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Vibration analysis by holographic interferometry Liem, Sing D. 1970

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VIBRATION ANALYSIS BY HOLOGRAPHIC INTERFEROMETRY by SING D. LIEM B . A . S c , 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 , 1968 A THESIS SUBMITTED IN PARTIAI FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t h e D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF A p r i l BRITISH COLUMBIA 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f t h e 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 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 , 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 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 t h e Head o f my Depar tment 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 not 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 . Depar tment o f Mpr -hani r - a l KngAnsn?i-ina u 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 a t e A p r ^ i l •^nJ iQ-jn ABSTRACT H o l o g r a p h i c i n t e r f e r o m e t r y o f f e r s t remendous p o t e n -t i a l i n t h e a n a l y s i s o f s m a l l a m p l i t u d e v i b r a t i o n s o f t h r e e d i m e n s i o n a l o b j e c t s . Due t o s t a b i l i t y r e q u i r e m e n t s and t h e n e c e s s i t y o f a c o h e r e n t l i g h t s o u r c e , t h e a n a l y s i s i s s t i l l l i m i t e d t o t h e l a b o r a t o r y I n t h i s e x p e r i m e n t t h e v a r i o u s h o l o g r a p h y s y s t e m s f o r v i b r a t i o n a n a l y s i s d e v e l o p e d t o d a t e were i n v e s t i g a t e d . S e c o n d l y , a r e a l t i m e h o l o g r a p h i c i n t e r f e r o m e t r y t e c h n i q u e was d e v e l o p e d f o r q u a n t i t a t i v e l y mapping t h e c o m p l e t e dynamic f i e l d o f a s u r f a c e , where t h e a m p l i t u d e s o f v i b r a t i o a r e on t h e o r d e r o f t h e w a v e l e n g t h o f l i g h t u s e d . T h i s t e c h n i q u e o f f e r s t h e f o l l o w i n g f e a t u r e s : (a) The p h a s e o f t h e o b j e c t c a n be d e t e r m i n e d . (b) The e x a c t r e s o n a n t f r e q u e n c y c a n be p i n p o i n t e d by o b s e r v i n g t h e ph a s e change o f t h e v i b r a t i n g o b j e c t (c) E n t i r e d i s p l a c e m e n t f i e l d s c a n be d e t e r m i n e d , e v e n t h o u g h t h e maximum a m p l i t u d e s o f v i b r a t i o n s a r e on t h e o r d e r o f o n l y a few m i c r o i n c h e s . (d) U nder c o n t i n u o u s i l l u m i n a t i o n t h e mode shape o f t h e o b j e c t c a n be r e a d i l y o b s e r v e d . i i i TABLE OF CONTENTS C h a p t e r Page 1 INTRODUCTION 1 1.1 P r e l i m i n a r y Remarks . . . . . 1 1.2 S t a t e m e n t o f P r o b l e m . 3 1.3 L i t e r a t u r e S u r v e y 4 2 THEORY 9 2.1 P r i n c i p l e o f H o l o g r a p h y 9 2.2 Time A v e r a g e d H o l o g r a m o f a V i b r a t i n g 2.3 R e a l Time Method U s i n g C o n t i n u o u s I l l u m i n a t i o n . . . . . 17 2.4 R e a l Time H o l o g r a p h y o f a V i b r a t i n g S u r f a c e w i t h o u t an I n i t i a l F r i n g e P a t t e r n U s i n g S t r o b o s c o p i c I l l u m i n a t i o n . . . . . . . . 19 2.5 R e a l Time H o l o g r a p h y o f a V i b r a t i n g S u r f a c e W i t h an I n i t i a l F r i n g e P a t t e r n U s i n g S t r o b o s c o p i c I l l u m i n a t i o n 23 3 EXPERIMENTAL APPARATUS 31 3.2 The O p t i c a l Bench W i t h O p t i c a l Components 31 3.3 The S t r o b i n g Mechanism and O b j e c t E x c i t a t i o n S ystem 37 3.4 The O b j e c t 41 4 EXPERIMENTAL PROCEDURE 42 C h a p t e r i v Page 5 EXPERIMENTAL RESULTS AND DISCUSSION OF RESULTS . . . . . 45 5.1 Geometry o f t h e H o l o g r a p h y Components 45 5.2 P o w e l l and S t e t s o n Time A v e r a g e d Method 46 5.3 R e a l Time Method U s i n g C o n t i n u o u s I l l u m i n a t i o n 49 5.4 R e a l Time Method W i t h o u t an I n i t i a l F r i n g e P a t t e r n U s i n g S t r o b o s c o p i c I l l u m i n a t i o n 51 5.5 R e a l Time Method W i t h an I n i t i a l F r i n g e P a t t e r n U s i n g S t r o b o s c o p i c I l l u m i n a t i o n 54 6 SUMMARY AND CONCLUSIONS ' . 59 6.1 Summary 5 9 6.2 C o n c l u s i o n s . . . . . 61 6.3 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 . . . . 62 BIBLIOGRAPHY 64 APPENDIX 1 . 67 V LIST OF FIGURES F i g u r e Page 2.1 Recording of the Hologram 10 2.2 R e c o n s t r u c t i o n of the Hologram 13 2.3 P l o t of [1 + cos (<j)2-tf>1)] Versus (4>2-<S>1) 20 2.4 Object Geometry as "Captured" by the Strobed L i g h t Source . . . . . . 21 2.5 Hologram and Object Reference System . . . . 24 2.6 F r i n g e P a t t e r n , w i t h Hologram Rotated by 6 1 . . . 25 2.7 F r i n g e P a t t e r n , With Hologram Rotated by 6^  and &2 26 2.8 Dynamic Object Geometry and F r i n g e P a t t e r n 27 2.9 P l o t to F i n d the Object Mode Shape 29 3.1 Schematic of the Holography System 32 3.2 Complete Holography System . 34 3.3 The Holography Table 34 3.4 Schematic of the S p a t i a l F i l t e r U n i t . . . . 35 3.5 Schematic of the Jodon P l a t e h o l d e r 37 3.6 Schematic of the S t r o b i n g Mechanism and Object E x c i t a t i o n System . 39 3.7 Schematic of the Phase Adjustment . . . . . . 39 3.8 S t r o b i n g Mechanism and Object E x c i t a t i o n System . . . . . . . . . . . 40 3.9 Membrane Ten s i o n i n g Device 40 5.1 Schematic of the Holography System 45 VI F i g u r e Page 5.2 Time Averaged F r i n g e P a t t e r n 47 5.3 F r i n g e P a t t e r n and Mode Shape a t 455 Hz . . . . 55 5.4 F r i n g e P a t t e r n and Mode Shape a t 475 Hz . . . . 57 A . l Time Averaged F r i n g e P a t t e r n s 68 A. 2 Mode Shape P a t t e r n s 69 A.3 Continuous I l l u m i n a t i o n F r i n g e P a t t e r n s . . . . 70 A.4 S t r o b o s c o p i c Real Time Without an I n i t i a l F r i n g e A.5 S t r o b o s c o p i c Real Time Without an I n i t i a l F r i n g e v i i ACKNOWLEDGEMENT The a u t h o r w i s h e s t o e x p r e s s h i s s i n c e r e t h a n k s t o Dr. C R . H a z e l l f o r t h e g u i d a n c e g i v e n t h r o u g h o u t t h e i n v e s -t i g a t i o n and p r e p a r a t i o n o f t h e t h e s i s . H i s h e l p and en c o u r a g e m e n t have been i n v a l u a b l e . The a u t h o r w o u l d a l s o l i k e t o t h a n k Mr. P h i l H u r r e n and Mr. J o h n Hoar 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 t u d y was made p o s s i b l e t h r o u g h r e s e a r c h g r a n t No. 67-3331, p r o v i d e d by t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada. NOMENCLATURE t o t a l amplitude of the l i g h t waves a t the pho t o p l a t e amplitude of the r e f e r e n c e wave amplitude of the o b j e c t wave diameter of the membrane v e c t o r i a l a d d i t i o n o f the o b j e c t i l l u m i n a t i o n and o b j e c t viewing v e c t o r amplitude of v i b r a t i o n of the o b j e c t o b j e c t v i b r a t i o n amplitude v e c t o r o b j e c t i l l u m i n a t i o n v e c t o r o b j e c t viewing v e c t o r amplitude of v i b r a t i o n of the o b j e c t as recorded by the F o t o n i c Sensor s t a t i c displacement v e c t o r o f the r e c o n s t r u c t e d image i n t e g e r number number of l i g h t f r i n g e s number of dark f r i n g e s r a d i u s of the membrane hologram r e f e r e n c e system o b j e c t r e f e r e n c e system wave t r a n s m i t t e d by the hologram displacement a s s o c i a t e d w i t h a f r i n g e order X (cos 8^ + cos 8 2 ) amplitude of the r e c o n s t r u c t e d wave i n t e n s i t y (brightness) of the r e c o n s t r u c t e d wave i n t e n s i t y of the l i g h t wave r e c o n s t r u c t e d by the hologram X X 1^ recorded waveform a t the photo p l a t e I. i n t e n s i t y d i s t r i b u t i o n o f the r e c o n s t r u c t e d image image J ' J zero order B e s s e l f u n c t i o n of the f i r s t k i n d o L expanding l e n s M f i r s t s u r f a c e m i r r o r 0 o b j e c t Ph photographic p l a t e S.F. s p a t i a l f i l t e r assembly V.B.S. v a r i a b l e beam s p l i t t e r X,Z holography t a b l e r e f e r e n c e system ax phase of the r e f e r e n c e wave a t p o i n t x a t the pho t o p l a t e 3 d e v i a t i o n angle of the prism 6^ r o t a t i o n of the hologram p a r a l l e l t o the x a x i s $2 r o t a t i o n of the hologram p a r a l l e l to the y a x i s 8^  angle between the o b j e c t i l l u m i n a t i o n and the d i r e c t i o n of v i b r a t i o n 02 angle between the d i r e c t i o n of viewing and the d i r e c t i o n o f v i b r a t i o n cj^-cj), phase d i f f e r e n c e of the o b j e c t wave a t the two p o s i t i o n s o f the o b j e c t cj> (x) phase of the o b j e c t wave A path l e n g t h d i f f e r e n c e of the o b j e c t wave a t the two p o s i t i o n s of the o b j e c t '1. INTRODUCTION 1.1 P r e l i m i n a r y Remarks Over the years v a r i o u s methods of v i b r a t i o n a n a l y s i s have been developed. One of the o l d e s t , y e t r a t h e r l i m i t e d method,has been t h a t o f i n t e r f e r o m e t r y . O p t i c a l i n t e r f e r -ometric systems show the v i b r a t i o n c h a r a c t e r i s t i c of the whole s u r f a c e and the p a r t i c u l a r mode shape. U n t i l r e c e n t l y these i n t e r f e r o m e t r i c systems were l i m i t e d to plane s u r f a c e s and the s u r f a c e of the o b j e c t had to be made o p t i c a l l y f l a t to make them s u i t a b l e f o r i n t e r f e r o m e t r i c study. With the r e c e n t developments of holography and i t s a p p l i c a t i o n to v i b r a t i o n study these l i m i t a t i o n s no longer e x i s t . * Osterberg [1] was the f i r s t one to develop an o p t i c a l i n t e r f e r o m e t r i c system to determine the modes and amplitudes of a v i b r a t i n g o b j e c t . V a r i o u s o p t i c a l systems to study v i b r a t i o n modes o f an o b j e c t have s i n c e been developed, but i t was not u n t i l the work of Powell and S t e t s o n [2] t h a t a r e v o l u t i o n a r y method of v i b r a t i o n a n a l y s i s became a v a i l a b l e . The p r i n c i p l e s of holography were f i r s t d i s c o v e r e d by Gabor [3] i n 19 48. Due to p r a c t i c a l l i m i t a t i o n s no f u r t h e r work was c a r r i e d out u n t i l the i n v e n t i o n of the l a s e r i n 1958 which p r o v i d e d the h i g h power coherent l i g h t source r e q u i r e d by holography. * The numbers i n square b r a c k e t s r e f e r to the r e f e r -ences l i s t e d i n the B i b l i o g r a p h y . 