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UBC Theses and Dissertations

Experimental investigation of nonlinear coupled vibrations of columns Johnson, Dale P. 1970

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EXPERIMENTAL INVESTIGATION OF NONLINEAR COUPLED VIBRATIONS OF COLUMNS by Dale P. Johnson B . A . S c , University of Brit ish Columbia, 1968 A'Thesis Submi tted-i-n Partial FulfiJJment of the Requirements for.the Degree <3f — Master of Applied Science In the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 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 that 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 . D a l e P. J o h n s o n Depar tment o f M e c h a n i c a l E n g i n e e r i n g 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 Date M a y 2 5 , 1 9 7 0 TABLE OF CONTENTS ABSTRACT ACKNOWLEDGEMENT... L I S T OF FIGURES... L I S T OF TABLES L I S T OF APPENDICES NOMENCLATURE CHAPTER I . INTRODUCTION P r e l i m i n a r y Remarks L i t e r a t u r e Review L i m i t a t i o n s o f I n v e s t i g a t i o n , . CHAPTER I I THEORY D i f f e r e n t i a l E q u a t i o n s o f M o t i o n f o r a Column.... T h e o r e t i c a l P r e d i c t i o n s . . T o r s i o n a l C o u p l i n g CHAPTER I I I APPARATUS AND INSTRUMENTATION G e n e r a l O u t l i n e E l e c t r o n i c S y stem . L o a d i n g Frame and Column D e s c r i p t i o n D e t a i l s o f M e a s u r i n g S y s t e m . . . CHAPTER I V TEST PROCEDURE C a l i b r a t i o n T e s t i n g . . . . P h o t o g r a p h y Page CHAPTER V RESULTS AND DISCUSSION. 3 8 I n t e r p r e t a t i o n o f F r e q u e n c y S p e c t r a 38 I d e n t i f i c a t i o n o f S t r a i n Peaks 39 R e s u l t s o f F l e x u r a l S t r a i n R e c o r d 42 R e s u l t s o f A x i a l S t r a i n R e c o r d 48 CHAPTER VI SUMMARY AND CONCLUSIONS 59 Summary 59 C o n c l u s i o n s 61 S u g g e s t i o n s f o r F u r t h e r Re-s e a r c h 64 BIBLIOGRAPHY 6 7 APPENDICES 68 ABSTRACT C o u p l i n g o f t h e f l e x u r a l , l o n g i t u d i n a l , and t o r s i o n a l v i b r a t i o n modes o f a column s u b j e c t e d t o p e r i o d i c a x i a l l o a d i n g was a n a l y t i c a l l y and e x p e r i m e n t a l l y i n v e s t i g a t e d . The i n i t i a l c r o o k e d n e s s o f t h e column and l o n g i t u d i n a l i n e r t i a g i v e r i s e t o c o u p l e d f l e x u r a l - l o n g i t u d m a l v i b r a -t i o n s . F u r t h e r , t h e Weber e f f e c t and l o n g i t u d i n a l i n e r t i a r e s u l t i n c o u p l i n g between l o n g i t u d i n a l and t o r s i o n a l o s c i l l a t i o n s . To a s s e s s t h e v a l i d i t y o f t h e t h e o r y , an e x p e r i m e n t a l a p p a r a t u s was s e t up t o a x i a l l y e x c i t e a column u s i n g a v i b r a t i o n c o n t r o l g e n e r a t o r and an e l e c t r o m a g n e t i c s h a k e r . The e x p e r i m e n t a l r e s u l t s were i n good agreement w i t h t h e t h e o r e t i c a l p r e d i c t i o n s . C o u p l e d l o n g i t u d i n a l v i b r a t i o n s e x h i b i t i n g a f r e q u e n c y r a t i o o f 1:2 were o b s e r v e d . C o u p l e d f l e x u r a l o s c i l l a t i o n s were a l s o o b s e r v e d , t h o u g h a f r e q u e n c y r a t i o was n o t e s t a b l i s h e d . " F u r t h e r , t h e e x p e r i m e n t a l r e s u l t s s u g g e s t t h a t c o u p l e d v i b r a t i o n s o t h e r t h a n t h o s e t h e o r e t i c a l l y e x p e c t e d were p r e s e n t . In p a r t i c u l a r , a l o n g i t u d i n a l c o u p l e d v i b r a t i o n w i t h a f r e q u e n c y r a t i o o f 1:3 was o b s e r v e d , and a c o r r e s p o n d i n g c o u p l e d f l e x u r a l o s c i l l a t i o n was p r e s e n t . A c o u p l e d t o r s i o n a l mode was e x p e r i m e n t a l l y o b s e r v e d when t h e a p p l i e d f r e q u e n c y was t w i c e t h e f u n d a m e n t a l t o r -s i o n a l f r e q u e n c y . A s e c o n d c o u p l e d t o r s i o n a l mode a p p e a r e d - 11 -when t h e e x c i t a t i o n f r e q u e n c y was t h r e e t i m e s t h e f u n d a m e n t a l t o r s i o n a l f r e q u e n c y . The p h a s e r e l a t i o n s h i p between t h e c o u p l e d v i b r a t i o n s was o b s e r v e d . The r e s o n a n t c o u p l e d v i b r a t i o n s were f o u n d t o be s i g n i f i c a n t a t c e r t a i n f r e q u e n c i e s . \ ACKNOWLEDGEMENT My p a r t i c u l a r g r a t i t u d e i s e x t e n d e d t o my f a c u l t y ad-v i s o r s , Dr. C R . H a z e l l and Dr. H. Ramsay, f o r i n v i t i n g me t o p a r t i c i p a t e i n t h i s work and f o r t h e i r a s s i s t a n c e t h r o u g h -o u t t h e d u r a t i o n o f t h e p r o j e c t . I a l s o w i s h t o th a n k t h e t e c h n i c a l s t a f f , and i n p a r t i -c u l a r , 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 i n v a l u a b l e c o - o p e r a t i o n i n s e t t i n g up t h e a p p a r a t u s . The c a r e and t i m e s p e n t by M i s s D i a n e J o h n s o n i n t y p i n g t h e t h e s i s , and by Mr. S i n g L e i m i n a s s i s t i n g w i t h t h e p h o t o g r a p h y , i s a p p r e c i a t e d . T h i s p r o j e c t 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. 66-9510 p r o v i d e d by t h e D e f e n s e R e s e a r c h B o a r d o f Canada. - i v -L I S T OF FIGURES F i g u r e Page Fig.. I I - l R e f e r e n c e A x i s F o r D i s p l a c e m e n t Measurements 9 F i g . I I - 2 E l a s t i c B a r Under A x i a l L o a d i n g 15 F i g . I I I - l S i g n a l F low C h a r t 19 F i g . I I I - 2 P h o t o g r a p h o f E x p e r i m e n t a l A p p a r a t u s 20 F i g . I I I - 3 S c h e m a t i c View o f Lower Column Mount 21 F i g . I I I - 4 Column M o u n t i n g and L o a d i n g S y s t e m . . 