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UBC Theses and Dissertations

Visualization of nodes, antinodes and lateral displacements in vibrating plates Niven, Robert D. 1967

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V I S U A L I Z A T I O N O F N O D E S , A N T I N O D E S A N D L A T E R A L D I S P L A C E M E N T S I N V I B R A T I N G P L A T E S b y R o b e r t D . N i v e n B . S c . , U n i v e r s i t y of A l b e r t a , 1964 A T h e s i s S u b m i t t e d i n P a r t i a l F u l f i l l m e n t of t h e R e q u i r e m e n t s f o r t h e D e g r e e of M a s t e r of A p p l i e d S c i e n c e In t h e D e p a r t m e n t of M e c h a n i c a l E n g i n e e r i n g W e a c c e p t t h i s t h e s i s a s c o n f o r m i n g to the r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A A p r i l , 1967 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada D e p a r t m e n t o f D a t e A B S T R A C T T h e u s e o f o p t i c a l t e c h n i q u e s f o r t h e s t u d y o f v i b r a t i n g s u r f a c e s h a s t h u s f a r b e e n l i m i t e d to m e a s u r i n g s m a l l a m p l i t u d e s o n t h e o r d e r o f a f e w h u n d r e d m i c r o i n c h e s . T o e x t e n d t h e m e a s u r i n g r a n g e to m u c h l a r g e r a m p l i t u d e s a c o m p l e t e l y n e w t e c h n i q u e i s s o u g h t . In t h i s t h e s i s o p t i c a l v i b r a t i o n m e t h o d s a r e p r e s e n t e d t h a t a l l o w a m p l i t u d e s o f . 0 0 1 " a n d u p to b e i n v e s t i g a t e d , t h e u p p e r l i m i t b e i n g d e t e r m i n e d s o l e l y b y t h e p r o h i b i t i v e s i z e a n d c o s t o f t h e e q u i p -m e n t . T h e s t u d y i s b a s e d o n a c o m b i n a t i o n o f t h e s h a d o w m o i r e d e f l e c t i o n m e a s u r i n g m e t h o d a n d t h e S a l e t - I k e d a s l o p e m e a s u r i n g m e t h o d w h i c h , as f a r as t h e a u t h o r k n o w s , h a v e b e e n a p p l i e d o n l y to t h e s t u d y o f s t a t i c s i t u a t i o n s . It i s s h o w n h o w t h e s e t w o m e t h o d s m a y b e a p p l i e d to the d y n a m i c c a s e to p e r m i t t h e d i r e c t v i s u a l i z a t i o n o f n o d a l a n d a n t i n o d a l l o c a t i o n s a n d d i s p l a c e m e n t s i n v i b r a t i n g p l a t e s . T h r e e s p e c i m e n s a r e s t u d i e d : a c a n t i l e v e r b e a m , a s q u a r e c a n t i l e v e r p l a t e a n d a c i r c u l a r f r e e p l a t e . C o m p l e t e p h o t o g r a p h i c r e s u l t s a l o n g w i t h t h e o r e t i c a l o r e x p e r i m e n t a l s o l u t i o n s a r e g i v e n f o r e a c h s p e c i m e n . A C K N O W L E D G E M E N T I w i s h to e x p r e s s m y s i n c e r e gra t i tude to m y a d v i s o r D r . C R . H a z e l l f o r h i s g e n e r o u s a s s i s t a n c e and g u i d a n c e throughout the i n v e s t i g a t i o n and to D r . J . P . D u n c a n w h o i n t r o d u c e d m e to the S a l e t - I k e d a technique and p r o v i d e d m e w i t h m a n y h e l p f u l s u g g e s t i o n s . I w o u l d a l s o l i k e to thank M r . P h i l H u r r e n and M r . John H o a r f o r t h e i r v a l u a b l e t e c h n i c a l a s s i s t a n c e . T h i s s tudy w a s m a d e p o s s i b l e t h r o u g h r e s e a r c h grant N o . A-3331 p r o v i d e d by the N a t i o n a l R e s e a r c h C o u n c i l of C a n a d a . i i i T A B L E O F C O N T E N T S PAGE A B S T R A C T i A C K N O W L E D G E M E N T S i i L I S T O F F I G U R E S vi N O M E N C L A T U R E v i i C H A P T E R I T H E P R O B L E M D E F I N E D 1 I n t r o d u c t i o n 1 S t a t e m e n t o f t h e P r o b l e m 3 R e v i e w o f t h e L i t e r a t u r e 3 V i b r a t i o n S t u d i e s 3 S t a t i c O p t i c a l T e c h n i q u e s 6 L i m i t a t i o n s o f t h e S t u d y 1 1 D e f i n i t i o n s 1 1 C H A P T E R II T H E O R E T I C A L C O N S I D E R A T I O N S 1 2 E v a l u a t i o n o f E x i s t i n g T e c h n i q u e s 1 2 D e f l e c t i o n 1 2 S l o p e 1 3 T h e o r y 15 S h a d o w M o i r e D e f l e c t i o n M e t h o d 1 5 S a l e t - I k e d a S l o p e M e t h o d 21 A n a l y s i s o f N o d e s a n d D e f l e c t i o n 25 A n a l y s i s o f A n t i n o d e s • • • 3 1 B e a m a n d P l a t e S p e c i m e n s to b e S t u d i e d . . . 3 7 iv PAGE C a l i b r a t i o n of the C a n t i l e v e r B e a m . . . . 37 C H A P T E R III E X P E R I M E N T A L A P P A R A T U S . 40 O p t i c a l S y s t e m s . . 40 S h a d o w M o i r e 40 S h a d o w M o i r e u s i n g S t r o b o s c o p y . . . . 43 S a l e t - I k e d a T e c h n i q u e 43 V i b r a t i o n A p p a r a t u s 47 T e s t S p e c i m e n s 49 D i s p l a c e m e n t C a l i b r a t i o n I n s t r u m e n t a t i o n . 50 C H A P T E R I V E X P E R I M E N T A L P R O C E D U R E 53 N o d e and D e f l e c t i o n A n a l y s i s . 53 S y s t e m A l i g n m e n t 53 P o s i t i o n i n g the M a s t e r G r i d 54 P o s i t i o n i n g of the C a m e r a and L e n s L ^ . 56 R e c o r d i n g N o d e s 56 R e c o r d i n g D i s p l a c e m e n t s 57 C a l i b r a t i o n of the C a n t i l e v e r B e a m . . 57 C h l a d n i P a t t e r n s 58 A n t i n o d e A n a l y s i s 59 S y s t e m A l i g n m e n t 59 C h o i c e of T a r g e t 60 R e c o r d i n g A n t i n o d e s 60 C H A P T E R V E X P E R I M E N T A L R E S U L T S 62 N a t u r a l F r e q u e n c i e s 62 PAGE T a b l e of Photographic Results . . . . . . . 63 C H A P T E R VI DISCUSSION O F R E S U L T S 82 Nodes 82 Antinodes 84 Displacement 86 Second O r d e r E f f e c t s 87 Ge n e r a l Observations 88 C H A P T E R VII S U M M A R Y A N D C O N C L U S I O N S 90 Summary 90 Conclusions 91 B I B L I O G R A P H Y 94 A P P E N D I X 97 v i L I S T O F F I G U R E S F I G U R E PAGE 1. M e t h o d s o f P r o d u c i n g M o i r e F r i n g e s . 16 2. . S h a d o w M o i r e A r r a n g e m e n t 17 3 . D e f l e c t i o n C a u s i n g O n e F r i n g e 18 4 . T y p i c a l M o i r e P a t t e r n . 21 5 . S a l e t - I k e d a A r r a n g e m e n t . . . 22 6. M o i r e F r i n g e M o t i o n In A V i b r a t i n g P l a t e . . . . . . . 27 7. T y p i c a l N o d a l P a t t e r n 29 8. F r i n g e S h i f t N e c e s s a r y f o r W a s h o u t . . . . . . . . . . 30 9 . M o t i o n o f L i g h t P a t h s R e f l e c t e d . O f f A V i b r a t i n g P l a t e . „ 32 10. T y p i c a l P a r t i a l A n t i n o d e P a t t e r n 34 1.1. M e t h o d of L o c a t i n g A n t i n o d e s . o 35 12. S t a t i c a n d D y n a m i c D e f l e c t i o n C u r v e s <>• 39 13. S h a d o w M o i r e L i g h t S o u r c e . . a . . a a . . . . . . . 40 14. S h a d o w M o i r e L i g h t S o u r c e . 42 15. G r i d a n d S u p p o r t F r a m e 44 16. .. S t r o b o s c o p e w i t h S p h e r i c a l M i r r o r . . . . < , < , . . . . . 44 17. S a l e t - I k e d a L i g h t S o u r c e 45 18. S a l e t - I k e d a L i g h t S o u r c e 46 19. V i b r a t i o n G e n e r a t i n g E q u i p m e n t . 48 2 0 . I n s t r u m e n t a t i o n . 50 2 1 . C a l i b r a t i o n E q u i p m e n t . . . . . . . . . . . . . . . . 51 v i i F I G U R E P A G E 2 2 . Shadow M o i r e A r r a n g e m e n t . 52 23 . S a l e t - I k e d a A r r a n g e m e n t . . . . . . . . . . . . . . . 52 24 . A l i g n i n g the L i g h t S o u r c e . . * * 54 2 5 . N o d e s In A C a n t i l e v e r B e a m . . . . . . . . . . . . . 66 26 . A n t i n o d e s In A C a n t i l e v e r B e a m . . . . . . . . . . . 67 , 2 7 . N o d e s In A S q u a r e P l a t e 68 28 . P a r t i a l A n t i n o d e s In A S q u a r e P l a t e . . e . . . . . . . 70 29 . S u p e r p o s i t i o n Of T w o P a r t i a l A n t i n o d e s T o L o c a t e T h e T r u e A n t i n o d e . . . . . . . . . . . . . . . . . 73 30 . N o d e s In A C i r c u l a r P l a t e . . 74 31. A n t i n o d e s In A C i r c u l a r P l a t e . . . . . . . «, 76 32 . D i s p l a c e m e n t s In A C a n t i l e v e r B e a m . . . . . . . . . 77 3 3 . D i s p l a c e m e n t s In A S q u a r e P l a t e . . . < > o . . . . . . 79 34. D i s p l a c e m e n t s In A C i r c u l a r P l a t e . . . . . . . . . . 80 3 5 . N o d e s A n d A n t i n o d e s In A P l a t e Of A r b i t r a r y S h a p e . . . 81 v i i i N O M E N C L A T U R E S Y M B O L a = l e n g t h of a s i d e of a s q u a r e p l a t e o r d i a m e t e r of a c i r c u l a r p l a t e f = f r e q u e n c y . h = i n c r e m e n t of d e f l e c t i o n o n the p l a t e s u r f a c e o c c u r r i n g b e t w e e n two a d j a c e n t f r i n g e s , p = p i t c h , c e n t r e to c e n t r e d i s t a n c e of e i t h e r two opaque o r two t r a n s p a r e nt l i n e s o n the m a s t e r g r i d , r = p e r p e n d i c u l a r d i s t a n c e f r o m the o p t i c a l a x i s to s o m e p a r t i c u l a r l i n e o r p o i n t o n a t a r g e t , t = p l a t e t h i c k n e s s . w = t o t a l d e f l e c t i o n of p l a t e s u r f a c e m e a s u r e d f r o m u n d e -f o r m e d p l a n e . w = t o t a l a i r gap t h i c k n e s s b e t w e e n m a s t e r g r i d a n d p l a t e s u r f a c e . D = p l a t e s t i f f n e s s - E t ^ 1 2 ( l - v > 2 ) E = m o d u l u s of e l a s t i c i t y . F = f o c a l l e n g t h of l e n s . I -- m o m e n t of i n e r t i a N = f r i n g e o r d e r . 0 - s l o p e at a p a r t i c u l a r p o i n t o n p l a t e s u r f a c e . P = m a s s / u n i t l e n g t h . = P o i s s o n ' s r a t i o S Y M B O L ^ - m a s s / u n i t v o l u m e . A l l u n i t s a r e i n the i n . - l b . - s e c . s y s t e m . T h e a b b r e v i a t i o n m e a n s l i n e s p e r i n c h . C H A P T E R I T H E P R O B L E M D E F I N E D I. I N T R O D U C T I O N In t h e a n a l y s i s of m a n y c o m p l e x v i b r a t i o n p r o b l e m s i t s o o n b e c o m e s o b v i o u s t h a t s o l u t i o n b y m e a n s of a m a t h e m e t i c a l m o d e l i s a n e x c e e d i n g l y c o m p l i c a t e d , i f n o t i m p o s s i b l e , u n d e r t a k i n g . E x p e r i -m e n t a l t e c h n i q u e s a n d i n s t r u m e n t s h a v e t h u s e v o l v e d f o r t h e m e a s u r -m e n t o f e v e r y p o s s i b l e t y p e of v i b r a t i o n f r o m s h i p ' s e n g i n e s to t h e m i n u t e f l u t t e r of a d i a p h r a g m . T h e m a j o r i t y o f t e c h n i q u e s , w h i l e e x -t r e m e l y d i v e r s i f i e d , s h a r e t w o t h i n g s i n c o m m o n : (1) t h e r e a d i n g s a r e r e p r e s e n t a t i v e of o n l y o n e p o i n t i n t h e f i e l d , u s u a l l y t h e p o i n t of a t -t a c h m e n t o f t h e t r a n s d u c e r a n d ; (2) t h e m e a s u r i n g d e v i c e m u s t b e p h y -s i c a l l y p l a c e d i n c o n t a c t w i t h t h e c o m p o n e n t u n d e r , s t u d y . A d i f f i c u l t y o f t e r p r e t a t i o n a r i s e s w i t h t h e f o r m e r m e t h o d . In o r d e r to g e t a n i d e a of t h e f u l l f i e l d o f v i b r a t i o n , n u m e r o u s r e a d i n g s m u s t b e t a k e n a n d s u b -s e q u e n t l y p l o t t e d . T h e r e a l s o e x i s t s t h e p o s s i b i l i t y t h a t a p o i n t o f s e v -e r e v i b r a t i o n m i g h t b e o v e r l o o k e d . T h e n e c e s s i t y f o r c o n n e c t i n g t h e v i b r a t i o n r e c o r d e r t o t h e s p e c i m e n m a y o r m a y not p r e s e n t a p r o b l e m d e p e n d i n g o n t h e p a r t i c u l a r c o m p o n e n t . If, f o r i n s t a n c e , a s t r a i n g a u g e i s a t t a c h e d to a s m a l l t h i n d i a p h r a g m t h e i n c r e a s e d m a s s a n d s t i f f n e s s o f f e r e d b y t h e g a \ i g e m a y s h i f t the r e s p o n s e s u f f i c i e n t l y to c a u s e m i s l e a d i n g r e s u l t s . T h e a p p e a r a n c e o f t h e q u a r t z c r y s t a l i n t h e e a r l y 1920 ' s 2 m a r k e d a n i n c r e a s e d i n t e r e s t i n t h e f i e l d - s t u d y o f v i b r a t i o n s . B e c a u s e o f t h e v e r y s m a l l a m p l i t u d e s i n v o l v e d a n d t h e s h o r t c o m i n g s o f e x i s t i n g m e a s u r i n g m e t h o d s at t h a t t i m e , a n e w t e c h n i q u e w a s s o u g h t . T h i s l e d to t h e i n t r o d u c t i o n o f i n t e r f e r o m e t r i c m e a s u r e m e n t s w h i c h a r e s t i l l u s e d a n d b e i n g i m p r o v e d u p o n t o d a y . T h e i r u s e f u l n e s s , h o w e v e r , i s l i m i t e d t o r e c o r d i n g a m p l i t u d e s i n the r a n g e o f a f e w h u n d r e d m i c r o -i n c h e s . S i n c e m a n y e n g i n e e r i n g a p p l i c a t i o n s r e q u i r e a m e a s u r e m e n t o f m u c h g r e a t e r v a l u e s , t h e r e r e m a i n s t h e n e c e s s i t y f o r i n v e s t i g a t i n g n e w f i e l d - m e a s u r i n g t e c h n i q u e s . In the l a s t t w e n t y y e a r s a v a r i e t y o f a p p l i c a t i o n s o f o p t i c a l p r i n c i p l e s h a v e a p p e a r e d t h a t a l l o w r e l a t i v e l y l a r g e s t a t i c a m p l i t u d e s to b e s t u d i e d . H o w e v e r , a s f a r a s t h e a u t h o r k n o w s , w i t h t h e e x c e p t i o n o f o n e v e r y r e c e n t c a s e n o n e o f t h e s e h a v e b e e n a p p l i e d t o t h e s t u d y of v i b r a t i o n s . S i n c e m o s t o f t h e m a r e n o t s u b j e c t to t h e d i s a d v a n t a g e s o u t l i n e d p r e v i o u s l y , t h e f e a s i b i l i t y of a p p l y i n g t h e s e o p t i c a l m e t h o d s to d y n a m i c s i t u a t i o n s w o u l d s e e m t h e n e x t l o g i c a l s t e p i n v i b r a t i o n a n a l -y s i s . It. w a s to t h i s e n d that, t h i s i n v e s t i g a t i o n w a s u n d e r t a k e n . II. S T A T E M E N T O F T H E P R O B L E M i It w a s t h e p u r p o s e of t h i s s t u d y to (1) d e v e l o p a t e c h n i q u e f o r t h e v i s u a l i z a t i o n o f n o d e s a n d a n t i n o d e s i n v i b r a t i n g p l a t e s a n d (2) d e t e r m i n e , b y m e a n s o f s t r o b o s c o p y , t h e d e f l e c t i o n at a n y p o i n t i n t h e p l a t e s . T h e s t u d y w a s b a s e d o n e x i s t i n g m o i r e a n d s p e c i a l i z e d o p t i c a l t e c h n i q u e s w h i c h h a v e t h u s f a r b e e n u s e d o n l y to s t u d y p l a t e s u n d e r s t a t i c l o a d i n g . III. R E V I E W O F T H E L I T E R A T U R E V i b r a t i o n . S t u d i e s T h e f i r s t e x p e r i m e n t a l i n v e s t i g a t i o n of v i b r a t i o n i n p l a t e s c a n p r o b a b l y b e a t t r i b u t e d to C h l a d n i (1) i n 1787 . B y u s i n g a f i n e s p r i n -k l i n g o f s a n d s p r e a d o v e r t h e s u r f a c e of a v i b r a t i n g p l a t e h e w a s a b l e t o o b s e r v e n o d a l p a t t e r n s f o r v a r i o u s m o d e s o f v i b r a t i o n . T h e p r i n c i p l e i n v o l v e d w a s s i m p l y t h a t a t e v e r y p o i n t , e x c e p t a t t h e n o d e s , t h e r e w a s s o m e f i n i t e a m p l i t u d e a n d c o n s e q u e n t l y the s a n d w a s . k e p t i n c o n s t a n t m o t i o n u n t i l i t c a m e t o r e s t a t a n o d e . T h i s t e c h n i q u e h a s b e e n a p p l i e d w i d e l y i n d e t e r m i n i n g e v e r y c o n c e i v a b l e n o d a l p a t t e r n i n p l a t e s a n d m e m -b r a n e s . M a r y W a l l e r (2} h a s m a d e a n o t a b l e c o n t r i b u t i o n i n t h i s f i e l d . A m e t h o d f o r o u t l i n i n g t h e a n t i n o d e s w a s d i s c o v e r e d i n t h e e a r l y 1 8 0 0 ' s b y S a v a r t ( 3 ) . H e f o u n d t h a t a v e r y f i n e p o w d e r , s u c h a s l y c o p o d i u m , w o u l d t e n d to a c c u m u l a t e at t h e a n t i n o d e s r a t h e r t h a n t h e n o d e s . T h i s p h e n o m e n o n w a s l a t e r d e s c r i b e d b y F a r a d a y (4) a n d a t -4 t r i b u t e d to the f l o w of s u r r o u n d i n g a i r c u r r e n t s . In t h e f i e l d of o p t i c a l v i b r a t i o n a n a l y s i s n u m e r o u s m e t h o d s h a v e e v o l v e d w h i c h a r e d e s c r i b e d b y W o o d (5). O n e o f p r i m a r y i n t e r -e s t w a s p e r f o r m e d b y D y e (6) i n w h i c h h e u s e d s t r o b o s c o p i c i n t e r f e r o -m e t r y to d e p i c t the s u r f a c e of a v i b r a t i n g q u a r t z c r y s t a l . T h e t e c h -n i q u e , u s i n g a M i c h e l s o n i n t e r f e r o m e t e r , r e q u i r e d t h a t a b e a m of m o n o -c h r o m a t i c l i g h t b e p a s s e d t h r o u g h a g l a s s p l a t e o n t o t h e c r y s t a l s u r -f a c e . T h e i n t e r f e r e n c e b e t w e e n t h e r e f l e c t e d b e a m s f r o m t h e t w o s u r -f a c e s r e s u l t e d i n f r i n g e s r e p r e s e n t i n g c o n t o u r s of a i r g a p t h i c k n e s s . W h e n t h e c r y s t a l w a s e x c i t e d t h e f r i n g e s r a p i d l y o s c i l l a t e d o v e r t h e s u r f a c e c a u s i n g a b l u r . H o w e v e r , b y u s i n g i n t e r r u p t e d l i g h t f r o m a s t r o b o s c o p e t h e m o t i o n c o u l d b e f r o z e n a n d t h e f r i n g e s o b s e r v e d at a n y p o s i t i o n . A l t h o u g h n o d e s a n d a n t i n o d e s w e r e n o t i n d i c a t e d d i r e c t l y t h e y w e r e e a s i l y l o c a t e d b y p r o p e r i n t e r p r e t a t i o n . T h e m e t h o d w a s e x -t r e m e l y s e n s i t i v e to s m a l l d e f l e c t i o n s o n t h e o r d e r of . 