2 L e i t h and Upatnieks [4] r e a l i z e d the p r a c t i c a l i m p l i c a t i o n s of holography and t h e i r r e s e a r c h work made holography a p r a c -t i c a l t o o l . I t was upon t h e i r s u g g e s t i o n t h a t Powell and Stet s o n a p p l i e d holography t o v i b r a t i o n a n a l y s i s . In t h e i r o r i g i n a l work Powell and S t e t s o n made time averaged holograms (to be e x p l a i n e d i n d e t a i l i n Chapter 2 ) o f a f i l m can bottom v i b r a t i n g a t v a r i o u s resonant f r e q u e n c i e s . The holograms upon r e c o n s t r u c t i o n showed the mode shapes of the o b j e c t and from the f r i n g e p a t t e r n the amplitude o f v i b r a t i o n c o u l d be determined. L a t e r more r e f i n e d methods were developed which o f f e r e d the p o s s i b i l i t y o f r e a l time a n a l y s i s . The purpose of t h i s i n v e s t i g a t i o n i s t o : (a) s e t up the v a r i o u s holography systems developed t o date; (b) make q u a n t i t a t i v e measurements of v i b r a t i o n amplitude f o r each system and compare the r e l a t i v e m e r i t s o f each system; (c) develop a r e a l time h o l o g r a p h i c i n t e r f e r o m e t r y technique f o r q u a n t i t a t i v e l y mapping the complex dynamic f i e l d of a v i b r a t i n g s u r f a c e where the amplitudes of v i b r a t i o n are on the order of the wavelength of l i g h t — a v ery important r e g i o n i n hig h frequency o s c i l l a t i o n s of e n g i n e e r i n g compon-ent s . The technique i s based on a method which was used s u c c e s s f u l l y i n the v i b r a t i o n a n a l y s i s of p l a t e s u s i n g the shadow moire method [ 5 ] . 3 1.2 Statement of Problem The o b j e c t t o be s t u d i e d i s a c i r c u l a r membrane three inches i n diameter made of .0005 i n c h t h i c k a l u m i n i z e d Mylar which i s e x c i t e d by a horn. The holography systems t o be s t u d i e d a r e : (a) Time averaged holography as developed by Powell and S t e t s o n . (b) Continuous i l l u m i n a t i o n r e a l time study by means of superimposing the r e c o n s t r u c t e d image of the s t a t i c o b j e c t on the o r i g i n a l o b j e c t and o b s e r v i n g the r e s u l t i n g f r i n g e s under continuous i l l u m i n -a t i o n . (c) Real time study by means of superimposing the r e c o n s t r u c t e d image of the s t a t i c o b j e c t e x a c t l y on the o r i g i n a l object.and o b s e r v i n g the r e s u l t i n g f r i n g e s w i t h a strobed l a s e r beam as the o r i g i n a l o b j e c t i s e x c i t e d . (d) Real time study by means o f superimposing the r e -c o n s t r u c t e d image of the s t a t i c o b j e c t on the o r i g i n a l o b j e c t i n such a way t h a t an i n i t i a l f r i n g e p a t t e r n i s c r e a t e d c o n s i s t i n g o f p a r a l l e l f r i n g e s r e p r e s e n t i n g a wedge shaped gap between image and o b j e c t . When the o b j e c t i s e x c i t e d and viewed w i t h a strobed l a s e r beam, the s t a t i c f r i n g e s w i l l deform 4 i n accordance w i t h the amplitude of v i b r a t i o n o f the o b j e c t . T h i s method i s p a r t i c u l a r l y u s e f u l f o r v e r y low amplitude o s c i l l a t i o n s , i . e . f o r amplitudes being a f r a c t i o n of the r e q u i r e d ampli-tude f o r the formation of an i n t e r f e r e n c e f r i n g e . . - . The f i r s t three systems are not o r i g i n a l and have been i n v e s t i g a t e d by s e v e r a l people. However, t o the bes t of the author's knowledge the f o u r t h system has never been 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 u s i n g h o l o g r a p h i c i n t e r f e r o m e t r y . 1. 3 L i t e r a t u r e Survey The a p p l i c a t i o n of holography to v i b r a t i o n a n a l y s i s was i n i t i a t e d by Powell and Stetson i n 1965 [ 2 ] . They made a time averaged hologram of a f i l m can bottom v i b r a t i n g a t i t s r esonant frequency and developed a formula f o r the f r i n g e f o r m a t i o n . T h i s method i s not a r e a l time method and the resonant frequency of the o b j e c t must be determined p r i o r t o making the hologram. To overcome t h i s shortcoming S t e t s o n and Powell [6] d i s c u s s e d a r e a l time method whereby a hologram o f the s t a t i o n a r y o b j e c t was f i r s t made and a c c u r a t e l y r e p o s i t i o n e d a f t e r d e v e l o p i n g . The hologram was a d j u s t e d t o g i v e a p a r a l l e l f r i n g e p a t t e r n and when the o b j e c t was e x c i t e d a t a resonant frequency and observed under continuous i l l u m i n -a t i o n , the l o c a l v i s i b i l i t y of these f r i n g e s v a r i e d as the 5 zero o r d e r B e s s e l f u n c t i o n of the amplitude of the v i b r a t i o n . A year l a t e r , S t etson and Powell [7] gave a d e t a i l e d t h e o r e t i c a l a n a l y s i s o f the f r i n g e f ormation i n h o l o g r a p h i c i n t e r f e r o m e t r y and developed the equations f o r q u a n t i t a t i v e l y determining the amplitude of v i b r a t i o n from r e a l time or time averaged f r i n g e p a t t e r n s . A f t e r these p u b l i c a t i o n s by Powell and S t e t s o n s e v e r a l people s t a r t e d working s i m u l t a n -eo u s l y and independently of one another i n t h i s f i e l d . In 1967 Monahan [ 8 ] , i n accordance w i t h the t h e o r -e t i c a l o u t l i n e g i v e n by Powell and S t e t s o n , developed the same formula f o r displacement o f the o b j e c t i n terms of the geometry of the system. To v e r i f y h i s c a l c u l a t i o n s a m o d i f i e d Twyman-Green I n t e r f e r o m e t e r was used t o determine the amplitude of v i b r a t i o n of the same o b j e c t e x c i t e d under the same c o n d i t i o n s . B r adford [9] d i s c u s s e d the p r a c t i c a l requirements f o r a holography system. He a p p l i e d the time averaged method t o study the o s c i l l a t i o n of a quartz bar and used a double exposed hologram to measure creep i n m a t e r i a l s . A r c h b o l d and Ennos [10] used both the time averaged and r e a l time technique of h o l o g r a p h i c i n t e r f e r o m e t r y . In t h e i r r e a l time technique they used a strobed l i g h t source i n s t e a d of continuous i l l u m i n a t i o n . The s t r o b i n g mechanism c o n s i s t e d of a d r i l l e d s h a f t d r i v e n by a t u r b i n e — a i r compressor arrangement. The e x c i t a t i o n s i g n a l was generated by a combination of the l a s e r l i g h t , a p o l a r i z e d p l a t e glued 6 to the end of the s h a f t , p o l a r i z i n g f i l t e r s and a photo t r a n s i s t o r . Phase adjustment was ob t a i n e d by r o t a t i n g the p o l a r o i d f i l t e r . The o b j e c t was an aluminum d i s c , 7 inches i n diameter d r i v e n by a p i e z o - e l e c t r i c t r a n s d u c e r b o l t e d a t the c e n t r e to the d i s c . The mode shapes and amplitudes of v i b r a t i o n c o u l d be determined i n r e a l time with t h i s technique. Watrasiewicz and S p i c e r [11] a l s o i n v e s t i g a t e d the r e a l time technique. They used a synchronous motor which drove a s l o t t e d d i s c t o s t r o b e the l a s e r beam. A s i g n a l generator was used t o e x c i t e both the o b j e c t and d r i v e the motor. Shajenko and Johnson [12] d i d an e x t e n s i v e study of s t r o b o s c o p i c h o l o g r a p h i c i n t e r f e r o m e t r y . They used an e l e c t r o - o p t i c a l arrangement c o n s i s t i n g of a p u l s e g e n e r a t o r , an a m p l i f i e r and a Pockels c e l l to strobe the l a s e r beam. The p u l s e was a m p l i f i e d t o d r i v e the Pockels c e l l which had a dc h a l f wave v o l t a g e of 1300 v o l t s . Waddell and Kennedy [13] used the time averaged method to study the v i b r a t i o n modes of t h i c k s t e e l p l a t e s and c y l i n d e r s and thereby proved the a p p l i c a b i l i t y o f holography to e n g i n e e r i n g components. W a l l [14] d i d a d e t a i l e d study of the time averaged method on l a r g e s t e e l p l a t e s . He d e r i v e d a formula showing the e f f e c t o f system geometry on the displacement and a l s o i n v e s t i g a t e d the v a r i o u s o u t s i d e e f f e c t s l i k e temperature and v i b r a t i o n on the system. 7 As a follow-up of t h e i r o r i g i n a l paper Powell [15] and S t e t s o n [16] d i s c u s s e d a more d e t a i l e d and r i g o r o u s proof of the v a r i o u s e f f e c t s o f displacement and r o t a t i o n of the o b j e c t on the f r i n g e f o r m a t i o n . However, these d i s -c u s s i o n s were not supported by experiment. M. L u r i e and M. Zambuto [17] a l s o c l a r i f i e d the c o r r e c t n e s s of assuming a zero order B e s s e l f u n c t i o n f r i n g e formation w i t h the use of a Twyman-Green I n t e r f e r o m e t e r . C.N. Z a i d e l , e t a l . (Soviet) [18] i n v e s t i g a t e d the s t r o b o s c o p i c technique and developed a c r i t e r i a f o r the r e l a t i o n between the pu l s e width and maximum a l l o w a b l e amplitude f o r sharp f r i n g e c o n t r a s t . The s t r o b i n g arrange-ment c o n s i s t e d o f a d i s c with s l o t s d r i v e n by an e l e c t r i c motor. Wuerker and H e f l i n g e r [19] used a pu l s e d l a s e r beam i n t h e i r study. T h i s enabled them to f r e e z e two d i f f e r e n t p o s i t i o n s of the o b j e c t . T h i s method was used to study t r a n s i e n t mechanical v i b r a t i o n i n a b r a s s p l a t e impacted with a s t e e l b a l l . A r c h b o l d , e t a l . [20] developed a method f o r v i s u a l o b s e r v a t i o n of the v i b r a t i o n modes by means of the speckle p a t t e r n caused by the use of a coherent l i g h t source. T h i s method pr o v i d e d o n l y a q u a l i t a t i v e a n a l y s i s of the mode shape, but no hologram had to be made. Pet e r , A.F. [21] used the time averaged method to study the v i b r a t i o n modes of t u r b i n e b l a d e s . He used the 8 mechanical impedance approach to f i n d the resonances of the tur b i n e blades and then made time averaged holograms of the various mode shapes. The mode shapes obtained by holography were i n very good agreement w i t h those obtained from the conventional mode p l o t s . W.C. Alvang et a l . [22] used the time averaged method to determine the mode shapes of t u r b i n e and compressor blades and vanes (100 Hz to 22 KHz), the mode shapes of c y l i n d r i c a l s h e l l type s t r u c t u r e s and the response of a c c o u s t i c a l l y d r i v e n f l a t panels. E x c e l l e n t p i c t u r e s were shown of the vari o u s components v i b r a t i n g at the resonant frequencies. 