22 F i g . I I I - 5 S e c t i o n a l D r a w i n g o f L o a d i n g and A l i g n m e n t Mechanism 2 4 F i g . I I I - 6 S c h e m a t i c View o f Column S p e c i m e n . . . 25 F i g . I I I - 7 P h o t o g r a p h o f Spe c i m e n L o a d i n g Frame 26 F i g . I I I - 8 S c h e m a t i c S t r a i n Gauge A r r a n g e m e n t F o r F l e x u r a l S t r a i n Measurement... 28 F i g . I I I - 9 S c h e m a t i c S t r a i n Gauge A r r a n g e m e n t F o r L o n g i t u d i n a l S t r a i n M e a s u r e -ment 29 F i g . V - l F o r c e d R e s o n a n t F r e q u e n c i e s P l o t t e d V e r s u s Mode Number F o r V a r i o u s End C o n d i t i o n s 40 F i g . V-2 F l e x u r a l S t r a i n R e c o r d O b t a i n e d A t M i d p o i n t o f Column 4 2 F i g . V-3 P o s s i b l e F l e x u r a l Mode Shapes F o r A x i a l l y E x c i t e d Column 4 3 F i g . V-4 F r e q u e n c i e s o f E x p e r i m e n t a l l y O b s e r v e d F l e x u r a l R e s o n a n t S t r a i n Peaks 45 F i g . V-5 Waveforms f o r O b s e r v e d C o u p l e d F l e x u r a l R e s o n a n c e s 46 - v -F i g u r e Page F i g . V-6 A x i a l S t r a i n R e c o r d O b t a i n e d a t M i d p o i n t o f Column. 49 F i g . V-7 A P o r t i o n o f a Column U n d e r g o i n g T w i s t i n g 51 F i g . V-8 F r e q u e n c i e s o f E x p e r i m e n t a l l y O b s e r v e d R e s o n a n t A x i a l S t r a i n Peaks 51 F i g . V-9 Waveforms a t F u n d a m e n t a l L o n g i t u d i -n a l Resonance 5 2 F i g . V-10 Waveforms a t S e c o n d C o u p l e d A x i a l Mode 5 4 F i g . V - l l Waveforms a t T h i r d C o u p l e d A x i a l Mode 5 5 F i g . V-12 Phase R e l a t i o n s h i p Between V i b r a t i o n Components a t S e c o n d C o u p l e d A x i a l Mode 56 F i g . V-13 Waveforms f o r C o u p l e d T o r s i o n a l Modes 5 7 F i g . A - l E x t e n s i o n and R o t a t i o n o f C e n t r a l P l a n e F i b e r 69 - v i -L I S T OF TABLES Page T a b l e V - l P e r m i s s a b l e A c c e l e r a t i o n L e v e l s 35 Over V a r i o u s F r e q u e n c y Ranges F o r Column S p e c i m e n - v i i -Page APPENDIX A S t r a i n E x p r e s s i o n f o r Column W i t h Some I n i t i a l C r o o k e d n e s s 6 8 APPENDIX B D e t a i l s o f E l e c t r o n i c E q u i p m e n t Used f o r V i b r a t i o n C o n t r o l S y s tem.. 7 1 APPENDIX C L i n e a r D i f f e r e n t i a l E q u a t i o n s o f M o t i o n f o r a Column 7 3 - v i i i -NOMENCLATURE Symbol P = t o t a l a p p l i e d a x i a l l o a d P0 = constant a p p l i e d a x i a l l o a d R. = amplitude of v a r i a b l e a p p l i e d a x i a l l o a d V" = frequency o f a p p l i e d a x i a l l o a d 6 X = s t r a i n of c e n t r a l plane f i b e r of column SIQ = bending s t r a i n 6 = t o t a l s t r a i n of a column f i b e r u - l o n g i t u d i n a l displacement w = i n i t i a l t r a n s v e r s e displacement w = dynamic t r a n s v e r s e displacement 0 = r o t a t i o n of c r o s s - s e c t i o n -x,y,z = co o r d i n a t e d i s t a n c e s V = s t r a i n energy T = k i n e t i c energy W = work done by e x t e r n a l l o a d i n g v - volume of column A = area o f c r o s s - s e c t i o n of column b = width of column h = he i g h t h o f column L = len g t h o f column s = f i b e r l e n g t h 1 = minimum moment of i n e r t i a of column c r o s s -s e c t i o n I = constant depending on dimenstions of c r o s s -s e c t i o n - i x -I 2 Io E G <t> t OC p i i w t g H, H 1 ( x ) , H 2 ( X ) , H 3 ( x ) X(x) , F(x) , G1(y.) t G 2 (x) wx, wx, u x A b b r e v i a t i o n s BAM CRO Hz RMS moment o f i n e r t i -3 a b o u t y a x i s p o l a r moment o f n e r t i a o f cross-s<. * t i o n e l a s t i c modulus s h e a r modulus d e n s i t y t i m e phase a n g l e f u n d a m e n t a l l o n g i t u d i n a l f r e q u e n c y r e s o n a n t t r a n s v e r s e f r e q i a n c i e s i n t e g e r r e s o n a n t t o r s i o n a l f r e q u e ;cy a c c e l e r a t i o n o f g r a v i t y f u n c t i o n s o f v a r i a b l e x c l y p a r t i a l d e r i v a t i v e o f a d s p l a c e m e n t w i t h r e s p e c t t o t h e v a r i a b l e x B r i d g e a m p l i f i e r and metex C a t h o d e r a y o s c i l l o s c o p e C y c l e s p e r s e c o n d Root-mean-square v a l u e o f a f u n c t i o n CHAPT ntrocSuctiort - 1 -CHAPTER I INTRODUCTION P r e l i m i n a r y Remarks In t h e modern day p r o d u c t i o n o f t u r b i n e s , a i r c r a f t , s p a c e c r a f t , and many o t h e r v a r i e t i e s o f h i g h s p e e d e q u i p -ment, t h e s t r u g g l e a g a i n s t u n d e s i r a b l e v i b r a t i o n i s b e i n g r a p i d l y i n t e n s i f i e d . H i g h f r e q u e n c y v i b r a t i o n s p r o v i d e d ynamic e x c i t a t i o n t o s t r u c t u r e s s u c h as columns. Thus t h e r e s p o n s e o f a column s u b j e c t e d t o p e r i o d i c a x i a l l o a d i n g becomes i m p o r t a n t f r o m a d e s i g n v i e w p o i n t . S i m p l e l i n e a r t h e o r y o f t e n does n o t p r o v i d e s u f f i c i e n t l y a c c u r a t e i n f o r -m a t i o n a t f r e q u e n c i e s o f o r d e r 10 KHz. The f o c u s o f t h i s i n v e s t i g a t i o n i s on s e c o n d o r d e r n o n l i n e a r c o u p l e d e f f e c t s i n column v i b r a t i o n s , w h i c h a t f r e q u e n c i e s o f s e v e r a l KHz can be q u i t e s i g n i f i c a n t . An e l a s t i c column s u s t a i n i n g a p e r i o d i c a x i a l l o a d o f t h e f o r m P = P D + P, c o s V t , where t i s t i m e and P has a c i r c u -l a r f r e q u e n c y V, may be e x c i t e d i n t o t r a n s v e r s e and t o r s i o n a l modes o f v i b r a t i o n . P a r a m e t r i c s t a b i l i t y s t u d i e s on t h e t r a n s v e r s e v i b r a t i o n s o f a column under dynamic p e r i o d i c l o a d i n g i