if necessary, towards t h e c e n t r o i d o f t h e v e r t i c e s a l r e a d y e s t a b l i s h e d u n t i l i t becomesaccessible. A f t e r t h e i n i t i a l Complex i s e s t a b l i s h e d , m i n i m i z a t i o n i s performed by e l i m i n a t i n g t h e l e a s t d e s i r a b l e v e r t e x ( i .e. t h e v e r t e x w i t h t h e l a r g e s t v a l u e o f t h e o b j e c t i v e f u n c t i o n f ) by r e f l e c t i o n o f t h i s v e r t e x an a r b i t r a r y d i s t a n c e through t h e c e n t r o i d o f t h e r e m a i n i n g B e - t i c e s , t e s t i n g t h e a c c e s s i b i l i t y o f t h e new v e r t e x , and moving i t , i f necessary, towards t h e c e n t r o i d the u n t i l i t becomes a c c e s s i b l e . T h i s process i s repeated u n t i l no f u r t h e r improvement i~ o b j e c t i v e f u n c t i o n f o ( v ) can be o b t a i n e d and t h e Complex c o l l a p s e s ( i n t h e l i m i t ) t o a s i n g l e v e r t e x c o r r e s p o n d i n g t o v*. A b a s i c p r o p e r t y of t h e R a y l e i g h q u o t i e n t i s t h a t o n l y a reasonable e s t i m a t e o f t h e modal v e c t o r y l i s r e q u i r e d t o y i e l d a good e s t i m a t e o f w.2. The Complex method was s e l e c t e d f o r a p p l i c a t i o n i n t h i s s t u d y s i n c e i t enabled us t o e x b l o i t t h i s p r o p e r t y t o reduce t h e computat i o n a l e f f o r t r e q u i r e d t o o b t a i n v*. We assume t h a t yt i s ( r e l a t i v e l y ) weak y coupled t o v and recompute yt o n l y p e r i o d i c a l l y d u r i n g t h e m i n i m i z a t i o n , n o t each t i m e uTAi s determined.
R. A . Rand and C . N .
Shen
We can do t h i s because o f t h e p r o p e r t y oC t h e d a y l e i g h q u o t i e n t d e s c r i b e d above and s i n c e t h e Complex method i s s e l f - c o r r e c t i n g ; a new v e r t e x i s chosen by r e j e c t i n g t r i a l v e r t i c e s u n t i l an improvement i s o b t a i n e d . I n a d d i t i o n , we recompute Ki, Mi d i r e c t l y o n l y when y x i s updated. P.t i n t e r m e d i a t e p o i n t s we assume t h a t K . = 6 6 6 K
1 2 3 9 1
1
M . = d 6 ~ ~ i 2 3 i
where
KT, MY a r e t h e cos 6
=-
4
( l a s t ) computed values o f Ki,
Mi
a t u0 and
Ul
4 cos u
0
1
Again, t h e Complex method a u t o m a t i c a l l y c o r r e c t s f o r any e r r o r i n t r o d u c e d by u s i n g e q u a t i o n ( 4 7 ) . W i t h these m o d i f i c a t i o n s t h e numerical o p t i m i z a t i o n proceeds v e r y q u i c k l y . RESULTS The fundamental frequency w f o r t r a n s v e r s e (bending) v i b r a t i o n o f t h e she1 1 shown i n F i g . 7 about t h e e l l i p s e m a j o r a x i T ( i . e . t h e v i b r a t o r y m o t i o n i s p a r a l l e l t o t h e e l l i p s e m i n o r a x i s ) was c a l c u l a t e d f o r d i s c r e t e values o f t h e m a t e r i a l d e s i g n parameters ul ( f i b e r o r i e n t a t i o n an l e ) , u2 ( l a m i n a t h i c k n e s s ) , and u3 ( f i b e r volume f r a c t i o n ) and t h e r e s u l t s used t o express qPas a c l o s e d form f u n c t i o n o f u. The c a l c u l a t e d values o f wT a r e summarized i n Table 2; t h e e x p r e s s i o n wT2(u) o b t a i n e d by l e a s t - s q u a r e s f i t t i n g o f e q u a t i o n ( 4 6 ) t o these r e s u l t s i s w T 2 ( u ) = 24520+1215100( Values o f w ' ( u ) T
o b t a i n e d u s i n g e q u a t i o n ( 4 9 ) a r e a l s o resented i n Table 2.
TABLE 2
1
4 u3cos U1 l+u3 1
( o r i e n t a t i o n angle a i n degrees)
R e s u l t s o f Frequency C a l c u l a t i o n
I
1
u7 (lamina thickness i n inches)
+calculated t ~ r o mf i t [equation (4911
!/
U3 ( f i b e r volume fraction)
+
wT2~1~-6 2 2 ( r a d /sec )
Distributed Mass and Stiffness Structural Systems
The weight of this shell and its associated attaching ring was minimized with res ect to the decision vector u defined by equation (10) subject to a (design) constraint o n w ~(u). Design constraints on stress and stiffness and manufacturing constraints were also imposed. Mathematical formulation of the problem is given in equations (12) through (20). The optimum decision vector u* which minimizes the weight was obtained by two methods. Using the approximate frequency relation equation (49) and solving the problem analytically using Lagrange mu1 tipliers yields
5
wo*(=f *) = w (u*) 0
0
=
3.639 lbs
and
u2*(=tL*)
=
u3*('vF*)
= .519
u4*(=b*)
=
.0073
.570 inches
where the superscript * denotes the value for optimal decision of u. Solving the problem using the modified Complex method and evaluating wT2(u) periodically using equation (33) yields w0*
=
3.650 lbs
ul*
=
10.05'
u2*
=
.00732 inches
u3*
=
.520
u4*
=
.572 inches
and
Complete results are summarized in Table 3.
a.
TABLE 3 Results for Optimum Decision of u Results at optimum using Lagrauge multipliers yi* (dimensionless) constraint
b.
Results at optimum using Complex
Xi* (Lagrange mu1 tipliers)
R. A. Rand and C. N. Shen
DISCUSSION From T a b l e 3 we n o t e t h a t t h e v a l u e o f u* o b t a i n e d n u m e r i c a l l y ' u s i n ( r t h e m o d i f i e d Complex method i s i n c l o s e agreement w i t h t h e a n a l y t i c a l s o l u t i o n o b t a i n e d u s i n g Lagrange m u l t i p l i e r s , v e r i f y i n g t h a t t h e m o d i f i e d Complex t e c h n i q u e developed i s acceptable. The a n a l y t i c a l r e s u l t s o b t a i n e d u s i n g Lagrange m u l t i p l i e r s can a l s o be used t o show by i m p l i c a t i o n t h a t t h e v a l u e o f u* o b t a i n e d i s indeed o p t i m a l , i . e . t h a t i t corresponds t o minimum weight. We n o t e f r o m T a b l e 3 t h a t a l l non-zero Lagrange m u l t i p l i e r s a r e l e s s t h a n zero. Then a Kuhn-Tucker theorum i n ~ p i i e st h a t u* cannot be a maximum. We can i n f a c t show t h a t u* corresponds t o a minimum by p e r t u r b i n g u*. Thus, a t u1*+6u1,
g3(u) i s violated;
a t ul*-6u1,
g5(u) i s violated;
a t u2*+6u2, g 4 ( u ) i s v i o l a t e d and f o ( u ) i n c r e a s e s ; a t u2*-6u2,
g2(u), g3(u) a r e violated;
a t u,*+6u3,
g4(u) i s v i o l a t e d and f o ( u ) i n c r e a s e s ;
J
a t u3*-6u3,
g2(u) i s v i o l a t e d ;
a t u4*+6u4, f c ( u ) increases; and a t u4*-du4,
g4(u) i s v i o l a t e d .