9 2. THEORY 2.1 P r i n c i p l e of Holography The p r i n c i p l e of holography was d i s c o v e r e d by Gabor [3] i n 1948. But i t was not u n t i l the d i s c o v e r y of the l a s e r beam and the p i o n e e r i n g work of L e i t h and Upatnieks [4,23,24] t h a t holography became a p r a c t i c a l p o s s i b i l i t y , opening up a completely new f i e l d i n i n t e r f e r o m e t r i c work. The b a s i c p r i n c i p l e of holography i s the i n t e r f e r e n c e e f f e c t between the beam d i s t o r t e d by the o b j e c t ( c o n t a i n i n g a l l the i n f o r m a t i o n of the o b j e c t ) and the r e f e r e n c e or u n d i s -t o r t e d beam. T h i s i n t e r f e r e n c e e f f e c t i s captured on a photographic p l a t e i n the form of a complex i n t e r f e r e n c e f r i n g e p a t t e r n . When the hologram i s r e i l l u m i n a t e d by the r e f e r e n c e beam o n l y , the l i g h t i s d i f f r a c t e d and c o n s t r u c t s the image of the o r i g i n a l o b j e c t with a l l i t s three dimen-s i o n a l c h a r a c t e r i s t i c s . The s e c r e t of holography i s not o n l y r e c o r d i n g the i n t e n s i t y d i s t r i b u t i o n of the l i g h t being r e f l e c t e d from an o b j e c t , as i n normal photography, but a l s o r e c o r d i n g the phase i n f o r m a t i o n of the r e f l e c t e d l i g h t . Because of the r a t h e r s p e c i a l i z e d nature of holography, i t i s f e l t t h a t a b r i e f d e s c r i p t i o n of the fundamental concepts i s i n o r d e r . To e x p l a i n the p r i n c i p l e s of holography the f o l l o w -ing s i m p l i f i e d treatment as developed by A.E. Ennos [25] i s used. 10 F i g u r e 2-1 Recording of the Hologram The l a s e r emits a coherent monochromatic c o l l i m a t e d l i g h t wave. T h i s beam i s focused, f i l t e r e d and then expanded by the s p a t i a l f i l t e r assembly and c o l l i m a t e d a g a i n by l e n s L. For the case of t r a n s m i s s i o n holography, the lower p a r t of the beam passes through a t r a n s p a r e n t o b j e c t and under-goes a phase s h i f t i n accordance wi t h the o b j e c t i n f o r m a t i o n . T h i s o b j e c t beam, can be rep r e s e n t e d by a = a e ' ^ ^ ^ . The upper p a r t of the beam remains unchanged and i s d e f l e c t e d by the prism through an angle 3 . T h i s r e f e r e n c e beam, can — i otx be represented by a = a e , where 11 &2n . . (2-1) The t o t a l l i g h t a m p l i t u d e a r r i v i n g a t t h e p h o t o p l a t e i s : a - ar ^ a s = are~ies3S* a3ei4**} .... (2-2) The p h o t o p l a t e i s s e n s i t i v e t o i n t e n s i t y I , w h i c h i s p r o p o r -t i o n a l t o t h e s q u a r e o f t h e a m p l i t u d e . The r e c o r d e d wave-f o r m c a n t h e r e f o r e be r e p r e s e n t e d by: a.© + aue'^'j * . . . . (2-3) D a r k f r i n g e s a r e f o r m e d on t h e p h o t o g r a p h i c p l a t e where d e s t r u c t i v e i n t e r f e r e n c e o c c u r s and l i g h t f r i n g e s a r e formed where c o n s t r u c t i v e i n t e r f e r e n c e o c c u r s . The f r i n g e s a r e f o r m e d p a r a l l e l t o t h e b i s e c t o r o f angle,.3 i n F i g u r e ( 2 - 1 ) . N o r m a l l y t h e s e f r i n g e s a r e i n v i s i b l e t o t h e naked eye and t h e h o l o g r a m i s i n f a c t a d i f f r a c t i o n g r a t i n g . The i n t e n s i t y o f the l i g h t from the o b j e c t i s recorded as a modulation of the c o n t r a s t of the i n t e r f e r e n c e f r i n g e s while the phase of the l i g h t i s recorded as a modulation i n spacing of the f r i n g e s . To r e c o n s t r u c t the image, the r e f e r e n c e beam — ictx a r = a^ e i s used t o i l l u m i n a t e the hologram and the wave t r a n s m i t t e d through the hologram can be re p r e s e n t e d by: A . « arl ^a^aj^^***"*]. . . . <2-4> The f i r s t term r e p r e s e n t s a pure i n t e n s i t y a t t e n u -a t i o n and does not c o n t a i n any i n f o r m a t i o n o f the o b j e c t . I t i s t r a n s m i t t e d without any d e v i a t i o n and r e p r e s e n t s the zerot h order d i f f r a c t i o n p a t t e r n of the wave form. The second term c o n t a i n s the term ^ which i s an exact r e p l i c a of the o b j e c t except f o r the s i g n change which means t h a t t h i s term w i l l reproduce the o b j e c t with *~ i (2 otx) a r e v e r s e c u r v a t u r e and the term e w i l l d i f f r a c t the image of the o b j e c t by an angle & so t h a t the angle between the r e a l o b j e c t and photographic p l a t e i s now 2 g . T h i s r e c o n s t r u c t i o n w i l l form the r e a l image of the o b j e c t and can be photographed without the use of any l e n s . The t h i r d term i s an exact r e p l i c a o f the o r i g i n a l o b j e c t except f o r some a t t e n u a t i o n . T h i s image i s the v i r t u a l image and can be seen by the eye when l o o k i n g through the hologram. Both the r e a l and v i r t u a l images are formed by f i r s t order d i f f r a c t i o n o n l y , s i n c e the i n t e r f e r e n c e f r i n g e s are o f s i n u s o i d a l i n t e n s i t y d i s t r i b u t i o n . 6 F i g u r e 2-2 R e c o n s t r u c t i o n of the Hologram 14 The f o r e g o i n g treatment was f o r a one dim e n s i o n a l change on the photographic p l a t e and a t r a n s p a r e n t o b j e c t . However, the a n a l y s i s holds t r u e f o r any three dimensional photographic emulsion and any three dimensional o b j e c t , not on l y i n the t r a n s m i s s i o n but a l s o the r e f l e c t i o n mode. 2.2 Time Averaged Hologram of a V i b r a t i n g Surface In t h i s method, which was f i r s t developed by Powell and Stetson [ 2 ] , a time averaged hologram i s made of the o b j e c t v i b r a t i n g a t i t s resonant frequency. When the hologram i s r e i l l u m i n a t e d by the o r i g i n a l r e f e r e n c e beam, a s e t of f r i n g e s can be seen superimposed on the o b j e c t . These f r i n g e s i n d i c a t e p o i n t s having the same amplitude o f v i b r a t i o n . The nodal areas can be e a s i l y determined s i n c e they show up as the b r i g h t e s t areas. From these holograms the mode shapes and amplitudes o f a v i b r a t i n g o b j e c t can be determined. The b i g g e s t drawback of the system i s t h a t the mode of v i b r a t i o n can o n l y be determined a f t e r d e v e l o p i n g and r e i l l u m i n a t i o n of the hologram. The n a t u r a l f r e q u e n c i e s of the o b j e c t have to be determined by some other means p r i o r to making the hologram. A s i m p l i f i e d treatment as e x p l a i n e d by Monahan [8] w i l l h e l p i n the understanding of the f r i n g e f o r m a t i o n . A t a resonant frequency and f o r a s i n u s o i d a l e x c i t a t i o n f o r c e , each p o i n t on the o b j e c t w i l l undergo s i n u s o i d a l o s c i l l a t i o n s . For every c y c l e , each p o i n t w i l l occupy an i n f i n i t e number 15 of p o s i t i o n s , but due to the nature of a s i n e motion a l l the p o i n t s w i l l spend a l a r g e r time a t the extreme amplitudes than a t any other p o s i t i o n . When a time exposure i s taken, each p o s i t i o n o f the o b j e c t w i l l c o n t r i b u t e t o the hologram, but o n l y the extreme p o s i t i o n s w i l l c o n t r i b u t e enough l i g h t energy t o expose the photographic emulsion. Because a coherent l i g h t source i s used, t h i s s u p e r p o s i t i o n w i l l produce f r i n g e s d e p i c t i n g c o n s t a n t amplitude p o i n t s . f ormation i s g i v e n by Powell and St e t s o n [ 2 ] . Monahan [ 8 ] , f o l l o w i n g the l i n e s " o f Powell and S t e t s o n , d e r i v e d the same r e l a t i o n between the amplitude of v i b r a t i o n and f r i n g e o r d e r . The e x p r e s s i o n f o r the r e c o n s t r u c t e d image i s : A r i g o r o u s treatment of the mathematics of the f r i n g e (Image of s t a t i o n a r y o b j e c t ) (2-5) where 9 1 = angle between the d i r e c t i o n o f i l l u m i n a t i o n and the d i r e c t i o n of v i b r a t i o n , 6 2 = angle between the d i r e c t i o n of viewing and the d i r e c t i o n o f v i b r a t i o n , m = amplitude of v i b r a t i o n of the o b j e c t a t (x / y ) X = wavelength of the l i g h t wave, JQ = zero order B e s s e l f u n c t i o n of the f i r s t k i n d , I , i , v = b r i g h t n e s s of the r e c o n s t r u c t i o n a t \x i y ) (x',y'). Thus one w i l l expect a dark f r i n g e superimposed on the o b j e c t a t (x',y') whenever the argument of J Q i s such t h a t t h i s zero o r d e r B e s s e l f u n c t i o n i s zero. The r e l a t i o n between a dark f r i n g e and amplitude of v i b r a t i o n becomes: (cos 0| 4- c o s 6 2 )m(x'9^) - Roots of J 0 f Roots of Jo I I w J ( 2 - 6 ) ^ , » n - . 2 3 5 ) 0 n = t92»3,.°* (2-7) £ 0 § © , cos0 2 ; T h e r e f o r e , by cou n t i n g the number of dark f r i n g e s and knowing the geometry of the system, the amplitude of v i b r a t i o n of a p a r t i c u l a r p o i n t can be c a l c u l a t e d . Due to the c h a r a c t e r i s t i c s of a zero order B e s s e l f u n c t i o n the maximum i n t e n s i t i e s w i l l reduce w i t h each 17 s u c c e s s i v e f r i n g e . T h i s method i s t h e r e f o r e l i m i t e d to small amplitudes and A.E. Ennos [10] mentioned a maximum number of 20 f r i n g e s f o r a c c e p t i b l e f r i n g e c o n t r a s t . 2.3 Real Time Method Using Continuous I l l u m i n a t i o n The disadvantage of the time averaged method i s t h a t the study i s done b l i n d l y , i . e . not u n t i l a f t e r the de v e l o p i n g and r e i l l u m i n a t i o n of the hologram can i t be determined a t what mode shape and amplitude the o b j e c t was e x c i t e d . S h o r t l y a f t e r t h e i r f i r s t p u b l i c a t i o n , Powell and Stetson [6] d i s -cussed a r e a l time method whereby a hologram was made of the s t a t i c o b j e c t and a f t e r d e v e l o p i n g , the hologram was r e p l a c e d at i t s o r i g i n a l p o s i t i o n . When the hologram was i l l u m i n a t e d by the o r i g i n a l r e f e r e n c e beam a s u p e r p o s i t i o n between the a c t u a l o b j e c t and the image r e c r e a t e d by the hologram was obtai n e d . Due to the e r r o r s i n v o l v e d i n the r e p o s i t i o n i n g of the hologram, a s t a t i c f r i n g e p a t t e r n was c r e a t e d . The hologram was ad j u s t e d to produce the g r e a t e s t p o s s i b l e p a r a l l e l i s m and s t r a i g h t n e s s of the f r i n g e s and when the o b j e c t was e x c i t e d a t i t s resonant frequency and observed under con-tinuous i l l u m i n a t i o n , the l o c a l v i s i b i l i t y o f the f r i n g e s was re p o r t e d t o have v a r i e d as the zero-order B e s s e l f u n c t i o n o f the amplitude of v i b r a t i o n . The equation f o r the i n t e n s i t y o f the image observed through the hologram, with the o b j e c t undergoing a s t a t i c • 1 8 displacement and harmonic v i b r a t i o n i s d e r i v e d by Brown, e t a l . [27] and g i v e n as: lima** = 2l*[l + cos k) X (rn-K)] • . (2-8) where I ^ m a g e _ i n t e n s i t y d i s t r i b u t i o n of