n t h e a x i a l d i r e c t i o n have been made by a number o f i n v e s t i g a t o r s . T h e s e i n v e s t i g a t i o n s have f o c u s e d p r i -m a r i l y on s m a l l f r e e l a t e r a l v i b r a t i o n s i n t h e f i r s t s p a t i a l made f o r an i n i t i a l l y s t r a i g h t column. As a c o n s e q u e n c e t h e s e r e s e a r c h e r s have i n t e r e s t e d t h e m s e l v e s i n r e l a t i v e l y - 2 -low a n g u l a r f r e q u e n c i e s o f e x c i t a t i o n . Towards t h e u p p e r e n d o f t h e a c o u s t i c f r e q u e n c y r a n g e e n c o u n t e r e d i n t h e p r e s e n t i n v e s t i g a t i o n a s i m p l e l i n e a r t h e o r y i s i n a c c u r a t e due t o l o n g i t u d i n a l i n e r t i a , p a r a m e t r i c c o u p l i n g phenomena and a h o s t o f o t h e r n o n l i n e a r i t i e s . However, a l i n e a r a n a l y s i s does p o i n t o u t t h a t t h e wave-l e n g t h s o f f l e x u r a l and l o n g i t u d i n a l v i b r a t i o n s become s i m i l a r i n o r d e r o f m a g n i t u d e a t t h e s e r e l a t i v e l y h i g h m e c h a n i c a l f r e q u e n c i e s . I t i s t h e r e f o r e r e a s o n a b l e t o e x p e c t i n t e r a c t i o n between t h e two modes o f o s c i l l a t i o n . S c h n e i d e r [9] h a s shown t h a t a t p a r t i c u l a r f r e q u e n c i e s o f t r a n s v e r s e e x c i t a t i o n c o u p l i n g c a u s e s l o n g i t u d i n a l o s c i l l a -t i o n s w i t h d i s c r e t e f r e q u e n c i e s i n a b a r . I n t h i s work, i t i s shown t h a t t h i s n o n l i n e a r c o u p l i n g e f f e c t works i n t h e r e v e r s e d i r e c t i o n a l s o ; namely, i f a column i s e x c i t e d l o n g i -t u d i n a l l y w i t h a f r e q u e n c y t h a t i s one h a l f o r one t h i r d t h e f u n d a m e n t a l r e s o n a n t l o n g i t u d i n a l f r e q u e n c y , i t i s p o s s i b l e t o e x c i t e t h e b a r i n t o f l e x u r a l o s c i l l a t i o n s w i t h s i g n i f i c a n t a m p l i t u d e s . T o r s i o n r e s u l t i n g i n r o t a t i o n o f t h e c r o s s s e c t i o n s o f a s t e e l column can r e s u l t i n a s h o r t e n i n g i n t h e a x i a l d i r e c t i o n o f t h e o r d e r o f t h e s q u a r e o f t h e r o t a t i o n . Thus t h e p o s s i b l e p a r a m e t r i c c o u p l i n g between t o r s i o n a l and l o n g i -t u d i n a l movements due t o t h e s h o r t e n i n g e f f e c t becomes a p p a r e n t . T s o [8] has shown t h a t i f a b a r i s u n d e r f o r c e d - 3 -l o n g i t u d i n a l e x c i t a t i o n w i t h a f r e q u e n c y w h i c h i s c l o s e t o t w i c e a n a t u r a l t o r s i o n a l f r e q u e n c y , i t i s p o s s i b l e t o e x c i t e t h e column i n t o t o r s i o n a l o s c i l l a t i o n s . The e q u a t i o n s o f m o t i o n a r e p r e s e n t e d w i t h o u t a c l o s e d -f o r m s o l u t i o n . The e q u a t i o n s a r e m a n i p u l a t e d , however, so t h a t t h e s e c o n d o r d e r c o u p l i n g t e r m s can be c o n v e n i e n t l y i n t e r p r e t e d f r o m a p h y s i c a l v i e w p o i n t . An e x p e r i m e n t a l s t u d y was made on t h e dynamic r e s p o n s e o f f l e x u r a l and t o r s i o n a l v i b r a t i o n s o f a column u n d e r p e r i o -d i c a x i a l l o a d i n g . S e v e r a l methods were a t t e m p t e d t o m o n i -t o r t h e column r e s p o n s e s and t o d e t e r m i n e t h e f r e q u e n c y s p e c t r u m f o r t h e s e c o u p l e d r e s o n a n c e s . O b s e r v e d r e s o n a n t f o r c e d v i b r a t i o n s and n o n l i n e a r c o u p l e d v i b r a t i o n s a r e compared w i t h t h o s e p r e d i c t e d by 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 . Of p a r t i c u l a r i m p o r t a n c e i n t h e s e e x p e r i m e n t s i s t h a t n o n -l i n e a r c o u p l i n g phenomena a r i s i n g f r o m a d m i s s i o n o f s t r a i n s due t o i n i t i a l c r o o k e d n e s s and l o n g i t u d i n a l i n e r t i a were c o n s i d e r e d . The e x p e r i m e n t a l a p p a r a t u s , w h i c h i s d e s c r i b e d i n C h a p t e r I I I , was a r r a n g e d so t h a t t h e p a r a m e t e r s P 0 , P, and Y can be c o n t r o l l e d i n d e p e n d e n t l y o f e a c h o t h e r . T h i s f e a t u r e a l l o w s f o r more r e a l i s t i c t h e o r e t i c a l m o d e l l i n g . S e v e r a l methods were a t t e m p t e d t o m o n i t o r c o u p l e d v i b r a t i o n s , and t o o b t a i n t h e f r e q u e n c y s p e c t r u m f o r t h e s e o s c i l l a t i o n s . L i t e r a t u r e Review P r e v i o u s r e s e a r c h c o n n e c t e d w i t h t h e p a r a m e t r i c r e s p o n s e - A -of bars and columns has focused primarily on parametric vibrations associated with small free lateral vibrations in the f i r s t spatial mode. The investigations have been restricted to relatively low angular frequencies of excita-t ion, not more than three or four times the fundamental lateral resonant frequency. Beliaev [2] was the f i r s t to analyze the parametric response of a column under time-dependent longitudinal ex-ci tat ion, and he reduced the equation of motion to the stan-dard Mathieu-Hill equation. He did not consider the influence of longitudinal iner t ia . Somerset [4] and Bolatin [6] experimentally verif ied that the s tabi l i ty of the column described by Beliaev could be analyzed by investigating the s tabi l i ty of the solutions of the Mathieu-Hill equation of motion for the column. An unstable region was found characterized by lateral column osci l lat ions at exactly one-half the excitation frequency. Experimentally observed ef-fects not anticipated by this equation were attributed to longitudinal inert