00005" p e r f r i n g e b u t w a s t o o s e n s i t i v e f o r l a r g e d e f l e c t i o n s . A s i m i l a r t e c h n i q u e b u t not r e q u i r i n g t h e u s e of a s t r o b o -s c o p e w a s p r e s e n t e d b y O s t e r b e r g (7) i n 1931. I n s t e a d of f r e e z i n g t h e m o t i o n of the f r i n g e s h e d e v e l o p e d a r e l a t i o n s h i p b e t w e e n t h e i n t e n s i t y o f t h e w a s h e d o u t f r i n g e p a t t e r n a n d t h e a m p l i t u d e o f v i b r a t i o n . In t h i s m a n n e r h e p r o d u c e d p a t t e r n s w h i c h r e a d i l y l o c a t e d t h e n o d e s , a n d t h r o u g h i n t e r p r e t a t i o n , t h e a n t i n o d e s . H o w e v e r , v i b r a t i o n s w i t h a m -p l i t u d e s i n e x c e s s o f o n e f r i n g e r e s u l t e d i n p o o r c o n t r a s t a n d u n s a t i s -5 factory patterns. A multiple beam analog of Osterberg's method y i e l d -ing much better r e s u l t s was given i n 1955 by Thornton and K e l l y (8). A f a r more recent v i b r a t i o n study using holography was employed by Powell and Stetson (9). The technique requires that a con-tinuous wave gas l a s e r having highly coherent and monochromatic l i g h t be used as the l i g h t source. The p e n c i l of emergent l i g h t from the la s e r was f i r s t s p l i t and expanded such that one.beam f e l l on the surface of the v i b r a t i o n model and the other, the reference beam, was mirrored onto i a photographic plate. Thus, a hologram for the time average of the coherent wavefronts scattered from the v i b r a t i n g model was formed. The image reconstructed by the hologram was found to contain a system of interference fringes which were contours of constant v i b r a t i o n amplitude. This method allows v i b r a t i n g objects with a r b i t r a r y surfaces to be analized. The use of stereo photography has recently been applied to v i b r a t i o n analysis by Wasil, Merchant and Del Vecchio (10). Two iden-t i c a l cameras are set up so that t h e i r o p t i c a l axes are p a r a l l e l and s l i g h t l y separated. They are then focused on a v i b r a t i n g plate and sim-ultaneous photographs taken, the motion being frozen by a stroboscope. The exposed negatives are then analyzed i n a steroscope with a r e s u l -t i n g three dimensional e f f e c t . Interpretation i s somewhat complicated and requires that d e f l e c t i o n contours be p l o t t e d . The very l a t e s t advance i n the f i e l d , and c e r t a i n l y the most 5 f a c t o r y p a t t e r n s . A m u l t i p l e b e a m a n a l o g o f O s t e r b e r g ' s m e t h o d y i e l d -i n g m u c h b e t t e r r e s u l t s w a s g i v e n i n 1955 b y T h o r n t o n a n d K e l l y (8 ) . A f a r m o r e r e c e n t v i b r a t i o n s s t u d y u s i n g i n t e r f e r o m e t r y w a s e m p l o y e d b y P o w e l l a n d S t e t s o n (9 ) . T h e t e c h n i q u e r e q u i r e s t h a t a c o n -t i n u o u s w a v e g a s l a s e r h a v i n g h i g h l y c o h e r e n t a n d m o n o c h r o m a t i c l i g h t b e u s e d a s the l i g h t s o u r c e . T h e p e n c i l of e m e r g e n t l i g h t f r o m t h e l a s e r i s f i r s t s p l i t a n d d i f f u s e d s u c h t h a t o n e b e a m f a l l s o n t h e s u r f a c e of the v i b r a t i o n m o d e l a n d t h e o t h e r , t h e r e f e r e n c e b e a m , i s m i r r o r e d o n t o a p h o t o g r a p h i c p l a t e . A t the m o d e l s u r f a c e t h e l i g h t i s s c a t t e r e d s u c h that, a c e r t a i n q u a n t a a l s o s t r i k e t h e p h o t o g r a p h i c p l a t e . T h e r e -s u l t i n g i n t e r e f e r e n c e b e t w e e n t h e r e f e r e n c e b e a m a n d t h e s c a t t e r e d l i g h t y i e l d s a h o l o g r a m . If the h o l o g r a m i s n o w i l l u m i n a t e d a n i m a g e of the i n t e r f e r e n c e f r i n g e s w i l l a p p e a r . T h e s e f r i n g e s a r e s e e n to m a p out c o n t o u r s of c o n s t a n t d e f l e c t i o n . T h e u s e of s t e r e o p h o t o g r a p h y h a s r e c e n t l y b e e n a p p l i e d to v i b r a t i o n a n a l y s i s b y W a s i l , M e r c h a n t a n d D e l V e c c h i o (10). T w o i d e n -t i c a l c a m e r a s a r e s e t u p s o t h a t t h e i r o p t i c a l a x e s a r e p a r a l l e l a n d s l i g h t l y s e p a r a t e d . T h e y a r e t h e n f o c u s e d o n a v i b r a t i n g plate , a n d s i m -u l t a n e o u s p h o t o g r a p h s t a k e n , t h e m o t i o n b e i n g f r o z e n b y a s t r o b o s c o p e . T h e e x p o s e d n e g a t i v e s a r e t h e n a n a l y z e d i n a s t e r e o s c o p e w i t h a r e s u l -t i n g t h r e e d i m e n s i o n a l e f f e c t . I n t e r p r e t a t i o n i s s o m e w h a t c o m p l i c a t e d a n d r e q u i r e s t h a t d e f l e c t i o n c o n t o u r s b e p l o t t e d . T h e v e r y l a t e s t a d v a n c e i n t h e f i e l d , a n d c e r t a i n l y t h e m o s t 6 c l o s e l y r e l a t e d t o t h i s s t u d y , h a s b e e n p r e s e n t e d i n a p a p e r b y N i c k o l a (11). H i s p r i m e i n t e r e s t i s t h e d y n a m i c r e s p o n s e of t h i n m e m b r a n e s b u t t h e p r i n c i p l e i n v o l v e d a p p l i e s e q u a l l y w e l l to p l a t e s . U s i n g t h e L i g t e n -b e r g m o i r e m e t h o d , to b e d e s c r i b e d l a t e r , h e h a s s u c c e e d e d i n r e c o r d -i n g t h e d y n a m i c p a r t i a l s l o p e s p r e s e n t i n a m e m b r a n e at r e s o n a n c e . In a l i k e m a n n e r he r e c o r d s i t s t r a n s i e n t r e s p o n s e w h e n s t r u c k b y a n i m -p a c t l o a d . In t h i s c a s e h i g h s p e e d p h o t o g r a p h y i s r e q u i r e d . S ta t ic . O p t i c a l T e c h n i q u e s T h e f i r s t p a r t o f t h i s i n v e s t i g a t i o n d e a l s w i t h t h e d i r e c t d e -t e r m i n a t i o n of n o d e s a n d a n t i n o d e s b y u s i n g s o m e of the m o r e r e c e n t l y d e v e l o p e d o p t i c a l t e c h n i q u e s . A t p r e s e n t t h e s e t e c h n i q u e s a r e b e i n g u s e d s o l e l y f o r s t a t i c a n a l y s i s a n d i t i s p r o p o s e d to a p p l y t h e m d i r e c t l y t o t h e d y n a m i c c a s e . A.s w i l l b e s h o w n l a t e r n o d e s a n d a n t i n o d e s a r e d e s c r i b e d i n t e r m s of d e f l e c t i o n s a n d s l o p e s . T h e r e f o r e , w i t h t h i s i n m i n d a n i n v e s t i g a t i o n of m o s t of t h e e x i s t i n g d e f l e c t i o n a n d s l o p e m e a s -u r i n g o p t i c a l t e c h n i q u e s w a s u n d e r t a k e n . It w a s o r i g i n a l l y b e l i e v e d t h a t m o i r e p a t t e r n t e c h n i q u e s w o u l d y i e l d t h e b e s t r e s u l t s a n d c o n s e q u e n t l y t h i s f i e l d w a s i n v e s t i g a t e d t h o r -o u g h l y . T h e o c a r i s (12) g i v e s a b r i e f s u m m a r y of s o m e 85 p a p e r s w r i t -t e n o n t h e s u b j e c t . A n e v e n m o r e c o m p l e t e r e f e r e n c e , i n c l u d i n g 173 p a p e r s , i s g i v e n i n a b i b l i o g r a p h y b y D u n c a n (13). O f t h e a v a i l a b l e m o i r e m e t h o d s , t h r e e f o r t h e m e a s u r e m e n t o f d e f l e c t i o n a n d t h r e e f o r t h e m e a s u r e m e n t o f s l o p e , w e r e f e l t to b e 7 a p p l i c a b l e to t h i s s t u d y . T h e d e f l e c t i o n t e c h n i q u e s w i l l b e c o n s i d -e r e d f i r s t . S h a d o w M o i r e T h i s m e t h o d , a l t h o u g h b r i e f l y m e n t i o n e d b y W e l l e r a n d S h e p a r d (14) as a p o s s i b l e d e f l e c t i o n m e a s u r i n g d e v i c e , s a w l i t t l e p r a c t i c a l a p p l i c a t i o n u n t i l q u i t e r e c e n t l y . T h e o c a r i s (15 a n d 16) s h o w s h o w the s h a d o w m o i r e i s a p p l i e d as a s t e p p i n g s t o n e i n t h e s o l u t i o n o f s t r e s s e s i n p l a t e s . T h e m e t h o d r e q u i r e s t h a t c o l l i m a t e d l i g h t p a s s t h r o u g h a r e f e r e n c e g r i d (100 - 500 l p i ) o n t o a s p e c i m e n a n d i n s o d o i n g c a s t s a s h a d o w o f t h e g r i d l i n e s . B y t h e n v i e w i n g t h e s p e c i m e n t h r o u g h t h e r e f e r e n c e g r i d at a n a n g l e d i f f e r e n t f r o m t h e a n g l e o f i n c i d e n c e o f t h e c o l l i m a t e d l i g h t a f a m i l y o f f r i n g e s w i l l b e o b s e r v e d , t h e n u m b e r b e i n g p r o p o r t i o n a l to the d e f l e c t i o n at t h a t p o i n t . T h e s e n s i t i v i t y i s o n t h e o r d e r o f . 0 0 5 " / f r i n g e f o r a 500 l p i g r i d . T h i s t e c h n i q u e w i l l b e d e t a i l e d i n t h e n e x t c h a p t e r . R e f l e c t i o n M o i r e M u c h l i k e the p r e v i o u s m e t h o d , t h i s t e c h n i q u e i n v o l v e s t h e v i e w i n g o f a s p e c i m e n t h r o u g h a r e f e r e n c e g r i d o f a p p r o x i m a t e l y 100 -500 l p i . H o w e v e r , i n t h i s c a s e , t h e s p e c i m e n i s p o l i s h e d s u c h t h a t t h e e y e s e e s t h e v i r t u a l i m a g e , r a t h e r t h a n t h e s h a d o w , o f t h e g r i d . T h e m e c h a n i c a l i n t e r f e r e n c e b e t w e e n t h e r e f l e c t i o n o f t h e r e f e r e n c e g r i d a n d t h e g r i d i t s e l f y i e l d s m o i r e f r i n g e s p r o p o r t i o n a l to t h e d e f l e c -t i o n . T h e s e n s i t i v i t y i s a p p r o x i m a t e l y t w i c e t h a t of the s h a d o w m o i r e m e t h o d . T h i s t e c h n i q u e i s p r e s e n t e d i n a p a p e r b y E b b e n i (17). / R e f l e c t i o n M o i r e U s i n g D i f f r a c t i o n T h r o u g h A F i n e G r i d A l t h o u g h i t i s n o t i n t e n d e d t o u s e d i f f r a c t i o n p h e n o m e n a f o r t h i s s t u d y it i s i n c l u d e d i n t h e r e v i e w b e c a u s e o f i t s m e r i t a n d p o s s i b l e f u t u r e u s e . I n t r o d u c e d a n d d e v e l o p e d b y M i d d l e t o n (18) t h e m e t h o d i s b a s e d u p o n t h e p h a s e d i s t u r b a n c e s u n d e r g o n e b y a b e a m of c o l l i m a t e d l i g h t p a s s i n g t h r o u g h a t r a n s p a r e n t d i f f r a c t i o n g r a t i n g a n d b e i n g r e f l e c -t e d b a c k t h r o u g h t h e g r a t i n g b y t h e s u r f a c e b e i n g s u r v e y e d . T h e r e s u l -t i n g f r i n g e s d u e to t h e p h a s e i n t e r f e r e n c e o f t h e c o m p o n e n t b e a m s c o m -p r i s i n g the e m e r g e n t b e a m i n d i c a t e e q u a l i n c r e m e n t s i n a i r g a p t h i c k -n e s s . S e n s i t i v i t y r a n g e s b e t w e e n . 0 0 0 0 5 " to . 0 1 0 " / f r i n g e . T h e f o l l o w i n g t e c h n i q u e s a r e f o r the m e a s u r e m e n t of s l o p e . L i g t e n b e r g (19) S i n c e i t s i n n o v a t i o n t h i s m e t h o d h a s p r o b a b l y b e c o m e o n e of t h e m o s t p o p u l a r , a n d n u m e r o u s p a p e r s h a v e b e e n w r i t t e n o n i t s v a r i o u s a p p l i c a t i o n s a n d r e f i n e m e n t s . B a s i c a l l y it i n v o l v e s p h o t o g r a p h i n g a r e l -a t i v e l y c o a r s e g r i d , f o r e x a m p l e 1/16" p i t c h , s e e n r e f l e c t e d i n t h e s u f -f a c e of t h e s p e c i m e n . T h e g r i d i s p l a c e d i n t h e p l a n e o f t h e c a m e r a l e n s . E x p o s u r e s are m a d e of t h e d e f o r m e d a n d u n d e f o r m e d s u r f a c e u s i n g t h e s a m e n e g a t i v e , w i t h t h e r e s u l t t h a t m o i r e f r i n g e s of c o n s t a n t s l o p e a p p e a r w h e n the n e g a t i v e i s d e v e l o p e d . It h a s the a d v a n t a g e t h a t 9 s u r f a c e d i s c o n t i n u i t i e s d o n o t s h o w u p b e c a u s e of c a n c e l l a t i o n i n t h e s u p e r p o s i t i o n . H o w e v e r , f o r t h e p u r p o s e o f t h i s s t u d y i t o f f e r s t h e p r o b l e m t h a t b e f o r e p h o t o g r a p h i c m a n i p u l a t i o n , f r i n g e s a r e n o t v i s i b l e to the e y e . R e f l e c t i o n A t t r i b u t e d to T h e o c a r i s (20) t h i s m e t h o d i n v o l v e s t h e u s e of t w o s e p a r a t e l i n e d g r i d s w h i c h m a y o r m a y n o t b e of t h e s a m e p i t c h . C o l l i m a t e d l ight , i s p a s s e d t h r o u g h a m a s t e r g r i d a n d i s r e f l e c t e d f r o m t h e p o l i s h e d s u r f a c e o f t h e s p e c i m e n , t h r o u g h a c o n d e n s i n g l e n s , o n t o a g r o u n d g l a s s s c r e e n . A n o t h e r g r i d i s t h e n a f f i x e d to t h e s c r e e n a n d b y s h i f t i n g i t to a s u i t a b l e p o s i t i o n o n t h e o p t i c a l a x i s a m o i r e p a t t e r n w i l l a p p e a r . If t h e s p e c i m e n i s n o w l o a d e d a n e w p a t t e r n w i l l b e c o m e v i s i b l e w h i c h , u p o n s u b t r a c t i o n of the i n i t i a l p a t t e r n , y i e l d s l i n e s o f c o n s t a n t s l o p e . A n o t h e r p o s s i b i l i t y e x i s t s } i n i t i a l l y if t h e g r o u n d g l a s s s c r e e n i s p e r f e c t l y a l i g n e d , t h e i m a g e of t h e m a s t e r g r i d a n d t h e s u p e r -i m p o s e d g r i d c a n b e m a d e to c o i n c i d e , e x a c t l y t h u s y i e l d i n g n o f r i n g e s at a l l . U p o n n o w l o a d i n g t h e s p e c i m e n j l i n e s o f c o n s t a n t s l o p e w i l l a p p e a r d i r e c t l y w i t h o u t t h e n e e d o f s u b t r a c t i n g t h e o r i g i n a l p a t t e r n . T h e s l o p e s r e c o r d e d a r e p a r t i a l s l o p e s i n a d i r e c t i o n n o r m a l to the g r i d l i n e s . O r -t h o g o n a l s l o p e s m a y b e m e a s u r e d b y r o t a t i n g b o t h g r i d s t h r o u g h 90 d e -g r e e s . R e f r a c t i o n A l t h o u g h q u i t e s i m i l a r to t h e a b o v e m e t h o d t h i s s y s t e m , s u g -gested by T