the recon-s t r u c t e d image, I = i n t e n s i t y o f the l i g h t wave recon-s t r u c t e d by the hologram, mg = s t a t i c displacement v e c t o r o f the o b j e c t , k = j— (m1 + m2) ->-m^ = o b j e c t i l l u m i n a t i o n v e c t o r , -> m2 = o b j e c t viewing v e c t o r , m = o b j e c t amplitude v e c t o r . In t h i s experiment (m • k) reduces to X (cos Oj + cos et) m E q u a t i o n [2-8] can be r e w r i t t e n as linage = 21 . [ l + C D S ( n V k) * J e { x (cos 6, + cos0 2 )m (2-9) 19 The term (m • k) i s due to the o r i g i n a l mismatch i n the r e -s = p o s i t i o n i n g . I t does not i n f l u e n c e the l o c a t i o n of the f r i n g e p a t t e r n but w i l l reduce the f r i n g e c o n t r a s t . Due to the u n i t y term, the f r i n g e c o n t r a s t i s reduced s u b s t a n t i a l l y i n t h i s method. The argument of J q i s e x a c t l y the same as i n the Powell and S t e t s o n time averaged method and as b e f o r e we expect a dark f r i n g e f o r the c o n d i t i o n : A  m " 2 ( C 0 S 6 , + €0S© 8) /Roots Of Jc , ( 2 _ 1 0 ) 2.4 Real Time Holography of a V i b r a t i n g Surface Without an  I n i t i a l F r i n g e P a t t e r n Using S t r o b o s c o p i c I l l u m i n a t i o n A f t e r the p u b l i c a t i o n o f the papers by Powell and S t e t s o n , v a r i o u s people s t a r t e d to take an i n t e r e s t i n the a p p l i c a t i o n of holography to v i b r a t i o n a n a l y s i s . In 1968, s e v e r a l papers [10,11,12,18] were p u b l i s h e d independently of one another d e s c r i b i n g a r e a l time method r e s u l t i n g i n the same f r i n g e p a t t e r n s as the Powell and S t e t s o n time averaged method. T h i s method a l s o i n v o l v e d the r e p o s i t i o n i n g of the hologram of the s t a t i c o b j e c t , but now i n s t e a d o f u s i n g continuous i l l u m i n a t i o n , the l a s e r beam was s t robed i n syn-c h r o n i z a t i o n w i t h the e x c i t a t i o n f o r c e . In e f f e c t , a super-p o s i t i o n of the s t a t i c o b j e c t and a p a r t i c u l a r p o s i t i o n o f . the deformed o b j e c t was o b t a i n e d . As the o b j e c t r e c o n s t r u c t e d by the hologram i s i d e n t i c a l i n a l l r e s p e c t s to the r e a l o b j e c t , the f r i n g e formation by t h i s method i s i d e n t i c a l t o a doubly exposed hologram w i t h the o b j e c t s l i g h t l y deformed du r i n g the two exposures. The formula f o r the f r i n g e formation i n the case of a doubly exposed hologram w i t h the o b j e c t undergoing a s l i g h t deformation d u r i n g the two exposures i s d e r i v e d i n d e t a i l by Brown [27] . The i n t e n s i t y d i s t r i b u t i o n i n the r e c o n s t r u c -ted image i s g i v e n as: kmage - 2|£.|* [l' + COS (+z - *j] • • • • <2-n> where |E Q| = amplitude of the r e c o n s t r u c t e d wave, (<j>2-4>-]_) = phase d i f f e r e n c e o f the wave between the two p o s i t i o n s o f the o b j e c t . Equation (2-11) shows t h a t the r e c o n s t r u c t e d image i s modulated by cos (<$>2-<$>^) t which i s a f u n c t i o n of the o b j e c t deformation. A p l o t of [1 + cos (c})_-<j)1) ] i s g i v e n i n F i g u r e 2-3. F i g u r e 2-3 P l o t o f [1 + cos ($--$,) 3 versus (c}>9-c}>,) Whenever &t) = «J5" . . . . (2-12) the i n t e n s i t y w i l l a l t e r n a t e between maximum and zero. The phase term ( c ^ - ^ ) can be expressed i n terms of the geometry of the system [ 8 ] . L e t the o b j e c t be a plane s u r f a c e and be deformed i n a t r a n s v e r s e d i r e c t i o n as shown i n F i g u r e 2-4. Fi g u r e 2-4 Obj e c t Geometry as Captured by the Strobed L i g h t Source AOB r e p r e s e n t s the wave due to the f i r s t exposure, A'O'B1 r e p r e s e n t s the wave due to the second exposure and m i s the displacement of the o b j e c t a t the p o i n t of i n t e r e s t . The r e s u l t a n t wave r e a c h i n g the hologram l o c a t e d a t C i s the v e c t o r i a l a d d i t i o n of these two waves and w i l l depend on m. The path d i f f e r e n c e o f the wave i s g i v e n by & » CUD (cOS Bg 4- C O S @ g j . . . . (2-13) The phase d i f f e r e n c e can be g i v e n by 8 - T V - 2sr) . . . . . (2-14) By combining equation (2-12) and (2-14), the amount of d e f o r -mation can be expressed i n terms of l i g h t and dark f r i n g e s , as f o l l o w s : PW • *»/rV*<.A . ^ e - z f t * . . . . (2-15) 5®, + C © S © 2 ) n = 0 , 2 , 4 , r e s u l t s i n a l i g h t f r i n g e , n = 1, 3, 5, . . . . , r e s u l t s i n a dark f r i n g e . L e t n^ be the number of l i g h t f r i n g e s and n 2 be the number of dark f r i n g e s a t the p o i n t of i n t e r e s t ; then n can be r e w r i t t e n as n = (2 n^ - 2) f o r l i g h t f r i n g e s and n = (2 n 2 - 1) f o r dark f r i n g e s . E x p r e s s i n g the displacement m i n terms of the number of l i g h t f r i n g e s (n^): 23 m = (cos®, +cos© 2) • rt'= l p 2 — • • • • • ( 2 " 1 6 ) Likewise i n terms o f the number of dark f r i n g e s ( n 2 ) , we get: (2nz-i) A m * 2(C034 + C O S Q , ) * " a - l p ^ - - . . (2-17) The r e f o r e by knowing the f r i n g e order (n.^ or n 2) a t the p o i n t of i n t e r e s t , the displacement m can be c a l c u l a t e d by u s i n g equation (2-16) or (2-17). 2.5 Real Time Holography of a V i b r a t i n g Surface With An I n i t i a l F r i n g e P a t t e r n Using S t r o b o s c o p i c I l l u m i n a t i o n A second method of r e a l time study i s now proposed by the author. I t i s the main c o n t r i b u t i o n of t h i s t h e s i s to the f i e l d o f r e a l time, f u l l f i e l d v i b r a t i o n v i s u a l i z a t i o n . The theory i s as f o l l o w s . A hologram i s made of the s t a t i c o b j e c t and a c c u r a t e l y r e p o s i t i o n e d . The hologram i s then g i v e n a s m a l l displacement so t h a t a f r i n g e p a t t e r n i s c r e a t e d between the o b j e c t and r e c o n s t r u c t e d image. Next, the change i n t h i s f r i n g e p a t t e r n i s observed under s t r o b o -s c o p i c i l l u m i n a t i o n when the o b j e c t i s v i b r a t e d . In e f f e c t , the procedure i s a frequency modulation of a s p a t i a l c a r r i e r wave. The approach i s s i m i l a r to t h a t used by H a z e l l [ 5 ] f o r the v i s u a l i z a t i o n of v i b r a t i n g p l a t e s by the shadow moire method. When the hologram i s a c c u r a t e l y r e p o s i t i o n e d a t i t s o r i g i n a l p o s i t i o n , the o b j e c t viewed through the hologram w i l l d i s p l a y no f r i n g e p a t t e r n a t a l l . F i g u r e (2.5) shows the hologram and a plane o b j e c t with t h e i r r e s p e c t i v e r e f e r e n c e systems. H@U0x&&AM PLAWEL O B J E C T F i g u r e 2.5 Hologram and Object Reference System When the hologram i s g i v e n a s m a l l r o t a t i o n 6^ about the x a x i s , the a c t u a l o b j e c t and r e c o n s t r u c t e d image w i l l form a wedge shaped gap and t h i s gap w i l l c r e a t e a f r i n g e p a t t e r n c o n s i s t i n g of p a r a l l e l , e q u i d i s t a n t l i n e s , p a r a l l e l to the a x i s of r o t a t i o n (x or x') as shown i n F i g u r e 2.6. For c l a r i t y , the gap between the r e c o n s t r u c t e d image and a c t u a l o b j e c t i s shown g r e a t l y exaggerated i n a l l the F i g u r e s . These f r i n g e s are contours o f equal displacement between the o b j e c t and image. F i g u r e 2.6 F r i n g e P a t t e r n , w i t h Hologram Rotated by 6^ S i m i l a r l y , when the hologram i s giv e n a smal l r o t a -t i o n 6 2 about the y a x i s , a p a r a l l e l , e q u i d i s t a n t f r i n g e p a t t e r n w i l l be c r e a t e d , but now the f r i n g e p a t t e r n w i l l be p a r a l l e l to y 1 . In g e n e r a l when the hologram i s giv e n both a s m a l l r o t a t i o n 6^ and o^/ t h e f r i n g e s w i l l again be p a r a l l e l and e q u i d i s t a n t , but now they w i l l not be p a r a l l e l t o e i t h e r x' 26 or y', but w i l l be i n c l i n e d a t some angle depending on the amount of r o t a t i o n <5^  and S^' Again these f r i n g e s are contours o f equal displacement between the o b j e c t and image. F i g u r e 2.7 F r i n g e P a t t e r n , with Hologram Rotated by 6^ and 6 By a d j u s t i n g the o r i e n t a t i o n of t h i s s t a t i c f r i n g e p a t t e r n , the v i b r a t i o n amplitude a c r o s s any d i a m e t r a l l i n e can be i n v e s t i g a t e d . L e t the hologram be r o t a t e d by 6^. The s t a t i c f r i n g e p a t t e r n and r e l a t i v e p o s i t i o n of the o b j e c t and r e c o n -s t r u c t e d image i s shown i n F i g u r e 2 . 6 . When the o b j e c t i s e x c i t e d and viewed under s t r o b o s c o p i c i l l u m i n a t i o n , the s t a t i c f r i n g e p a t t e r n w i l l be modulated i n accordance with the displacement of the v i b r a t i n g o b j e c t . For s i m p l i c i t y l e t the o b j e c t v i b r a t e a t i t s fundamental frequency and l e t the l a s e r beam be strobed i n s y n c h r o n i z a t i o n w i t h the maximum displacement of the o b j e c t . F i g u r e 2.8 shows the o b j e c t geometry as "captured" by the strobed l a s e r beam and the corresponding s t a t i c and dynamic f r i n g e p a t t e r n superimposed on one another. F i g u r e 2.8 Dynamic Object Geometry and F r i n g e P a t t e r n 28 To f i n d t h e a m p l i t u d e a c r o s s t h e d i a m e t r a l l i n e AA, b o t h t h e l o c a t i o n o f t h e s t a t i c and dynamic f r i n g e s must be p l o t t e d w i t h r e s p e c t t o t h e d i s p l a c e m e n t t h e y r e p r e s e n t . By p l o t t i n g t h e s t a t i c f r i n g e p a t t e r n , t h e o r i e n t a t i o n o f t h e s t a t i c o b j e c t and r e c o n s t r u c t e d image c a n be f o u n d . By p l o t t i n g t h e dynamic f r i n g e p a t t e r n on t o t h e same p l o t , t h e a m p l i t u d e o f t h e o b j e c t c a n be d e t e r m i n e d . F i g u r e 2.9 shows t h e mode shape o f t h e o b j e c t a c r o s s t h e d i a m e t r a l l i n e AA. The numbers on t h e y' and z 1 a x i s r e p r e s e n t t h e f r i n g e o r d e r . The y 1 a x i s r e p r e s e n t s t h e l o c a t i o n o f t h e f r i n g e a c r o s s t h e d i a m e t r a l l i n e AA, w h i l e t h e z 1 a x i s r e p r e s e n t s t h e c o n s t a n t d i s p l a c e m e n t a s s o c i a t e d w i t h e v e r y f r i n g e . The p r o c e d u r e i s t o p l o t t h e l o c a t i o n o f t h e s t a t i c f r i n g e s a c r o s s AA w i t h r e s p e c t t o t h e a s s o c i a t e d d i s p l a c e m e n t s . The r e l a t i v e p o s i t i o n o f t h e r e c o n s t r u c t e d image i s f o u n d by t h i s p l o t . N e x t t h e l o c a t i o n o f t h e dynamic f r i n g e s a c r o s s AA i s p l o t t e d and w i t h t h e r