ia , l inear and nonlinear damping, non-linear e las t i c i ty , rotary iner t ia , and internal f r i c t ion . Somerset and Evan-Iwanowski [ 7 ] theoretically and ex-perimentally investigated the parametric ins tabi l i ty of straight columns sustaining a periodic axial load of low circular frequency. They also brief ly considered the i n -fluence of damping and of nonlinearities which are amplitude dependent, such as nonlinear e las t ic i ty and longitudinal i n e r t i a . They f o u n d t h a t t h e column i s u n s t a b l e o v e r c e r t a i n r e g i o n s o f t h e ( P Q , P I , V " ) p a r a m e t r i c s p a c e , and t h a t t h e p r e s e n c e o f n o n l i n e a r f a c t o r s may t e n d t o r e n d e r t h e column i n c r e a s i n g l y u n s t a b l e as t h e a m p l i t u d e o f l a t e r a l v i b r a t i o n i n c r e a s e s . Numerous o t h e r r e f e r e n c e s on t h e t o p i c o f p a r a -m e t r i c v i b r a t i o n s can be f o u n d i n a s u r v e y a r t i c l e by E v a n -Iwanowski [5 ] . T s o [ 8 ] has s t u d i e d a companion p r o b l e m o f t h e p a r a -m e t r i c t o r s i o n a l s t a b i l i t y o f an e l a s t i c c a n t i l e v e r o f r e c t a n g u l a r c r o s s - s e c t i o n u n d e r dynamic a x i a l l o a d i n g . He showed a n a l y t i c a l l y t h a t i f a b a r i s u n d e r f o r c e d l o n g i t u d i -n a l e x c i t a t i o n w i t h a f r e q u e n c y w h i c h i s c l o s e t o t w i c e a n a t u r a l t o r s i o n a l f r e q u e n c y , i t i s p o s s i b l e t o e x c i t e t h e b a r i n t o t o r s i o n a l o s c i l l a t i o n s . F u r t h e r , when t h e f r e q u e n c y o f t h e a p p l i e d l o a d i n g becomes c l o s e t o t h e f u n d a m e n t a l l o n g i t u d i n a l f r e q u e n c y , t h e e f f e c t o f l o n g i t u d i n a l i n e r t i a on t h e t o r s i o n a l s t a b i l i t y o f t h e b a r has t o be t a k e n i n t o c o n s i d e r a t i o n . The p r e s e n t work c o n s i d e r s thfe r e s p o n s e o f a column c l a m p e d a t b o t h ends w i t h t h e l o w e r end a x i a l l y e x c i t e d . The e q u a t i o n s o f m o t i o n a c c o u n t i n g f o r i n i t i a l c r o o k e d n e s s and l o n g i t u d i n a l i n e r t i a a r e c o n s i d e r e d a t f r e q u e n c i e s up t o and above t h e f u n d a m e n t a l r e s o n a n t l o n g i t u d i n a l f r e q u e n c y . To t h e k nowledge o f t h e a u t h o r , c o u p l e d v i b r a t i o n s i n b a r s r e -s u l t i n g from a d m i s s i o n o f s t r a i n s i n t h e c e n t r a l p l a n e h a v e been t r e a t e d o n l y by S c h n e i d e r [9] i n t h e p a s t . He e x c i t e d a b a r t r a n s v e r s e l y and i n v e s t i g a t e d t h e c o u p l e d l o n g i t u d i n a l - 6 -o s c i l l a t i o n s . He f o u n d t h a t r e s o n a n t l o n g i t u d i n a l o s c i l l a -t i o n s o c c u r a t 1 and 1/2 t i m e s t h e t r a n s v e r s e e x c i t a t i o n a t t h e f u n d a m e n t a l l o n g i t u d i n a l r e s o n a n t v i b r a t i o n . L i m i t a t i o n s o f I n v e s t i g a t i o n The i n v e s t i g a t i o n h as s e v e r a l i m p o r t a n t l i m i t a t i o n s . F i r s t l y , t h e t h e o r e t i c a l e q u a t i o n s d e r i v e d a r e n o t e x a c t . A c l o s e d f o r m s o l u t i o n i s n o t a v a i l a b l e , s o o n l y t h e f r e -q u e n c i e s o f c o u p l e d v i b r a t i o n s a r e i n d i c a t e d . A n a l y t i c a l l y , n o t h i n g i s s a i d a b o u t t h e a m p l i t u d e s o f c o u p l e d r e s o n a n t o s c i l l a t i o n s . The B e r n o u l l i - E u l e r t h e o r y o f f l e x u r e i s u s e d , l i m i t i n g t h e a n a l y s i s t o a n e a r l y s t r a i g h t e l a s t i c i s o t r o p i c column u n i f o r m a l o n g i t s l e n g t h and c a p a b l e o f d e v e l o p i n g b e n d i n g s t r e s s e s . The c r o s s - s e c t i o n i s assumed t o be s y m m e t r i c a l a b o u t t h e p l a n e o f l o a d i n g so t h a t b e n d i n g o c c u r s i n t h e same p l a n e . The m a t e r i a l p r o p e r t i e s a r e assumed c o n -s t a n t a l o n g t h e l e n g t h o f t h e column and s y m m e t r i c a l w i t h r e s p e c t t o t h e p l a n e o f l o a d i n g . B e n d i n g s t r a i n i s assumed l i n e a r l y p r o p o r t i o n a l t o d i s t a n c e f r o m t h e c e n t r a l p l a n e . The e f f e c t s o f s h e a r d e f o r m a t i o n and o f r o t a r y i n e r t i a on t r a n s v e r s e v i b r a t i o n s a r e i g n o r e d . F o r most e n g i n e e r i n g a p p l i c a t i o n s , were r e l a t i v e l y s t i f f columns a r e u s e d , t h e s e a r e n o t s e r i o u s l i m i t a t i o n s a t t h e l o w e r modes o f o s c i l l a t i o n , I n t e r n a l and e x t e r n a l damping and t r a n s v e r s e l o a d i n g a r e n o t c o n s i d e r e d . S e c o n d l y , t h e e x p e r i m e n t a l s t u d y has numerous r e s t r i c -t i o n s . I t i s . n o t p o s s i b l e t o r e a l i z e i n p r a c t i c e t h e b o u n d a r y c o n d i t i o n s u s e d i n t h e a n a l y t i c a l d e v e l o p m e n t . Clamped column ends a r e a t b e s t o n l y a p p r o x i m a t e d i n an e x p e r i m e n t a l s e t - u p . F u r t h e r , a l i g n m e n t i s a l w a y s a p p r o x i -mate, meaning t h a t t h e column e x c i t a t i o n i s n o t p u r e l y a x i a l . S t r a i n gauges were t h e o n l y s e n s i t i v e means f o u n d t o m o n i t o r n o n l i n e a r c o u p l e d v i b r a t i o n s . A c o n s i d e r a b l e amount o f c a u t i o n was r e q u i r e d t o m i n i m i z e e l e c t r o n i c n o i s e and i n d u c e d m agnetism s i n c e t h e s t r a i n l e v e l s were v e r y s m a l l . A s s e s s i n