So u* does indeed correspond t o a minimum o f t h e o b j e c t i v e f u n c t i o n f ( u ) . The d i r e c t c o m p u t a t i o n a l c o s t o f t h e Complex m i n i m i z a t i o n i s o n l y about two t i m e s t h e c o s t of g e n e r a t i n g t h e d a t a t o o b t a i n t h e approximate e x p r e s s i o n f o r w 2 ( u ) used i n t h e a n a l y t i c a l m i n i m i z a t i o n . Thus, t h e Complex m i n i m i z a t i o n i s c o m p e t i t i v e wien t h e manpower c o s t s i n v o l v e d i n t h e a n a l y t i c a l m i n i m i z a t i o n a r e a l s o considered. Several aspects o f t h e techniques p r e s e n t e d i n t h i s paper a r e new. The q u a d r i l a t e r a l s h e l l element developed t o d i s c r e t i z e t h e d i s t r i b u t e d parameter s h e l l s t r u c t u r e can be a p p l i e d t o s h e l l s which a r e l a m i n a t e d composites. The mass m a t r i x f o r t h e element i s a h y b r i d which combines t h e c o m p u t a t i o n a l speed o f a lumped mass m a t r i x w i t h t h e accuracy o f a c o m p a t i b l e mass m a t r i x . The s p e c i a l R a y l e i g h q u o t i e n t t e c h n i q u e f o r frequency c a l c u l a t i o n i s d i r e c t l y c o m p a t i b l e w i t h t h e d i s c r e t i z e d model and v e r y g e n e r a l . The e f f e c t o f i n e r t i a l s t i f f e n i n g i s d i r e c t l y i n c l u d e d . S p e c i f i c f r e q u e n c i e s , n o t n e c e s s a r i l y t h e lowest, w i t h e a s i l y approximated modal v e c t o r s a r e o b t a i n e d by u n c o n s t r a i n e d m i n i m i z a t i o n o f t h e m o d i f i e d R a y l e i g h q u o t i e n t . The m o d i f i e d Complex method developed f o r t h e optimum d e c i s i o n o f u i s c o m p a t i b l e w i t h t h e f r e q u e n c y c a l c u l a t i o n t e c h n i q u e i n t h a t i t assumes t h a t t h e c o n s t r a i n e d f r e q u e n c y w, i s r e l a t i v e l y i n s e n s i t i v e t o s m a l l changes i n u. Thus t h e modal v e c t o r yt c o r r e s p o n d i h g t o wT i s recomputed o n l y p e r i o d i c a l l y . The t e c h n i q u e i s s e l f - c o r r e c t i n g . I n addition, the t e c h n i q u e does n o t r e q u i r e e v a l u a t i o n o f t h e g r a d i e n t o f e i t h e r t h e o b j e c t i v e f u n c t i o n f o r t h e c o n s t r a i n t s gi, so t h a t n o i s e problems encountered i n an e a r l i e r problem, when g r a d i g n t techniques were employed i n c o m b i n a t i o n w i t h t h e p e n a l t y f u n c t i o n method o f F i a c c o and McCormick (Ref. 15) t o m i n i m i z e t h e w e i g h t o f a c y l i n d r i c a l s h e l l s u b j e c t t o a c o n s t r a i n t on i t s c i r c u m f e r e n t i a l e x t e n s i o n mode frequency (Ref. 10) a r e e l i m i n a t e d . Para1 l e l s o l u t i o n o f t h e o p t i m a l d e c i s i o n problem by an " a n a l y t i c a l " t e c h n i q u e p r o v i d e s d i r e c t v e r i f i c a t i o n of t h e m o d i f i e d Complex method developed and p e r m i t s an e s t i m a t i o n o f t h e e f f i c i e n c y o f t h e numerical method. Such a method seems f e a s i b l e f o r automated d e s i g n o f complex s t r u c t u r e s s u b j e c t t o c o n s t r a i n t s based upon t h i s study. F i n a l l y , p r e v i o u s s t r u c t u r a l o p t i m i z a t i o n s t u d i e s have c o n c e n t r a t e d p r i m a r i l y on s a t i s f y i n g e i t h e r dynamic ( f r e q u e n c y ) o r s t a t i c ( s t r e s s ) c o n s t r a i n t s ; i n t h i s s t u d y t h e f e a s i b i l i t y o f combining s t r u c t u r a l a n a l y s i s techniques and o p t i m i z a t i o n techniques d i r e c t l y t o o b t a i n optimum s t r u c t u r a l designs under simultaneous frequency and s t r e s s c o n s t r a i n t s i s demonstrated. The techniques developed a r e g e n e r a l and a p p l i c a b l e t o o t h e r t y p e s o f problems. Acknowledgement P a r t o f t h i s paper i s s u b m i t t e d t o t h e School o f E n g i n e e r i n g a t Rensselaer P o l y t e c h n i c I n s t i t u t e i n p a r t i a l f u l f i l l m e n t o f t h e requirements o f t h e degree o f D o c t o r o f Engineering.
Distributed Mass and Stiffness Structural Systems
REFERENCES (1)
Kraus, H. (1967) T h i n E l a s t i c S h e l l s , Wiley, New York.
(2)
Z i e n k i e w i c z , 0. C. (1971) The F i n i t e - E l e m e n t Method i n E n g i n e e r i n g Science, McGraw-Hill, New York.
(3)
P r z e m i e n i e c k i , J. S . (1967) Theory o f M a t r i x S t r u c t u r a l A n a l y s i s , McGraw-Hill, New York
(4) Schmit, L. A., The s t r u c t u r a l s y n t h e s i s concept and i t s p o t e n t i a l r o l e i n d e s i g n w i t h composites, Proceedin s, General E l e c t r i c I O f f i c e o f Naval Research Composites Symposicm, P d (1967). (5)
Turner, M. J., O p t i m i z a t i o n o f s t r u c t u r e s t o s a t i s f y f l u t t e r requirements, J. AIAA 7, 945 (1969).
(6) Rubin, C. P., Minimum w e i g h t d e s i g n o f complex s t r u c t u r e s s u b j e c t t o a frequency cons t r a i n t , J. AIAA 8, 923 (1970).
(7) McCart, R. B. Haug, E. J., and S t r e e t e r , T. D., Optimal d e s i g n of s t r u c t u r e s w i t h cons t r a i n t s on n a t u r a l f r e q u e n c i e s , Vol. o f Tech. Papers, AIAA S t r u c t u r a l Dynamics and A e r o e l a s t i c i t y S p e c i a l i s t Conference, New Orleans (1969). (8)
Beveridge, G. S. and Schechter, R. S . (1972) O p t i m i z a t i o n : Theory and P r a c t i c e , McGrawH i l l , New York.
(9)
Deak, A.L. and Pian, T. H. H., A p p l i c a t i o n o f t h e smooth-surface i n t e r p o l a t i o n t o t h e f i n i t e - e l e m e n t a n a l y s i s , J. AIAA 5, 187 (1967).
(10)
Rand, R. A. and Shen, C. N., Optimum d e s i g n o f composite s h e l l s s u b j e c t t o n a t u r a l frequency c o n s t r a i n t s , J. Computer and S t r u c t u r e s 3, 247 (1973).
(11)
Rand, R. A., Optimum d e s i g n o f s h e l l s t r u c t u r e s s u b j e c t t o c o n s t r a i n t s , D o c t o r ' s Thesis, Rensselaer P o l y t e c h n i c I n s t i t u t e , Troy, New York (1977).
(12)
Guyan, R. J.,
(13)
Temple, G.
(14)
Fox, R. L. and Kapoor, M. P., A m i n i m i z a t i o n method f o r t h e s o l u t i o n o f t h e e i g e n problem a r i s i n g i n s t r u c t u r a l dynamics, Proceedings, W r i q h t - P a t t e r s o n Conference on M a t r i x Methods, Ohio (1968).
(15)
Fiacco, A. V. and McCormick, G. P. (1968) N o n l i n e a r Programming: s t r a i n e d M i n i m i z a t i o n Techniques, Wiley, New York.
(16)
N o v o z h i l o v , V. V.
Reduction o f S t i f f n e s s and Mass M a t r i c e s , J. AIAA 3, 380 (1965).
and B i c k l e y , W . G. (1956) R a y l e i g h ' s P r i n c i p l e , Dover, New York.
S e q u e n t i a l Uncon-
(1959) Theory o f T h i n S h e l l s ( T r a n s l a t i o n ) , P. Noordhoff.
5,,c2 r,
= meridmmal CoordiMtes
-
"0-1
cmrdinate
VF =
F i g . 1.
FIBER WW4E
F i g . 2. D e f i n i t i o n o f l a m i n a t e parameters a , tL, v
D i f f e r e n t i a l s h e l l element
The response o f t h i s element i s d e s c r i b e d by a s e t o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s c o n s i s t i n g of f i v e e q u i l i b r i u m e q u a t i o n s , s i x f o r c e - s t r a i n r e l a t i o n s , and s i x s t r a i n - d i s p l a c e m e n t r e l a t i o n s . A complete d e r i v a t i o n o f these e q u a t i o n s i s g i v e n by Kraus ( 1 ) . The response o f t h e complete s h e l l s t r u c t u r e i s o b t a i n e d by i n t e g r a t i n g t h i s s e t o f e q u a t i o n s o v e r t h e e n t i r e s h e l l surface and a p p l y i n g a p p r o p r i a t e i n i t i a l and boundary c o n d i t i o n s . The r e s u l t i n o e q u a t i o n s a r e solved, f o r a p a r t i c u l a r c o n f i g u r a t i o n and l o a d i n g , e i t h e r by f i n i t e - d i f f e r e n c e techniaues o r by d i r e c t numerical i n t e g r a t i o n .
*Now w i t h General E l e c t r i c N u c l e a r Energy D i v i s i o n , San Jose, C a l i f o r n i a , U.S.A. 511