e c o n s t r u c t e d image as r e f e r e n c e t h e d i s p l a c e m e n t a s s o c i a t e d w i t h t h e f r i n g e o r d e r i s p l o t t e d p a r a l l e l t o t h e z' a x i s r e s u l t i n g i n a p l o t o f t h e a c t u a l o b j e c t g e o m e t r y a c r o s s t h e d i a m e t r a l l i n e AA. The a m p l i t u d e o f v i b r a t i o n i s now f o u n d by s u b t r a c t i n g , p a r a l l e l t o t h e a b s c i s s a , t h e s t a t i c p o s i t i o n o f t h e o b j e c t f r o m t h e o b j e c t mode s h a p e . T h i s method o f f e r s t h e p o s s i b i l i t y o f c h a n g i n g t h e s e n s i t i v i t y o f t h e s y s t e m . I n t h e s t r o b o s c o p i c r e a l t i m e method w i t h o u t an i n i t i a l f r i n g e p a t t e r n , t h e minimum d i s p l a c e -ment r e q u i r e d f o r a f r i n g e t o f o r m depends on t h e g e o m e t r y 29 • 4 - 3 - 2 - I O I 2 3 4 5 6 7 S © I O I I 12 Z* DISPLACEMENT C jjin/order Figure 2.9 P l o t to Fi n d the Object Mode Shape = s t a t i c f r i n g e s = dynamic f r i n g e s . C = A (cos 8, + cos 8„) of the system. For 0^ = 60° and 6 2 = 0°, the minimum d i s p l a c e -ment req u i r e d i s 16.6 y i n . However, by using t h i s c a r r i e r wave method, even f r a c t i o n a l displacements can be recorded by using a high d e n s i t y c a r r i e r . Another advantage i s t h a t the phase of the d i s p l a c e -ment f i e l d a t any p o i n t i n the o b j e c t can be determined by observing'the change i n the d i r e c t i o n of c u r v a t u r e of the s t a t i c f r i n g e p a t t e r n . For example i n F i g u r e 2.8, the s t a t i c f r i n g e p a t t e r n i s s h i f t e d downwards. By knowing the r e l a t i v e p o s i t i o n of the r e c o n s t r u c t e d image and a c t u a l o b j e c t , the o n l y p o s s i -b i l i t y f o r the f r i n g e s to be s h i f t e d downwards i s f o r the o b j e c t to be strobed a t the p a r t of the c y c l e as shown i n F i g u r e 2.9. The above i n f o r m a t i o n i s obtained by s t r o b i n g the f r i n g e p a t t e r n a t the same frequency of v i b r a t i o n as t h a t of the o b j e c t or an i n t e g e r m u l t i p l e t h e r e o f . By s t r o b i n g the f r i n g e p a t t e r n a t a frequency s l i g h t l y d i f f e r e n t from the f r e -quency of v i b r a t i o n of the o b j e c t , i t should be p o s s i b l e to s e t up a beat phenomenon showing the behaviour of the o b j e c t as f r i n g e s sweeping back and f o r t h across the o b j e c t i n slow motion. P r o v i d i n g the v i b r a t i o n i s t h a t o f steady s t a t e , the f r i n g e motion can then be recorded by an o r d i n a r y movie camera. Otherwise a high speed camera i s r e q u i r e d . 31 3. EXPERIMENTAL APPARATUS B a s i c l y the system can be d i v i d e d i n t o f o u r p a r t s : 1. The l a s e r 2. The o p t i c a l bench w i t h o p t i c a l components 3. The s t r o b i n g mechanism and o b j e c t e x c i t a t i o n system 4. The o b j e c t to be s t u d i e d ( c i r c u l a r membrane). 3 .1 The Laser The l a s e r used i n t h i s study was the S p e c t r a - P h y s i c s Model 125 A Continuous Wave Helium Neon Gas L a s e r . I t emits a coherent c o l l i m a t e d monochromatic beam, 2 mm i n diameter a t o a wavelength of 6 32 8 A and has a measured maximum power output of 6 8 mw a f t e r a warmup p e r i o d of three hours. The l a s e r i s very s e n s i t i v e to temperature changes and to i n s u r e maximum performance the room temperature had to be kept a t approximately 65°F. 3.2 The O p t i c a l Bench With O p t i c a l Components Because of the s t r i c t s t a b i l i t y requirement i t was necessary to mount the o p t i c a l components on a v i b r a t i o n i s o -l a t e d o p t i c a l bench (Gaertner Jeong Holography Table [28]). The o p t i c a l bench i s composed of nine p a r a l l e l s t e e l r a i l s , which are mounted on a 2 i n c h t h i c k s t e e l t a b l e . T h i s t a b l e i s mounted i n a r i g i d l y c o n s t r u c t e d metal frame and pneumatic-a l l y supported by f o u r i n n e r tubes t o reduce shock and v i b r a t i o n to a minimum. The o p t i c a l components are r i g i d l y f i x e d to the bench by magnetic clamps. The o r i g i n a l Garetner Jeong Holography System was s l i g h t l y m o d i f i e d . The o r i g i n a l f i x e d r a t i o (20% t r a n s m i s s i v e ) beam s p l i t t e r and photo p l a t e h o l d e r were r e p l a c e d by a v a r i a b l e d e n s i t y beam s p l i t t e r and an a d j u s t a b l e p l a t e h o l d e r , both made by Jodon E n g i n e e r i n g A s s o c i a t e s . An a d d i t i o n a l s p a t i a l f i l t e r was used so t h a t the r e f e r e n c e and o b j e c t beam were f i l t e r e d i n d i v i d u a l l y . A schematic of the holography system used i s shown i n F i g u r e 3.1. F i g u r e 3.1 Schematic of the Holography System 33 The i n c o m i n g beam was s p l i t by t h e v a r i a b l e beam s p l i t t e r V.B.S. i n t o two beams. The t r a n s m i t t e d beam, w h i c h i n t h i s c a s e was t h e r e f e r e n c e beam, was r e f l e c t e d by m i r r o r and p a s s e d t h r o u g h t h e s p a t i a l f i l t e r S.F.^ and l e n s t o t h e p h o t o g r a p h i c p l a t e Ph. The r e f l e c t e d beam, w h i c h i n t h i s c a s e was t h e o b j e c t beam, was r e f l e c t e d by m i r r o r M 2 and p a s s e d t h r o u g h t h e s p a t i a l f i l t e r S . F ^ and l e n s t o t h e o b j e c t 0, where i t was d i f f u s e l y r e f l e c t e d t o t h e p h o t o g r a p h i c p l a t e Ph. A p h o t o g r a p h o f t h e o v e r a l l v i e w o f t h e h o l o g r a p h y s y s t e m and o f t h e h o l o g r a p h y t a b l e i t s e l f i s shown i n F i g u r e 3.2 and F i g u r e 3.3. The p u r p o s e o f t h e v a r i a b l e beam s p l i t t e r V.B.S., M o d e l VBA-200 made by J o d o n E n g i n e e r i n g A s s o c i a t e s , was t o s p l i t t h e beam i n t o two beams w i t h t h e c o r r e c t beam i n t e n s i t y r a t i o s . By r o t a t i n g t h e beam s p l i t t e r g l a s s t h r o u g h 360° i n i t s own p l a n e t h e t r a n s m i t t e d beam i n t e n s i t y v a r i e d f r o m 0.7% t o 90% o f t h e i n c i d e n t beam w h i l e t h e r e f l e c t e d beam v a r i e d f r o m 85% t o 8.5% o f t h e i n c i d e n t beam. F o r optimum r e c o r d i n g o f t h e h o l o g r a m t h e r e f e r e n c e and o b j e c t beam r a t i o s were a p p r o x i m a t e l y f i v e t o one [ 2 9 ] , w h i l e d u r i n g r e c o n s t r u c t i o n t h e beam s p l i t t e r was a d j u s t e d t o g i v e c o m p a r a b l e o b j e c t and image b r i g h t n e s s t o o b t a i n maximum f r i n g e c o n t r a s t . The p u r p o s e o f t h e s p a t i a l f i l t e r was t o expand and f i l t e r o u t a l l unwanted s p a t i a l f r e q u e n c i e s f r o m t h e l a s e r beam. I t c o n s i s t e d Of a c o n v e r g e n t l e n s and a p i n h o l e 10 t o 25 m i c r o n s i n d i a m e t e r . The l e n s c o n v e r g e d t h e beam i n t o t h e p i n -F i g u r e 3.3 The Holography Table hole where i t was d i f f r a c t e d by the p i n h o l e i n t o c o n c e n t r i c c i r c l e s of a l t e r n a t i n g b r i g h t and dark areas. F i g u r e 3.4 Schematic of the S p a t i a l F i l t e r U n i t For t h i s purpose the lens was a d j u s t e d i n the long t u d i n a l d i r e c t i o n and the p i n h o l e was ad j u s t e d i n a v e r t i c a l plane i n two p e r p e n d i c u l a r d i r e c t i o n s as shown i n F i g u r e 3.4 The procedure f o r a d j u s t i n g the s p a t i a l f i l t e r was t o remove the p i n h o l e and a d j u s t the lens by eye so t h a t the beam h i t the c e n t r e of the l e n s and c o i n c i d e d w i t h the o p t i c a l a x i s of the l e n s . The p i n h o l e was then r e p l a c e d and by proper adjustment of the two p e r p e n d i c u l a r c o n t r o l s the p i n h o l e was centered on the beam. The i n t e n s i t y of the beam was then 36 v a r i e d by a d j u s t i n g the l e n s . The f i r s t b r i g h t area at the centre i s c a l l e d the A i r y d i s c and only t h i s p a r t of the beam was used. The s p a t i a l f i l t e r S.F.^ c o n s i s t e d of a 10 x micro-scopic o b j e c t i v e and a pinhole of about 25 microns, while the s p a t i a l f i l t e r S.F. c o n s i s t e d of a 20 x microscopic o b j e c t i v e and a pinhole of about 10 microns. The d i v e r g i n g lenses and of 10 inches f o c a l length expanded the beam to the req u i r e d c r o s s - s e c t i o n a l area to cover the whole photographic p l a t e and o b j e c t , r e s p e c t i v e l y . The photographic p l a t e holder i s a very c r i t i c a l component i n r e a l time s t u d i e s , where accurate r e p o s i t i o n i n g of the hologram i s r e q u i r e d . The Jodon P l a t e holder, model number PH45 has two microheads i n the x and y d i r e c t i o n . As these microheads are lo c a t e d o f f center (Figure 3.5) , a d j u s t -ment of these microheads w i l l cause a t r a n s l a t i o n i n the x and y d i r e c t i o n and simultaneously a r o t a t i o n 6^ and °f the hologram. The hologram i s sp r i n g loaded a g a i n s t a v e r t i c a l base which i s perpendicular to the z d i r e c t i o n , so tha t p r a c t i c a l l y no adjustment i s req u i r e d i n the z d i r e c t i o n . This base i s attached to the main base by four small s t e e l studs and adjustment of the microheads w i l l cause minute deformation of these s t e e l studs. 37 F i g u r e 3.5 Schematic of the Jodon P l a t e Holder 3.3. The S t r o b i n g Mechanism and Object E x c i t a t i o n System The s t r o b i n g mechanism c o n s i s t e d o f an 0.1 in c h t h i c k aluminum d i s c d r i v e n by a v a r i a b l e speed (0-3000 rpm) 3/16 H.P. D.C. Motor. To st r o b e the l a s e r beam, 60 holes w i t h a diameter of 0.031 inches were d r i l l e d a t a r a d i u s of s i x inc h e s . T h i s arrangement gave a 1/20 duty c y c l e ( r a t i o o f on time t o o f f ti m e ) . Two convergent l e n s e s of s i x inches f o c a l l e n g t h were used r e s p e c t i v e l y to focus the l a s e r beam i n t o the hole and to c o l l i m a t e the beam again a f t e r the ho l e . 