g t h e e x a c t c o n t r i b u t i o n o f n o i s e t o t h e m e a s u r e d s i g n a l i s d i f f i c u l t . To m a i n t a i n t h e s i g n a l l e v e l as h i g h as p o s s i b l e , f r e q u e n c y bands o f c o n s t a n t l e v e l s i n u s o i d a l a c c e l e r a t i o n were u s e d t o a c h i e v e an ' a p p r o x i m a t e c o n s t a n t power' e x c i t a t i o n s p e c t r a . F i n a l l y , t h e t e s t s were l i m i t e d t o an u p p e r f r e q u e n c y o f 1 0 KHz by t h e a u t o m a t i c v i b r a t i o n e x c i t e r c o n t r o l . CHAPT I T h e o r y - 8 -CHAPTER I I THEORY D i f f e r e n t i a l E q u a t i o n s o f M o t i o n f o r a Column L i n e a r d i f f e r e n t i a l e q u a t i o n s o f m o t i o n commonly u s e d f o r columns ( A p p e n d i x C) assume no s t r a i n s e x i s t i n t h e n e u t r a l p l a n e when t h e a x i a l l o a d i n g i s z e r o . C o n s i d e r i n g p l a n e m o t i o n , t h i s means t h a t t h e n e u t r a l a x i s f o r t h e un-l o a d e d column i s a s t r a i g h t l i n e segment. T h i s a s s u m p t i o n has l i m i t a t i o n s i n p r a c t i c e s i n c e a l l columns e x h i b i t some i n i t i a l c r o o k e d n e s s . The n o n l i n e a r i t i e s i n t h e d i f f e r e n t i a l e q u a t i o n s d e r i v e d h e r e i n a r i s e f r o m a s t r a i n e x p r e s s i o n w h i c h t a k e s i n t o a c c o u n t any i n i t i a l c r o o k e d n e s s o r s t a t i c d i s -p l a c e m e n t o f a column o r b a r . The L a g r a n g i a n d e f i n i t i o n o f s t r a i n i s u s e d ; t h a t i s , dynamic d e f l e c t i o n s a r e m e a s u r e d r e l a t i v e t o an i n i t i a l d e f l e c t i o n c u r v e . L o v e [ 1 ] d e r i v e d an a n a l a g o u s E u l e r i a n d e s c r i p t i o n o f s t r a i n . M e t t l e r [ 3 ] d e v e l o p e d t h e s t r a i n e x p r e s s i o n f o r a column w i t h some i n i t i a l c r o o k e d n e s s (Appen-d i x A ) . He o b t a i n e d t h e s t r a i n e x p r e s s i o n a t t h e c e n t r a l p l a n e f i b e r o f t h e column as £ x = u x + * x w x + (V2) w x2 ( 2 - i ) where, r e f e r r i n g t o F i g . I I - l u = l o n g i t u d i n a l d i s p l a c e m e n t w = i n i t i a l t r a n s v e r s e d i s p l a c e m e n t w - dynamic t r a n s v e r s e d i s p l a c e m e n t - 9 -and t h e s u b s c r i p t s d e n o t e p a r t i a l d i f f e r e n t i a t i o n w i t h r e s -p e c t t o t h a t v a r i a b l e . z,w -INITIAL DEFLECTION CURVE •DYNAMIC DEFLECTION CURVE F i g . I I - l . R e f e r e n c e A x i s f o r D i s p l a c e m e n t Measurements From s i m p l e b e n d i n g t h e o r y , t h e a d d i t i o n a l s t r a i n o f a f i b e r a t a d i s t a n c e z f r o m t h e n e u t r a l a x i s i s 6 b = " z w x x (2-2) T h i s e x p r e s s i o n i s o b t a i n e d a s s u m i n g t h a t p l a n e c r o s s -s e c t i o n s r e m a i n p l a n e d u r i n g b e n d i n g ; t h a t i s , s h e a r d e f o r -m a t i o n i s n e g l e c t e d . The t o t a l s t r a i n i s t h e r e f o r e 6 = 6 x + 6 b (2-3) The c o u p l e d d i f f e r e n t i a l e q u a t i o n s o f m o t i o n c a n now be d e r i v e d , as was done by M e t t l e r [ 3 ] . A s l e n d e r column ( s l e n d e r n e s s r a t i o g r e a t e r t h a n 50) i s . c o n s i d e r e d so t h a t r o t a r y i n e r t i a c a n be n e g l e c t e d , and damping i s I g n o r e d . O n l y a x i a l l o a d i n g i s c o n s i d e r e d . - 10 -From usual beam bending theory, assuming that stresses occur only in the x-direction, the e last ic strain energy is V = | £ e d v (2-4) where, E = modulus of e las t i c i ty v = volume Substituting equations {2-1) and (2-2) into equation (2-3), substituting the resultant expression into equation (2-4) and integrating part ia l ly over the cross-section of the column, the strain energy becomes V = f J0L(ux + *x wx +(v2)w xVdx EI'JLwxx2 d x (2-5) + 2. J0 where, A = cross-sectional area of column I = minimum moment of inert ia of column cross-section The approximate kinetic energy is T = ~ JqL ( u t 2 + w t 2 ) dx (2-6) where, 0 = density of column material t = t ime Hamilton's principle for a conservative system states that (V - T) dt = 0 '1 Substituting equations (2-5) and (2-6) into (2-7) and employ-ing calculus of variations provides the coupled differential equations of plane motion for the free vibrations of a column. EA(u x + (l/2)wx2 + wxwx) x + A u t t = 0 (2-8) E l w x x x x " EA[(u x + ( 1 / 2) W x 2 + w ^ x ) ( ^ + _ ^ } + d>Awtt = 0 . (2-9). - 11 -I f t h e n o n l i n e a r t e r m s i n e q u a t i o n (2-8) a r e d r o p p e d , t h e f a m i l i a r e q u a t i o n f o r t h e l o n g i t u d i n a l v i b r a t i o n o f columns r e s u l t s . S i m i l a r i l y , l i n e a r i z i n g e q u a t i o n (2-9) p r o v i d e s t h e common e q u a t i o n f o r l a t e r a l v i b r a t i o n o f beams. T h e o r e t i c a l P r e d i c t i o n s A q u a l i t a t i v e t r e a t m e n t o f t h e e q u a t i o n s o f m o t i o n a l l o w s some p r e d i c t i o n s t o be made c o n c e r n i n g t h e r e s o n a n t o s c i l l a t i o n s o f t h e column. I n p a r t i c u l a r , t h e f r e q u e n c i e s a t w h i c h c o u p l e d terms a r e l i k e l y t o be most s i g n i f i c a n t a r e r e v e a l e d . E x p a n d i n g , d i f f e r e n t i a t i n g and r e a r r a n g i n g e q u a t i o n (2-9) g i v e s E I w x x x x " E A [ u x x w x + u x w x x ] + A 4 w t t = E A [ u x x w x + u x W x x ] (2-10) The f o r m o f t h e a x i a l e x c i t a t i o n i n t h e t e s t s i s t h a t o f c o n s t a n t l e v e l s i n u s o i d a l a c c e l e r a t i o n . A d o u b l e i n t e g r a -t i o n o f t h e a c c e l e r a t i o n w i t h r e s p e c t t o t i m e i n d i c a