R. A . Rand and C . N .
Shen
An a l t e r n a t i v e t o t h i s c l a s s i c a l approach has e v o l v e d o v e r t h e p a s t s e v e r a l y e a r s . Before f o r m u l a t i o n o f t h e g o v e r n i n g e q u a t i o n s t h e c o n t i n u o u s s t r u c t u r e , w h i c h i n t h e c l a s s i c a l app r o a c h has an i n f i n i t e number o f degrees o f freedom, i s i d e a l i z e d as an assembly o f d i s c r e t e elements, o f f i n i t e dimension, w i t h assumed response c h a r a c t e r i s t i c s . These elements a r e i n t e r c o n n e c t e d a t a f i n i t e number o f nodal p o i n t s . The d i s p l a c e m e n t components o f t h e nodal p o i n t s become t h e degrees o f freedom f o r t h e s t r u c t u r e . The response of t h e s t r u c t u r e i s obt a i n e d by combining t h e response o f t h e d i s c r e t e elements such t h a t e q u i l i b r i u m and compatib i l i t y c o n d i t i o n s a r e s a t i s f i e d a t a l l nodal p o i n t s . However, s i n c e t h e s i z e of t h e elements remains f i x e d and t h e number o f degrees o f freedom i s f i n i t e , i n t e g r a t i o n s o v e r t h e e n t i r e s t r u c t u r e a r e r e p l a c e d by f i n i t e summations and t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n s o f t h e cont i n u o u s s t r u c t u r e a r e r e p l a c e d by a system o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s o f l a r g e dimens i o n , w h i c h a r e i d e a l l y s u i t e d f o r s o l u t i o n on l a r g e s c a l e d i g i t a l computers. The m a j o r d i f f e r e n c e between t h i s new approach, now commonly r e f e r r e d t o as t h e f i n i t e - e l e m e n t method, and t h e c l a s s i c a l approach i s t h a t a p p r o x i m a t i o n s a r e i n t r o d u c e d i n t h e f o r m u l a t i o n o f t h e g o v e r n i n g e q u a t i o n s , r a t h e r t h a n i n t h e i r s o l u t i o n . The f i n i t e - e l e m e n t method has been a p p l i e d t o v a r i o u s t y p e s o f s t r u c t u r e s w i t h e x c e l l e n t r e s u l t s ( R e f s . 2,3). There i s c o n s i d e r a b l e i n t e r e s t i n d e v e l o p i n g automated t o o l s f o r s e e k i n g optimum s t r u c t u r a l c o n f i g u r a t i o n s ( R e f . 4 ) . The s t r u c t u r a l o p t i m i z a t i o n problem i s g e n e r a l l y posed as t h a t o f d e t e r m i n i n g t h e v a l u e o f a s e t o f v a r i a b l e d e s i g n parameters such t h a t an o b j e c t i v e f u n c t i o n i s n ~ i n i m i z e d( o r maximized) s u b j e c t t o a p p r o p r i a t e d e s i o n c o n s t r a i n t s . The o b j e c t i v e f u n c t i o n i s d e f i n e d i n such a way t h a t i t i s a measure o f t h e "goodness" o f t h e s t r u c t u r e under c o n s i d e r a tion. Because o f t h e i r h i g h s t r u c t u r a l e f f i c i e n c y , d i s t r i b u t e d parameter s h e l l s t r u c t u r e s a r e used i n h i g h performance a p p l i c a t i o n s where w e i g h t i s most o f t e n t h e p r i m e measure o f t h e "goodness" o f t h e d e s i g n . O t h e r measures such as f a b r i c a t i o n c o s t s and r e f u r b i s h a b i l i t y c o u l d a l s o e n t e r i n t o t h e e v a l u a t i o n o f t h e d e s i g n , b u t i n t h i s s t u d y o n l y t h e s t r u c t u r a l w e i g h t w i l l be i n c l u d ed i n t h e o b j e c t i v e f u n c t i o n and t h e terms "minimum w e i g h t d e s i g n " and "optimum d e s i g n " w i l l be used i n t e r c h a n g e a b l y . To f u r t h e r i n c r e a s e t h e s t r u c t u r a l e f f i c i e n c y , s h e l l s t r u c t u r e s a r e o f t e n c o n s t r u c t e d o f a f i b e r - r e i n f o r c e d composite. Composite m a t e r i a l s have been employed w i t h e v e r - i n c r e a s i n g f r e quency i n r e c e n t y e a r s , e s p e c i a l l y i n h i g h performance s h e l l s t r u c t u r e s . A l t h o u g h a composite m a t e r i a l i s , i n g e n e r a l , d e f i n e d as a m a t e r i a l system composed of a m i x t u r e o r c o m b i n a t i o n of two o r more c o n s t i t u e n t s t h a t d i f f e r i n f o r m a n d / o r c o m p o s i t i o n and t h a t a r e e s s e n t i a l l y i n s o l u b l e i n one a n o t h e r , t h i s d i s c u s s i o n d e a l s e x c l u s i v e l y w i t h h i g h performance f i b e r comp o s i t e s , i . e . composite m a t e r i a l s i n w h i c h h i g h s t r e n g t h f i b e r s a r e embedded w i t h p r e f e r r e d o r i e n t a t i o n s i n a ( r e l a t i v e l y ) l o w s t r e n g t h m a t r i x . Such composites a r e f a b r i c a t e d by s t a c k i n g l a m i n a e c o n s i s t i n g o f s t i f f f i b e r s embedded i n a s o f t m a t r i x , as shown s c h e m a t i c a l l y i n F i g . 2. The f i b e r o r i e n t a t i o n a n g l e a , l a m i n a t h i c k n e s s tL, and f i b e r volume f r a c t i o n v F f o r each lami n a a r e ( m a t e r i a l ) d e s i g n parameters. However, i n h i g h performance a p p l i c a t i o n s t h e s h e l l s a r e l i k e l y t o be s u b j e c t e d t o e x t e r n a l e x c i t a t i o n s . To a v o i d h a r m f u l resonance e f f e c t s , t h e s h e l l s must be designed s u b j e c t t o cons t r a i n t s on t h e i r n a t u r a l f r e q u e n c i e s ( R e f s . 5, 6, 7 ) . These c o n s t r a i n t s a r e u s u a l l y o f t h e form
n a t u r a l frequency o f t h e s h e l l i n q u e s t i o n and w A , w a r e d e s i g n conwhere w i s t h e ith 'i s t r a i n t $ d e t e r m i n e d from t h e f r e q u e n c i e s o f t h e e x t e r n a l e x c i t a t i o n s . 7 C o n s t r a i n t s o f t h e f o r m ( I ) , ( 2 ) a r e imposed on t h e n a t u r a l f r e q u e n c i e s t o m a i n t a i n s t r u c t u r a l i n t e g r i t y under t h e time-dependent l o a d i n g s . To i n s u r e s t r u c t u r a l i n t e g r i t y under any t i m e independent l o a d i n g s , s t r e s s c o n s t r a i n t s o f t h e f o r m
where oi i s t h e a p p l i e d s t r e s s and si imposed.
i s t h e l a m i n a t e s t r e n g t h i n t h e ith direction, are also
Distributed Mass and Stiffness Structural Systems
I n l i Y i h ~O T tile above d i s c u s s i o n , t h e o b j e c t i v e s of t h i s s t u d y were t o develop a t e c h n i o u e t o determine t h e n a t u r a l frequencies o f g e n e r a l composite s h e l l s ; develop a t e c h n i q u e t o o p t i m i z e such s h e l l s s u b j e c t t o n a t u r a l frequency and s t r e s s c o n s t r a i n t s ; and demonstrate t h e techniques by a p p l i c a t i o n t o a t y p i c a l problem. To accomplish these o b j e c t i v e s , a frequency c a l c u l a t i o n method c o u p l i n g t h e f i n i t e - e l e m e n t t e c h n i q u e f o r d i s c r e t i z a t i o n of c o n t i n u o u s s t r u c t u r a l systems w i t h a s p e c i a l R a y l e i g h q u o t i e n t t e c h n i q u e i s developed. A m i n i m i z a t i o n t e c h n i q u e der i v e d by m o d i f y i n g t h e "Complex Method" o f Box(8) i s a l s o developed. T h i s m i n i m i z a t i o n techn i q u e i s c o m p a t i b l e w i t h t h e frequency c a l c u l a t i o n techniaue. A s o l u t i o n i s p r e s e n t e d i n which t h e optimum c o n f i g u r a t i o n f o r a d i s t r i b u t e d parameter s h e l l s t r u c t u r e i s o b t a i n e d t h r o u g h o p t i m a l d e c i s i o n o f a v e c t o r o f d e s i g n parameters which j n c l u d e s b o t h m a t e r i a l and geometric v a r i a b l e s . The s h e l l i s s u b j e c t e d t o simultaneous frequency and s t r e s s c o n s t r a i n t s . T h i s paper d e a l s p r i m a r i l y w i t h t h e frequency c a l c u l a t i o n technique. The o p t i m i z a t i o n p o r t i o n o f the study i s described i n less d e t a i l . ANALYSIS Problem D e f i n i t i o n A d i s t r i b u t e d parameter system c o n s i s t i n g o f a s l e n d e r c o n i c a l s h e l l o f e l l i p t i c a l c r o s s s e c t i o n w i t h a m e t a l l i c end r i n g i s r i g i d l y a t t a c h e d t o a f o u n d a t i o n and s u b j e c t e d t o i n e r t i a l l o a d i n g a l o n g i t s a x i s and p e r i o d i c t r a n s v e r s e ( e x t e r n a l ) l o a d i n g normal t o i t s a x i s and p a r a l l e l t o t h e m i n o r e l l i p t i c a l a x i s , as shown s c h e m a t i c a l l y i n F i g . 3. Such l o a d i n g s would o c c u r on a s h e l l used as t h e p r i m a r y s t r u c t u r a l member o f a h e l i c o p t e r r o t o r blade, as shown schema t i c a l l y i n F i g . 4.
C~
al shell structure ( s c h m t ~ c l
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, b) ring g-tv and load on d l f f e m t l a l c l m t
',
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F i g . 3.