38 Because the o b j e c t e x c i t a t i o n s i g n a l had to be synchronized with the l i g h t p u l s e , the same d i s c was used to generate the e x c i t a t i o n s i g n a l . S i x t y h o l e s w i t h a diameter of 0.25 inches were d r i l l e d a t a r a d i u s o f 4.775 inches to g i v e an equal o n - o f f r a t i o . The s i g n a l was obtained by u s i n g a l i g h t bulb-photo t r a n s i s t o r arrangement as shown i n F i g u r e 3.6. The l i g h t bulb and photo t r a n s i s t o r were p o s i t i o n e d on e i t h e r s i d e of the d i s c . When the d i s c was r o t a t e d , the l i g h t from the bulb was strobed by the h o l e arrangement and a square wave was generated a t the output o f the photo t r a n s i s t o r (Motorola, type MRD 300). T h i s s i g n a l was f e d i n t o a Krohn-Hite Model 335 (0.02-20 KHZ) band pass f i l t e r and by a d j u s t i n g the appro-p r i a t e low pass band r e g i o n the fundamental s i n e wave was obtained from the square wave. T h i s s i g n a l was a m p l i f i e d and used to d r i v e the horn e x c i t e r . The phase s h i f t between the l i g h t p u l s e and e x c i t a t i o n s i g n a l was obtained by changing the l o c a t i o n of the photo-t r a n s i s t o r with r e s p e c t to the l i g h t b u l b . F i g u r e 3.7 shows a c r o s s - s e c t i o n a l view of the d i s c and the r e l a t i v e p o s i t i o n s of the l i g h t bulb and photo t r a n s i s t o r . For c l a r i t y the s i z e and l o c a t i o n s of the holes were not drawn to s c a l e . By moving the photo t r a n s i s t o r along the a r c AA the phase between the l i g h t p u l s e and e x c i t a t i o n s i g n a l was a d j u s t e d . A photograph of the s t r o b i n g mechanism i s shown i n F i g u r e 3.8. 39 F i g u r e 3.6 Schematic of the S t r o b i n g Mechanism and Object E x c i t a t i o n System S T R O B I N G ' D I S K 5 E C T 1 Q N B-B F i g u r e 3.7 Schematic of the Phase Adjustment F i g u r e 3.8 S t r o b i n g Mechanism and O b j e c t E x c i t a t i o n System F i g u r e 3.9 Membrane T e n s i o n i n g D e v i c e 41 3.4 The O b j e c t ( C i r c u l a r Membrane) The o b j e c t was a t h r e e i n c h d i a m e t e r c i r c u l a r membrane made o f 0 . 0 0 0 5 i n c h e s t h i c k a l u m i n i z e d M y l a r . To p r e s t r e s s t h e M y l a r and o b t a i n a u n i f o r m t e n s i o n t h e d e v i c e as shown i n F i g u r e 3.9 was u s e d [ 3 0 ] . The M y l a r (3) was c l a m p e d between t h e b o t t o m (1) and t h e c o v e r r i n g (4) by e i g h t s c r e w s ( 9 ) . Compre s s e d a i r a t a p r e s s u r e o f one p s i s t o r e d i n a 20 g a l l o n drum, was c o n n e c t e d t o t h e a i r i n l e t (2) so t h a t t h e M y l a r was p u t u n d e r t e n s i o n . The c i r c u m f e r e n c e o f t h e membrane h o l d e r (5) was c o a t e d w i t h a t h i n l a y e r o f epoxy g l u e and s c r e w e d i n t o t h e b a s e o f t h e t e n s i o n i n g d e v i c e ( 6 ) . T h i s whole u n i t ( 6 , 7 , 8 ) was t h e n p o s i t i o n e d on t o p o f t h e M y l a r and by a d j u s t -i n g t h e t e n s i o n s c r e w (8) t h e membrane h o l d e r was p r e s s e d a g a i n s t t h e M y l a r and l e f t i n p o s i t i o n u n t i l t h e epoxy g l u e had s e t (24 h o u r s ) . I t was i m p o s s i b l e t o g e t a u n i f o r m l y t e n s i o n e d mem-b r a n e as n o t a l l t h e r i p p l e s i n t h e M y l a r were e l i m i n a t e d when t h e M y l a r was s t r e t c h e d by t h e c o m p r e s s e d a i r . I t was a l s o i m p o s s i b l e t o d e p o s i t a u n i f o r m l a y e r o f g l u e on t h e membrane h o l d e r . 42 4. EXPERIMENTAL PROCEDURE The f i r s t step was to l i n e up the beam through the s t r o b i n g d i s c by a d j u s t i n g the two convergent l e n s e s . The o p t i c a l components on.the o p t i c a l bench were arranged as shown i n F i g u r e 3.1. Because o f the l i m i t e d coherence l e n g t h of the l a s e r (approximately 7 inches [18 cm.] f o r the S p e c t r a P h y s i c s Model 125 A L a s e r ) , i t was important t h a t the o b j e c t and r e f e r e n c e beam path l e n g t h were approximately the same. The v a r i o u s components were then a d j u s t e d as d i s c u s s e d i n S e c t i o n 3.2. At the p o s i t i o n o f the photo p l a t e h o l d e r a s o l a r c e l l , (N 210 CG made by Hoffman, U.S.A.), was p o s i t i o n e d to measure the r e l a t i v e i n t e n s i t i e s of the two beams. The face of the s o l a r c e l l was p a r a l l e l to the photo p l a t e and by re a d i n g the v o l t a g e output on an o s c i l l o s c o p e the v a r i a b l e beam s p l i t t e r V.B.S. was a d j u s t e d to g i v e a r e f e r e n c e t o o b j e c t beam r a t i o of 5:1 [29]. The photographic p l a t e s used were Agfa Gevaert type 10E70 4" x 5" g l a s s p l a t e s which had a r e s o l u t i o n of 2800 lines/mm and were twenty times as f a s t as the Kodak.649 F s p e c t r o -s c o p i c p l a t e s which had been used almost e x c l u s i v e l y f o r h o l o -graphy up u n t i l a couple of years ago. The c o r r e c t exposure time was found by t r i a l and e r r o r . A v o l t a g e r e a d i n g o f 4 0 mv of the s o l a r c e l l would r e q u i r e an exposure time of 1/8 seconds and a v o l t a g e r e a d i n g of 20 mv an exposure time o f 1/4 seconds. 43 T h e s e f i g u r e s i n d i c a t e d an i n v e r s e l i n e a r r e l a t i o n s h i p between t h e v o l t a g e o u t p u t and e x p o s u r e t i m e , i . e . h a l v i n g t h e v o l t a g e o u t p u t r e q u i r e d a d o u b l i n g o f t h e e x p o s u r e t i m e . F o r any v o l t a g e v a l u e up t o 20 0 mv ( a p p r o x i m a t e l i m i t o f t h e l i n e a r r a n g e o f t h e s o l a r c e l l ) t h e r e q u i r e d e x p o s u r e t i m e was d e t e r -mined by i n t e r p o l a t i n g t h e above v a l u e s . A f t e r t h e p h o t o p l a t e was e x p o s e d , i t was d e v e l o p e d i n c o m p l e t e d a r k n e s s i n M e t i n o l U a t a t e m p e r a t u r e o f 6 8°F f o r f o u r m i n u t e s . I t was t h e n p u t i n a s t o p b a t h , c o n s i s t i n g o f 0.2 8 a c e t i c a c i d w h i c h was d i l u t e d by a r a t i o o f 1 t o 8, f o r 30 s e c o n d s and a f t e r w a r d s f i x e d i n a Hypo s o l u t i o n f o r f o u r m i n u t e s . A t t h i s s t a g e t h e l i g h t was s w i t c h e d o n . The p l a t e was washed b r i e f l y i n a Hypo e l i m i n a t o r s o l u t i o n and t h e n washed i n r u n n i n g w a t e r f o r t e n m i n u t e s a f t e r w h i c h i t was p u t i n a s o l u t i o n o f P h o t o f l o and l e f t t o d r y f o r a c o u p l e o f h o u r s . The d e v e l o p e d h o l o g r a m was t h e n r e p o s i t i o n e d i n t h e p h o t o p l a t e h o l d e r and t h e v a r i o u s e x p e r i m e n t s were c a r r i e d o u t . A n o n - c o n t a c t i n g d i s p l a c e m e n t t r a n s d u c e r ( F o t o n i c S e n s o r , MTI I n s t r u m e n t s D i v i s i o n , Latham, New York) b a s e d on t h e p r i n c i p l e o f f i b r e o p t i c s was u s e d t o d e t e r m i n e t h e a m p l i t u d e o f v i b r a t i o n a t t h e c e n t r e o f t h e membrane. T h i s r e a d i n g was t h e n compared w i t h t h e c a l c u l a t e d a m p l i t u d e o b t a i n e d by t h e f r i n g e t h e o r y . The s t a b i l i t y o f t h e e x c i t a t i o n f r e q u e n c y was v e r y i m p o r t a n t t o o b t a i n s h a r p f r i n g e p a t t e r n s . U n f o r t u n a t e l y t h e 44 speed of the D.C. Motor tended t o d r i f t c o n s t a n t l y . (± 10 Hz) and sharp f r i n g e p a t t e r n s were o n l y o b t a i n e d f o r s h o r t p e r i o d s of time. To r e c o r d these f r i n g e s h i g h speed f i l m had to be used. Three d i f f e r e n t f i l m s were t r i e d : a) 4 by 5 P o l a r o i d 3 000 ASA sheet f i l m ; b) 35 mm Kodak High Speed Recording f i l m , and c) 35 mm Kodak T r i X f i l m . I t was found t h a t the P o l a r o i d f i l m was q u i t e i n -o s e n s i t i v e to r e d l i g h t (6328A) and although i t had a speed of 3000 ASA exposure times of 30 seconds a t a l e n s stop of f5.6 were r e q u i r e d . The Kodak High Speed Recording f i l m had a speed of 1000 ASA and r e q u i r e d an exposure time of 10 seconds a t a lens stop of f5.6. U n f o r t u n a t e l y the speckle e f f e c t due to the l a s e r beam was a m p l i f i e d by the coarse g r a i n s i z e of the f i l m emulsion. The Kodak T r i X f i l m had an o r i g i n a l speed of 400 ASA, but by u s i n g ACU-1 developer t h i s f i l m can be used a t a speed of 12 00 ASA. T y p i c a l exposure times were 10 seconds a t a lens stop of f5.6. T h i s f i l m was found t o g i v e the b e s t r e s u l t s . I t was as f a s t as the Kodack High Speed Recording f i l m and had a f i n e g r a i n s i z e . C olour s l i d e s were a l s o taken of the f r i n g e p a t t e r n . The f i l m used was 35 mm Ektachrome d a y l i g h t f i l m developed a t a speed of 32 0 ASA. T y p i c a l exposure times were 10 seconds a t a l e n s stop of f 4 . 5. EXPERIMENTAL RESULTS AND DISCUSSION OF RESULTS 5.1 Geometry o f t h e H o l o g r a p h y Components The r e l a t i o n between f r i n g e p a t t e r n and a m p l i t u d e o f v i b r a t i o n i s a f u n c t i o n o f t h e o b j e c t i l l u m i n a t i o n a n g l e 9 t h e v i e w i n g a n g l e and t h e w a v e l e n g t h o f t h e l i g h t u s e d . F i g u r e 5.1 shows a s c h e m a t i c o f t h e h o l o g r a p h y s y s t e m , w i t h t h e s p a t i a l f i l t e r s and e x p a n d i n g l e n s e s o m i t t e d F i g u r e 5.1 S c h e m a t i c o f t h e H o l o g r a p h y S y s t e m The c o o r d i n a t e s o f t h e components a r e as f o l l o w s : V.B.S. = (0,975, 5) M x = (0.975, 20) M 2 = (4.1, 0) 0 = (31.6, 12.5) Ph = (31.6, 0). The o b j e c t i l l u m i n a t i o n angle 9 , : tan ft OPh 31.6- 4.1 = 2.2 ©r 0, • 6%6 8 The viewing angle @2 = ®° The wavelength used A <= 6328 A9 - 2<4.9 yUJrt. 5.2 Powell and St e t s o n Time Averaged Method A time averaged hologram was made of the membrane v i b r a t i n g a t i t s v a r i o u s resonant f r e q u e n c i e s . The resonant f r e q u e n c i e s of the membrane were determined by o b s e r v i n g the maximum output of the F o t o n i c Sensor on the o s c i l o s c o p e . F i g u r e 5.2 shows the membrane v i b r a t i n g a t i t s f u n -damental frequency a t 450 Hz. The amplitude of v i b r a t i o n a t the c e n t r e of the membrane as determined by the F o t o n i c Sensor was: 66.