h e o c a r i s and K o u t s a b e s s i s (21), involves the use of a t r a n s -parent specimen and depends upon the r e f r a c t i o n , not the r e f l e c t i o n , of li g h t . A c o l l i m a t e d monochromatic beam i s p a s s e d through a m a s t e r g r i d , through the s p e c i m e n and then imaged on a ground g l a s s s c r e e n . A l l l i e on the same op t i c a l a x i s . A s before, another g r i d i s p l a c e d on the s c r e e n and by pr o p e r adjustment a suitable moire'pattern can be formed. Loading the specimen y i e l d s constant slope l i n e s i n a d i r e c -tion n o r m a l to the g r i d l i n e s . Salet-Ikeda The above methods have a l l depended upon the m o i r e pheno-menon, but these are not th<e only, nor are they n e c e s s a r i l y the best methods. Sab in (22), i n an evaluation of numerous o p t i c a l techniques o f f e r s s e v e r a l systems worthy of m e r i t . One that is of p r i m e i n t e r e s t i s the Salet-Ikeda slope m e a s u r i n g technique which has been used ex-tens i v e l y by Duncan and B r o w n (23) for the determination of s t r e s s e s i n plates. A c o a r s e l i n e d target on the or d e r of 10 l p i i s located at the focus of a lens. A sheet of ground glass is p l a c e d behind the target and is i l l u m i n a t e d by a high intensity light source. The diffused light passes through the lens onto the specimen which then r e f l e c t s it to another r e c -eiving l e n s . A pinhole just l a r g e enough to provi d e sufficient intensity of i l l u m i n a t i o n is p l a c e d at the f o c a l point of this lens. F o r a p e r f e c t l y flat s u rface the i n i t i a l o b s e r v e d pattern w i l l be completely black or com-pletely white depending on the exact p o s i t i o n of the pinhole and type of 11 t a r g e t g r i d u s e d . T h e p a r t i a l s l o p e s r e c o r d e d a r e i n a d i r e c t i o n n o r m a l to t h e t a r g e t l i n e s . A m o r e c o m p l e t e d e s c r i p t i o n of t h i s m e t h o d i s g i v e n i n t h e f o l l o w i n g c h a p t e r . I V . L I M I T A T I O N S O F T H E S T U D Y T h e i n v e s t i g a t i o n i s l i m i t e d to t h e s t u d y o f f l a t o r n e a r l y f l a t s u r f a c e s . T h e v i b r a t i o n i s s t e a d y s t a t e i n w h i c h o n l y s t a n d i n g w a v e s e x -i s t . V . D E F I N I T I O N S A n o d e i s a p o i n t , l i n e o r s u r f a c e i n a s t a n d i n g w a v e w h e r e s o m e c h a r a c t e r i s t i c o f t h e w a v e f i e l d h a s z e r o a m p l i t u d e . A n a n t i n o d e i s a p o i n t , l i n e o r s u r f a c e i n a s t a n d i n g w a v e w h e r e s o m e c h a r a c t e r i s t i c of t h e w a v e f i e l d h a s m a x i m u m a m p l i t u d e . o r A n a n t i n o d e i s a p o i n t , l i n e o r s u r f a c e i n a s t a n d i n g w a v e w h e r e s o m e c h a r a c t e r i s t i c o f the w a v e f i e l d h a s z e r o s l o p e i n e v e r y d i r e c t i o n . C H A P T E R II T H E O R E T I C A L C O N S I D E R A T I O N S I. E V A L U A T I O N O F E X I S T I N G T E C H N I Q U E S S e v e r a l t e c h n i q u e s h a v e b e e n p r e s e n t e d f o r t h e m e a s u r e m e n t of d e f l e c t i o n a n d s l o p e . S i n c e a l l of t h e s e a r e s u b j e c t t o v a r i o u s s h o r t -c o m i n g s i t w a s n e c e s s a r y to e v a l u a t e t h e m i n d i v i d u a l l y a n d i n s o d o i n g c h o o s e t h e t w o ( i . e . o n e f o r d e f l e c t i o n m e a s u r e m e n t a n d o n e f o r s l o p e m e a s u r e m e n t ) m o s t s u i t a b l e f o r t h i s s t u d y . T h i s w a s d o n e m o s t l y b y c o n s i d e r i n g p r a c t i c a l l i m i t a t i o n s o f t h e s y s t e m s . D e f l e c t i o n T h r e e t e c h n i q u e s w e r e d e e m e d to b e f e a s i b l e f o r m e a s u r i n g d y n a m i c d i s p l a c e m e n t s . T h e d i f f r a c t i o n t e c h n i q u e , h o w e v e r , w a s n o t c o n s i d e r e d to b e s u i t a b l e f o r t h e s t u d y b e c a u s e of i t s e x t r e m e s e n s i t i v -i t y a n d v a r i o u s c o m p l e x i t i e s . T h e s h a d o w a n d r e f l e c t i o n m o i r e m e t h o d s b o t h w e r e s e t u p i n t h e l a b o r a t o r y and t h e i r i n d i v i d u a l m e r i t s a s s e s s e d . T h e s h a d o w m o i r e w a s o b s e r v e d to g i v e b r i g h t , h i g h c o n t r a s t p a t t e r n s w i t h s p e c i m e n s of l i g h t s u r f a c e f i n i s h a n d i t w a s f o u n d t h a t t h e g r i d c o u l d b e m o v e d u p to 3" a w a y f r o m t h e s p e c i m e n s b e f o r e t h e f r i n g e s b e c a m e i n d i s t i n g u i s h a b l e . W i t h d a r k s u r f a c e f i n i s h e s t h e c o n t r a s t w a s c o n s i d e r -a b l y r e d u c e d . W i t h t h e r e f l e c t i o n m o i r e the c o n t r a s t w a s f o u n d to b e p o o r u n l e s s a w h i t e i l l u m i n a t e d s c r e e n w a s p r o v i d e d a s a b a c k g r o u n d . I n a l i k e m a n n e r i t w a s p o s s i b l e to m o v e t h e g r i d a b o u t 3" f r o m the s p e c i m e n w i t h o u t l o s i n g f r i n g e c l a r i t y . T h e s u r f a c e f i n i s h i n t h i s c a s e h a d to b e r e f l e c t i v e . A f t e r s o m e d e l i b e r a t i o n it w a s f e l t t h a t t h e s h a d o w m o i r e w o u l d b e t h e b e t t e r o f t h e two t e c h n i q u e s a n d w a s h e n c e c h o s e n f o r v i s u a l i z i n g n o d e s a n d d y n a m i c d i s p l a c e m e n t s . T w o o t h e r f a c t o r s / a l s o a i d e d i n t h i s d e c i s i o n . O n e w a s t h e f a c t t h a t t h e s h a d o w m o i r e t h e o r y w a s m o r e s t r a i g h t f o r w a r d a n d d i d n o t i n v o l v e t h e n u m e r o u s a p -p r o x i m a t i o n s a s s o c i a t e d w i t h the . r e f l e c t i o n m o i r e . It w a s a l s o e a s i e r to u s e s t r o b o s c o p y to s t o p t h e f r i n g e m o t i o n s i n c e a m o r e i n t e n s e l i g h t f i e l d c o u l d b e a c h i e v e d . Its m a j o r d i s a d v a n t a g e w a s t h a t i t s s e n s i t i v i t y w a s o n l y h a l f t h a t of t h e r e f l e c t i o n s y s t e m . Slope It h a s b e e n s h o w n t h a t f o u r s u i t a b l e t e c h n i q u e s e x i s t f o r m e a s u r i n g s l o p e s . T h e r e f r a c t i o n m o i r e / m e t h o d w a s i m m e d i a t e l y d i s -r e g a r d e d s i n c e i t r e q u i r e d t h a t s p e c i a l t r a n s l u c e n t m o d e l s b e m a d e , w h i c h i n i t s e l f w a s a d r a w b a c k to w h i c h the o t h e r t e c h n i q u e s w e r e n o t s u b j e c t e d . T h e L i g t e n b e r g m o i r e m e t h o d i n i t s o r i g i n a l f o r m w a s a l s o r e j e c t e d s i n c e f r i n g e s c o u l d n o t b e o b s e r v e d b e f o r e the g r i d l i n e s w e r e s u p e r i m p o s e d . T h i s c o u l d h a v e b e e n o v e r c o m e b y i n s e r t i n g a s h e e t o f g l a s s i n f r o n t of t h e p l a t e s p e c i m e n t h e r e b y c r e a t i n g a v i s i b l e m o i r e p a t t e r n b u t b e c a u s e of the l a r g e p h y s i c a l s i z e o f the c o m p o n e n t s i n the L i g t e n b e r g m e t h o d the i d e a w a s a b a n d o n e d . T h e m a i n r e a s o n f o r t h i s i s t h a t i t i s h o p e d t h a t t h e o p t i c a l e q u i p m e n t u s e d i n t h i s s t u d y m a y , i n the f u t u r e , b e i n c o r p o r a t e d i n t o a m e a s u r i n g i n s t r u m e n t a n d t h e r e f o r e s i z e b e c o m e s a m a j o r f a c t o r . T h e t w o r e m a i n i n g m e t h o d s w e r e s e t u p i n the laboratory and analyzed. It was hoped that the r e f l e c t i o n method described by Theocaris would give the desired r e s u l t s but this was not the case. The main problem lay i n not being able to get s u f f i c i e n t l y bright c l e a r fringes and consequently low contrast photographs resulted. This was probably due to the l i m i t a t i o n s of the equipment used. The Salet-Ikeda method on the other hand gave sharp high contrast patterns and the o p t i c a l system was found to be very easy to a l i g n . This then was chosen as the slope technique f o r v i s u a l i z i n g antinodes. 1 5 II. T H E O R Y S h a d o w M o i r e D e f l e c t i o n M e t h o d F r o m m o i r e t h e o r y i t i s o b s e r v e d t h a t i f t w o l i n e d g r i d s of u n l i k e p i t c h a r e s u p e r i m p o s e d s u c h t h a t t h e i r l i n e s a r e a p p r o x i m a t e l y p a r a l l e l a n e t w o r k o f m o i r e f r i n g e s w i l l a p p e a r . T h e a p p e a r a n c e o f t h e s e f r i n g e s i s , i n f a c t , t h e c o m m o n " b e a t " p h e n o m e n o n t h a t o c c u r s w h e n a n y t w o s i m i l a r f r e q u e n c i e s a r e a d d e d t o g e t h e r . I n t h i s c a s e , h o w e v e r , the f r e q u e n c i e s a r e r e p r e s e n t e d b y the l i n e d e n s i t i e s . F o r e x a m p l e , i f o n e g r i d of 100 l p i i s s u p e r i m p o s e d , o n a n o t h e r g r i d o f 101 l p i , o n e m o i r e f r i n g e p e r i n c h w i l l a p p e a r . A n o t h e r w a y o f s t a t i n g t h i s / w o u l d b e to s a y t h a t f o r e v e r y m o i r e f r i n g e t h a t i s o b s e r v e d t h e r e m u s t n e c e s s a r i l y b e a d i f f e r e n c e o f o n e p i t c h b e t w e e n t h e t w o a d j a c e n t g r i d s . A f a r m o r e d e t a i l e d a n d m a t h e m a t i c a l e x p l a n a t i o n o f the m o i r e ' f r i n g e p h e n o m e n o n m a y b e f o u n d i n m a n y of the r e f e r e n c e s g i v e n b y T h e o c a r i s (12) a n d D u n c a n (13). M o i r e f r i n g e s a r e n o t a l w a y s p r o d u c e d b y s u p e r i m p o s i n g t w D g r i d s o f u n l i k e p i t c h . M o r e c o m m o n l y , i n f a c t , b o t h g r i d s a r e of i d e n -t i c a l p i t c h a n d f r i n g e s a r e d e v e l o p e d i n o n e of two w a y s : (1) l i n e a r s t r e t c h a n d / o r r o t a t i o n o f o n e g r i d , t h e o t h e r b e i n g h e l d c o n s t a n t o r ; (2) r o t a t i o n o f o n e g r i d i n a d i r e c t i o n n o r m a l to t h e o t h e r a s s h o w n i n F i g . 1 ( a ) . In t h e l a t t e r of t h e s e t w o m e t h o d s t h e p i t c h of t h e l o w e r g r i d a p p e a r s to e x p a n d b e c a u s e of i t s r e l a t i v e r o t a t i o n a n d i n t h i s m a n n e r a c h i e v e s t h e p i t c h d i f f e r e n c e n e c e s s a r y to p r o d u c e f r i n g e s . A s s u m e n o w 16 !E'o Observer To Observer (a) (b) FIGURE 1 METHODS OP PRODUCING MOIRE FRINGES t h a t t h e l o w e r g r i d i s r e m o v e d a n d i n i t s p l a c e i s s u b s t i t u t e d a p l a t e w i t h a m a t t e s u r f a c e . If c o l l i m a t e d l i g h t i s t h e n p a s s e d t h r o u g h t h e u p -p e r g r i d , s h a d o w s w i l l b e c a s t o n the p l a t e s u r f a c e g i v i n g t h e i m p r e s -s i o n of a s e c o n d g r i d . T h i s i s s h o w n i n F i g . 1 ( b ) . W h e n n o w v i e w e d f r o m a b o v e t h e i n t e r f e r e n c e b e t w e e n t h e m a s t e r g r i d a n d t h e s h a d o w s i s s e e n to y i e l d m o i r e f r i n g e s . T h i s t h e n b e c o m e s the b a s i s of t h e t e c h -n i q u e n o w t o b e d i s c u s s e d . C o n s i d e r t h e s y s t e m i n F i g . 2. L i g h t s o u r c e L i ! ^ i s p l a c e d at t h e f o c a l p o i n t of l e n s L ^ i n s u c h a m a n n e r t h a t t h e e m e r g e n t l i g h t i s c o l l i m a t e d p a r a l l e l to t h e A B a x i s . In t r a v e l l i n g a l o n g t h i s p a t h t h e l i g h t p a s s e s t h r o u g h a m a s t e r g r i d of a p p r o x i m a t e l y 100 t o 500 l p i , t h u s c a s t i n g a s h a d o w o f t h e l i n e s o n the s p e c i m e n . A n u p p e r b o u n d o f 500 l p i i s p l a c e d u p o n t h e g r i d b e c a u s e of t h e a p p e a r a n c e of d i f f r a c t i o n e f f e c t s a n d l o w f r i n g e c o n t r a s t at h i g h e r l i n e d e n s i t i e s . A s s u m e t h e s p e c i m e n i s n o w v i e w e d a l o n g the a x i s B C a n d i s i m a g e d o n t h e g r o u n d g l a s s s c r e e n o f t h e c a m e r a . T h e g r o u n d 18 g l a s s w i l l " s e e " t h e s u p e r p o s i t i o n of t w o g r i d s , t h a t i s , t h e m a s t e r g r i d a n d i t s s h a d o w . If i n i t i a l l y t h e s p e c i m e n i s u n d e r f o r m e d t h e p i t c h of the s h a d o w g r i d i s t h e s a m e a s t h a t of t h e m a s t e r a n d c o n s e q u e n t l y n o f r i n g e s w i l l b e o b s e r v e d . I n t r o d u c t i o n o f d e f o r m a t i o n , h o w e v e r , c a u s e s t h e a p p a r e n t p i t c h o f t h e s h a d o w to c h a n g e w i t h the r e s u l t t h a t m o i r e f r i n g e s a r e g e n e r a t e d . F o r t h e p u r p o s e o f a n a l y s i s i t i s i n t e r -e s t i n g t o s t u d y t h e d e f o r m a t i o n r e q u i r e d to p r o d u c e o n e f r i n g e . R e f e r to F i g . 3. FIGURE 3 DEFLECTION CAUSING ONE FRINGE 19 L e t t h e s p e c i m e n b e v i e w e d a l o n g s o m e l i g h t p a t h p a r a l l e l to t h e o p t i c a l a x i s B C s u c h t h a t d a r k m o i r e f r i n g e s a r e cb s e r v e d a l o n g t h e l i n e s M M ' a n d N N ' a s s h o w n i n F i g . 3. It w a s p r e v i o u s l y s t a t e d t h a t f o r o n e m o i r e f r i n g e to b e g e n e r a t e d t h e r e m u s t be a d i f f e r e n c e o f o n e p i t c h b e t w e e n t h e t w o g r i d s . T h e r e f o r e i t m u s t b e a s s u m e d t h a t b e -t w e e n l i n e s M M 1 a n d N N 1 t h e n u m b e r of p i t c h e s o n t h e m a s t e r g r i d a n d t h e n u m b e r of p i t c h e s o n t h e s h a d o w g r i d m u s t v a r y b y o n e p i t c h . If h i s t h e d i s t a n c e r e q u i r e d to p r o d u c e t h e m i s m a t c h of o n e p i t c h i t c a n b e s e e n f r o m g e o m e t r y t h a t : p = h t a n i + h t a n o h = P (2.1) t a n i + t a n o o r f o r s m a l l a n g l e s , h = p (2.2) i + o W i t h t h i s i n f o r m a t i o n i t i s n o w p o s s i b l e to d e t e r m i n e t h e r e l -a t i v e d e f l e c t i o n b e t w e e n a n y two p o i n t s o n t h e s u r f a c e of t h e s p e c i m e n . T o e a s e i n t e r p r e t a t i o n i t i s n e c e s s a r y to a s s i g n e a c h f r i n g e a v a l u e c a l -l e d t h e f r i n g e o r d e r N . F o r c o n v e n i e n c e t h e f i r s t f r i n g e to a p p e a r w h e n t h e p l a t e d e f l e c t s i s d e s i g n a t e d t h e f i r s t o r d e r f r i n g e . In a n a l y z i n g a d e f o r m e d p l a t e i t b e c o m e s o b v i o u s t h a t e i t h e r t h e d a r k o r l i g h t f r i n g e s m a y b e u s e d a s t h e r e f e r e n c e p a t t e r n . If t h e o b j e c t i v e i s to d e t e r m i n e t h e c h a n g e i n d e f l e c t i o n b e t w e e n t w o p a r t i c u l a r p o i n t s o n the s u r f a c e the c h o i c e of f r i n g e s i s p u r e l y a m a t t e r of c o n v e n i e n c e . H o w e v e r , i f t h e t o t a l d e f l e c t i o n i s r e q u i r e d t h e c h o i c e of f r i n g e s b e c o m e s of t h e u t -20 m o s t i m p o r t a n c e . T h i s i s a f f e c t e d p r i m a r i l y b y w h e t h e r o r n o t t h e i n -i t i a l u n l o a d e d f i e l d i s d a r k o r l i g h t . If t h e f i e l d i s c o m p l e t e l y l i g h t t h e d e f l e c t i o n w i s g i v e n b y : w = N h l i g h t f r i n g e s (2 .3) w = ( N - l / 2 ) h d a r k f r i n g e s ( 2 . 4 ) o r i f t h e i n i t i a l f i e l d i s c o m p l e t e l y d a r k t h e d e f l e c t i o n b e c o m e s : w = ( N - l / 2 ) h l i g h t f r i n g e s ( 2 . 