S h e l l s t r u c t u r e and l o a d i n g
F i g . 4. A p p l i c a t i o n o f t a p e r e d e l l i p t i c a l s h e l l as p r i m a r y s t r u c t u r a l member o f r o t o r b l a d e (schematic)
For t h e case shown i n F i g . 4 t h e i n e r t i a l l o a d i n g a r i s e s due t o t h e r o t a t i o n o f t h e s h e l l about an a x i s normal t o i t s c e n t e r l i n e and t h e t r a n s v e r s e l o a d i n g i s t h e l i f t generated by t h i s r o t a t i o n . The r i n g i s used so t h a t r e a c t i o n s a t t h e attachment p o i n t w i l l n o t r e s u l t i n h i g h l o c a l stresses i n the s h e l l wall. The she:l i s assumed t o be a f i b e r - r e i n f o r c e d l a m i n a t e c o n s t r u c t e d by s t a c k i n g f i b e r g l a s s - e p o x y laminae. The f i b e r o r i e n t a t i o n a n g l e a, t h i c k n e s s t , and f i b e r volume f r a c t i o n v f o r each lamina a r e ( m a t e r i a l ) d e s i g n parameters. F o r a n a l y s \ s purposes we assume t h e l a m i k a t e c o n s i s t s o f f o u r laminae. The v a l u e s o f vF, t and t h e magnitudes of a a r e equal f o r each laminae; t h e s i g n of a v a r i e s such t h a t t h e l a m ~ n a k ei s balanced. The l a m i n a t e geometry i s shown i n F i g . 5. M a t e r i a l p r o p e r t i e s o f t h e f i b e r g l a s s and epoxy c o n s t i t u e n t s o f t h e l a m i n a t e a r e shown i n Table 1 . The s i z e of t h e m e t a l l i c a t t a c h i n g r i n g i s c h a r a c t e r i z e d by a s i z e parameter b, where b i s a ( g e o m e t r i c ) d e s i g n parameter. The geometry o f t h e r i n g i s shown i n F i g . 6. TABLE 1 M a t e r i a l P r o p e r t i e s o f Composite C o n s t i t u e n t s
Fiberglass EPOXY
e 2 ( E l a s t i c modulus, l b / i n )
v (Poissons r a t i o )
(Weight d e n s i t y , l b s / i n 3 )
16.0 X lo6 0 . 5 X lo6
0.20 0.35
.0914 .0500
P
R. A. Rand and C . N. Sllen
F i a . 5.
Laminate geometry
F i g . 6.
A t t a c h i n g r i n g geometry
For t h e h e l i c o p t e r b l a d e a p p l i c a t i o n c o n s i d e r e d i n t h i s problem t h e o v e r a l l s h e l l geometry and l o a d i n g s a r e as shown i n F i g . 7. The a x i a l l o a d i n g f v a r i e s a l o n g t h e l e n g t h o f t h e s h e l l b u t i s assumed t o be independent of t i m e T . The t r a n s v e r s e l o a d i n g n i s assumed t o be u n i f o r m a l o n g t h e l e n g t h of t h e s h e l l and t o c o n s i s t o f a l a r g e time-independent component no due t o t h e l i f t generated by b l a d e r o t a t i o n and a time-dependent component n ' a r i s i n g f r o m unsteady aerodynamics ( f l u t t e r ) . The e x c i t a t i o n f r e q u e n c y o f n ' i s assumed t 8 v a r y randomly between 0 and w ' To a v o i d t h e harmful resonance e f f e c t s o f laPge t i p d e f l e c t i o n s and l a r g e bending s t r e s e e s i n t h e s h e l l which would o c c u r i f t h e s h e l l was e x c i t e d a t c l o s e t o a n a t u r a l frequenc y by n o o ' , t h e fundamental t r a n s v e r s e (bending) frequency wT o f t h e s h e l l i s c o n s t r a i n e d such that
.
where
w 0
= kw' 0
The f a c t o r k i s chosen such t h a t a p p l i c a t i o n o f t h e c o n t r a i n t ( 4 ) w i l l i n s u r e t h a t t h e response o f t h e s h e l l t o n ' w i l l be s m a l l r e l a t i v e t o t h e response t o no. I n o r d e r t o i n s u r e s t r u c t u r a l i n t e g r i t y undep t h e ( t i m e - i n d e p e n d e n t ) l o a d i n g s f, n and t o i n s u r e t h a t t h e r i n g serves i t s purpose o f p r e v e n t i n g h i g h l o c a l s t r e s s e s a t t h e a t t a c h 8 e n t p o i n t , s t r e s s c o n s t r a i n t s o f t h e form
and a s t i f f n e s s c o n s t r a i n t o f t h e form
a r e a l s o imposed. I n e q u a t i o n ( 6 ) uL i s t h e a p p l i e d s t r e s s i n t h e l a m i n a t e i n t h e f i b e r d i r e c t i o n and sL !s t h e l a m i n a t e s t r e n g t h i n t h e f i b e r d i r e c t i o n . S i m i l a r l y , i n e q u a t i o n ( 7 ) o N i s t h e a p p l i e d s t r e s s i n t h e l a m i n a t e i n t h e d i r e c t i o n t r a n s v e r s e t o t h e f i b e r s and s i s t h e l a m i n a t e s t r e n g t h i n t h a t d i r e c t i o n . C o n s t r a i n t s ( 6 ) , ( 7 ) a r e a p p l i e d a t t h e poin! o f maximum s t r e s s i n t h e s h e l l . I n e q u a t i o n (8) kR i s t h e r i n g a x i a l s t i f f n e s s p e r u n i t l e n g t h o f circumference and k i s a l o w e r l i m i t f o r kR. R0
Since t h e s h e l i i s used i n an a i r b o r n e a p p l i c a t i o n i t must have minimum weight. The d e s i g n problem becomes t h a t o f d e t e r m i n i n g t h e m a t e r i a l d e s i g n parameters a , t L , v F and t h e geometric d e s i g n parameter b such t h a t t h e s h e l l has minimum w e i g h t and s a t i s f i e s t h e frequency cons t r a i n t ( 4 ) and t h e s t r e s s and s t i f f n e s s c o n t r a i n t s ( 6 ) , ( 7 ) , ( 8 ) . I n a d d i t i o n t o these d e s i g n c o n s t r a i n t s i t i s a l s o r e q u i r e d t h a t t be g r e a t e r t h a n .0325" (a m a n u f a c t u r i n g c o n s t r a i n t ) , t h a t v be g r e a t e r t h a n . 4 b u t l e s s t h k n . 8 ( a m a n u f a c t u r i n g c o n s t r a i n t ) , and t h a t a be g r e a t e r t h a n l & o b u t l e s s t h a n 800 (a m a n u f a c t u r i n g c o n s t r a i n t ) .
Distributed Mass and Stiffness Structural Systems
f = axla1 load a t : n = n o m l load at z a , = 6 a , = ? b , = d b2=2.66' i = 60
ROTATION
"dl'
F i g . 7.
S h e l l geometry and l o a d i n g
Problem F o r m u l a t i o n The d e s i g n parameters a r e assembled t o form a v e c t o r u o f problem v a r i a b l e s . The v e c t o r u can i n c l u d e b o t h g e o m e t r i c v a r i a b l e s which d e s c r i b e t h e s i z e o r shape o f t h e s h e l l o r a s s o c i a t e d components and m a t e r i a l v a r i a b l e s which d e s c r i b e t h e e l a s t i c p r o p e r t i e s o r g e o m e t r i c c o n f i g u r a t i o n o f t h e l a m i n a t e c o m p r i s i n g t h e s h e l l w a l l . As i n d i c a t e d above, t h e r e a r e f o u r d e s i g n parameters f o r t h e problem c o n s i d e r e d i n t h i $ paper: t h e f i b e r o r i e n t a t i o n a n g l e a (measured w i t h r e s p e c t t o t h e s h e l l a x i a l c o o r d i n a t e s ) , t h e lamina t h i c k n e s s tL ( n o t e t h a t t h e t o t a l s h e l l t h i c k n e s s can be o b t a i n e d once tL i s determined), t h e f i b e r volume f r a c t i o n vF, and t h e r i n g s i z e parameter b. Thus u becomes
where
ul
= f i b e r o r i e n t a t i o n a n g l e a (degrees)
u2 = lamina t h i c k n e s s tL ( i n c h e s ) u3 = f i b e r volume f r a c t i o n vF
u4 = r i n g s i z e parameter b ( i n c h e s )
1
The w e i g h t w o f t h e s h e l l and i t s a s s o c i a t e d components and t h e d e s i g n (and m a n u f a c t u r i n g ) c o n s t r a i n t s g r e t h e n expressed i n terms o f u. The s p e c i f i c c o n s t r a i n t s depend upon t h e s p e c i f i c problem u n d e r r c o n s i d e r a \ i o n b u t w i l j i n c l u d e i n e q u a l i t y c o n s t r a i n t s o f t h e form
2 where wi, 'I , 'I a r e as d e f i n e d p r e v i o u s l y . The magnitude o f wi(u) must be determined numer*i Bi u s i n g t h e s p e c i a l R a y l e i g h q u o t i e n t t e c h n i q u e t o be d e s c r i b e d below. Thus ically g . ( u ) and i t s d e r i v a t e s , i f r e q u i r e d , must be e v a l u a t e d n u m e r i c a l l y . T h i s c o u l d c r e a t e comput a t i o n a l d i f f i c u l t i e s during the minimization o f the objective function. I n l i g h t o f t h i s d i s c u s s i o n , t h e s p e c i f i c problem under c o n s i d e r a t i o n becomes t h a t o f m i n i m i z i n g t h e o b j e c t i v e f u n c t i o n f o ( u ) , where fo(u) = wo(u) s u b j e c t t o t h e frequency c o n s t r a i n t g,(u)
=
a 0
'I2-'?
T
0
R. A. Rand and C. N. Shen the stress constraints g2(u) = sL(u)-oL(u)
0
the r i n g s t i f f n e s s constraint
the f i b e r o r i e n t a t i o n constraint
t h e lamina t h i c k n e s s c o n s t r a i n t
t h e f i b e r volume f r a c t i o n c o n s t r a i n t
and t h e r i n g s i z e c o n s t r a i n t
Equations (12) through (20) a r e d e r i v e d i n d e t a i l elsewhere (Ref. 11). We n o t e t h a t t h e problem i s a g e n e r a l methematical programming problem w i t h o n l y i n e q u a l i t y c o n s t r a i n t s .