^ /jdn. 47 F i g u r e 5.2 Time Averaged F r i n g e P a t t e r n E x c i t a t i o n frequency - 450 Hz F o t o n i c Sensor output a t c e n t r e - 38 mv (p-p) F o t o n i c Sensor c a l i b r a t i o n f a c t o r - 3.5 u in/mv. The boundary of the membrane i s f i x e d and r e p r e s e n t s the zero order f r i n g e . Counting from the boundary inwards, a t the c e n t r e a l i g h t f r i n g e of the 8th order i s seen. The argument of the 8th peak of the zero order B e s s e l f u n c t i o n of the f i r s t k i n d can be found by i n t e r p o l a t i n g the arguments of the 8th and 9th zeros of the B e s s e l f u n c t i o n which are g i v e n by Brown [27]. I n t e r p o l a t i n g these v a l u e s , the 8th l i g h t f r i n g e g i v e s a value of 25.92. S u b s t i t u t i n g t h i s v a l u e i n t o 48 formula (2-6) the amplitude of v i b r a t i o n i s gi v e n as, Roots Of Jo m 2(cos@i -f COSGz) * Tt 24.9 2(.413 + l)V Tt 123 /Jan. The % d i f f e r e n c e between mp g and m i s ( 1 2 ' ^ ~ 5 6 6 ' 5 ) x 100% = 7.8%. The formula f o r the amplitude as developed by Powell and S t e t s o n has been checked e x p e r i m e n t a l l y by v a r i o u s people as being c o r r e c t . The F o t o n i c Sensor as used i n t h i s s e t up can t h e r e f o r e be used to check the formula f o r the other methods and w i l l be used as the r e f e r e n c e . F i g u r e A . l shows the v a r i o u s mode p a t t e r n s obtained by t h i s method. The decrease i n f r i n g e c o n t r a s t w i t h i n c r e a s i n g amplitude can be observed by comparing F i g u r e 5.2 and F i g u r e A . l a . F i g u r e A . l b i s a s t r i k i n g example of the disadvantage of t h i s method. The probe o f the F o t o n i c Sensor was l o c a t e d o f f c e n t r e a t the p o i n t of maximum amplitude. When the hologram was made, the output of the F o t o n i c Sensor showed a s i n u s o i d a l waveform 49 i n d i c a t i n g the membrane t o be v i b r a t i n g a t a resonant frequency. As we can see the mode shape has not developed f u l l y y e t , but t h i s c o u l d not be determined u n t i l a f t e r d e v e l o p i n g the hologram. In the r e a l time method the d e v e l o p i n g o f the mode shape can be seen and the exact n a t u r a l frequency can be d e t e r -mined . 5.3 Real Time Method Using Continuous I l l u m i n a t i o n The hologram of the s t a t i c o b j e c t was r e p o s i t i o n e d at the p h o t o p l a t e h o l d e r and the two microheads were a d j u s t e d t o cr e a t e a p a r a l l e l f r i n g e p a t t e r n . The membrane was e x c i t e d and observed under continuous i l l u m i n a t i o n . The change i n f r i n g e v i s i b i l i t y as suggested by Powell and St e t s o n was not observed by the author of t h i s t h e s i s . Instead, t h i s method r e s u l t e d i n the o b s e r v a t i o n o f the mode shape o f the o b j e c t . At the nodal areas, the o b j e c t remained s t a t i o n a r y and t h e r e f o r e the nodal areas o f the o b j e c t were v i s i b l e as u n d i s t u r b e d s e c t i o n s of the s t a t i c f r i n g e p a t t e r n . For c l e a r , w e l l d e f i n e d nodal p a t t e r n s , the o b j e c t must have a minimum amplitude of v i b r a t i o n f o r f r i n g e washout to occur. A c c o r d i n g t o [26], the amplitudes must be a t l e a s t s u f f i c i e n t to cause a f r i n g e movement of one q u a r t e r of the f r i n g e s p a c i n g . F i g u r e A-2 shows the mode shape of the membrane v i b r a t i n g a t 885 and 1400 Hz. _ A s l i g h t m o d i f i c a t i o n to the above method r e s u l t e d i n the o b s e r v a t i o n of the same•fringe 50 p a t t e r n s as o b t a i n e d by the Powell and S t e t s o n time averaged method. Instead of d e l i b e r a t e l y c r e a t i n g a mismatch between the o b j e c t and r e c o n s t r u c t e d image, the hologram was a d j u s t e d to minimize the s t a t i c f r i n g e p a t t e r n . Complete e l i m i n a t i o n of the s t a t i c f r i n g e p a t t e r n was i m p o s s i b l e and u s u a l l y one or two f r i n g e s were v i s i b l e a c r o s s the o b j e c t . The o b j e c t was e x c i t e d and again observed under continuous i l l u m i n a t i o n . F i g u r e A-3 shows the f r i n g e p a t t e r n of the membrane v i b r a t i n g a t 455 and 885 Hz. For the membrane v i b r a t i n g a t 455 Hz, the amplitude a t the c e n t r e of the membrane as measured by the F o t o n i c Sensor was: The f r i n g e p a t t e r n as shown i n F i g u r e A-3b was not w e l l d e f i n e d and i t was very d i f f i c u l t to determine the exact number of f r i n g e s . The 8th order l i g h t f r i n g e i s seen a t the c e n t r e of the membrane. S u b s t i t u t i n g the corresponding v a l u e of J Q i n t o equation (2-10), the amplitude of v i b r a t i o n i s o b t a i n e d as, Roots o f J « 7c 51 The v a l u e obtained by the f r i n g e theory was almost e x a c t l y h a l f the a c t u a l amplitude as determined by the F o t o n i c Sensor. T h i s method was repeated s e v e r a l times w i t h d i f f e r e n t amplitudes, but there was a c o n s i s t a n t d i f f e r e n c e by a f a c t o r of two. T h i s d i s c r e p e n c y c o u l d be due to the i n i t i a l mis-alignment. The s t a t i c displacement modulates the f r i n g e p a t t e r n by the f a c t o r cos (m • k) , equation ( 2 - 8 ) . i n t h i s experiment s -> -> 3 6 0 ° ° (m • k) reduces t o ^  = (—r (cos 8 + cos 8 „ ) m ) = [ ( 2 0 . 4 ) m ] S A X z s s The s t a t i c f r i n g e p a t t e r n i n F i g u r e A . 3 a shows th r e e dark f r i n g e s across the membrane, which means t h a t across the membrane m ' s v a r i e s from 0 to (2) ( 1 7 . 6 ) u i n . T h e r e f o r e a c r o s s the membrane i j j v a r i e s from 0 to 7 2 0 ° . As the dynamic f r i n g e s are formed a c c o r d i n g t o the zero order B e s s e l f u n c t i o n of the f i r s t k i n d , the f r i n g e c o n t r a s t w i l l reduce w i t h i n c r e a s i n g amplitude. Cos \}> w i l l reduce the f r i n g e c o n t r a s t even f u r t h e r and a c o n d i t i o n might a r i s e where the o r i g i n a l f r i n g e s cannot be separated, r e s u l t i n g i n the d i s c r e p e n c y of the c a l c u l a t i o n . The p i c t u r e s show the i n f e r i o r i t y of the f r i n g e p a t t e r n obtained by t h i s method. A l s o note the d i s c o n t i n u i t y of the f r i n g e s a t some p l a c e s , which c o u l d be due to the i n i t i a l misalignment. 5.4 Real Time Method Without An I n i t i a l F r i n g e P a t t e r n Using  S t r o b o s c o p i c I l l u m i n a t i o n The hologram of the s t a t i c o b j e c t was r e p o s i t i o n e d at the photo p l a t e h o l d e r . The two microheads were ad j u s t e d to 52 e l i m i n a t e the i n i t i a l misalignment, but p e r f e c t s u p e r p o s i t i o n of the o b j e c t and image c o u l d not be obtained r e s u l t i n g i n one or two r e s i d u a l f r i n g e s . At a resonant frequency the photo-t r a n s i s t o r was ad j u s t e d so t h a t the peak amplitude of the membrane c o i n c i d e d with the l a s e r p u l s e . F i g u r e A.4 shows the f r i n g e p a t t e r n s obtained by t h i s method, of the membrane v i b r a t i n g a t 458 and 875 Hz. The s t a t i c f r i n g e p a t t e r n showed one f r i n g e across the membrane. At 458 Hz, (Figure A.4b), the f r i n g e s c o n s i s t e d of c o n c e n t r i c c i r c l e s and the amplitude a t the c e n t r e of the membrane as measured by the F o t o n i c Sensor was: <TW - (0.5X9QX3.*) - 171.? /itrt. Counting from the boundary inwards a l i g h t f r i n g e of the 10th order i s seen a t the c e n t r e . S u b s t i t u t i n g the f r i n g e order i n -to equation (2-16), A  - (V-O (.4134-l) if3.4 yum. 53 The d i f f e r e n c e = 111.5 a 100 % » 7.7 % The d i f f e r e n c e can be a t t r i b u t e d to the i n i t i a l f r i n g e p a t t e r n the f r i n g e p a t t e r n , the amplitude of the F o t o n i c Sensor was f l u c t u a t i n g s l i g h t l y so t h a t an exact r e a d i n g c o u l d not be taken. F i g u r e A.4c shows the membrane v i b r a t i n g a t i t s second resonant frequency w i t h a d i a m e t r a l l i n e a t 875 Hz. The maximum amplitude f o r t h i s c o n d i t i o n i s : In F i g u r e A.4d a dark f r i n g e occurs a t the maximum amplitude, and from equation (2-17), the maximum amplitude of v i b r a t i o n i s , t h i s method. At 875 Hz, one h a l f of the o b j e c t was v i b r a t i n g 180° out of phase wi t h the other h a l f , but the f r i n g e p a t t e r n f o r both halves were i d e n t i c a l . and to the i n s t a b i l i t y of the system. During the r e c o r d i n g o f nn = (6Jfi7.6) = lOS.e /An. (I3jfd.e) = H4.4 yuirt. The phase of the o b j e c t cannot be determined by 54 5.5 Real Time Method With An I n i t i a l F r i n g e P a t t e r n Using  S t r o b o s c o p i c I l l u m i n a t i o n The hologram of the s t a t i c o b j e c t was r e p o s i t i o n e d a t the p h o t o p l a t e h o l d e r . The two micro heads were adjus t e d to c r e a t e a p a r a l l e l f r i n g e p a t t e r n w i t h a d e n s i t y of 4 f r i n g e s per i n c h . The o b j e c t was then e x c i t e d and observed under s t r o b o s c o p i c i l l u m i n a t i o n . F i g u r e A-5 shows the f r i n g e p a t t e r n s of the s t a t i c membrane and the membrane v i b r a t i n g a t 455, 475 and 885 Hz. The r e l a t i v e p o s i t i o n o f the o b j e c t w i t h r e s p e c t t o the r e c o n s t r u c t e d image was determined by a p p l y i n g a f o r c e a t the top o f the o b j e c t and ob s e r v i n g the change i n d e n s i t y of the s t a t i c f r i n g e p a t t e r n . In t h i s experiment the f o r c e was a p p l i e d towards the hologram r e s u l t i n g i n an i n c r e a s e i n f r i n g e d e n s i t y . T h i s i n d i c a t e d t h a t the top of the image was s l a n t e d away from the o b j e c t . F i g u r e 5.3a shows a p l o t o f the s t a t i c f r i n g e p a t t e r n and the dynamic f r i n g e p a t t e r n of the membrane v i b r a t i n g a t 4 55 Hz, superimposed on one another. By f o l l o w i n g the procedure as d i s c u s s e d i n S e c t i o n 2-5, a graph of the mode shape of the membrane acro s s the d i a m e t r a l l i n e AA i s obtained (Figure 5.3b). On the same graph the fundamental e i g e n f u n c t i o n normalized a t the c e n t r e i s p l o t t e d . The e i g e n f u n c t i o n f o r the fundamental mode i s giv e n by: fc404 Jfe) [31] 0 F i g u r e 5.3 F r i n g e Pattern: and Mode Shape a t 4 5 5 Hz r a d i u s o f t h e p o i n t o f i n t e r e s t d i a m e t e r o f t h e membrane The a m p l i t u d e a t t h e c e n t r e o f t h e membrane as r e c o r d e d by t h e F o t o n i c S e n s o r i s : m „ _ = ( 0 . 5 ) ( 3 8 ) ( 3 . 5 ) = 6 6 . 5 u i n . From r • b . F i g u r e 5 . 3 , t h e a m p l i t u d e a t t h e c e n t r e o f t h e membrane i s f o u n d t o be m = 6 1 . 8 u i n . The % d i f f e r e n c e b a s e d on t h e F o t o n i c S e n s o r i s 7 . 1 % . F i g u r e 5.4 shows t h e f r i n g e p a t t e r n and mode shape o f t h e membrane v i b r a t i n g a t 4 7 5 Hz. The F o t o n i c S e n s o r i n d i c a t e d an a m p l i t u d e m „ = ( 0 . 5 ) ( 4 2 ) ( 3 . 5 ) = 7 3 . 5 u i n . a t t h e c e n t r e o f t h e membrane. From F i g u r e 5.4 an a m p l i t u d e o f m = 6 1 . 