5 ) w = N h d a r k f r i n g e s ( 2 . 6 ) S i n c e i n m o s t c a s e s a c o m p l e t e l y d a r k o r l i g h t f i e l d i s i m -p o s s i b l e to o b t a i n d u e t o d e v i a t i o n s f r o m f l a t n e s s , i t i s o f t e n m o r e c o n -v e n i e n t to c o v e r t h e e n t i r e s p e c i m e n w i t h a n i n i t i a l f r i n g e p a t t e r n . i S u c h a p a t t e r n i s p r o d u c e d b y p l a c i n g t h e p l a t e at a s l i g h t a n g l e to t h e g r i d a n d t h u s c r e a t i n g f r i n g e s " a r t i f i c i a l l y " . T h i s p r e s e n t s t h e a d d e d d i f f i c u l t y t h a t a f t e r d e f o r m a t i o n t h e i n i t i a l p a t t e r n m u s t b e s u b t r a c t e d f r o m t h e f i n a l p a t t e r n i n o r d e r to d e t e r m i n e t h e d e f l e c t i o n . H o w e v e r , t h e r e i s a d i s t i n c t a d v a n t a g e to u s i n g t h i s p r o c e d u r e , i n t h a t , a d e t a i l e d a n a l y s i s of t h e e n t i r e f i e l d i s o b t a i n e d . In o t h e r w o r d s , a n i n i t i a l " m i s -m a t c h " i s p r o d u c e d . T h i s p r o c e d u r e i s u s e d f r e q u e n t l y f o r t h e s t a t i c a n a l y s i s of p l a t e s l o a d e d l a t e r a l l y . I n t h e p r e v i o u s d i s c u s s i o n i t i s s e e n t h a t e a c h f r i n g e o r d e r r e p r e s e n t s a p a r t i c u l a r c h a n g e i n h e i g h t a n d t h e r e f o r e t h e f r i n g e p a t t e r n i s a c o n t o u r m a p o f t h e p l a t e s u r f a c e . C o n s e q u e n t l y t h e s h a p e of t h e d e -f l e c t i o n c u r v e m a y b e f o u n d a l o n g a n y p a r t i c u l a r l i n e o n the s u r f a c e of 2 1 1 2 1 1 1 0 Q N=8 F r i n g e s on P l a t e Surface 1 0 5-_j 111 0 / ^ \ T / / / / w 5 6 7 8 9 1 0 1 1 1 2 1 3 FIGURE 4 TYPICAL MOIRE PATTERN t h e s p e c i m e n a s i s s h o w n i n F i g . 4 . A g a i n t h e f r i n g e s m a y b e e i t h e r d a r k o r l i g h t d e p e n d i n g o n t h e i n i t i a l p a t t e r n u s e d . F r o m t h i s d i a g r a m i t i s e a s i l y s e e n t h a t t h e s l o p e m a y a l s o b e e v a l u a t e d a n y w h e r e a l o n g the c u r v e b y a g r a p h i c a l d i f f e r e n t i a t i o n . S a l e t - I k e d a S l o p e M e t h o d T h e S a l e t - I k e d a t e c h n i q u e c a n b e s t b e u n d e r s t o o d b y c o n s i -d e r i n g t h e p h y s i c a l a r r a n g e m e n t o f t h e a p p a r a t u s a s s h o w n i n F i g . 5. C o n s i d e r t h e s p e c i m e n t o h a v e a h i g h l y r e f l e c t i v e s u r f a c e a n d to b e i n i t i a l l y l o c a t e d a t t h e i n t e r s e c t i o n of the o p t i c a l a x e s o f t h e t w o 23 lens systems. Let it be adjusted to such an angle that any light travel-ling along the axis AB will be reflected back along B C , through the pin-hole and into the camera. A target, which may consist of straight lines, circles, dots or any suitable configuration is located in the focal plane of lens L i and is illuminated by a high intensity light source LSj . Since it is desirable to have the light evenly distributed over the target surface a sheet of ground glass is mounted behind the target to act as a diffusing screen. Diffused light then passes through the target into lens L^ where it is collimated parallel to some axis defined by its point of emission on the target. To study the effect of the deformed and unde-formed plate on the paths of incident light consider two typical rays and R£ emitted from point T 1 on the target. First consider the plate in the undeformed state and assume that it is perfectly flat. Since T 1 is not on the optical axis^R^ and R2 will not travel along a path parallel to the axis but rather will be inclined at an angle a . Therefore upon striking the plate at B\ and B 2 they, will be reflected off at an angle a to the BC axis and hence will not be focused at the pinhole.. If, in the same manner, every single ray emitted from T' is traced through the system it will be found that they will all be reflected at an angle a and thus it is not possible in any way to observe T 1 at the pinhole. In fact, the only point on the target that may be observed is the point T on the optical axis. If T is a black dot or line the whole field will appear completely dark: if it is a light space the field will appear completely light. 24 If t h e s p e c i m e n i s n o w d e f o r m e d t h e s l o p e at e v e r y p o i n t w i l l c h a n g e w i t h t h e r e s u l t t h a t t h e r e f l e c t e d r a y s o f l i g h t w i l l t a k e u p n e w p a t h s . A g a i n c o n s i d e r r a y s a n d R2 e m i t t e d f r o m T 1 o n t h e t a r g e t . B e f o r e s t r i k i n g t h e s p e c i m e n at p o i n t s a n d D2 t h e r a y s w i l l f o l l o w t h e i d e n t i c a l p a t h t h e y p r e v i o u s l y t o o k . H o w e v e r , b e c a u s e t h e s l o p e s at a n d D% a r e n o w n o t t h e s a m e t h e r a y s w i l l b e r e f l e c t e d i n d i f f e r e n t d i r e c t i o n s . In s u c h a m a n n e r i t i s p o s s i b l e , b y v a r y i n g t h e s l o p e , to r e f l e c t o n e of t h e r a y s a l o n g s o m e p a t h t h a t l i e s p a r a l l e l to a x i s B C , i n t h i s c a s e R p S i n c e a l l t h e l i g h t t h a t t r a v e l s p a r a l l e l to B C m u s t b e b r o u g h t to f o c u s b y l e n s L2 i t c a n b e s e e n t h a t R^ w i l l p a s s t h r o u g h t h e p i n h o l e a n d b e r e c o r d e d b y t h e c a m e r a . L i k e w i s e a n y o t h e r p o i n t o n t h e t a r g e t w h i c h h a s a n e m e r g e n t l i g h t r a y p a r a l l e l to B C w i l l a l s o b e r e c o r d e d . S i n c e t h e p a t h of t h e e m e r g i n g r a y s i s c o n t r o l l e d s o l e l y b y t h e a n g l e o f r o t a t i o n $ i t i s s e e n t h a t this i s a s l o p e m e a s u r i n g m e t h o d , w i t h the l i n e s r e c o r d e d b y the c a m e r a d e n o t i n g c o n t o u r s o f c o n s t a n t s l o p e . T a k i n g r n to b e t h e d i s t a n c e b e t w e e n T o n t h e o p t i c a l a x i s a n d s o m e n*^ 1 l i n e o n t h e t a r g e t , a n d F ^ t o b e t h e f o c a l l e n g t h o f l e n s L ^ , a f u n c t i o n a l r e l a t i o n s h i p i s s e e n t o e x i s t f o r d e t e r m i n i n g $ . D u n c a n a n d B r o w n (23) h a v e s h o w n t h e r e l a t i o n s h i p to b e : $ = 1/2 t a n " ' r n ( 2 . 7 ) F l S i n c e $ i s d i r e c t i o n d e p e n d e n t t h e c h o i c e o f t a r g e t s a n d t h e i r l i n e o r i e n t a t i o n i s o f t h e u t m o s t i m p o r t a n c e . H o w e v e r , r e g a r d -l e s s of w h a t t y p e of t a r g e t i s u s e d i t s h o u l d b e n o t e d t h a t t h e d i r e c t i o n 25 of t h e s l o p e i s a l w a y s n o r m a l to the d i r e c t i o n of t h e t a r g e t l i n e s . F o r e x a m p l e , a c i r c u l a r t a r g e t y i e l d s c o n t o u r s w h i c h j o i n p o i n t s of e q u a l s l o p e i n t h e r a d i a l d i r e c t i o n . In t h e p r e v i o u s d i s c u s s i o n i t w a s a s s u m e d that t h e p l a t e w a s p e r f e c t l y f l a t b e f o r e l o a d i n g . T h i s d o e s n o t l i m i t t h e t e c h n i q u e , h o w e v e r . If t h e u n l o a d e d p l a t e h a s s o m e i n i t i a l c u r v a t u r e i t w i l l y i e l d a v e r y d e f -i n i t e p a t t e r n w h e n v i e w e d t h r o u g h t h e p i n h o l e . U p o n l o a d i n g , h o w e v e r , t h e p a t t e r n w i l l b e o b s e r v e d to c h a n g e . If t h e i n i t i a l p a t t e r n i s t h e n s u b -t r a c t e d f r o m t h e l o a d e d p a t t e r n t h e s l o p e s w i l l b e g i v e n d i r e c t l y . T h i s i s o f t e n v e r y n e c e s s a r y s i n c e t h e t e c h n i q u e i s e x t r e m e l y s e n s i t i v e to t h e s l i g h t e s t d e v i a t i o n f r o m f l a t n e s s . A n a l y s i s o f N o d e s a n d D e f l e c t i o n s F r o m t h e d e f i n i t i o n g i v e n i n C h a p t e r II i t i s s e e n t h a t f o r a p o i n t to b e a n o d e t h e a m p l i t u d e at t h a t p o i n t m u s t r e m a i n z e r o f o r a l l t i m e . T h e r e f o r e , a n y s y s t e m t h a t h a s t h e a b i l i t y to s c a n a v i b r a t i n g p l a t e a n d p i c k o u t the z e r o d e f l e c t i o n p o i n t s w o u l d b e a s u i t a b l e m e a s -u r i n g d e v i c e . S i n c e t h e s h a d o w m o i r / m e t h o d h a s t h e a d v a n t a g e of b e i n g a b l e to d e p i c t t h e c o n t o u r s o f d e f l e c t i o n e v e r y w h e r e i n a d e f o r m e d s u r f a c e a n d s i n c e n o m e c h a n i c a l a t t a c h m e n t to t h e s u r f a c e i s n e c e s s a r y i t w o u l d s e e m t o b e a v e r y d e s i r a b l e t e c h n i q u e . Its f e a s i b i l i t y f o r v i s -u a l i z i n g n o d a l p a t t e r n s w i l l n o w b e i n v e s t i g a t e d . It h a s a l r e a d y b e e n s h o w n i n F i g . 3 t h a t a s a p l a t e d e f o r m s 26 a n e w m o i r e f r i n g e w i l l a p p e a r f o r e a c h i n c r e m e n t o f d e f l e c t i o n h t h a t t h e p l a t e u n d e r g o e s . T h e r e f o r e i f t h e p l a t e i n F i g . 6 (a) i s d e l i b e r a t e -l y p l a c e d at a n a n g l e to t h e g r i d a n i n i t i a l f r i n g e p a t t e r n w i l l b e c r e a t e d o n i t s s u r f a c e . C o n s i d e r n o w t h a t t h e p l a t e i s s e t i n t o a s t a t e o f r e s o n -a n c e a n d a s s u m e s t h e m o d e s h o w n i n F i g . 6 (b) . N o d e s a r e i n d i c a t e d at A a n d B . T o s t u d y t h e m o t i o n o f t h e f r i n g e s as t h e p l a t e o s c i l l a t e s b e t w e e n i t s t w o m a x i m u m a m p l i t u d e s i t i s f i r s t c o n v e n i e n t to i n v e s t i g a t e o n e p a r t i c u l a r f r i n g e o c c u r i n g at p o i n t P . T h i s f r i n g e i s a c o n t o u r l i n e r e p r e s e n t i n g t h e d e f l e c t i o n w a n d r e g a r d l e s s o f w h a t h a p p e n s to o t h e s h a p e o f t h e p l a t e t h i s f r i n g e i s o b l i g e d to s h i f t to a n y p o i n t h a v i n g t h i s d e f l e c t i o n . T h e r e f o r e as t h e p l a t e d e f l e c t s u p w a r d f r o m i t s u n d e -f o r m e d p o s i t i o n 1 to s o m e n e w p o s i t i o n 2, t h e P f r i n g e w i l l m i g r a t e a l o n g t h e W g c o n t o u r to P ' . S i m i l a r l y i f t h e m o t i o n o f e v e r y o t h e r f r i n g e i s i n v e s t i g a t e d i t w i l l b e o b s e r v e d to b e h a v e i n a l i k e m a n n e r . T h u s t h e e n t i r e f r i n g e f i e l d a p p e a r s to s w e e p ' a c r o s s t h e p l a t e i n t h e d i r e c t i o n s i n d i c a t e d b y t h e a r r o w s . A t t h e n o d e s , h o w e v e r , t h e d e f l e c -t i o n i s c o n s t a n t . T h e r e f o r e f r i n g e s a p p e a r i n g t h e r e r e m a i n f i x e d . If t h e m o t i o n i s n o w i n v e s t i g a t e d as t h e p l a t e d e f l e c t s d o w n w a r d to p o s i t i o n 3 , t h e P f r i n g e i s s e e n to s h i f t a l o n g the w c o n t o u r to P " . i o A s i n the a b o v e c a s e t h e f r i n g e f i e l d s w e e p s o v e r t h e s u r f a c e i n s t e p w i t h t h e P f r i n g e , w i t h t h e e x c e p t i o n t h a t i t i s n o w m o v i n g i n t h e o p -p o s i t e d i r e c t i o n . A g a i n t h e f r i n g e s o c c u r r i n g at t h e n o d e s r e m a i n s t a t i o n a r y . T h e r e f o r e , as d e f l e c t i o n o c c u r s t h e f r i n g e s e i t h e r a p p r o a c h o r l e a v e t h e n o d e s b u t at n o t i m e e v e r c r o s s t h e m . 27 PLATE ;AT REST PLATE AT RESONANCE FIGURE 6 MOIRE FRINGE MOTION IN A VIBRATING PLATE 28 D u r i n g v i b r a t i o n t h e f r i n g e s w e e p i n g a c t i o n i s g r e a t l y i n c r e a -s e d s i n c e t h e f r i n g e s a r e m o v i n g a t a v e r y h i g h v e l o c i t y . T h i s r a p i d o s -c i l l a t i o n o f t h e p a t t e r n r e s u l t s i n a b l u r r e d i m a g e a n d i s t e r m e d " w a s h -o u t " . If t h e e y e i s u s e d to v i e w t h e v i b r a t i n g p a t t e r n a l o w e r l i m i t i s p l a c e d o n t h e a b i l i t y to o b s e r v e f r i n g e b l u r r i n g . T h i s a r i s e s o u t of t h e f a c t t h a t t h e e y e r e t a i n s a n i m a g e f o r 1 /30 s e c . A n y t h i n g m o v i n g s l o w e r t h a n t h i s w i l l n o t b e b l u r r e d . T h e c a m e r a , h o w e v e r , i s f a r l e s s r e s t r i c -t e d s i n c e b y u s i n g a t i m e e x p o s u r e e v e n the s l o w e s t m o v i n g p a t t e r n w i l l l e a d to w a s h o u t . T h e p r i n c i p l e s a n d fac ts o u t l i n e d e n a b l e t h e n o d a l p a t t e r n s a n d d e f l e c t i o n s to b e o b s e r v e d a n d r e c o r d e d e v e r y w h e r e i n t h e p l a t e . T h e p a t t e r n o f t h e n o d a l l i n e s a r e r e v e a l e d d i r e c t l y a s i n F i g . : 7 a n d n e e d n o f u r t h e r i n t e r p r e t a t i o n . T o d e t e r m i n e t h e d e f l e c t i o n s i t i s n e c e s s a r y to " s t o p " t h e m o t i o n o f t h e f r i n g e s b y e i t h e r of t h e f o l l o w i n g t e c h n i q u e s : (1) a s t r o b o s c o p e f l a s h i n g at the s a m e f r e q u e n c y a s t h e p l a t e o r ; (2) h i g h s p e e d p h o t o g r a p h y . In t h i s m a n n e r i t i s p o s s i b l e to r e c o r d t h e f r i n g e p a t t e r n at a n y p a r t i c u l a r i n s t a n t a n d t h e n " s u b t r a c t . " t h e i n d i c a t e d d e f l e c -t i o n f r o m t h e s t a t i c d e f l e c t i o n to o b t a i n the d e f l e c t i o n d u e to v i b r a t i o n . U p to t h i s p o i n t n o t h i n g h a s b e e n s a i d r e g a r d i n g t h e e f f e c t of a m p l i t u d e o n t h e c h a r a c t e r o f t h e o b s e r v e d n o d a l p a t t e r n . T h i s f a c t o r i s of m a j o r i n t e r e s t s i n c e i t i m p o s e s b o t h a n u p p e r a n d l o w e r l i m i t o n t h e c a p a c i t y o f t h e t e c h n i q u e to r e v e a l n o d e s . If the a m p l i t u d e of v i b r a -t i o n i s s m a l l t h e f r i n g e s h i f t o n t h e p l a t e s u r f a c e w i l l l i k e w i s e b e s m a l l . FIGURE 7 TYPICAL NODAL PATTERN 30 (a) DARK FRINGE NOT WASHED OUT (b) DARK FRINGE WASHED OUT FIGURE 8 FRINGE SHIFT NECESSARY FOR WASHOUT If the shift is on the o r d e r of 1/10 of a f r i n g e width as shown i n F i g . 8 (a)this is i n s u f f i c i e n t to washout the d a r k f r i n g e s and t h e r e f o r e i n e f -f e c t i v e i n def ining the node. . T h e m i n i m u m f r i n g e shift n e c e s s a r y f o r washout is shown i n F i g . 8 (b) and is found to be 1/2 a f r i n g e width or i n t e r m s of d e f l e c t i o n : w m i r . = h ( 2 ' 8 ) 4 The upper bound is m o r e d i f f i c u l t to eva luate . It has a l r e a d y been shown i n F i g . 