t e s t h a t t h e d i s p l a c e m e n t t i m e - v a r i a t i o n a t t h e e x c i t e d e n d o f t h e column i s a l s o s i n u s o i d a l . I t i s t h e r e f o r e r e a s o n a b l e t o e x p e c t t h e a x i a l d i s p l a c e m e n t a t any p o i n t a l o n g t h e column t o v a r y s i n u s o i d a l l y i n t i m e , u = U(x) c o s ( r t + OC ) (2-11) where V i s t h e a p p l i e d f r e q u e n c y and i s some ph a s e a n g l e . A p p r o p r i a t e l y d i f f e r e n t i a t i n g e q u a t i o n (2-11) and s u b s t i t u -t i n g i n t o e q u a t i o n (2-10) y i e l d s - 12 -E I w x x x x - E A [ u x x w x + u x w x x ] + A^ w t t = F(x ) cos(Yt+CQ (2-12) where F(x) i s some f u n c t i o n o f t h e c o o r d i n a t e x d e t e r m i n e d by t h e s t a t i c d i s p l a c e m e n t o f t h e column. The r i g h t s i d e o f e q u a t i o n (2-12) can be r e g a r d e d as a f o r c i n g f u n c t i o n . Then t h e column e f f e c t i v e l y h as a s i n u s o i d a l t r a n s v e r s e i n -p u t o f f r e q u e n c y V . ' L i n e a r ' r e s o n a n t f l e x u r a l o s c i l l a t i o n s a r e t h e r e f o r e e x p e c t e d t o be e x c i t e d . S t r i c t l y s p e a k i n g , t h e s e f l e x u r a l o s c i l l a t i o n s a r e p a r a m e t r i c a l l y e x c i t e d , s i n c e p a r a m e t r i c o s c i l l a t i o n s o c c u r i n d i r e c t i o n s n o r m a l t o t h e e x c i t a t i o n , w h i l e f o r c e d o s c i l l a t i o n s a r e u n d e r s t o o d t o o c c u r i n d i r e c t i o n s p a r a l l e l t o t h e e x c i t a t i o n . However, t o a v o i d c o n f u s i n g t h e s e r e s o n a n c e s w i t h c o u p l e d f l e x u r a l r e s o n a n t modes, t h e y w i l l be r e f e r r e d t o as ' l i n e a r ' r e s o n a n t f l e x u r a l o s c i l l a t i o n s , s i n c e t h e y can be p r e d i c t e d f r o m l i n e a r t h e o r y f o r a column t r a n s v e r s l y e x c i t e d . N e g l e c t i n g t h e i n f l u e n c e o f t h e c o u p l e d t e r m s f o r t h e t i m e b e i n g , e q u a t i o n (2-12) i n d i c a t e s t h a t t h e t r a n s v e r s e r e s p o n s e i s s i n u s o i d a l . A s i n u s o i d a l t r a n s v e r s e r e s p o n s e can be s u b s t i t u t e d b a c k i n t o e q u a t i o n (2-8) d e s c r i b i n g t h e a x i a l m o t i o n t o g e t an i m p r o v e d m a t h e m a t i c a l d e s c r i p t i o n o f t h e l o n g i t u d i n a l b e h a v i o r . D i f f e r e n t i a t i n g and r e a r r a n g i n g e q u a t i o n (2-8) y i e l d s " u x x + i - u t t = w x W x x + w xw x x + w xw x x (2-13) where c f> 1 3 t i l e v e l o c i t y o f l o n g i t u d i n a l waves a l o n g t h e l e n g t h o f t h e column. S u b s t i t u t i n g a s i n u s o i d a l t r a n s -- 13 -verse response i n t o the coupled terms on the r i g h t s i d e of equation (2-13), and remembering t h a t w x and w x x are f u n c t i o n s of x only, g i v e s w x w x x = H(x)cos2tft (2-14) w Xxw x = H 1 ( x ) c o s ^ t (2-15) ^xWxx = H 2 ( x ) c o s t f t (2-16) Since cos2 Yi = + c o s 2 y t ) , equation (2-14) becomes w xw x x = H 3(x) [1 + cos2tft] (2-17) S u b s t i t u t i n g e x p r e s s i o n s (2-15), (2-16) and (2-17) i n t o e q uation (2-13) , one o b t a i n s ~ uxx + ~ 2 u t t ={H!(x) + H 2 ( X ) } costft + H 3 (x)(H-cos2*t) (2-18) T h i s equation i n d i c a t e s t h a t the l o n g i t u d i n a l v i b r a t i o n i s the sum of two s i n u s o i d a l o s c i l l a t i o n s w i t h a frequency r a t i o o f 1:2. These are the n o n l i n e a r coupled v i b r a t i o n s , and two d i f f e r e n t resonant f r e q u e n c i e s are expected. The f i r s t coupled resonant l o n g i t u d i n a l v i b r a t i o n i s at the same frequency t as the e x c i t a t i o n frequency. The second coupled resonant l o n g i t u d i n a l v i b r a t i o n i s at twice the frequency of the t r a n s v e r s e v i b r a t i o n o f the column. I t occurs at an e x c i t a t i o n frequency o f j & V where oo, i s the e x c i t a t i o n frequency f o r the f i r s t coupled l o n g i t u d i n a l v i b r a t i o n . To continue t h i s somewhat i t e r a t i v e procedure, the a x i a l response can be used to o b t a i n a more p r e c i s e t r a n s -verse response. The f i g h t s i d e of equation (2-18) can be - 14 -r e g a r d e d as two l o n g i t u d i n a l s i n u s o i d a l f o r c i n g f u n c t i o n s w i t h f r e q u e n c i e s Y and . T h e s e e x p r e s s i o n s c a n i n t u r n be s u b s t i t u t e d i n t o t h e c o u p l e d t e r m s on t h e r i g h t s i d e o f e q u a t i o n (2-10) . As a c o n s e q u e n c e t w o . . f l e x u r a l c o u p l e d v i b r a t i o n s can be p r e d i c t e d o f t h e f o r m • G 1 ( x ) c o s t + G 2 ( x ) c o s 2 J T t (2-19) f r o m w h i c h two r e s o n a n t t r a n s v e r s e f r e q u e n c i e s a t a>x and jit), a r i s e . The j u s t i f i c a t i o n f o r c o n t i n u i n g t h e i t e r a t i o n s f u r t h e r i s q u e s t i o n a b l e , s i n c e t h e c o u p l i n g phenomena i s a s e c o n d o r d e r e f f e c t t o s t a r t w i t h . The a d d i t i o n a l t e r m s p r o v i d e d by e a c h i t e r a t i o n become d e c r e a s i n g l y s i g n i f i c a n t . The t h e o r e t i c a l a n a l y s i s does n o t i n d i c a t e t h e s i g n i f i c a n c e o f t h o s e c o u p l e d t e r m s d i s c u s s e d . The 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 i s o f o b v i o u s v a l u e . T o r s i o n a l C o u p l i n g C o u p l i n g between l o n g i t u d i n a l and t o r s i o n a l m o t i o n e x i s t s due t o t h e " s h o r t e n i n g e f f e c t " . An i n i t i a l l y un-t w i s t e d s t e e l b a r o r column s u b j e c t e d t o t o r q u e s u n d e r g o