...
The o b j e c t i v e f u n c t i o n fo(u) and t h e c o n s t r a i n t s g . ( u ) , i = 2, , 8 a r e c l o s e d form f u n c t i o n s of u, b u t s i n c e w ( u ) must be determined n u m e r i c a l l y , t h e c o n s t r a i n t g ( u ) i s n o t a c l o s e d form f u n c t i o n o f u. ~ 6 ter a n s v e r s e frequency w T ( u ) i s determined by f i r s t b i s c r e t i z i n g t h e c o n t i n u ous s h e l l u s i n g t h e f i n i t e - e l e m e n t t e c h n i q u e and t h e n m i n i m i z i n g a s p e c i a l R a y l e i g h t q u o t i e n t d e f i n e d f o r t h e d i s c r e t e model. The techniques a r e d e s c r i b e d below. Thus t h e s o l u t i o n t o t h e problem formulated i n e q u a t i o n s (12) t h r o u g h (20) i n v o l v e s a m i n i m i z a t i o n w i t h i n a m i n i m i z a t i o n . D i s c r e t i z a t i o n of D i s t r i b u t e d Parameter S h e l l For a s h e l l s t r u c t u r e , d i s c r e t i z a t i o n i s performed i n b o t h t h e a x i a l and c i r c u m f e r e n t i a l d i r e c t i o n s , as shown s c h e m a t i c a l l y i n F i g . 8. The number o f elements r e q u i r e d t o model t h e s h e l l i s a f u n c t i o n o f t h e response c h a r a c t e r i s t i c s o f t h e element b e i n g used and t h e accuracy d e s i r e d . I n g e n e r a l , accuracy i n c r e a s e s w i t h i n c r e a s i n g number o f elements, as shown s c h e m a t i c a l l y i n F i g . 9. However, no f i r m r u l e s f o r t h e number o f elements r e q u i r e d f o r a p a r t i c u l a r problem can be f o r m u l a t e d . The response o f any element i i n a d i s c r e t i z e d s h e l l model i s d e s c r i b e d by t h e r e l a t i o n
where Ki = s t i f f n e s s m a t r i x f o r element i Mi = mass m a t r i x f o r element i
..
q . = v e c t o r o f nodal p o i n t displacement components f o r element i 1
A fi
=
v e c t o r o f a p p l i e d nodal p c i n t f o r c e components f o r element i
2 denotes d / d l Z , where 7 i s time. The f o r m u l a t i o n s o f Ki, M. i n e q u a t i o n and t h e s u p e r s c r i p t doubly(21) depend upon t h e t y p e o f element used i n t h e d i s c r e t i z a t i o n . For t h i s s t u d y curved q u a d r i l a t e r a l s h e l l element i s developed u t i l i z i n o t h e smooth s u r f a c e i n t e r p o l a t i o n
a
Distributed Mass and Stiffness Structural Systems
ERROR
\
F i g . 8.
Discretization o f shell structure u s i n g f i n i t e elements
F i g . 9. Accuracy o f f i n i t e - e l e m e n t c a l c u l a t i o n as a f u n c t i o n o f number o f elements
f u n c t i o n s o f P i a n ( 9 ) . A d e s c r i p t i o n o f t h e element and d e t a i l s o f t h e d e r i v a t i o n s o f Ki, a r e presented elsewhere (Refs. 10, 1 1 ) .
Mi
The response o f t h e ( c o m p l e t e ) she1 1 s t r u c t u r e i s d e s c r i b e d by assembling t h e response equat i c n s o f t h e i n d i v i d u a l elements. T h i s assembly y i e l d s a l a r g e s e t of e q u a t i o n s of t h e form
where
#
~ e n o t e sa m a t r i x c o m p o s i t i o n r a t h e r t h a n a d i r e c t summation. An example o f m a t r i x c o m p o s i t i o n f o r a s i m p l e two element mcdel i s p r e s e n t e d elsewhere (Ref. 1 1 ) . We s i m p l i f y e q u a t i o n (22) b y w r i t i n g 1
where K = (assembled) s t i f f n e s s m a t r i x f o r t h e s h e l l M = (assembled) mass m a t r i x f o r t h e s h e l l
q = v e c t o r o f nodal p o i n t displacement components f o r e n t i r e s h e l l f
A
= v e c t o r o f a p p l i e d nodal p o i n t f o r c e components f o r e n t i r e s h e l l
T y p i c a l l y , t h e s e t (23) c o n t a i n s s e v e r a l hundred e q u a t i o n s and t h e s o l u t i o n i s o b t a i n e d u s i n g m a t r i x techniques developed f o r l a r g e s e t s o f e q u a t i o n s . Frequency C a l c u l a t i o n Using R a y l e i g h Q u o t i e n t The o r d i n a r y d i f f e r e n t i a l e q u a t i o n s (23) o b t a i n e d by d i s c r e t i z i n g t h e s h e l l can be d i r e c t l y i n t e g r a t e d w i t h r e s p e c t t o t i m e t o o b t a i n t h e complete s h e l l response h i s t o r y . Imposing cons t r a i n t s on t h i s response r a t h e r t h a n on t h e n a t u r a l f r e q u e n c i e s would r e s u l t i n a t r u e o p t i m a l c o n t r o l problem, which c o u l d be t r e a t e d w i t h i n t h e framework o f t h e techniques developed. However, t h e computation c o s t s f o r a l l b u t t h e s i m p l e s t c o n f i g u r a t i o n s and l o a d i n g s would be p r o h i b i t i v e . We make t h e problem one o f o p t i m a l d e c i s i o n r a t h e r t h a n o p t i m a l c o n t r o l by imposi n g c o n s t r a i n t s o n l y 'on t h e n a t u r a l f r e q u e n c i e s and n o t on t h e a c t u a l response. The n a t u r a l frequencies o f t h e s h e l l d e s c r i b e d m a t h e m a t i c a l l y by e q u a t i o n (23) a r e o b t a i n e d by setting fA = 0
i n e q u a t i o n (23) and s o l v i n g t h e r e s u l t i n g f r e e v i b r a t i o n e q u a t i o n
R . A . Rand a n d C. N . Shen
The c l a s s i c a l method f o r s o l v i n g e q u a t i o n ( 2 4 ) t o o b t a i n t h e s h e l l n a t u r a l f r e q u e n c i e s i s t o assume a s o l u t i o n o f t h e form j w-r q = ye
(25)
where
and s u b s t i t u t e t h i s s o l u t i o n i n t o e q u a t i o n ( 2 4 ) .
l21
The r e s u l t i n g frequency e q u a t i o n becomes
-wM+K y = 0
(26)
E q u a t i o n (26) i s s o l v e d by numerical techniques f o r n 3 values o f n a t u r a l frequency wi and c o r r e s p o n d i n g nodal v e c t o r yl, where n i s t h e number o f degrees o f freedom i n t h e f i n i t e element system. However, i n such syst&rrs t h e o r d e r o f t h e m a t r i c e s M and K i s so l a r g e t h a t c o m p u t a t i o n a l c o s t s and s t o r a g e requirements become excessive. One way t o a v o i d these problems i s t o s e l e c t "key" degrees o f freedom and reduce t h e o r d e r o f system o f e q u a t i o n s u s i n g a r e d u c t i o n o f t h e t y p e proposed by Guyan ( 1 2 ) . T h i s t e c h n i q u e i s a r b i t r a r y i n t h e method of s e l e c t i n g t h e "key" nodes. I n t h i s s t u d y t h e f r e q u e n c i e s a r e o b t a i n e d f r o m K, M u s i n g a R a y l e i g h - t y p e technique, which r e s u l t s i n s i g n i f i c a n t l y l o w e r computational c o s t s . The t e c h n i q u e i s based upon R a y l e i g h ' s p r i n c i p l e , which s t a t e s t h a t i n t h e fundamental mode o f v i b r a t i o n o f an e l a s t i c system t h e d i s t r i b u t i o n o f k i n e t i c and p o t e n t i a l e n e r g i e s i s such as t o make t h e frequency a minimum (Ref. 1 3 ) . By d e f i n i t i o n , t h e R a y l e i g h q u o t i e n t r ( q ) a s s o c i a t e d w i t h q i s g i v e n by 7
T T I n e q u a t i o n (27) q Kq i s p r o p o r t i o n a l t o t h e p o t e n t i a l ( s t r a i n ) energy i n t h e model and q Mq i s p r o p o r t i o n a l t o t h e k i n e t i c energy i n t h e model. Thus, as a consequence o f R a y l e i g h ' s p r i n c i p l e , i f r ( q ) i s m i n i m i z e d w i t h r e s p e c t t o a, t h e n as t h e minimum i s approached
when w i s t h e fundamental f r e q u e n c y o f t h e she1 1, y1 i s t h e c o r r e s p o n d i n g model v e c t o r , and c l i s a sEale f a c t o r which a r i s e s due t o t h e f a c t t h a t i f y1 i s a s o l u t i o n t o e q u a t i o n (24) f o r can o n l y be determined t o w i t h i n an a r b i t r a r y conw12, t h e n clyl i s a l s o a s o l u t i o n . Thus yl stant. T h i s method can be extended t o f i n d successive v a l u e s o f wi2 and yi f o r t h e h i g h e r modes b y i n t r o d u c i n g o r t h o g o n a l i t y c o n s t r a i n t s . Thus, i f r ( q ) i s m i n i m i z e d s u b j e c t t o t h e c o n s t r a i n t t h a t q be o r t h o g o n a l t o y l , so
then
I n general, i f r ( q ) i s minimized subject t o t h e cciistraint; ni modal v e c t o r s , so
then
t h a t q be o r t h o g o n a l t o t h e f i r s t
D i s t r i b u t e d Mass a n d S t i f f n e s s S t r u c t u r a l S y s t e m s
M a t h e m a t i c a l l y , these r e s u l t s a r e expressed as
w
ni+l
=
min r ( q ) 9
I
,2
qT~yi = 0, i = 1.2
G n.+l G n 1
,..., ni
o
\
1
Bv t h e use o f R a v l e i q h ' s ~ r i n c i ~ tl hee frequencv c a l c u l a t i o n i s t r a n s f o r m e d from a d i r e c t ( n u m e r i c a l ) c a l c u l a t i o n t o an lrnconstrained ( f o r w 1 2 ) o r a c o n s t r a i n e d ( f o r wi2, 2 < i < n ) A princip?e m i n i m i z a t i o n . A s i m i l a r scheme has been p r e s e n t e d b Fox and Kapoor (Ref. 14) reason f o r t h e t r a n s f o r r i a t l o n o f t h e frequency c a l c u f a t i o n from a d i r e c t t o an i n d i r e c t one i s t h a t t h e assembly o f t h e ( l a r g e ) m a t r i c e s K,M can be r e p l a c e d by t h e assembly o f v e c t o r s Ka and Mq. T h i s assembly proceeds a n a l o g o u s l y t o t h e assembly o f K, M.