6 that the ampli tude has n o effect on the m o v e m e n t of the f r i n g e s at the n o d e . It d o e s , h o w e v e r , a l t e r the width of the n o d a l b a n d . A s the a m p l i t u d e i s i n c r e a s e d the node w i l l appear to b e c o m e p r o g r e s s i v e l y n a r r o w e r u n t i l f i n a l l y it r e a c h e s a point at w h i c h there 3 1 is i n s u f f i c i e n t contrast to make it c l e a r l y v i s i b l e . T h i s can only be ap-p r e c i a t e d by act u a l l y viewing the pattern while v a r y i n g the amplitude. A N A L Y S I S O F A N T I N O D E S F r o m the second definition of an antinode given in Chapter II it is apparent that i f a point is an antinode the slope of a l l li n e s p a s s i n g through the point must be equal to zero. Stated mathemati-c a l l y this becomes: (2.9) where n is the d i r e c t i o n of some a r b i t r a r y line o c c u r i n g at the anti-node. Although i t would be most d e s i r a b l e to investigate p e r f e c t l y flat: plates this places a l i m i t a t i o n on the analysis and t h e r e f o r e the plates w i l l be assumed t o have some i n i t i a l warp. The p e r f e c t l y f l a t p l a t e w i l l then "be t r e a t e d as a s p e c i a l case* The Salet-Ikeda technique w i l l now be investigated as a means of locating antinodes. In the previous d i s c u s s i o n of the Salet-Ikeda technique it was shown that i f a deformed or warped surface was p l a c e d in the o p t i c a l system, a pattern of lines would appear on the photographic screen. E a c h line was found to be a constant slope contour on the plate surface, the d i r e c t i o n of the slope being m e a s u r e d n o r m a l to the d i r e c t i o n of the target l i n e s . T h i s situation for a s m a l l element of plate i s shown gr e a t l y exaggerated in F i g . 9 (a). H e r e the r e f l e c t e d rays are a l l p a r a -Z-w 32 FIGURE 9 MOTION OF LIGHT PATHS REFLECTED OFF A VIBRATING PLATE 33 l l e l to t h e o p t i c a l a x i s B C a n d h e n c e r e c o r d e d b y t h e c a m e r a . L e t t h e p l a t e n o w b e e x c i t e d at s o m e n a t u r a l f r e q u e n c y a n d a s s u m e t h a t it t a k e s t h e s h a p e s h o w n i n F i g . 9 ( b ) . If the p l a t e i s e x a m i n e d w h e n i t r e a c h e s t h e p o s i t i o n 2 i t w i l l b e o b s e r v e d t h a t n e a r l y e v e r y r e f l e c t e d r a y t h a t w a s i n i t i a l l y p a r a l l e l to B C i s n o w r e f l e c t e d i n s o m e r a n d o m d i r e c t i o n w h i c h i s n o t p a r a l l e l to B C . In f a c t t h e o n l y r e f l e c t e d r a y t h a t m a i n t a i n s i t s o r i g i n a l d i r e c t i o n i s the o n e at the a n t i n o d e . T h e s a m e c o n d i t i o n a p p l i e s as t h e a n t i n o d a l c o n t o u r l i n e t r a v e l s t o p o s i t i o n 3, T h u s as t h e p l a t e o s c i l l a t e s r a p i d l y b e t w e e n i t s m a x i m u m a m p l i -t u d e s e v e r y r a y , w i t h t h e e x c e p t i o n o f t h a t at t h e a n t i n o d e , s w e e p s b a c k a n d f o r t h a n d i n s o d o i n g c a n n o t b e d e t e c t e d as a c o n t i n u o u s s i g n a l at t h e p i n h o l e . T h e a n t i n o d e r a y , h o w e v e r , a p p e a r s to r e m a i n f i x e d w i t h r e s p e c t to t h e p l a t e s o t h a t a n y p a t t e r n o r l i n e t h a t o c c u r s at t h a t p o i n t w i l l n e c e s s a r i l y r e m a i n m o t i o n l e s s . V i b r a t i n g p l a t e s r e p r e s e n t a c o n d i t i o n o f two d i m e n s i o n a l w a v e m o t i o n a n d h e n c e t h e r e e x i s t s t h e p o s s i b i l i t y o f h a v i n g t w o d i m -e n s i o n a l s t a n d i n g w a v e s . P a r t i a l w a v e s w i l l b e d e t e c t e d i n p a r t i c u l a r d i r e c t i o n s , e a c h w i t h n o d e s a n d a n t i n o d e s . T h e a n t i n o d e s f o r a n y o n e p a r t i c u l a r w a v e o r i e n t e d i n the \ d i r e c t i o n a r e d e f i n e d b y s O H o w e v e r , b e c a u s e t h i s w a v e i s i n t e r a c t i n g w i t h w a v e s o r i e n t e d i n o t h e r d i r e c t i o n s the s l o p e i n t h e o r t h o g o n a l d i r e c t i o n i s n o t n e c e s s a r i l y z e r o a n d h e n c e t h e c o n d i t i o n f o r a n a n t i n o d e g i v e n i n e q n . 2 .9 i s n o t s a t i s -f i e d . T h u s i t i s c o n v e n i e n t to r e f e r to t h e d i r e c t i o n d e p e n d e n t a n t i n o d e as a " p a r t i a l a n t i n o d e " i n t h e ^ d i r e c t i o n . If t h e ^ w a v e i s c o n t i n u -o u s a c r o s s t h e e n t i r e p l a t e the p a r t i a l a n t i n o d e w i l l l i k e w i s e t r a v e r s e 34 G r i d D i r e c t i o n P a r t i a l Antinode i n Y D i r e c t i o n (a) STATIC PATTERN (b) DYNAMIC PATTERN FIGURE 10 TYPICAL PARTIAL ANTINODE PATTERN the e n t i r e p l a t e . A t y p i c a l p a t t e r n is shown i n F i g . 10. In the S a l e t - I k e d a technique it was just shown that antinodes m a y be r e a d i l y detec ted by the fact that the p a t t e r n or l i n e at that point does not washout when v i b r a t i o n i s i n t r o d u c e d . T h e q u e s t i o n now a r i s e s , what k i n d of antinode does it m e a s u r e - - the t rue antinode d e s c r i b e d i n eqn. 2 .9 or a p a r t i a l antinode d e s c r i b e d i n the l a s t p a r a g r a p h ? Since only s t ra ight l ine targets are u s e d the technique m e a s u r e s s lopes i n only one d i r e c t i o n and thus under v i b r a t i o n is seen to m e a s u r e p a r t i a l (a) PARTIAL ANTINODE IN; Y DIRECTION! (b) PARTIAL. ANTINODE IN f DIRECTION ANTINODE SUPERPOSITION OP PARTIAL ANTINODES FIGURE 11 METHOD OP LOCATING ANTINODES 3 6 a n t i n o d e s . It is c o m p l e t e l y i n s e n s i t i v e to s lopes i n the o r t h o g o n a l d i r -e c t i o n and hence cannot detect t r u e a n t i n o d e s . S i n c e the antinode cannot be o b s e r v e d d i r e c t l y a m e t h o d f o r l o c a t i n g it m u s t be f o u n d . A n antinode by d e f i n i t i o n has z e r o slope i n e v e r y d i r e c t i o n and t h e r e f o r e it m u s t n e c e s s a r i l y l i e on a p a r t i a l a n t i -n o d e . T h u s if two p a r t i a l ant inodes of d i f f e r e n t d i r e c t i o n are s u p e r -i m p o s e d the t rue antinode m u s t l i e at their i n t e r s e c t i o n point as shown i n F i g . 11. T o extend the technique to the study of p e r f e c t l y f la t s u r f a c e s it is only n e c e s s a r y to shift the p i n h o l e a long the o p t i c a l ax is to s o m e point that i s not at the f o c a l point of lens L ^ . In this m a n n e r the p i n -hole is c a p a b l e of v i e w i n g a m u c h g r e a t e r a r e a of the target and thus has the effect of c r e a t i n g an i n i t i a l p a t t e r n on the plate s u r f a c e . A l -though this is a d e p a r t u r e f r o m exact S a l e t - I k e d a t h e o r y it is of no i m -p o r t a n c e s i n c e quanti tat ive r e s u l t s a r e not r e q u i r e d . If the plate is now v i b r a t e d exac t ly the s a m e effect w i l l be o b s e r v e d as o c c u r r e d i n the c a s e of the w a r p e d p l a t e . A n o t h e r m e t h o d that c o u l d be a p p l i e d to f la t p la tes w o u l d be to locate the p i n h o l e at the f o c a l point of l e n s and then by m a n i p u l a t i n g the target, p r o d u c e a c o m p l e t e l y d a r k f i e l d on the p l a t e . U p o n e x c i t i n g the plate antinodes wo u l d appear d i r e c t l y as b l a c k l i n e s . In the p r e v i o u s a n a l y s i s only s t ra ight l i n e targets w e r e d i s -c u s s e d . A l t h o u g h the technique is not l i m i t e d to the use of this type of 3 7 t a r g e t a l o n e i t i s f o u n d t h a t t h e r e s u l t s a r e e a s i e r to i n t e r p r e t . B e a m A n d P l a t e S p e c i m e n s T o B e S t u d i e d In the i n v e s t i g a t i o n t h r e e p a r t i c u l a r s h a p e s a r e e x a m i n e d : (1) a c a n t i l e v e r b e a m ; (2) a s q u a r e c a n t i l e v e r p l a t e a n d ; (3) a c i r c u l a r f r e e p l a t e . S i n c e t h e s o l u t i o n o f t h e c a n t i l e v e r b e a m i s w e l l k n o w n a n d m a n y t a b l e s r e l a t i n g to i t s c h a r a c t e r i s t i c s h a v e b e e n p u b l i s h e d t h i s i s c h o s e n a s t h e p r i m a r y s p e c i m e n a n d i s h e n c e u s e d i n d e t e r m i n i n g t h e e x a c t l o c a t i o n of t h e n o d e s a n d a n t i n o d e s a s w e l l a s t h e d e f l e c t i o n s . S i m i l a r l y t a b l e s e x i s t f o r the s o l u t i o n of the s q u a r e a n d c i r c u l a r p l a t e s (24 , 25 a n d 26) b u t a s a n a l t e r n a t e s o l u t i o n t h e n o d e s a r e d e t e r m i n e d b y m e a n s of C h l a d n i s a n d p a t t e r n s . N o s o l u t i o n i s g i v e n f o r t h e l o c a t i o n o f t h e a n t i n o d e s i n t h e s e t w o p l a t e s . It i s c o n s i d e r e d s u f f i c i e n t t o s h o w t h e t h e o r e t i c a l l o c a t i o n o f the a n t i n o d e s i n a c a n t i l e v e r b e a m . A l l t h e p e r t i n e n t e q u a t i o n s r e l a t i n g to t h e s p e c i m e n s a r e g i v e n i n A p p e n d i x . A . C a l i b r a t i o n O f T h e C a n t i l e v e r B e a m I n o r d e r to e v a l u a t e t h e s h a d o w m o i r e ' m e t h o d a s a d y n a m i c d i s p l a c e m e n t m e a s u r i n g t e c h n i q u e i t i s n e c e s s a r y to p r o v i d e a m e a n s of c o m p a r i n g t h e o b t a i n e d r e s u l t s w i t h k n o w n v a l u e s of d i s p l a c e m e n t . T h e c a n t i l e v e r b e a m i s c o n s i d e r e d a n i d e a l s p e c i m e n f o r . t h i s p u r p o s e b e c a u s e of i t s w e l l k n o w n a n d s i m p l y a p p l i e d s o l u t i o n s . If t h e b e a m i s v i b r a t e d at i t s f u n d a m e n t a l f r e q u e n c y it i s f o u n d t h a t i t s d e f l e c t i o n v s 38 x/L curve due to i n e r t i a loading i s almost i d e n t i c a l to the curve produced by loading a s i m i l a r weightless beam with a point s t a t i c load applied at i t s free end. These curves are shown i n F i g . 12. I t can be seen that the maximum deviation between the curves occurs at about x/L = .5 and i s roughly 3 per cent. However, very close to the fixed end the difference becomes n e g l i g i b l e . I f a s t r a i n gauge i s placed as close to this end as possible i t w i l l y i e l d approximately the same output for both the dynamic and s t a t i c d e f l e c t i o n at the free end. Since the s t a t i c d e f l e c t i o n can be e a s i l y determined i t i s only necessary to apply known loads to the beam and note the out-put of the s t r a i n gauge on some recording device. I f the dynamic case i s then analyzed, by comparing i t s s t r a i n gauge output to that of the s t a t i c case the d e f l e c t i o n may be determined d i r e c t l y . This, however, i s for the free end only and to f i n d the d e f l e c t i o n at any other point the dynamic curve must be used. DYNAMIC CURVE STATIC CURVE X/L FIGURE 12 STATIC AND DYNAMIC DEFLECTION CURVES C H A P T E R III E X P E R I M E N T A L A P P A R A T U S I. O P T I C A L S Y S T E M S Shadow M o i r e The g e o m e t r i c a l placement of the componentshias a l r e a d y been given i n F i g . 2. Thi s was somewhat of an i d e a l i z e d s y s t e m and it was n e c e s s a r y and convenient to make c e r t a i n m o d i f i c a t i o n s . The point light source, for instance, is by definition an in f i n i t e l y s m a l l point of light, to which tungsten f i l a m e n t and m e r c u r y vapor lamps a r e a poor approximation. To improve this situation it was n e c e s s a r y to introduce a light condenser followed by an i r i s diaphragm as shown i n F i g . 13. LIGHT SOURCE CONDENSER IRIS DIAPHRAGM FIGURE 13 / SHADOW MOIRE LIGHT SOURCE 41 The lenses used i n this condenser were 5" i n diameter and had a f o c a l length of 7". The i r i s diaphragm had a m i n i m u m aperature of 1/16". In o r d e r to obtain the m aximum light intensity an O s r a m H B O 200 W/2 m e r c u r y vapor lamp was used as the light source. Because of the danger of explosion and u l t r a v i o l e t r a d i a t i o n the lamp was enclosed i n a thin a luminum tube 3" i n diameter by 6" long. T h i s also acted as a draft tube and p r o v i d e d sufficient cooling. The light emitted by the lamp was found to contain 3 major wave lengths, green being the most prominent. F o r the purpose of m i n i m i z i n g d i f f r a c t i o n effects at the grid a l l the wavelengths except g r e e n were el i m i n a t e d by using a Kodak #58 (B) f i l t e r . The power r e q u i r e d to operate the lamp was p r o v i d e d by a Mo d e l P-210D power supply available f r o m George W. Gates and Co. A l l the equipment i n c o r p o r a t e d i n the light source arrangement is shown i n F i g . 14. L e n s e s L^ and L2 as shown in F i g . 2 were both high quality f i e l d lenses each having f o c a l lengths of 60", Lens L^was 12" i n d i a -meter, lens 10" i n diameter. Both were mounted in adjustable supports. Two m a s t e r g r i d s were used in the investigation, one of 242 l p i and one of 500 l p i . Both were produced photographically f r o m etched m a s t e r s by contact p r i n t i n g onto 8" x 10" gla s s plates. To sup-42 FIGURE 14 / SHADOW MOIRE LIGHT SOURCE LIST OF COMPONENTS (1) p o w e r s u p p l y (2) l i g h t s o u r c e (3) c o n d e n s e r (4) f i l t e r ( 5 ) i r i s d i a p h r a g m 43 p o r t the g r i d a s t u r d y f r a m e h a v i n g 3 d e g r e e s of m o v e m e n t was c o n -s t r u c t e d . T h e g r i d m o u n t e d in its f r a m e is shown in F i g . 15. T h e c a m e r a u s e d in the study was a 4" x 5" C a l u m e t with a 240/480 m m . S c h n e i d e r l e n s . T h e e n t i r e shadow m o i r e a r r a n g e m e n t is shown i n F i g . 22. Shadow M o i r e U s i n g S t r o b o s c o p y F o r the d e t e r m i n a t i o n of d y n a m i c d e f l e c t i o n s it was n e c e s s a r y to f r e e z e the m o t i o n of the f r i n g e s . T h i s was c a r r i e d out b y u s i n g a G e n e r a l R a d i o M o d e l 1531 -A s t r o b o s c o p e . B e c a u s e of the i n t e r m i t -tent i n t e n s i t y p r o d u c e d b y this i n s t r u m e n t the o v e r a l l i n t e n s i t y for the p u r p o s e of taking p i c t u r e s was quite l o w . In o r d e r to use as m u c h l ight as p o s s i b l e the c o n d e n s i n g l e n s e s and i r i s d i a p h r a g m w e r e d i s -p e n s e d with and the l a m p i t s e l f was p l a c e d at the f o c a l point of lens L ^ . T h e s t r o b o s c o p e c a m e c o m p l e t e with its own r e f l e c t o r w h i c h h a d to be r e m o v e d . In its p l a c e was subst i tuted a l a r g e 8" s p h e r i c a l m i r r o r with a f o c a l length of 2 1/2" . T h e m i r r o r was adjusted u n t i l the s t robe l a m p was l o c a t e d at its center of c u r v a t u r e . In this m a n n e r a l l the l ight c o l l e c t e d b y the m i r r o r was r e f l e c t e d b a c k t h r o u g h the s o u r c e . T h e a r r a n g e m e n t is shown i n F i g . 16. S a l e t - I k e d a T e c h n i q u e If the S a l e t - I k e d a a r r a n g e m e n t i n F i g . 5 is c o m p a r e d to that of the shadow m o i r e ' i n F i g . 2 it is o b s e r v e d that, with the except ion . FIGURE 1 5 GRID AND SUPPORT FRAME FIGURE 16 STROBOSCOPE WITH SPHERICAL MIRROR 45 of the t a r g e t , and the p inhole i n f r o n t of the c a m e r a , the two s y s t e m s a r e i d e n t i c a l . F o r th is r e a s o n the s a m e l e n s e s and l ight s o u r c e as w e r e d e s c r i b e d f o r the shadow m o i r e w e r e u s e d f o r both t e c h n i q u e s . T h e S a l e t - I k e d a technique d i d not r e q u i r e the u s e of a point l ight s o u r c e but i n s t e a d r e q u i r e d that a w e l l i l l u m i n a t e d target be p l a c e d i n the f o c a l plane of l e n s L ^ . T o a c h i e v e the r e q u i r e d i l l u m i n a t i o n the s o u r c e and c o n d e n s i n g s y s t e m were set up as p r e v i o u s l y d e s c r i b e d . H o w e v e r , the i r i s d i a p h r a g m was r e m o v e d and r e p l a c e d by a sheet of g r o u n d g l a s s . T h e g r o u n d g l a s s was not p l a c e d at the f o c a l point of the c o n d e n s i n g l e n s , h o w e v e r , but was shif ted to a point at w h i c h it was c o v e r e d by a f a i r l y u n i f o r m l ight f i e l d as is shown i n F i g . 