e s a x i a l s h o r t e n i n g i n a d d i t i o n t o r o t a t i o n o f t h e c r o s s -s e c t i o n s . T s o [8] shows a n a l y t i c a l l y t h a t i f a b a r i s u n d e r f o r c e d l o n g i t u d i n a l , v i b r a t i o n s w i t h a f r e q u e n c y w h i c h i s c l o s e t o t w i c e a n a t u r a l t o r s i o n a l f r e q u e n c y , i t i s p o s s i b l e t o p a r a m e t r i c a l l y e x c i t e t h e b a r i n t o t o r s i o n a l o s c i l l a t i o n s . F o r s i m p l i c i t y , t o r s i o n a l and f l e x u r a l c o u p l i n g a r e t r e a t e d d i s s o c i a t e d f r o m e a c h o t h e r i n t h e p r e s e n t work. F o l l o w i n g T s o ' s d e v e l o p m e n t and r e f e r r i n g t o F i g . I I - 2 , t h e l o n g i t u d i n a l d i s p l a c e m e n t o f a p o i n t a t a d i s t a n c e y f r o m t h e c e n t -r o i d a l a x i s c a n be w r i t t e n as u ( x , y , t ) = U ( x , t ) -P(y.t) F i g . I I - 2 . E l a s t i c B a r Under A x i a l L o a d i n g W a r p i n g as i n S t . V e n a n t t o r s i o n i s n o t t a k e n i n t o a c c o u n t i n e q u a t i o n ( 2 - 2 0 ) . I t i s c o n s i d e r e d l a t e r i n t h e f o r m u -l a t i o n , however. The a x i a l s t r a i n i s g i v e n by € = r j x - \ y 2 O x 2 (2-21) where U = l o n g i t u d i n a l d i s p l a c e m e n t o f c r o s s s e c t i o n 0 = r o t a t i o n o f c r o s s s e c t i o n The e l a s t i c s t r a i n e n e r g y becomes GAh" e x 2}dx (2-22) - 16 -where A = c r o s s - s e c t i o n a l area I 2 = moment of i n e r t i a about 2 - a x i s 1-^  = constant depending on dimensions of c r o s s - s e c t i o n 1 2 7 The term 3- GAh 6 X accounts f o r S t . Venant t o r s i o n . The k i n e t i c energy i s T = IJQ [ A U t + - I « e t 2 ] d x (2-23) where I Q = p o l a r moment of i n e r t i a o f c r o s s - s e c t i o n . The work done by the a p p l i e d loads at the end i s W = f ^ p ^ t j u d ^ t j h d y (2-24) where p(y,t) r e p r e s e n t s the end l o a d on the column. Assume t h a t the mathematical form of the end l o a d i s sep a r a b l e , p(y,t) = P(t)*My) (2-25) where ^(y) 1 S a n even f u c t i o n of y. S u b s t i t u t i n g equations (2-20) and (2-25) i n t o e quation (2-24) y i e l d s W = R l A P ( t ) U ( l ) - j R 2 I 2 P ( t ) j L © x 2 d x { 2 _ 2 6 ) 1 f b/2 0 where R-,= - j h^(y) dy 1 A j - b / 2 R = r f 2 hyV(y)dy Using Hamilton's P r i n c i p l e , S f t 2(T - V + W)dt = 0 (2-27) with equations ( 2 - 2 2 ) , (2-23) and ( 2 - 2 6 ) , and employing a v a r i a t i o n a l procedure, the coupled d i f f e r e n t i a l , equations o f motion f o r t o r s i o n a l and l o n g i t u d i n a l v i b r a t i o n s r e s u l t ; < D A U T T - E A U X X - E I 2 0 x e x x = 0 (2-28) <H>©tt - t^GAh 2 + R 2 i 2 p ( t ) ] e x x + E i 2 ( U x e x ) x - |^Ii©xx©x 2 * 0 ( 2" 2 9> - 1 7 -These equations of motion describe the l o n g i t u d i n a l and t o r s i o n a l response of a uniform Column of t h i n r ectangular cross s e c t i o n b by h and of f i x e d length L , loaded symmet-r i c a l l y about the OX-axis and uniformly over the thickness of the s t r i p . The column m a t e r i a l i s assumed l i n e a r and i s o -t r o p i c . I f the n o n l i n e a r terms are neglected i n equations (2-28) and (2-29) the equations become uncoupled. Equation (2-2 8) takes the form of the f a m i l i a r equation f o r l o n g i t u d i -n a l v i b r a t i o n s of a rod, while equation (2-29) becomes O l ^ t t " [GAh2/3 + R 2 l 2 P ( t ) ] 0 x x = 0 ( 2- 3 0> This equation i s d i f f e r e n t from the usual form f o r t o r s i o n a l v i b r a t i o n only i n the s t i f f n e s s term, which accounts f o r the a x i a l end load. I t i s seen t h a t a compressive end load such as i s used i n t h i s experiment w i l l decrease the t o r s i o n a l s t i f f n e s s of the column. Using appropriate end c o n d i t i o n s , the approximate resonant t o r s i o n a l frequencies can be c a l c u -l a t e d (Appendix C ) . Tso points out the existence of two c r i t i c a l frequency ranges where t o r s i o n a l coupling i s most l i k e l y to occur. The f i r s t range appears when the e x t e r n a l l y a p p l i e d frequency i s close to twice the n a t u r a l frequency of a p a r t i c u l a r t o r s i o n a l mode. The second c r i t i c a l range appears when the a p p l i e d frequency i s c l o s e to the l o n g i t u d i n a l frequency, p a r t i c u -l a r l y i f the dimensions of the column are such that the fun-damental l o n g i t u d i n a l frequency i s close to the n a t u r a l frequency f o r a t o r s i o n a l mode. CHAPTI Apparatu CHAPTER I I I APPARATUS AND INSTRUMENTATION G e n e r a l O u t l i n e A s i g n a l f l o w d i a g r a m o f t h e a p p a r a t u s and i n s t r u m e n -t a t i o n i s shown i n F i g . I I I - l . F i g . I I I - 2 i s a p h o t o g r a p h o f t h e a c t u a l a p p a r a t u s and i n s t r u m e n t a t i o n u s e d i n t h i s s t u d y . The e x p e r i m e n t a l a p p a r a t u s c an be c o n s i d e r e d i n two p a r t s . The f i r s t p a r t i s t h e v i b r a t i o n c o n t r o l s y s t e m , w h i c h i n c l u d e s t h e f e e d b a c k c i r c u i t , and c o n s i s t s o f a s i g n a l g e n e r a t o r , e l e c t r o m a g n e t i c s h a k e r , a m p l i f i e r s and an a c c e l e -r o m e t e r . The a u t o m a t i c v i b r a t i o n e x c i t e r c o n t r o l i s p r o -grammed t o d e l i v e r an e l e c t r o n i c s i g n a l t o t h e s h a k e r c o r r e s -p o n d i n g t o t h e t y p e o f e x c i t a t i o n wanted. The s h a k e r r e s p o n d s by t r a n s m i t t i n g a f o r c e t h r o u g h t h e s h a k e r t a b l e t o t h e s p e c i m e n . F e e d b a c k i s e s s e n t i a l s i n c e t h e v i b r a t i o n c o n t r o l e