.
D e t e r m i n a t i o n o f S h e l l Transverse Frequency
WT
I n g e n e r a l , u n c o n s t r a i n e d m i n i m i z a t i o n o f r ( q ) y i e l d s t h e l o w e s t n a t u r a l frequency o f t h e d i s c r e t i z e d model. I f t h e c o n s t r a i n e d frequency i s n o t t h e l o w e s t frequency o f t h e s h e l l , t h e c a l c u l a t i o n o f t h e c o n s t r a i n e d frequency r e q u i r e s a c o n s t r a i n e d m i n i m i z a t i o n o f r ( q ) . The o p t i m a l d e c i s i o n o f u i n such problems t h u s r e q u i r e s a c o n s t r a i n e d m i n i m i z a t i o n ( t o o b t a i n t h e f r e q u e n c y ) w i t h i n a c o n s t r a i n e d m i n i m i z a t i o n ( t o o b t a i n u*) and t h e c o m p u t a t i o n a l c o s t s c o u l d become 1a r g e . I n t h i s s t u d y we develop a t e c h n i q u e by which t h e c a l c u l a t i o n o f t h e s h e l l t r a n s v e r s e b e n d i n r f r e q u e n c y w , i n c l u d i n g i n e r t i a l s t i f f e n i n g r e s u l t i n g from t h e a x i a l l o a d f due t o r o t a t i o n w R o f t h e shelT, i s t r a n s f o r m e d t o an u n c o n s t r a i n e d m i n i m i z a t i o n r e g a r d l e s s o f i t s o r d e r i n t h e ( o r d e r e d ) v e c t o r o f modal f r e q u e n c i e s . The t r a n s f o r m a t i o n i s performed by d e f i n i n g a s p e c i a l R a y l e i g h q u o t i e n t r ' ( q ' ) , where -r
.-
I n e q u a t i o n (33) q ' i s o b t a i n e d f r o m q as d e s c r i b e d below and uf i s t w i c e t h e p o t e n t i a l energy c o n t r i b u t e d t o t h e t r a n s v e r s e ) v i b r a t i o n mode c o r r e s p o n d i n g t o w T by t h e ( a x i a l ) i n e r t i a l l o a d f. I n t h i s s t u d y u i s g i v e n by (Ref. 11)
i
where ~ ( z i) s t h e t r a n s v e r s e displacement o f t h e s h e l l c e n t r o i d a t any c r o s s - s e c t i o n , as shown s c h e m a t i c a l l y i n F i g . 10. I n e q u a t i o n (33), t h e m o d i f i e d nodal p o i n t displacement v e c t o r q ' i s o b t a i n e d from q by s e t t i n g t o 0 those components i n q which, due t o symmetry o r r i g i d body m o t i o n c o n s i d e r a t i o n s , do n o t c o n t r i b u t e t o t h e e n e r g i e s i n t h e t r a n s v e r s e mode F o r example, a t node n o f element i shown and a r e s e t t o z e r o t o s a t i s f v symmetry i n F i g . 10, t h e ( r o t a t i o n a l ) degrees o f freedom requirements. For t h e s e f i v e degrees o f freedom, t h e t r a n s f o r m a t i o n from q t o q ' i s as f o l l o w s :
@
2
The corresponding components of t h e g r a d i e n t v r ' ( q l ) of r ' a r e a l s o s e t t o 0. The v a l u e o f w T 2 i s now o b t a i n e d by m i n i m i z a t i o n o f r ' ( q ' ) as g i v e n by e q u a t i o n ( 3 3 ) . The m i n i m i z a t i o n i s u n c o n s t r a i n e d r e g a r d l e s s of t h e a c t u a l o r d e r of wT, and t h e g r a d i e n t V r ' ( q t ) o f r ' ( q l ) can be o b t a i n e d b y d i f f e r e n t i a t i o n o f e q u a t i o n ( 3 3 ) , so any o f s e v e r a l g r a d i e n t techn i q u e s can be used. I n t h i s s t u d y t h e c o n j u g a t e g r a d i e n t method i s used (Ref. 8). T h i s s p e c i a l R a y l e i g h q u o t i e n t t e c h n i q u e can be a p p l i e d t o any c o n s t r a i n e d f r e q u e n c y f o r which t h e modal v e c t o r can be r e a s o n a b l y e s t i m a t e d . Optimum D e c i s i o n The d e c i s i o n ( m i n i n ~ i z a t i o n ) nroblem f o r m u l a t e d i n e q u a t i o n s (12) through (20) i s a n o n l i n e a r mathematical programming problem w i t h on1y inequal it y c o n s t r a i n t s . The optimum d e c i s i o n v e c t o r u* i s o b t a i n e d u s i n g two techniques: an " a n a l y t i c a l " s o l u t i o n i s o b t a i n e d u s i n g t h e c l a s s i c a l
R. A. Rand and C. N. Shen method o f Lagrange m u l t i p l i e r s and conpared w i t h a numerical s o l u t i o n o b t a i n e d u s i n g a m o d i f i e d Complex method. I n o r d e r t o e l i m i n a t e dimensional i n c o n s i s t e n c i e s i n t h e a n a l y t i c a l s o l u t i o n and t o e l i m i n a t e s c a l i n g ( n u m e r i c a l ) problems i n t h e numerical s o l u t i o n , t h e problem i s f i r s t transformed t o nondimensional c o o r d i n z t e space by d e f i n i n g a new d e c i s i o n v e c t o r v w i t h nondimensional components
F i g . 10.