17. LIGHT SOURCE CONDENSER GROUND GLASS AND TARGET FIGURE 17 SALET-IKEDA LIGHT SOURCE FIGURE 18 SALET-IKEDA LIGHT SOURCE LIST OF COMPONENTS (1) power supply (2) l i g h t source (3) condenser (4) f i l t e r (5) t a r g e t 47 A target was then affixed to the surface of the ground glass downstream of the light source. T h r e e m a i n targets were used, 4 l p i , 7 l p i , and 20 l p i . These were made by drawing p a r a l l e l l i n e s on a sheet of paper and photographing them. The negative was then p l a c e d in the enlarger and by v a r y i n g the degree of enlargement and exposing a sheet of high contrast f i l m , targets of any suitable density could be made. Pin h o l e s of v a r i o u s diameters were investigated but it was found that the lowest f/stop on the camera, approximately f/60, y i e l d e d sufficient c l a r i t y of f r i n g e image and was t h e r e f o r e used throughout the experiment. The light source with target attached i s shown i n F i g . 18. The entire Salet-Ikeda arrangement i s shown i n F i g . 23. II. V I B R A T I O N A P P A R A T U S V i b r a t i o n Generating Equipment The apparatus used to set up v i b r a t i o n i n the specimen is shown in F i g . 19. It c o n s i s t e d of a Goodmans V 47 - 3 JTL v i b r a t i o n generator attached to a s o l i d steel base. The backing plate on which the generator was mounted was made adjustable so that contact be-tween the specimen and the generator could be set to any suitable p o s i -tion. A heavy steel bar was attached to the top of the base to which the v i b r a t i o n specimens were clamped. 48 FIGURE 19 VIBRATION GENERATING EQUIPMENT LIST OF COMPONENTS (1) v i b r a t i o n generator and base (2) a m p l i f i e r (3) audio generator 49 P o w e r for the g e n e r a t o r was s u p p l i e d by a 10 watt a m p l i f i e r d r i v e n by a H e a t h k i t audio g e n e r a t o r . V o l t a g e and c u r r e n t m e t e r s w e r e i n s e r t e d i n the c i r c u i t to m o n i t o r o v e r l o a d . T e s t S p e c i m e n s T h r e e s p e c i m e n s w e r e s t u d i e d : (a) s q u a r e plate - 4 " x 4 " x . 0 5 8 " t h i c k p l e x i g l a s s (b) c a n t i l e v e r b e a m - 4 " x 1/2" x . 0 5 8 " t h i c k p l e x i g l a s s (c) c i r c u l a r plate - 4 " d i a . x . 030" t h i c k p l e x i g l a s s In the p r e v i o u s d i s c u s s i o n it was seen that two s u r f a c e f i n -i shes w e r e r e q u i r e d on the s p e c i m e n s , r e f l e c t i v e and m a t t e . T h e r e f -l e c t i v e f i n i s h was p r o d u c e d by paint ing the b a c k s u r f a c e of the s p e c i -m e n s w i t h a f lat b l a c k paint . In a l i k e m a n n e r , the matte s u r f a c e was p r o d u c e d by s p r a y i n g the f r o n t s u r f a c e w i t h f lat white p a i n t . In the tests the s q u a r e plate and b e a m w e r e both c l a m p e d s o l i d l y to the s t e e l b a s e but w e r e not r i g i d l y c o n n e c t e d to the v i b r a -t i o n g e n e r a t o r . T h e c i r c u l a r p l a t e , h o w e v e r , was f r e e on the edge and had to be s u p p o r t e d by the g e n e r a t o r s p i n d l e . T h i s m e a n t that s o m e f o r m of a t tachement h a d to be d e v i s e d . T h i s was done quite s u c c e s -s f u l l y by d r i l l i n g and tapping a p i e c e of 1/8" p l e x i g l a s s to the s i z e of the spindle s c r e w and then g r i n d i n g the edges d o w n to a s m a l l c i r c u l a r d i s c about 3/16" i n d i a m e t e r . T h e s m a l l d i s c was then bonded to the centre of the plate with a c r y l i c c e m e n t . A f t e r d r y i n g , the spindle s c r e w was f i r m l y s c r e w e d into the d i s c and the plate s e c u r e l y h e l d . D i s p l a c e m e n t C a l i b r a t i o n I n s t r u m e n t a t i o n T h e a p p a r a t u s shown i n F i g . 20 was u s e d to d e t e r m i n e the d i s p l a c e m e n t s i n d u c e d i n the v i b r a t i n g c a n t i l e v e r b e a m . S t r a i n Gauges LA C a n t i l e v e r Beam BAM FIGURE 20 INSTRUMENTATION O s c i l l i s c o p e Weights v a r y i n g f r o m 1/16 o z . up to 1/4 o z . w e r e a t tached by m e a n s of a f ine t h r e a d to the end of the c a n t i l e v e r . T h e d e f l e c t i o n i n t r o d u c e d by the l o a d i n g was r e c o r d e d by L i m a - B a l d w i n C D - 8 s t r a i n gauges , one ac t ive and one d u m m y . T h i s i n f o r m a t i o n was then f e d into a B r i d g e A m p l i f i e r M e t e r M o d e l B A M - 1 and af ter a m p l i f i c a t i o n into a T e k t r o n i x M o d e l 502A o s c i l l i s c o p e . The e q u i p m e n t is shown i n F i g . 21. 51 FIGURE 21 CALIBRATION EQUIPMENT LIST OF COMPONENTS (1) Tibration generator and specimen (2) BAM (3) oscilliscope 5 2 FIGURE 2 2 SHADOW MOIRE ARRANGEMENT FIGURE 2 3 SALET-IKEDA ARRANGEMENT C H A P T E R IV E X P E R I M E N T A L P R O C E U D R E I. N O D E A N D D E F L E C T I O N A N A L Y S I S USING T H E S H A D O W M O I R E M E T H O D System Alignment It has p r e v i o u s l y been shown i n Chapter II that f or sharp c l e a r shadows to exist underneath the master g r i d p e r f e c t l y p a r a l l e l light is r e q u i r e d . The quality of the m o i r e patterns t h e r e f o r e is v e r y m u c h dependent upon how a c c u r a t e l y the opti c a l s y s t e m i s aligned. In p r a c t i c e , however, pe r f e c t alignment is i m p o s s i b l e due to the imp e r -fections i n the lense s and the in a b i l i t y to obtain a point light source. These stray effects may none the l e s s be m i n i m i z e d . Since the l o c a -t i o n of the camera, lens L2 and the g r i d have no effect on the p a r a l l e l i s m of the incident light on the s p e c i m e n these components w i l l not enter i n -to the following d i s c u s s i o n . The f i r s t step i n the aligning procedure-was to roughly p o s i -tion the g e o m e t r i c centre of a l l the components along a common ax i s . T h i s was done quite e a s i l y by m e a s u r i n g f r o m a common datum to the centre line of each component. In aligning the light source, condenser and diaphragm the quantity of light p assing through the diaphragm into •the f i e l d lens L] was m a x i m i z e d by placing the light source at a d i s -tance f r o m the condenser somewhat l e s s than twice the condenser f o c a l length, as shown in F i g . 24. With the light source in place the i r i s was (a) (b) FIGURE 24 ALIGNING THE LIGHT SOURCE then c l o s e d and moved along the op t i c a l axis u n t i l the image of the light s ource was seen c l e a r l y on the i r i s . These components were then a l l l o c k e d in place. With the i r i s s t i l l c l o s e d down to its s m a l l e s t aperature the f i e l d lens was r e l o c a t e d such that the f o c a l point of the lens and i r i s aperature were coincident. A plane m i r r o r was then p l a c e d i n the c o l l i m a t e d light f i e l d , downstream of the f i e l d lens but facing back toward the light source. F i n a l l y , the f i e l d dens was adjusted u n t i l the r e f l e c t e d image of the l i g h t - f i l l e d aperature f e l l d i r e c t l y on the aperature i t s e l f . T h i s procedure guaranteed highly c o l l i m a t e d light e m e rging f r o m the f i e l d lens. P o s i t i o n i n g the Ma s t e r G r i d In the pr e v i o u s section it was noted that the quality of the m o i r e patterns was d i r e c t l y dependent upon the degree of c o l l i m a t i o n of the incident light. However, before m o i r e patterns are actually ob-served, the light must pass through the ma s t e r g r i d and thus it can be seen that the g r i d may be used as a method of c o n t r o l l i n g the type of pattern p l a c e d on the plate. Consequently the positioning of the g r i d is of the utmost importance. In the inve s t i g a t i o n it was found that the g r i d could be made to v a r y three parameters: (a) contrast (b) f r i n g e density (c) f r i n g e o r i e n t a t i o n with r e s p e c t to the specimen Unfortunately, the best of a l l three of these p a r a m e t e r s could not be obtained at the same time. F o r instance, high f r i n g e density r e s u l t e d in low contr a s t and vice v e r s a . The deciding factor i n most of the studies was f r i n g e orientation. F o r the :nodal studies it was d e s i r -able to cover the plate with an i n i t i a l f i e l d of straight f r i n g e s at approx-imately 45 degrees to the h o r i z o n t a l . T h i s was achieved by f i r s t p l a c i n g the g r i d such that it was everywhere p a r a l l e l to the plate.; no f r i n g e s c o u l d be observed. The g r i d was then ti l t e d s l i g h t l y with re s p e c t to the plate u n t i l a d e s i r e d density of h o r i z o n t a l f r i n g e s appeared. By means of the fine adjustment screws the g r i d was then rotated i n the h o r i z o n t a l plane with the r e s u l t that the f r i n g e s rotated. T h i s was con-tinued u n t i l the lin e s were at approximately 45 degrees to the h o r i z o n -t a l . The r e a s o n f o r choosing 45 degrees was that it gave a f a i r l y u n i -f o r m coverage of the plate and nodes were r e a d i l y detected. F o r the displacement study using stroboscopy it was not d e s i r a b l e to have the fr i n g e s at 45 degrees but rather to have them p a r a l l e l to one of the boundaries. T h i s was to aid i n i n t e r p r e t a t i o n of the f r i n g e patterns. To b r i n g the f r i n g e s around p a r a l l e l to some p a r t i c u l a r edge it was only n e c e s s a r y to adjust the g r i d i n one plane or the other. D u r i n g these adjustments i t i s assumed that the viewer was either near the f o c a l point of lens L2 o r su f f i c i e n t l y far away f r o m the plate that the light r e a c h i n g his eye was approximately p a r a l l e l . Up to this point nothing has been s a i d r e g a r d i n g the p h y s i c a l o r i e n t a t i o n of the g r i d l i n e s , that i s , h o r i z o n t a l or v e r t i c a l . U s u a l l y this is det e r m i n e d by the p h y s i c a l layout of the system. If the op t i c a l axis of the s y s t e m is i n the h o r i z o n t a l plane the g r i d lines w i l l be v e r -t i c a l . Such was the case i n this study. P o s i t i o n i n g Of The C a m e r a And Lens L2 L e n s was set at an a r b i t r a r y distance f r o m the specimen. The c a m e r a lens was located approximately at the f o c a l point of L £ . The exact p o s i t i o n was adjusted to: (1) v a r y f r i n g e density, (2) m i n i m i z e r e f l e c t i v e effects at the plate s u r f a c e and (3) v a r y plate image siz e on the c a m e r a scre e n . R e c o r d i n g Nodes Once the i n i t i a l or static f r i n g e p a t t e r n had been established on the plate and the v i b r a t i o n generator adjusted to allow good contact the nodes were ready to be r e c o r d e d . With the audio generator adjusted 57 to the d e s i r e d natural frequency the amplitude was i n c r e a s e d u n t i l sharp nodes were observed. These were r e c o r d e d photographically using A S A 400 f i l m at a setting of f/22. Depending on the brightness of the p a r t i c u -l a r pattern being o b s e r v e d the exposure time v a r i e d f r o m 5 to 30 seconds. R e c o r d i n g D i s p l a c e m e n t s The light source was r e p l a c e d by the stroboscope as p r e v i o u s l y d e s c r i b e d . The strobe was then adjusted to the same frequency as that of the n a t u r a l frequency of the plate. A f t e r ensuring that a suitable i n -i t i a l pattern c o v e r e d the surface the plate was set into motion. Since the light was f l a s h i n g at the same frequency as the plate no f r i n g e motion was observed, however, as the frequency of the strobe was v a r i e d a beating phenomenon was encountered and the f r i n g e s t r a v e l l e d a c r o s s the plate. The v e l o c i t y of the fringe movement depended upon the di f f e r e n c e between the strobe and v i b r a t i o n frequencies and since both instruments were r e l a t i v e l y d r i f t f r e e it could be c o n t r o l l e d with ease. To r e c o r d the d i s -placements at any time the c a m e r a was t r i g g e r e d at the d e s i r e d instant. The f i l m was A S A 3000 P o l a r o i d and the setting f / l l . The approximate exposure time was 2 seconds. C a l i b r a t i o n Of The C a n t i l e v e r B e a m The method of c a l i b r a t i n g the c a n t i l e v e r beam has been f u l l y d i s c u s s e d in Chapter I I and only the p h y s i c a l details w i l l be dealt with here. 5 8 To r e c o r d the strai n i n the beam one active and one dummy s t r a i n gauge were used. The active gauge was bonded as cl o s e as pos-sible to the clamped end on the back of the beam. The dummy was mounted on a piece of p l e x i g l a s s and attached to the s t e e l base c l o s e to the beam. The two gauges acting together f o r m e d half of a Wheatstone B r i d g e c i r c u i t , the other half being inside the B A M . The a m p l i f i e d signal f r o m the B A M was r e c o r d e d on the o s c i l l i s c o p e . Since it was d e s i r e d to r e a d dynamic displacements d i r e c t l y off the o s c i l l i s c o p e a known static l oad was applied to the end of the beam. The r e s u l t i n g s i g n a l r e p r e s e n t e d a ca l c u l a b l e static displacement any-where along the length of the beam. By then adjusting the B A M gain and choosing a suitable range on the scope, the scope could be made to rea d the exact displacement. The scale chosen i n this study was . 005"/cm. d i v i s i o n on the s c a l e . T h i s allowed sufficient latitude i n choice of i n -put amplitudes. B ecause of the low frequencies involved and also the need to balance the c i r c u i t r y the o s c i l l i s c o p e was operated i n the DC mode. It was found also, that i n or d e r to m i n i m i z e noise a m p l i f i c a t i o n the steel base had to be grounded. Chlad n i P a t t e r n s To provide e x p e r i m e n t a l proof of the actual existence of nodes Chlad n i patterns were used. These were f o r m e d quite e a s i l y by p l a c i n g 5 9 the plates i n a h o r i z o n t a l position, exciting them at resonance and then-s p r i n k l i n g them with sand. The sand gravitated toward the nodes and y i e l d e d highly definable patterns. II. A N T I N O D E A N A L Y S I S USING T H E S A L E T - I K E D A M E T H O D System Alignment The alignment of the Salet-Ikeda apparatus was a much s i m -ple r undertaking than for the shadow m o i r e , e s p e c i a l l y since quantitative r e s u l t s were not sought. The light source, with target i n place, was set up as shown i n F i g . 17. By shifting the ground glass along the op-t i c a l axis a f a i r l y u n i f o r m and intense light f i e l d could be made to cover the entire target. Once this had been achieved the components were l o c k e d i n place. F i e l d lens L]^ was then moved to a p o s i t i o n at which its f o c a l point c o i n c i d e d with the centre of the target. T h i s completed alignment of the incident light. In o r d e r to a l i g n the r e m a i n d e r of the sy s t e m one of the plate specimens had to be p l a c e d at some convenient p o s i t i o n along the op-t i c a l axis of the incident light. By then rotating the plate through some suitable angle the r e f l e c t e d light could be d i r e c t e d into lens L2. The c a m e r a then, with its diaphragm c l o s e d right down, was p l a c e d at the f o c a l point of lens L £ . A f t e r slight v e r t i c a l and h o r i z o n t a l adjustment a c l e a r image of the target could be observed. The system was then ready to r e c o r d antinodes. 