8 u i n . was o b t a i n e d a t t h e c e n t r e o f t h e membrane r e p r e s e n t i n g a 1 6 % d i f f e r e n c e b a s e d on t h e F o t o n i c S e n s o r . A t 4 5 5 and 4 7 5 Hz, t h e membrane was v i b r a t i n g a t e i t h e r s i d e o f t h e e x a c t . f u n d a m e n t a l f r e q u e n c y o f 4 6 0 Hz. The change i n d i r e c t i o n o f t h e c u r v a t u r e s o f t h e f r i n g e p a t t e r n s between 4 55 and 4 75 Hz i n d i c a t e d t h a t t h e membrane u n d e r w e n t a 1 8 0 ° p h a s e s h i f t when i t p a s s e d i t s r e s o n a n t f r e q e u n c y o f 4 6 0 Hz. A t 4 5 5 Hz, t h e c a l c u l a t e d a m p l i t u d e a g r e e s q u i t e w e l l w i t h t h e a m p l i t u d e i n d i c a t e d by t h e F o t o n i c S e n s o r . A t 4 7 5 Hz t h e r e was a 1 6 % d i f f e r e n c e . T h i s d i f f e r e n c e was due t o t h e f a c t t h a t t h e l a s e r was n o t s t r o b e d e x a c t l y a t t h e maximum a m p l i t u d e o f t h e membrane ( f i n e c o n t r o l was v e r y d i f f i c u l t ) and t h e r e f o r e t h e l o w e r v a l u e o b t a i n e d f r o m t h e p l o t . The non symmetry.of t h e membrane mode shape c o u l d be a t t r i b u t e d t o 57 - 5 - 4 -3 - 2 -1 0 1 2 3 4 5 <S> 7 S 9 10 II 12 DISPLACEMENT C /im./ondsr F i g u r e 5.4 F r i n g e P a t t e r n and Mode Shape a t 4 7 5 Hz the f a c t t h a t the membrane was not s t r e s s e d u n i f o r m l y by the membrane t e n s i o n i n g d e v i c e . F i g u r e A.5d shows the f r i n g e p a t t e r n o f the membrane v i b r a t i n g a t i t s second resonant frequency o f 885 Hz. T h i s f r i n g e p a t t e r n c l e a r l y i n d i c a t e s the 180° phase d i f f e r e n c e between the two v i b r a t i n g h a l v e s of the membrane. 59 6. SUMMARY AND CONCLUSIONS 6.1 Summary I t was the purpose o f t h i s i n v e s t i g a t i o n to s e t up and e v a l u a t e the v a r i o u s holography systems f o r v i b r a t i o n a n a l y s i s developed t o date and to develop a r e a l time h o l o g r a p h i c i n t e r f e r o m e t r y technique f o r q u a n t i t a t i v e l y mapping the complete dynamic f i e l d o f a v i b r a t i n g s u r f a c e , where the amplitudes o f v i b r a t i o n are on the order o f the wavelength o f l i g h t used. The o b j e c t chosen f o r study was a c i r c u l a r membrane, three inches i n diameter, made of 0.0005 inches t h i c k a luminized Mylar. The Powell and St e t s o n time averaged method i s the s i m p l e s t method to use f o r v i b r a t i o n v i s u a l i z a t i o n , as no e l a b o r a t e r e p o s i t i o n i n g of the the hologram and no e l a b o r a t e s t r o b i n g arrangement i s r e q u i r e d . T h i s method i s not a r e a l time method and the amplitudes o f v i b r a t i o n are l i m i t e d to about 20 f r i n g e s . For the r e a l time methods a c c u r a t e r e p o s i t i o n i n g of the hologram i s r e q u i r e d . The v i b r a t i o n c h a r a c t e r i s t i c s o f the o b j e c t can be s t u d i e d and observed i n r e a l time. The continuous i l l u m i n a t i o n r e a l time method does not r e q u i r e the s t r o b i n g of the l a s e r beam. The f r i n g e con-t r a s t i s very poor and h i g h l y dependent on the i n i t i a l m i s a l i g n -ment. No accurate q u a n t i t a t i v e amplitude measurements were ob t a i n e d . 60 The s t r o b o s c o p i c r e a l time method r e q u i r e s the s t r o b -i n g of the l a s e r . The s t r o b i n g mechanism c o n s i s t e d of an alum-inum d i s c d r i v e n by a v a r i a b l e speed D.C. motor. T h i s s t r o b i n g mechanism was not too s a t i s f a c t o r y as the speed o f the motor v a r i e d c o n t i n u o u s l y and the r e was a l s o a c o n s i d e r a b l e l o s s o f l i g h t through the s t r o b i n g system. In the s t r o b o s c o p i c r e a l time method without an i n i t i a l f r i n g e p a t t e r n , the mode shape and amplitude o f v i b r a -t i o n o f the o b j e c t can be determined d i r e c t l y from the f r i n g e p a t t e r n . The minimum amplitude r e q u i r e d f o r the formation of a f r i n g e i s dependent on the geometry of the holography system, wh i l e the maximum amplitude i s determined by the c a p a b i l i t y of the eye or photodensitometer to d i s t i n g u i s h the f r i n g e s . The f r i n g e p a t t e r n does not i n d i c a t e any phase r e l a t i o n . The s t r o b o s c o p i c r e a l time method with an i n i t i a l f r i n g e p a t t e r n , r e q u i r e s a c c u r a t e c o n t r o l of the hologram r e -p o s i t i o n i n g to c r e a t e the i n i t i a l s t a t i c f r i n g e p a t t e r n . The a n a l y s i s of the f r i n g e p a t t e r n i s more complicated and a p l o t of the s t a t i c and dynamic f r i n g e p a t t e r n s must be made to determine the mode shape and amplitude o f v i b r a t i o n . However, the phase of the o b j e c t can be determined. In a d d i t i o n , the mode shape of the o b j e c t can a l s o be determined by us i n g con-tinuous i l l u m i n a t i o n i n s t e a d of s t r o b o s c o p i c i l l u m i n a t i o n s i n c e the nodal l i n e s become v i s i b l e as und i s t u r b e d p o r t i o n s of the i n i t i a l f r i n g e p a t t e r n w h i l e the remainder of the p a t t e r n v/ashes out. In the course of the experiment f u l l f i e l d coverage 61 o f t h e o b j e c t was o b t a i n e d , e v e n when t h e maximum a m p l i t u d e o f v i b r a t i o n was o n l y a f r a c t i o n o f t h e minimum a m p l i t u d e r e -q u i r e d f o r a f r i n g e t o f o r m i n t h e p r e v i o u s method. 6.2 C o n c l u s i o n s H o l o g r a p h i c i n t e r f e r o m e t r y i s a v e r y u s e f u l t o o l i n v i b r a t i o n a n a l y s i s o f t h r e e d i m e n s i o n a l o b j e c t s . Due t o t h e s t a b i l i t y r e q u i r e m e n t s and t h e u s e o f a c o h e r e n t l i g h t s o u r c e , t h e a n a l y s i s i s l i m i t e d t o t h e l a b o r a t o r y . The P o w e l l and S t e t s o n t i m e a v e r a g e d method i s t h e s i m p l e s t method, b u t has l i m i t e d u s e as t h e v i b r a t i o n a n a l y s i s c a n n o t be done i n r e a l t i m e . The c o n t i n u o u s i l l u m i n a t i o n r e a l t i m e method a l l o w s t h e q u a l i t a t i v e a n a l y s i s o f t h e v i b r a t i n g o b j e c t i n r e a l t i m e . When t h e h o l o g r a m i s s l i g h t l y d i s p l a c e d t h e mode shape o f t h e o b j e c t c a n be r e a d i l y o b s e r v e d . The s t r o b o s c o p i c r e a l t i m e method i s t h e most power-f u l method, b u t r e q u i r e s an e l a b o r a t e s t r o b i n g mechanism. W i t h o u t an i n i t i a l f r i n g e p a t t e r n , t h e a m p l i t u d e o f v i b r a t i o n and mode shape c a n be r e a d i l y d e t e r m i n e d . The a c c u r a c y o f t h e c a l c u l a t e d a m p l i t u d e d e pends on t h e i n i t i a l m i s a l i g n m e n t o f t h e h o l o g r a m . I n t h i s e x p e r i m e n t a minimum a m p l i t u d e o f 17.6 u i n . i s r e q u i r e d f o r t h e f o r m a t i o n o f a f r i n g e . W i t h an i n i t i a l f r i n g e p a t t e r n t h e mode shape and a m p l i t u d e o f v i b r a t i o n c a n o n l y be d e t e r m i n e d a f t e r p l o t t i n g both the static and dynamic fringe pattern. The phase of the object can be determined and the exact resonant frequency can be pinpointed by observing the phase change of the vibrating object when i t passes through the resonant frequency. Amplitudes in the order of a couple of micro inches can be determined. 6.3 Suggestions for Future Research The holography table isolated by the four a ir tubes was not sufficient to completely eliminate low frequency vibra-t ion. It is recommended to build a heavy concrete table with a steel top on a foundation isolated from the building and supported by layers of damping material to eliminate this low frequency vibration. Because of the c r i t i c a l requirements of the strobing mechanism i t is recommended to replace the mechanical strobing system by an electro-optical system consisting of a Pockels-c e l l and the required electronics to trigger the Pockels-cell, provide the required signal synchronization and variable pulse width. It is recommended to look into the poss ib i l i ty of using a recording medium which can be developed by a dry process on s i te . Hughes Aircraft Company are in the process of develop-ing Polymer plates which are sensitive to l ight and are fixed by the ul tra violet part of the spectrum [ 3 2 ] . Haines and Hildebrand [ 3 3 ] are the only ones to investigate experimentally the formation of interference fringes in holographic interferometry. I t i s recommended to i n v e s t i g a t e the theory of the f r i n g e f o r -mation f u r t h e r and e s t a b l i s h e x p e r i m e n t a l l y the e f f e c t s of pure t r a n s l a t i o n , pure r o t a t i o n or a combination of t r a n s l a t i o n and r o t a t i o n on the f r i n g e f ormation and l o c a t i o n . I t i s e s s e n t i a l t h a t the hologram be developed on s i t e so t h a t no i n i t i a l misalignment i s i n t r o d u c e d . I t i s recommended to i n v e s t i g a t e the p o s s i b i l i t y o f us i n g the continuous i l l u m i n a t i o n r e a l time method to analyze t r a n s i e n t and random v i b r a t i o n . Again i t i s e s s e n t i a l t h a t the hologram be developed on s i t e . The holography systems developed so f a r r e q u i r e d a f i x e d datum f o r the o b j e c t . I t i s recommended to look i n t o the p o s s i b i l i t y of e l i m i n a t i n g t h i s requirement by a l l o w i n g the r e f e r e n c e beam to monitor the l i n e a r motion of the o b j e c t . 64 BIBLIOGRAPHY 1. O s t e r b e r g , H., "An I n t e r f e r o m e t e r Method o f S t u d y i n g t h e V i b r a t i o n s o f 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  o f t h e O p t i c a l S o c i e t y o f A m e r i c a , V o l . 22, No. 19, 1932, pp. 19-35. 2. P o w e l l , R.L. and S t e t s o n , 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 W a v e f r o n t R e c o n s t r u c t i o n , " J o u r n a l o f t h e  O p t i c a l S o c i e t y o f A m e r i c a , V o l . 55, No. 12, 1965, pp. 1593-1598. 3. 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Lord R a y l e i g h , (J.W. S t r u t t ) , The Theory of Sound, V o l . 1, 1945, Dover p u b l i c a t i o n s . 32. Radio and E l e c t r o n i c s , V o l . 41, No. 2, 1970, p. 4. 33. Haines, K.A. and H i l d e b r a n d , B.P., "Surface - Deformation Measurement Using the Wavefront R e c o n s t r u c t i o n Technique," A p p l i e d O p t i c s , V o l . 5, No. 4, 1966, pp. 595-602. 67 A P P E N D I X I 68 (a) 450 Hz. (c) 1400 Hz. Figure A - l Time Averaged (b) 885 Hz. (d) 2370 Hz. Patterns 70 (a) S t a t i c 71 (c) 875 Hz. (d) 875 Hz. F i g u r e A-4 S t r o b o s c o p i c Real Time Without an I n i t i a l F r i n g e P a t t e r n (a) S t a t i c (b) 4 5 5 Hz. mF < s > ~ 3 8 m v (P-P) F i g u r e A-5 S t r o b o s c o p i c Real Time With an I n i t i a l F r i n g e P a t t e r n 

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