S h e l l m o t i o n i n t r a n s v e r s e mode (schematic)
Then t h e o p t i m a l d e c i s i o n problem becomes t h a t o f m i n i m i z i n a f o ( v ) = w0(v) s u b j e c t t o t h e nondimensional c o n s t r a i n t s yl(v)
=
2
nT ( v ) - 1 > 0
>0
y2(v) = sL(v)-uL(v)
y3(v) = sN(v)-aN(v) > O Y ~ ( v )= kR(V)-kR ( v )
>
0
0
y 5 ( v ) = (1-vl)(vl-.125) y ( v ) = v2-1 6
>0
>0
y 7( v ) = (1-v3)(v3-. 5) 2 0 y 8 ( v ) = v4-1
0
w h e r e n T i s a nondimension frequency g i v e n by
The a n a l y t i c a l s o l u t i o n t o t h i s problem i s o b t a i n e d u s i n g Lagrange m u l t i p l i e r s (Ref. 8 ) . we seek t h e s t a t i o n a r y p o i n t s o f a new o b j e c t i v e f u n c t i o n f ' ( v ) d e f i n e d as
Thus,
3
f l ( v ) = fo(v)+
=y
( v )
i= 1
where A . a r e t h e Lagrange m u l t i p l i e r s . posing the conditions
and s e l e c t i n g hi such t h a t
The s t a t i o n a r y p o i n t s f o r f ' ( v ) a r e determined by im-
Distributed Mass and Stiffness Structural Systems
Equations (40) and (41) comprise a s e t o f 12 e q u a t i o n s i n 12 unknowns. T h i s s e t i s s o l v e d t o d e t e r m i n e t h e s t a t i o n a r y p o i n t s o f f ' ( v ) . The s t a t i o n a r y p o i n t r e s u l t i n g i n t h e minimum v a l u e of f o ( v ) corresponds t o t h e optimum d e c i s i o n v e c t o r v*. Before t h e a n a l y t i c a l s o l u t i o n can proceed, t h e c o n s t r a i n t y ( v ) must be expressed as a f u n c t i o n o f v. T h i s i n t u r n r e q u i r e s t h a t o~ be expressed a4 an a n a l y t i c a l f u n c t i o n o f u. g e n e r a l f u n c t i o n a l form o f w T 2 ( u ) i s
The
where G r e p r e s e n t s t h e s h e l l e x t e r n a l geometry. I n t h i s problem wR,C a r e f i x e d and we c o u l d f i t a g e n e r a l power s e r i e s i n u., i = 1,2,3,4 t o an a r r a y of ( n u m e r i c a l l y c a l c u l a t e d ) values o f wT2. However, f o r t h i n s h e l l s we can s i m p l i f y t h e problem by assuming t h a t we can s e p a r a t e i n e r t i a l and bending e f f e c t s . Then e q u a t i o n (42) becomes WTL;
1
fl “l ( w R > ~ ) t f 2W '( ~ 9 ~ ) For f i x e d wR, C t h i s reduces t o
where a. i s a c o n s t a n t . The bending c o n t r i b u t i o n fZw(u) i n e q u a t i o n (44) i s now assumed t o be p r o p o r t i o n a l t o k /ms, where ks, ms a r e measures of s t i f f n e s s and mass, r e s p e c t i v e l y , f o r t h e s h e l l . Thus equa%ion ( 4 4 ) becomes
where a i s a c o n s t a n t . F i n a l l y , f o r t h i n l a m i n a t e d s h e l l s w i t h f i x e d c o n s t i t u e n t p r o p e r t i e s , k s which c o n t r o l s w T i s ( a p p r o x i m a t e l y ) g i v e n by
and
and e q u a t i o n ( 4 5 ) becomes w T 2 ( u ) = ao +a 1
U3COS
U1
l+u3
2 The c o n s t a n t s a a a r e determined by c a l c u l a t i n g w ( u ) f o r d i s c r e t e values of u!. u2, u3, and f i t t i n g equgiioA ( 4 6 ) t o t h e r e s u l t s u s i n g a l e a i t - s q u a r e s technique. The ana y t i c a l s o l u t i o n thus u t i l i z e s an approximate e x p r e s s i o n f o r LJ ( u ) . T The numerical s o l u t i o n t h e problem posed by e q u a t i o n s (36) and ( 3 7 ) i s o b t a i n e d u s i n g a modif i e d Complex method (Ref. 8 ) . T h i s method i s a t t r a c t i v e f o r numerical o p t i m i z a t i o n problems of t h e t y p e c o n s i d e r e d i n t h i s s t u d y because i t does n o t r e q u i r e e v a l u a t i o n o f t h e g r a d i e n t s of e i t h e r f ( v ) o r y . ( v ) , i = 1, . . . , 8. The b a s i c Complex method i s s i m i l a r t o t h e Simplex method eRcept t h a t t h e shape, as w e l l as t h e s i z e , o f t h e Complex v a r i e s . The i n i t i a l Complex i s e s t a b l i s h e d by t a k i n g any a c c e s s i b l e p o i n t as t h e s t a r t i n g v e r t e x . Each o f t h e r e m a i n i n g v e r t i c e s i s found by randomly d e f i n i n g a v e r t e x , t e s t i n g i t s a c c e s s i b i l i t y , and moving i t , if necessary, towards t h e c e n t r o i d o f t h e v e r t i c e s a l r e a d y e s t a b l i s h e d u n t i l i t becomesaccessible. A f t e r t h e i n i t i a l Complex i s e s t a b l i s h e d , m i n i m i z a t i o n i s performed by e l i m i n a t i n g t h e l e a s t d e s i r a b l e v e r t e x ( i .e. t h e v e r t e x w i t h t h e l a r g e s t v a l u e o f t h e o b j e c t i v e f u n c t i o n f ) by r e f l e c t i o n o f t h i s v e r t e x an a r b i t r a r y d i s t a n c e through t h e c e n t r o i d o f t h e r e m a i n i n g B e - t i c e s , t e s t i n g t h e a c c e s s i b i l i t y o f t h e new v e r t e x , and moving i t , i f necessary, towards t h e c e n t r o i d the u n t i l i t becomes a c c e s s i b l e . T h i s process i s repeated u n t i l no f u r t h e r improvement i~ o b j e c t i v e f u n c t i o n f o ( v ) can be o b t a i n e d and t h e Complex c o l l a p s e s ( i n t h e l i m i t ) t o a s i n g l e v e r t e x c o r r e s p o n d i n g t o v*. A b a s i c p r o p e r t y of t h e R a y l e i g h q u o t i e n t i s t h a t o n l y a reasonable e s t i m a t e o f t h e modal v e c t o r y l i s r e q u i r e d t o y i e l d a good e s t i m a t e o f w.2. The Complex method was s e l e c t e d f o r a p p l i c a t i o n i n t h i s s t u d y s i n c e i t enabled us t o e x b l o i t t h i s p r o p e r t y t o reduce t h e computat i o n a l e f f o r t r e q u i r e d t o o b t a i n v*. We assume t h a t yt i s ( r e l a t i v e l y ) weak y coupled t o v and recompute yt o n l y p e r i o d i c a l l y d u r i n g t h e m i n i m i z a t i o n , n o t each t i m e uTAi s determined.
R. A . Rand and C . N .
Shen
We can do t h i s because o f t h e p r o p e r t y oC t h e d a y l e i g h q u o t i e n t d e s c r i b e d above and s i n c e t h e Complex method i s s e l f - c o r r e c t i n g ; a new v e r t e x i s chosen by r e j e c t i n g t r i a l v e r t i c e s u n t i l an improvement i s o b t a i n e d . I n a d d i t i o n , we recompute Ki, Mi d i r e c t l y o n l y when y x i s updated. P.t i n t e r m e d i a t e p o i n t s we assume t h a t K . = 6 6 6 K
1 2 3 9 1
1
M . = d 6 ~ ~ i 2 3 i
where
KT, MY a r e t h e cos 6
=-
4
( l a s t ) computed values o f Ki,
Mi
a t u0 and
Ul
4 cos u
0
1
Again, t h e Complex method a u t o m a t i c a l l y c o r r e c t s f o r any e r r o r i n t r o d u c e d by u s i n g e q u a t i o n ( 4 7 ) . W i t h these m o d i f i c a t i o n s t h e numerical o p t i m i z a t i o n proceeds v e r y q u i c k l y . RESULTS The fundamental frequency w f o r t r a n s v e r s e (bending) v i b r a t i o n o f t h e she1 1 shown i n F i g . 7 about t h e e l l i p s e m a j o r a x i T ( i . e . t h e v i b r a t o r y m o t i o n i s p a r a l l e l t o t h e e l l i p s e m i n o r a x i s ) was c a l c u l a t e d f o r d i s c r e t e values o f t h e m a t e r i a l d e s i g n parameters ul ( f i b e r o r i e n t a t i o n an l e ) , u2 ( l a m i n a t h i c k n e s s ) , and u3 ( f i b e r volume f r a c t i o n ) and t h e r e s u l t s used t o express qPas a c l o s e d form f u n c t i o n o f u. The c a l c u l a t e d values o f wT a r e summarized i n Table 2; t h e e x p r e s s i o n wT2(u) o b t a i n e d by l e a s t - s q u a r e s f i t t i n g o f e q u a t i o n ( 4 6 ) t o these r e s u l t s i s w T 2 ( u ) = 24520+1215100( Values o f w ' ( u ) T
o b t a i n e d u s i n g e q u a t i o n ( 4 9 ) a r e a l s o resented i n Table 2.
TABLE 2
1
4 u3cos U1 l+u3 1
( o r i e n t a t i o n angle a i n degrees)
R e s u l t s o f Frequency C a l c u l a t i o n
I
1
u7 (lamina thickness i n inches)
+calculated t ~ r o mf i t [equation (4911
!/
U3 ( f i b e r volume fraction)
+
wT2~1~-6 2 2 ( r a d /sec )
Distributed Mass and Stiffness Structural Systems
The weight of this shell and its associated attaching ring was minimized with res ect to the decision vector u defined by equation (10) subject to a (design) constraint o n w ~(u). Design constraints on stress and stiffness and manufacturing constraints were also imposed. Mathematical formulation of the problem is given in equations (12) through (20). The optimum decision vector u* which minimizes the weight was obtained by two methods. Using the approximate frequency relation equation (49) and solving the problem analytically using Lagrange mu1 tipliers yields
5
wo*(=f *) = w (u*) 0
0
=
3.639 lbs
and
u2*(=tL*)
=
u3*('vF*)
= .519
u4*(=b*)
=
.0073
.570 inches
where the superscript * denotes the value for optimal decision of u. Solving the problem using the modified Complex method and evaluating wT2(u) periodically using equation (33) yields w0*
=
3.650 lbs
ul*
=
10.05'
u2*
=
.00732 inches
u3*
=
.520
u4*
=
.572 inches
and
Complete results are summarized in Table 3.
a.
TABLE 3 Results for Optimum Decision of u Results at optimum using Lagrauge multipliers yi* (dimensionless) constraint
b.