6 0 Choice of T a r g e t Throughout the tests only straight line targets were used. T h e r e was, however , no p a r t i c u l a r r e a s o n for this other than to aid i n i n t e r p r e t a t i o n . Because of the nature of the technique quite a r b i t r a r y patterns of li n e s could be made to y i e l d equally good r e s u l t s . . The den-sity of the target l i n e s , however, was most important and it was found that a target of 20 l p i or m o re could not be used since poor contrast r e -sulted. T h e r e f o r e a l l the tests were run with either one of the two r e -maining targets. The method adopted was to set up the plate with one p a r t i c u l a r target and then check the image of the target on the ground glass of the c a m e r a f o r con t r a s t and density. . If the pattern wasn't sat-i s f a c t o r y the other g r i d was substituted. It was found that a reasonable pattern could always be achieved with one g r i d or the other. R e c o r d i n g Antinodes It has p r e v i o u s l y been shown that the antinode l i e s at the i n -t e r s e c t i o n of any two p a r t i a l antinodes and therefore, f o r any one nat-u r a l frequency being investigated photographs must be taken with the target l i n e s i n two different d i r e c t i o n s . In this study it was found that p l a c i n g the target v e r t i c a l l y f o r one photograph and h o r i z o n t a l l y f or the other was the easiest to p e r f o r m . In the case of the beam only one d i m e n s i o n a l wave motion existed and hence it was only n e c e s s a r y to use one g r i d . T h i s was p l a c e d h o r i z o n t a l l y so as to be able to r e c o r d slopes taken i n a d i r e c t i o n along the beam. 61 When the plate was excited at resonance and the antinodes were being studied it was noticed that quite often the p a r t i a l antinodes o c c u r r e d at a point or line at which the pattern was rather poor. In fact, it o c c a s i o n a l l y happened that a p a r t i a l antinode f e l l a c r o s s a point at which there was no pattern at. a l l , i n which case the p a r t i a l antinode went undetected. To remedy this situation it was n e c e s s a r y to " s c a n " the plate by making minute h o r i z o n t a l or v e r t i c a l adjustments i n the p o s i t i o n of the pinhole. T h i s was done quite e a s i l y by using the v e r n i e r adjustments on the c a m e r a t r i p o d . In this manner a suitable pattern could be found that would locate a l l the p a r t i a l antinodes. Another p r o b l e m encountered in studying the patterns was the appearance of wide bands of yellow and blue light between the dark f r i n -ges. T h i s tended to grey the negatives and hence give poor p i c t u r e s . The situation was g r e a t l y i m p r o v e d by using a Kodak #58 (B) f i l t e r . C H A P T E R V E X P E R I M E N T A L R E S U L T S I. N A T U R A L F R E Q U E N C I E S The c a l c u l a t e d and o b s e r v e d natural f r e q u e n c i e s of the beam and plates are given i n Table 1. In determining these f r e q u e n c i e s three m a t e r i a l p r o p e r t i e s E, <^  and ^  were r e q u i r e d . T h e s e w e r e found e x p e r i m e n t a l l y s E and.^ by simple tensions t e s t s and ^ » by weighing a known volume c The v a l u e s were: E = 560,000 p s i v> = 0,39 (O = 1*01 x IO" 4 l b s e c 2 ^ ~ " i n * MODE C A N T I L E V E R . B E A M S Q U A R E P L A T E " " ' " C I K C U L A R : PLATE C A L C U L A T E D O B S E R V E D C A L C U L A T E D O B S E R V E D . C A L C U L A T E D O B S E R V E D 1 44 Hz 48 Hz 46 Hz 43 Hz 147 Hz 143 Hz 2 281 " 310 w 113 " 108 " 3 780 w 850 " 283 w 270 " 4 1530 " - 363 " 346 " 576 w 620 M 5 2520 • - 413 " 394 w 8 1680 M 1650 " T A B L E 1 N A T U R A L F R E Q U E N C I E S 63 II. T A B L E O F P H O T O G R A P H I C R E S U L T S The following pages include a l l the p i c t u r e s that were taken of the beam and two plate specimens. F o r the purpose of c o m p a r i s o n a t h e o r e t i c a l solution or sand pattern is given d i r e c t l y opposite each photograph of the nodal patterns. F o r the antinodes, however, a theo-r e t i c a l solution i s given only for the beam. At the end of the r e s u l t s an a r b i t r a r y shape is investigated to locate its nodes and antinodes for p a r t i c u l a r n a t u r a l f r e q u e n c i e s . The photographs are l i s t e d as follows: NODES SPECIMEN MODE FREQ. FIGURE. CANTILEVER: BEAM S t a t i c - 25 (a) 1 48 Hz 25 (b) 2 310 " 25 (c) 3 850 " 25 (d) SQUARE PLATE S t a t i c - 27 (a) 1 43 Hz 27 (b) 2 108 " 27 (c & d) 3 270 " 27 (e & f ) 4 346 » 27 (g & h) CIRCULAR PLATE S t a t i c - 30 (a) 1 143 Hz 30 (b & c) 4 620 " 30 (d & e) ARBITRARY SHAPE 420 " 35 (a) 4 6 4 ANTINODES SPECIMEN MODE FREQ. FIGURE CANTILEVER BEAM S t a t i c - 26 (a) 1 48 Hz 26 (b) 2 310 rr 26 (c) 3 850 fi 26 (d) SQUARE PLATE ( g r i d i n x d i r e c t i o n ) S t a t i c 1 43 Hz 28 28 (a) (c) 2 108 ti 28 (e) 3 270 ti 28 ( s ) 4 346 rt 28 ( i ) 5 394 n 28 00 SQUARE PLATE ( g r i d i n 7 d i r e c t i o n ) S t a t i c 1 43 Hz 28 28 (b) (d) 2 108 ri 28 ( f ) 3 270 11 28 (n) 4 346 rt 28 (3) 5 394 rt 28 (m) CIRCULAR PLATE S t a t i c - 31 (a) 4 620 Hz 31 (b) 8 1650 n 31 (c) ARBITRARY SHAPE ( g r i d i n n d i r e c t i o n ) 600 w 35 (b) ARBITRARY SHAPE (gr i d ! i n x d i r e c t i o n ) 600 35 (c) S u p e r p o s i t i o n of p a r t i a l j antinodes (.square p l a t e ) 29 65 DISPLACEMENTS SPECIMEN MODE FREQ. FIGURE CANTILEVER BEAM P o s i t i v e Displacement 1 48 Hz 32 (a) S t a t i c 32 (b) Negative Displacement, II II 32 (c) Actual. Arrangement^ 32 (d) Dynamic Displacement Curve 32 (e) SQUARE PLATE S t a t i c 33 (a) P o s i t i v e Displacement 4 346 Hz 33 (b) Negative Displacement. it Hi 33 (o) CIRCULAR PLATE S t a t i c : 34 (a) P o s i t i v e Displacement; 1 143 Hz 34 (b) Negative Displacement: II H 34 (c) \\\\\\\\ (a) STATIC PATTERN \ ,W\\ , \ \ (c) 2nd MODE (310 Hz) (b) l e t M O D S ( 48 Hz) \\\\\\\ (d) 3rd MODE (850 Hz) FIGURE 25 NODES IN A CANTILEVER BEAM (a) STATIC PATTERN /////// (c) 2nd MODE (310 Hs) /////// (b) 1 s t MODE ( 48 Hz) / / / / / / (d) 3rd MODE (850 Hz) FIGURE 2 6 ANTINODES IN A CANTILEVER BEAM 68 (a) STATIC PATTERN (c) 2nd MODE - SAND (108 Hz) (d) 2nd MODE (108 Hz) FIGURE 27 NODES IN A SQUARE PLATE (e) 3rd MODE - SAND (270 Hz) («) 4th MODE - SAND (346 Hz) 00 4th MODE (346 Hz) NODES FI&URE 27 IN A SQUARE PLATE FIGURE 28 PARTIAL ANTINODES IN A SQUARE PLATE 71 TARGET DIRECTION TARGES DIRECTION 3rd MODE (270 Hz) FIGURE 28 PARTIAL ANTINODES IN A SQUARE PLATE 72 TARGET DIRECTION TARGET DIRECTION FIGURE 2 8 PARTIAL ANTINODES IN A SQUARE PLATE 73 ////////////// / t /////(( f/// tj 3rd MODE,, PARTIAL ANTINODE IN Y DIRECTION 3rd MODE, PARTIAL ANTINODE IN X DIRECTION / / / / / / / / / / / / / / ANTINODE ( c ) SUPERPOSITION OP PARTIAL ANTINODES FIGURE 23 SUPERPOSITION OP TWO PARTIAL ANTINODES TO LOCATE THE TRUE ANTINODE (a) STATIC PATTERN FIGURE 30 NODES IN A CIRCULAR PLATE (d) 4th MODE - SAND (620 Hz) (e) 4th MODE (620 Hz) FIGURE 3 0 NODES IN A CIRCULAR PLATE TABGET DIRECTION (a) STATIC PATTERN (b) ( c ) 4th MODE 8th MODE (620 Hz) (1620 Hz) FIGURE 31 ANTINODES IN A CIRCULAR PLATE DISPLACEMENTS IN A CANTILEVER BEAM ^ = EXPERIMENTAL RESULTS THEORETICAL CURVE -I .2 .3 A .5 .6 .7 (E) DYNAMIC DISPLACEMENT CURVE FIGURE 32 DISPLACEMENTS IN A CANTILEVER BEAM 8 9 LO X/L (b) (c) POSITIVE DISPLACEMENT, 4th MODE NEGATIVE DISPLACEMENT, 4th MODE (346 Hz) (346 Hz) FIGURE 33 DISPLACEMENTS IN A SQUARE PLATE (a) STATIC PATTERN (b) (o) POSITIVE DISPLACEMENTj 1st MODE NEGATIVE.DISPLACEMENT, 1st MODE (143 H«) (143 Hz) FIGURE 34 DISPLACEMENTS IN A CIRCULAR PLATE (a) TYPICAL NODAL PATTERN (420 Hz) FIGURE 35 NODES AND ANTINODES IN A PLATE OF ARBITRARY SHAPE C H A P T E R VI DISCUSSION O F R E S U L T S I. N O D E S C o m p a r i s o n of the sand p a t t e r n or t h e o r e t i c a l solutions with the m o i r e patterns obtained i n the experiments c l e a r l y show that the technique does, i n fact, definitely locate the nodal positions. The pat-t e r n s a r e c l e a r and di s t i n c t and in spite of the vague c r o s s hatch that c o v e r s most of the specimens there i s no doubt as to which l i n e s a ctually t r a c e out the nodes. T h e i r d i s t i n c t n e s s , however, was found to be v e r y dependent upon the density and o r i e n t a t i o n of the g r i d and it was thus de-s i r a b l e to apply a rough f i g u r e of m e r i t to the system. A good i n d i c a -t i o n was shown to be given by the e m p i r i c a l r e l a t i o n : _ ^ a _ = k (6.1) P F o r sharp c l e a r nodes a value of k=50 was found to be applicable while a value k=100 y i e l d e d v i s i b l e but r a t h e r i n f e r i o r patterns. F o r instance if a 200 l p t g r i d was used and sharp patterns were sought the g r i d could be p l a c e d no f u r t h e r than 1/4" f r o m the plate. F r o m eqn. 6,1 it is seen that if l a r g e amplitudes are to be studied and yet m a i n t a i n good f r i n g e quality it i s n e c e s s a r y to employ a much c o a r s e r g r i d . Thus there i s no upper l i m i t p l a c e d on the technique other than the p h y s i c a l l i m i t a t i o n s on the size of the apparatus components. T h i s , of course, i s subject to the upper bound mentioned i n Chapter II i n which contrast between the node and r e m a i n i n g washed out pattern becomes important. 83 The m i n i m u m deflection that could be detected with a 500 l p i grid, angles of roughly i=10 degrees and o = 20 degrees was f r o m eqn. 2.1; h = - 0 0 2 tan 10 + tan 20 h = . 004"/fringe and f r o m eqn. 2.8, w m i n = i l 4 = .001" Thus the technique can detect nodes f o r any plate having an amplitude of v i b r a t i o n g r e a t e r than . 001". D i f f r a c t i o n effects begin to become s i g -nificant when grids of s m a l l e r p i t c h are us ed. It was shown i n the last chapter that if the i n i t i a l p a t t e r n on the plate is or i e n t e d at 45 degrees to the h o r i z o n t a l good plate coverage r e s u l t s . However, in many of the photographs the pattern is not at 45 degrees but r a t h e r at some other angle. In fact, i n some cases it is either completely h o r i z o n t a l or v e r t i c a l . In these cases it was found that even though a 45 degree pattern y i e l d e d a good nodal pattern it could be g r e a t l y i m p roved upon by rotating the g r i d . A s i n a l l techniques involving the use of fringes there a r i s e s the p r o b l e m of determining the exact centre of the f r i n g e l i n e s . In this case, however, not one f r i n g e but rather a band of a f a m i l y of f r i n g e s is involved and therefore methods of fr i n g e sharpening such as the one given by P o s t (27) cannot be applied here. Since the width of the node l i n e i s amplitude dependent, u s u a l l y sufficiently good r e s u l t s were ob-tained by i n c r e a s i n g the amplitude to a point just before contrast d i s -appeared. However, when sufficient amplitude c o u l d not be i m p a r t e d to the beam as in F i g . 25 (d) the patterns r e q u i r e d c o n s i d e r a b l e i n t e r -pretation. II. A N T I N O D E S The antinodes d e t e r m i n e d i n the case of the c a n t i l e v e r beam compare favourably with theory and it is assumed that the plates do l i k e w i s e . If i n the case of the square plate the nodal patterns a r e s u p e r i m p o s e d on top of the antinode patterns they are seen to f i l l i n the gaps as would be expected. F o r the c i r c u l a r plates this may only be done f o r the 4th mode. The i n i t i a l patterns on the plates are not n e c e s s a r i l y an i n -d i c a t i o n of the i n i t i a l c u r v a t u r e s since the c a m e r a was often not p l a c e d exactly at the f o c a l point of lens L^- None the l e s s c e r t a i n surface d i s c o n t i n u i t i e s do show up. F o r instance, both the c i r c u l a r and square plates show a s m a l l c i r c u l a r pattern on the surface. T h i s was due to shrinkage brought about by cementing generator attachments to the back surfac e s of the plates. The method shown for determining the true antinode by s u p e r p o s i t i o n of the two p a r t i a l antinodes as in F i g . 29 works v e r y w e l l but the question a r i s e s , why could the negatives not be super-85 i m p o s e d to achieve the same r e s u l t photographically ? Unfortunately this was not thought of un t i l after a l l the photographs had been taken• :i Since many of these were taken f r o m different c a m e r a positions they could not be superimposed,. T h i s , however, i s c o n s i d e r e d to be the most ide a l method for v i s u a l i z i n g antinodes and i s suggested to anyone p e r f o r m i n g the study i n the future. Another question also a r i s e s , if one p a r t i a l antinode i s determined with the target l i n e s i n one d i r e c t i o n and another p a r t i a l antinode with the target lines i n some other d i r e c -t i o n why c o u l d not both sets of li n e s be superimposed on the target and i n such a manner r e c o r d both p a r t i a l antinodes s i m u l t a n e o u s l y ? T h i s was attempted but was found to be subject to definite p h y s i c a l l i m i t a -tions in viewing the pattern. T h i s arose f r o m the fact that the c u r v a -tures of the two waves at the antinode were not n e c e s s a r i l y the same and hence pattern washout o c c u r r e d at a much lower amplitude in one d i r -e c t i o n than i t did i n the other. T h i s had the following effect: at a m p l i -tude 1, a p a r t i a l antinode i n say the y d i r e c t i o n appeared, the x pattern r e m a i n i n g unaffected. At amplitude 2, which was greater than a m p l i -tude 1, the x p a t t e r n washed out and produced a p a r t i a l antinode i n the x d i r e c t i o n . However, amplitude 2 was so l a r g e that the p a r t i a l a n t i -node i n the y d i r e c t i o n had been reduced to such a thin line that it was no longer v i s i b l e . Thus i n most cases only one p a r t i a l antinode was v i s i b l e at a t i m e . T h i s p r o b l e m did not, of course, occur if the c u r v a -tu r e s at the antinode were the same. The l i m i t s on the technique a r e v e r y s i m i l a r to those d i s -86 / c u s s e d f o r the shadow m o i r e . The upper l i m i t i s di f f i c u l t to determine since it is a matter of contrast between the p a r t i a l antinode and the washed out pattern. The lower l i m i t , however, may be dete r m i n e d f r o m eqn. 2.7. A s in the case of the nodes the n e c e s s a r y condition f o r washout i s that the frin g e shift be greater than 1/2 a f r i n g e width or i n other words the m i n i m u m slope change i s : ^min = $ 4 = 1 tan"-*- r 8 ? ! Taking t y p i c a l values as r = . 1", F|=60" it can be seen that $ m ± a = . 0002 r a d s . T h i s thenBeconeDes the lower l i m i t on the technique using the above apparatus. III. D I S P L A C E M E N T The displacements i n the ca n t i l e v e r beam are given i n F i g . 32 (a) - (c) along with m e a s u r e d angles of the sys t e m i n F i g . 32. (d), The peak to peak input displacement at the f r e e end of the beam i n this p a r t i c u l a r case was .025" as r e a d off the o s c i l l i s c o p e . By counting f r i n g e s to the nearest 1/4 fr i n g e it is seen that the static pattern has 13 f r i n g e s , the p o s i t i v e d e f l e c t i o n pattern 10 3/4 f r i n g e s and the negative de f l e c t i o n pattern 15 1/4 f r i n g e s . T h i s r e p r e s e n t s a peak to peak fringe change of 4 1/2 f r i n g e s and the total displacement i s thus given by: w = 4 , 5 h - J L = 4 . 5 2 4 2 tan 8 ° + tan 3 1 ° = i 0252'» T h i s value i s exceptionally c l o s e to the input amplitude and serves to demonstrate the v a l i d i t y of the technique as a dynamic displacement m e a s u r i n g technique. The fact that they;;ar;e so^clase is p a r t l y c o i n c i d e n t a l because of the v a r i o u s e r r o r s i n both the m e a s u r i n g and c a l i b r a t i o n methods. The major e r r o r s are as follows: E r r o r i n reading angles 1/2° i n 40° 1.2% E r r o r in reading f r i n g e s 1/4 i n 13 2.4% E r r o r i n c a l i b r a t i o n .001" i n . 