Results at optimum using Complex
Xi* (Lagrange mu1 tipliers)
R. A. Rand and C. N. Shen
DISCUSSION From T a b l e 3 we n o t e t h a t t h e v a l u e o f u* o b t a i n e d n u m e r i c a l l y ' u s i n ( r t h e m o d i f i e d Complex method i s i n c l o s e agreement w i t h t h e a n a l y t i c a l s o l u t i o n o b t a i n e d u s i n g Lagrange m u l t i p l i e r s , v e r i f y i n g t h a t t h e m o d i f i e d Complex t e c h n i q u e developed i s acceptable. The a n a l y t i c a l r e s u l t s o b t a i n e d u s i n g Lagrange m u l t i p l i e r s can a l s o be used t o show by i m p l i c a t i o n t h a t t h e v a l u e o f u* o b t a i n e d i s indeed o p t i m a l , i . e . t h a t i t corresponds t o minimum weight. We n o t e f r o m T a b l e 3 t h a t a l l non-zero Lagrange m u l t i p l i e r s a r e l e s s t h a n zero. Then a Kuhn-Tucker theorum i n ~ p i i e st h a t u* cannot be a maximum. We can i n f a c t show t h a t u* corresponds t o a minimum by p e r t u r b i n g u*. Thus, a t u1*+6u1,
g3(u) i s violated;
a t ul*-6u1,
g5(u) i s violated;
a t u2*+6u2, g 4 ( u ) i s v i o l a t e d and f o ( u ) i n c r e a s e s ; a t u2*-6u2,
g2(u), g3(u) a r e violated;
a t u,*+6u3,
g4(u) i s v i o l a t e d and f o ( u ) i n c r e a s e s ;
J
a t u3*-6u3,
g2(u) i s v i o l a t e d ;
a t u4*+6u4, f c ( u ) increases; and a t u4*-du4,
g4(u) i s v i o l a t e d .
So u* does indeed correspond t o a minimum o f t h e o b j e c t i v e f u n c t i o n f ( u ) . The d i r e c t c o m p u t a t i o n a l c o s t o f t h e Complex m i n i m i z a t i o n i s o n l y about two t i m e s t h e c o s t of g e n e r a t i n g t h e d a t a t o o b t a i n t h e approximate e x p r e s s i o n f o r w 2 ( u ) used i n t h e a n a l y t i c a l m i n i m i z a t i o n . Thus, t h e Complex m i n i m i z a t i o n i s c o m p e t i t i v e wien t h e manpower c o s t s i n v o l v e d i n t h e a n a l y t i c a l m i n i m i z a t i o n a r e a l s o considered. Several aspects o f t h e techniques p r e s e n t e d i n t h i s paper a r e new. The q u a d r i l a t e r a l s h e l l element developed t o d i s c r e t i z e t h e d i s t r i b u t e d parameter s h e l l s t r u c t u r e can be a p p l i e d t o s h e l l s which a r e l a m i n a t e d composites. The mass m a t r i x f o r t h e element i s a h y b r i d which combines t h e c o m p u t a t i o n a l speed o f a lumped mass m a t r i x w i t h t h e accuracy o f a c o m p a t i b l e mass m a t r i x . The s p e c i a l R a y l e i g h q u o t i e n t t e c h n i q u e f o r frequency c a l c u l a t i o n i s d i r e c t l y c o m p a t i b l e w i t h t h e d i s c r e t i z e d model and v e r y g e n e r a l . The e f f e c t o f i n e r t i a l s t i f f e n i n g i s d i r e c t l y i n c l u d e d . S p e c i f i c f r e q u e n c i e s , n o t n e c e s s a r i l y t h e lowest, w i t h e a s i l y approximated modal v e c t o r s a r e o b t a i n e d by u n c o n s t r a i n e d m i n i m i z a t i o n o f t h e m o d i f i e d R a y l e i g h q u o t i e n t . The m o d i f i e d Complex method developed f o r t h e optimum d e c i s i o n o f u i s c o m p a t i b l e w i t h t h e f r e q u e n c y c a l c u l a t i o n t e c h n i q u e i n t h a t i t assumes t h a t t h e c o n s t r a i n e d f r e q u e n c y w, i s r e l a t i v e l y i n s e n s i t i v e t o s m a l l changes i n u. Thus t h e modal v e c t o r yt c o r r e s p o n d i h g t o wT i s recomputed o n l y p e r i o d i c a l l y . The t e c h n i q u e i s s e l f - c o r r e c t i n g . I n addition, the t e c h n i q u e does n o t r e q u i r e e v a l u a t i o n o f t h e g r a d i e n t o f e i t h e r t h e o b j e c t i v e f u n c t i o n f o r t h e c o n s t r a i n t s gi, so t h a t n o i s e problems encountered i n an e a r l i e r problem, when g r a d i g n t techniques were employed i n c o m b i n a t i o n w i t h t h e p e n a l t y f u n c t i o n method o f F i a c c o and McCormick (Ref. 15) t o m i n i m i z e t h e w e i g h t o f a c y l i n d r i c a l s h e l l s u b j e c t t o a c o n s t r a i n t on i t s c i r c u m f e r e n t i a l e x t e n s i o n mode frequency (Ref. 10) a r e e l i m i n a t e d . Para1 l e l s o l u t i o n o f t h e o p t i m a l d e c i s i o n problem by an " a n a l y t i c a l " t e c h n i q u e p r o v i d e s d i r e c t v e r i f i c a t i o n of t h e m o d i f i e d Complex method developed and p e r m i t s an e s t i m a t i o n o f t h e e f f i c i e n c y o f t h e numerical method. Such a method seems f e a s i b l e f o r automated d e s i g n o f complex s t r u c t u r e s s u b j e c t t o c o n s t r a i n t s based upon t h i s study. F i n a l l y , p r e v i o u s s t r u c t u r a l o p t i m i z a t i o n s t u d i e s have c o n c e n t r a t e d p r i m a r i l y on s a t i s f y i n g e i t h e r dynamic ( f r e q u e n c y ) o r s t a t i c ( s t r e s s ) c o n s t r a i n t s ; i n t h i s s t u d y t h e f e a s i b i l i t y o f combining s t r u c t u r a l a n a l y s i s techniques and o p t i m i z a t i o n techniques d i r e c t l y t o o b t a i n optimum s t r u c t u r a l designs under simultaneous frequency and s t r e s s c o n s t r a i n t s i s demonstrated. The techniques developed a r e g e n e r a l and a p p l i c a b l e t o o t h e r t y p e s o f problems. Acknowledgement P a r t o f t h i s paper i s s u b m i t t e d t o t h e School o f E n g i n e e r i n g a t Rensselaer P o l y t e c h n i c I n s t i t u t e i n p a r t i a l f u l f i l l m e n t o f t h e requirements o f t h e degree o f D o c t o r o f Engineering.
Distributed Mass and Stiffness Structural Systems
REFERENCES (1)
Kraus, H. (1967) T h i n E l a s t i c S h e l l s , Wiley, New York.
(2)
Z i e n k i e w i c z , 0. C. (1971) The F i n i t e - E l e m e n t Method i n E n g i n e e r i n g Science, McGraw-Hill, New York.
(3)
P r z e m i e n i e c k i , J. S . (1967) Theory o f M a t r i x S t r u c t u r a l A n a l y s i s , McGraw-Hill, New York
(4) Schmit, L. A., The s t r u c t u r a l s y n t h e s i s concept and i t s p o t e n t i a l r o l e i n d e s i g n w i t h composites, Proceedin s, General E l e c t r i c I O f f i c e o f Naval Research Composites Symposicm, P d (1967). (5)
Turner, M. J., O p t i m i z a t i o n o f s t r u c t u r e s t o s a t i s f y f l u t t e r requirements, J. AIAA 7, 945 (1969).
(6) Rubin, C. P., Minimum w e i g h t d e s i g n o f complex s t r u c t u r e s s u b j e c t t o a frequency cons t r a i n t , J. AIAA 8, 923 (1970).
(7) McCart, R. B. Haug, E. J., and S t r e e t e r , T. D., Optimal d e s i g n of s t r u c t u r e s w i t h cons t r a i n t s on n a t u r a l f r e q u e n c i e s , Vol. o f Tech. Papers, AIAA S t r u c t u r a l Dynamics and A e r o e l a s t i c i t y S p e c i a l i s t Conference, New Orleans (1969). (8)
Beveridge, G. S. and Schechter, R. S . (1972) O p t i m i z a t i o n : Theory and P r a c t i c e , McGrawH i l l , New York.
(9)
Deak, A.L. and Pian, T. H. H., A p p l i c a t i o n o f t h e smooth-surface i n t e r p o l a t i o n t o t h e f i n i t e - e l e m e n t a n a l y s i s , J. AIAA 5, 187 (1967).
(10)
Rand, R. A. and Shen, C. N., Optimum d e s i g n o f composite s h e l l s s u b j e c t t o n a t u r a l frequency c o n s t r a i n t s , J. Computer and S t r u c t u r e s 3, 247 (1973).
(11)
Rand, R. A., Optimum d e s i g n o f s h e l l s t r u c t u r e s s u b j e c t t o c o n s t r a i n t s , D o c t o r ' s Thesis, Rensselaer P o l y t e c h n i c I n s t i t u t e , Troy, New York (1977).
(12)
Guyan, R. J.,
(13)
Temple, G.
(14)
Fox, R. L. and Kapoor, M. P., A m i n i m i z a t i o n method f o r t h e s o l u t i o n o f t h e e i g e n problem a r i s i n g i n s t r u c t u r a l dynamics, Proceedings, W r i q h t - P a t t e r s o n Conference on M a t r i x Methods, Ohio (1968).
(15)
Fiacco, A. V. and McCormick, G. P. (1968) N o n l i n e a r Programming: s t r a i n e d M i n i m i z a t i o n Techniques, Wiley, New York.
(16)
N o v o z h i l o v , V. V.
Reduction o f S t i f f n e s s and Mass M a t r i c e s , J. AIAA 3, 380 (1965).
and B i c k l e y , W . G. (1956) R a y l e i g h ' s P r i n c i p l e , Dover, New York.
S e q u e n t i a l Uncon-
(1959) Theory o f T h i n S h e l l s ( T r a n s l a t i o n ) , P. Noordhoff.