025" 4. 0% T o t a l e r r o r 7.6% The e x p e r i m e n t a l displacements i n the square and c i r c u l a r plates have not been evaluated but are included for completeness. In the c i r c u l a r plate no i n i t i a l pattern i s p l a c e d on the surface so that d i s -placements: m a y be evaluated d i r e c t l y without the need of subtracting the i n i t i a l pattern. IV. S E C O N D O R D E R E F F E C T S In both the nodal and antinodal patterns if can be seen that a secondary background f r i n g e pattern of low intensity appears on most plates. Since the response of a generic p o i n t i n the p l a t e i s har-monic, i.e« x = A s i n and, - ~ — = Aw cos urfc dt it can be seen that as wt approaches w/2 the ve l o c i t y of the plate ap-88 proaches zero. Since the f r i n g e v e l o c i t y is d i r e c t l y r e l a t e d to the plate v e l o c i t y it is obs e r v e d that it must likewise apprach zero. Hence a mo t i o n l e s s f r i n g e condition, i . e . f r i n g e "dwell", r e s u l t s at both am-plitude peaks. Although this condition is momentary i t is sufficient to leave a faint i m p r e s s i o n on the eye or c a m e r a and thus the secondary pattern on the plate is the su p e r p o s i t i o n of the two ma x i m u m amplitude f r i n g e patterns on the plate. In some instances the secondary f r i n g e pat-terns a r e of high.enough density to actually produce a secondary m o i r e pattern, or a "moire' of a m o i r e " . T h i s can be c l e a r l y seen i n F i g . 27 (h) i n which co n c e n t r i c horseshoe shaped patterns are e n c l o s e d by the p r i m a r y nodal pattern. V. G E N E R A L O B S E R V A T I O N S One point which g e n e r a l l y plagued the study throughout its entirety was the method of introducing v i b r a t i o n into the plates. T h i s d i f f i c u l t y a r o s e f r o m the fact that, because of stiffening effects, no p h y s i c a l attachment could be p r o v i d e d between the plate and generator spindle. T h i s meant that to excite.the plate it was n e c e s s a r y to place the generator spindle i n c l o s e contact with the plate surface. Two major problems o c c u r r e d because of this: (1) if a l a r g e amplitude was introduced the spindle would tend to "buzz" on the plate and introduce second o r d e r h a r m o n i c s which washed out the entire pattern; (2) i i the plate was d r i v e n at a nodal point it was i m p o s s i b l e to excite that p a r t -i c u l a r mode. The f i r s t of these p r o b l e m s was elim i n a t e d by-keeping 89 the amplitude low enough so that buzzing did not occur. T h i s presented a l i m i t a t i o n on the technique because i n many cas e s , f o r instance F i g . 25 (d), the amplitude could not be i n c r e a s e d s u f f i c i e n t l y to y i e l d sharp definable nodes. The second p r o b l e m was o v e rcome by shifting the spindle to different points on the plate and thus ensuring that it was not being d r i v e n at a node. F o r the c i r c u l a r plate a different situation a r o s e . Since the spindle was p h y s i c a l l y attached to the plate no buzzing r e s u l t e d and the plate could be d r i v e n to any d e s i r e d amplitude. However, be-cause it was being d r i v e n at the centre no modes c o n t a i n i n g z ' a d i a l n o d e s c o u l d be excited,, C H A P T E R VII S U M M A R Y A N D C O N C L U S I O N S I. S U M M A R Y It was the intention of this investigation to extend the use of o p t i c a l techniques to the study of v i b r a t i n g plates subject to r e l a t i v e -l y l a r g e displacements. T h r e e v i b r a t i o n a l c h a r a c t e r i s t i c s were sought: (1) the l o c a t i o n of the nodes; (2) the l o c a t i o n of the antinodes and; (3) the displacement at any point in the plate. Nodes and antinodes are d e s c r i b e d in t e r m s of deflection and slope r e s p e c t i v e l y . With this in mind s e v e r a l e x i s t i n g o p t i c a l systems capable of m e a s u r i n g deflections and slopes were i n v e s t i -gated. With the exception of one study by N i c k o l a (11) these had been applied only to static studies and it was hoped that they would be ex-tended to include the dynamic case. These systems were a l l set up in the l a b o r a t o r y and evaluated and i t was found that the shadow m o i r e method for deflection measurement and the Salet-Ikeda method for slope measurement yielded the best r e s u l t s . These were hence chosen for the study. T h r e e plate specimens were investigated; a ca n t i l e v e r beam, a square ca n t i l e v e r plate and a free c i r c u l a r plate. A l l were made of thin p l e x i g l a s s . F o r the analysis of nodes the shadow m o i r e s y s t e m us i n g a 91 high intensity light source was set up. The specimens were s p r a y e d with f l a t white paint to give a matte f i n i s h . Upon adjusting the g r i d and exciting the plate it was found that highly definable nodal patterns were r e v e a l e d . These patterns were, r e c o r d e d photographically. T o locate the antinodes the Salet-Ikeda s y s t e m was set up. T h i s time the specimens were spr a y e d on the back surface so as to give a r e - , f l e c t i v e f ront surface. B y exciting the plate and using the c a m e r a as the pinhole p a r t i a l antinodes i n a p a r t i c u l a r d i r e c t i o n were r e c o r d e d . B y superimposing the p a r t i a l antinodes the true antinodes were r e v e a l e d d i r e c t l y . D isplacements were found by using the shadow m o i r e ' d e f l e c -tion method with a stroboscope as the light source. In this manner f r i n g e motion could be "stopped" at any point i n the v i b r a t i o n c y c l e and the displacement r e c o r d e d photographically. As a means of ev-aluating the opti c a l r e s u l t s t h e o r e t i c a l solutions and sand patterns were of-f e r e d for locating the nodes. Antinode locations were compared with theory. II. C O N C L U S I O N S F r o m the inve s t i g a t i o n the following conclusions were drawn: (1) Nodes may be located i n any v i b r a t i n g fla t or n e a r l y flat plate having an amplitude of v i b r a t i o n greater than . 001" . (2) Antinodes may be lo c a t e d i n any vi b r a t i n g flat 92 or n e a r l y f l a t plate having a m i n i m u m change i n dynamic slope of .0002 rads. (typical) (3) Dynamic displacements may be determined i n any v i b -rating flat or n e a r l y flat plate having an amplitude of v i b r a t i o n greater than . 001" . III. S U G G E S T I O N S F O R F U T U R E R E S E A R C H V e r y l i t t l e needs to be s a i d r e g a r d i n g improvement of the ac-tual equipment or arrangements u s e d i n the experiments. However, as it was pointed out i n the last chapter a better means of inducing v i b r a t i o n i s needed. T h e r e f o r e it is recommended that an a c o u s t i c a l generator be used i n the future. The inv e s t i g a t i o n undertaken i n this thesis was l i m i t e d to the study of flat plates under steady state conditions. T h e r e f o r e two pos-sible f i e l d s of f u r t h e r investigation a r e suggested: (1) the ap p l i c a t i o n of the technique to s p h e r i c a l and a r b i t r a r y surfaces and; (2) the study of t r a n s i e n t conditions. It i s anticipated that both these p r o b l e m s may be solved by slight m o d i f i c a t i o n s to the existing equipment. In o r d e r to " c a p t u r e " the transient waves it is suggested that high speed photo-graphy be used. Since the shortest exposure time in this study, using A S A 3000 f i l m , was 2 seconds it i s apparent that a m o r e intense light s ource w i l l have to be found. The second o r d e r f r i n g e s o b s e r v e d i n most of the photo-93 graphs are an in t e r e s t i n g phenomenon. Since it is suspected that they a r e the moire'patterns of the two m aximum amplitude f r i n g e patterns it is wondered if the same effect c o u l d not be achieved with just one g r i d attached to the vi b r a t i n g specimen. T h i s is worthy of i n v e s t i -gation. The opt i c a l systems f o r the shadow m o i r e method and the Salet-Ikeda method are almost i d e n t i c a l . It i s t h e r e f o r e suggested that they be i n c o r p o r a t e d together i n a v i b r a t i o n a n a l y s i s instrument. 94 B I B L I O G R A P H Y 1. Chla d n i , E . F . , "Entdeckung i n d. T h e o r i e d. Klanges," 1787, only a v a i l a b l e i n f o r m a t i o n . 2. W a l l e r , M. D. , C h l a d n i F i g u r e s , A Study i n Symmetry, G. B e l l and Sons L t d . , London, 1961. 3. Savart, M. , "Sur l a Communication des Mouvemens V i b r a t o r i e s entre les C o r p s Solides ;, " Annales de C h i m i e , v o l . 14, 1820, p.p. 113 - 167. 4. F a r a d a y , M. , "On a P e c u l i a r C l a s s of A c o u s t i c a l F i g u r e s ' , " P h i l i s o p h i c a l T r a n s a c t i o n s , 1831, p.p. 299 - 340. 5. Wood, A. B. , A Textbook of Sound, G. B e l l and Sons L t d . , London, 1946. 6. Dye, D. W. , N. P. L. Reports, 1928. 7. O s t e r b e r g , H. , "An Interferometer Method of Studying the V i b r a t i o n s of an O s c i l l a t i n g Q u a r t z P l a t e J o u r n a l of the O p t i c a l Society  of A m e r i c a , v o l . 22, no. 19( 1932, p.p. 19 - 35. 8. Thornton, B. S. and.Kelly, C. J . , " M u l t i p l e - B e a m I n t e r f e r o m e t r i c Method of Studying. S m a l l V i b r a t i o n s , M J o u r n a l of the. O p t i c a l  Society of A m e r i c a , v o l . 46, no. 3, 1956, p.p. 191 - 194. 9. P o w e l l , R. L . and Stetson, K. A. , " I n t e r f e r o m e t r i c V i b r a t i o n A n a l y s i s by Wavefront R e c o n s t r u c t i o n J o u r n a l of the O p t i c a l  Society of A m e r i c a , v o l . 55, no. 12, 1965 (p.p. 1593 - 1598. 10. W a s i l , B. A., Merchant, D. C. and. D e l Vecchio, J . J ; , "Photo-g r a m m e t r i c M e asurements of Dynamic Displacements „" E x p e r i m e n t a l M e c h a n i c s , v o l . 5, no. 10, 1965, p. p. 332 - 339. 11. N i c k o l a , W. E . , "The Dynamic Response of T h i n Membranes by the Moire* Methodj " paper presented at the s p r i n g meeting of the SESA, May, 1966. 12. T h e o c a r i s , P. S. , " M o i r e F r i n g e s : A P o w e r f u l M e a s u r i n g Device A p p l i e d M e chanics Review,, v o l . 15, no. 5, 1962, p.p. 333 - 339. 13. Duncan, J . P., "The O p t i c a l Survey of C u r v e d Surfaces depart-mental paper, Dept. of M e c h a n i c a l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, 1966. 9 5 14. W e l l e r , R. and Shepard, B. M. , "Displacement Measurement by-M e c h a n i c a l Interferometry - j P r o c e e d i n g s of the SESA, v o l . 6, no. 1, 1948, p.p. 35 - 38. 15. T h e o c a r i s , P. S. , "Moire"'Method in P l a t e s , " P r o c e e d i n g s of the C o l l o q u i a m of the International A s s o c i a t i o n of S h e l l Structures, Warsaw, 1963. 16. T h e o c a r i s , P. S. , "Isopachic P a t t e r n s by the M o i r e Method f E x p e r i m e n t a l Mechanics, v o l . 4, no. 6, 1964, p.p. 153 - 159. 17. Ebbeni, J . , " O b s e r v a t i o n en Incidence Oblique des Phenomenes de M o i r e par R e f l e x i o n sur une Plaque Gauchie y n the B u l l e t i n of the R o y a l B e l g i u m Academy (Science D i v i s i o n ) , v o l . 50, 1964, p.p. 114 - 124. 18. Middleton, E . , "A R e f l e c t i o n Technique for the Survey of the De-f l e c t i o n of F l a t P l a t e s paper presented at the M e c h a n i c a l Engineering. Dept. , U n i v e r s i t y of Sheffield, Jan. 1966. 19. L i g t e n b e r g , F . K. , "The M o i r e Method - A New E x p e r i m e n t a l Method f o r the D e t e r m i n a t i o n of Moments in. S m a l l Slab Models ;» P r o c e e d i n g s of the SESA, v o l . 12, p.p. 83 - 89, 1954. 20. T h e o c a r i s , P. S. , " M o i r e P a t t e r n s of Slope Contours in F l e x e d P l a t e s E x p e r i m e n t a l Mechanics, v o l . 6, no. 4, 1966, p.p. 212 - 217. 21. T h e o c a r i s , P. S. and K o u t s a b e s s i s , A., "Slope Measurement by means of M o i r e F r i n g e s TV J o u r n a l of S c i e n t i f i c Instruments, vo l . 42, 1965, p.p. 607 - 6lC~. 22. Sabin, P. G. , " E x p e r i m e n t a l Techniques for the Study of E l a s t i c F l e x u r e M. Eng. thes i s , U n i v e r s i t y of Sheffield, 1964. 23. Duncan, J . P. and Brown, C. J.. E . , "Slope Contours i n F l e x e d E l a s t i c P l a t e s by Salet-Ikeda Technique P r o c e e d i n g s of the F i r s t International C o n g r e s s on. E x p e r i m e n t a l Mechanics, P e r g a m o n P r e s s , New York, 1963. 24. Young, D. , " V i b r a t i o n of Rectangular P l a t e s by the R i t z Method ?," J o u r n a l of A p p l i e d Mechanics, v o l . 17, 1950, p.p. 448 - 453. 25. Barton, M. V., " V i b r a t i o n of Rectangular and Skew C a n t i l e v e r P l a t e s J o u r n a l of A p p l i e d Mechanics, v o l . 18, 1951, p.p. 129 -134. 96 H a r r i s , C. M. and Crede, C. E . , Shock and V i b r a t i o n Handbook, v o l . 1, M c G r a w - H i l l Book Co. , 1961. Post, D. , "Sharpening and M u l t i p l i c a t i o n of M o i r e F r i n g e s »»paper pre s e n t e d at the M e c h a n i c a l E n g i n e e r i n g Dept. , U n i v e r s i t y of Sheffield, Jan. 1966. Young, D. and F e l g a r , R. P. , " T a b l e s of C h a r a c t e r i s t i c Functions Representing N o r m a l Modes of V i b r a t i o n of A Beam," B u l -l e t i n of the U n i v e r s i t y of Texas, 1949. A P P E N D I X A B E A M AND P L A T E E Q U A T I O N S C a n t i l e v e r B e a m A v e r y complete set of tables of the c h a r a c t e r i s t i c functions of a v i b r a t i n g beam is given by D. Young and R. P. F e l g a r (28). These values were c a l c u l a t e d using the R i t z method and v a r y f r o m the exact method by no more than 1. 5% which is quite adequate for this study. The following values were hence taken f r o m these tables. The n a t u r a l frequency of the n mode i s given by: fn = Pn2 / EI 2n v ja where 0 nL is tabulated below i n Table 2. a 0nL 1 1 . 8 7 5 2 4 . 6 9 4 3 7o855 4 1 0 . 9 9 6 5 1 4 . 1 3 7 TABLE 2 F o r any p a r t i c u l a r mode, n the deflection is given by: w n = A $ n Where A i s the amplitude coefficient and $n the c h a r a c t e r i s t i c func-tion that must satisfy the r e q u i r e d boundary conditions. S i m i l a r l y for slope: O b v i o u s l y the nodes and antinodes may be r e a d i l y d etermined by eq-uating w and w 1 to zero r e s p e c t i v e l y . Thus for a node: *n - 0 and f o r an antinode, In the r e f e r e n c e tables, values f o r $ n and $^ a r e tabulated i n i n -crements of 1/50 of the beam length and hence to find the zero values a c e r t a i n amount of in t e r p o l a t i o n is involved. Table 3 shows the r e l a -tive positions of the nodes and antinodes as a rat i o of X / L • These may be compared to values given i n the Shock and V i b r a t i o n Handbook (26) on page 7 - 1 4 . 99 S q u a r e C a n t i l e v e r P l a t e F r o m . Y o u n g ' s p a p e r ( 2 4 ) t h e n a t u r a l f r e q u e n c y i s s e e n t o b e g i v e n b y : ^ = Q T T Q 2 / w h e r e g i v e n i n T a b l e 4 . C i r c u l a r F r e e P l a t e T h e n a t u r a l f r e q u e n c y o f a c i r c u l a r f r e e p l a t e i s g i v e n o n p a g e 1 - 1 5 o f t h e S h o c k a n d V i b r a t i o n H a n d b o o k ( 2 6 ) a n d i s : w h e r e i s g i v e n i n T a b l e 5 . 100 n Pa n 1 3 . 4 9 4 1 6 . 0 9 2 8 . 5 4 7 2 1 0 . 5 3 3 2 1 . 4 4 3 1 4 . 1 9 4 2 7 . 4 6 4 2 3 . 8 0 5 3 1 . 1 7 5 4 0 . 8 8 6 4 4 . 6 8 T A B L E 4 7 § 1 . - 3 8 8 6 9 . 4 4 T A B L E 5 

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