x c i t e r u n i t i s c a p a b l e o f ' c h e c k i n g ' t h e e x c i t a t i o n i t i s p r o v i d i n g and making s u i t a b l e c o r r e c t i o n s t o t h e o u t p u t s i g n a l . T h i s i s a c c o m p l i s h e d u s i n g an a c c e l e r o m e t e r f e e d b a c k c i r c u i t t o t h e e x c i t e r c o n t r o l u n i t . The a c c e l e r o m e t e r m o n i t o r s t h e e x c i t a t i o n l e v e l , and the. e x c i t e r c o n t r o l u n i t compares t h i s w i t h t h e programmed e x c i t a t i o n so t h a t i t c a n compensate f o r any d i f f e r e n c e between t h e two. The s e c o n d p a r t o f t h e a p p a r a t u s c o n s i s t s o f m o n i t o r i n g i n s t r u m e n t s n e c e s s a r y t o g a t h e r i n f o r m a t i o n on t h e b e h a v i o r o f t h e t e s t s p e c i m e n . S t r a i n gauges a t t a c h e d t o t h e s p e c i m e n A ACCELEROMETER PREAMPLIFIER GENERATOR (Feedback and Synchronization) Feedback FOUR BEAM OSCILLOSCOPE L U FOTONIC SENSOR Jb-o-LEVEL RECORDER POWER AMPLIER For Synchronization 1> Fig. IH - 1. Signal Flow Diagram - 20 -F i g . I I I - 2 . P h o t o g r a p h o f E x p e r i m e n t a l A p p a r a t u s 1. F r e q u e n c y C o u n t e r 2. Time S w i t c h 3. A m p l i f i e r and S p e c t r u m A n a l y z e r 4. CRO 5. L e v e l R e c o r d e r 6 . V i b r a t i o n E x c i t e r C o n t r o l 7. F o t o n i c S e n s o r U n i t 8. BAM 9. Wavetek S i g n a l G e n e r a t o r 10. Power A m p l i f i e r 11. A c c e l e r o m e t e r P r e a m p l i f i e r 12. S p e c i m e n Frame 13. Column Sp e c i m e n 14. S p r i n g 15. F o t o n i c S e n s o r H o l d e r 16. S h a k e r 17. R o o t s B l o w e r i n d i c a t e f l e x u r a l and l o n g i t u d i n a l s t r a i n l e v e l s . The a c c e l -e r o m e t e r u s e d i n t h e f e e d b a c k c i r c u i t a l s o m o n i t o r e d t h e a c c e -l e r a t i o n l e v e l a t t h e l o w e r c o l u m n mount. T h e s e s i g n a l s w e r e d i s p l a y e d on a CRO and l e v e l r e c o r d e r , and w h e r e a p p r o p r i a t e , s p e c t r a l a n a l y s e s o f t h e s i g n a l s w e r e p e r f o r m e d . E l e c t r o n i c S y s t e m The a u t o m a t i c v i b r a t i o n e x c i t e r c o n t r o l ( h e r e a f t e r c a l l e d t h e c o n t r o l g e n e r a t o r ) p r o v i d e d a p r e - p r o g r a m m e d s i g n a l d e t e r -m i n e d by t h e d e s i r e d v i b r a t i o n l e v e l t o a power a m p l i f i e r . The a m p l i f i e d s i g n a l was u s e d t o p r o d u c e m e c h a n i c a l v i b r a t i o n by t h e s h a k e r . An a c c e l e r o m e t e r r i g i d l y m o u n t e d on t h e l o w e r s p e c i m e n mount ( F i g . I I 1 - 3 ) m o n i t o r e d t h e a c c e l e r a t i o n o f t h e a c t u a l v i b r a t i o n a t t h a t p o i n t . F i g . I I I - 3 . S c h e m a t i c V i e w o f L o w e r Column Mount - 22 -F o l l o w i n g a m p l i f i c a t i o n and impedance matching of the a c c e l e -rometer p r e a m p l i f i e r , the v i b r a t i o n l e v e l was compared with the programmed s e t - p o i n t i n the c o n t r o l generator. The c o n t r o l generator then completed the feedback c i r c u i t by c o r r e c t i n g any d i f f e r e n c e between the accelerometer s i g n a l and the preprogrammed s i g n a l . A more d e t a i l e d s p e c i f i c a t i o n of the instruments used i n the v i b r a t i o n c o n t r o l loop i s conta i n e d i n Appendix B. Loading Frame and Column D e s c r i p t i o n The column mounting and l o a d i n g system i s shown i n F i g . I I I - 4 . N CONCRETE BASE Column Mounting and Loading System - ACCELEROMETER -LOWER MOUNT BLOWER SHAKER BED STRAIN GAUGES FRAME UPPER MOUNT COLUMN SPECIMEN SPRING Fig. I I I - 4 . - 23 -The column s p e c i m e n was l o a d e d v e r t i c a l l y i n t h e frame. I t was a r r a n g e d so t h a t P0 , Pt and Y c o u l d be v a r i e d i n d e p e n -d e n t l y . The s t a t i c l o a d P„ was a p p l i e d t o t h e s p e c i m e n by two p a r a l l e l s p r i n g s mounted on e i t h e r s i d e and a t t a c h e d between t h e u p p e r and l o w e r mounts, and was a l w a y s o f s u f f i -c i e n t m a g n i t u d e t o m a i n t a i n t h e column i n c o m p r e s s i o n . The v a r y i n g component o f t h e l o a d P = P 0 + P! c o s Y t was a p p l i e d by t h e s h a k e r t h r o u g h t h e l o w e r mount. The c y c l i c d i s p l a c e -ments o f t h e column and s p r i n g s were v e r y s m a l l ( o f t h e o r d e r o f 200 u i n ) s o t h e v a r i a t i o n o f t h e s t a t i c f o r c e P Q was q u i t e n e g l i g i b l e . S i n c e t h e t e s t s were c o n d u c t e d a t f r e q u e n -c i e s above f o u r o r f i v e h u n d r e d Hz, t h e o p e r a t i n g f r e q u e n c i e s were al w a y s much g r e a t e r t h a n t h e n a t u r a l f r e q u e n c y o f t h e c o m p l e t e s p r i n g - m a s s s y s t e m , c o n s i s t i n g o f t h e column, s p r i n g s , l o w e r mount, s h a f t and s h a k e r t a b l e . The l o w e r end o f t h e column was c o n s t r a i n e d t o move v e r t i c a l l y . To a c c o m p l i s h t h i s , t h e a r r a n g e m e n t shown i n F i g . I I I - 5 was u s e d . - 24 -F i g . 111-5 S e c t i o n a l D r a w i n g o f L o a d i n g and A l i g n m e n t Mechanism 3 A — i n c h d i a m e t e r A t l a s S u p e r i o r s h a f t c o n n e c t e d t h e s h a k e r t a b l e t o t h e l o w e r s p e c i m e n mount. T r a n s v e r s e m o t i o n o f t h e l o w e r mount was p r e v e n t e d u s i n g a Thompson a d j u s t a b l e b a l l b u s h i n g t o g u i d e t h e s h a f t . The s h a k e r was s i t u a t e d b e l o w t h e s p e c i m e n so t h a t t h e moving s h a k e r h e a d was p o s i t i o n e d d i r e c t l y i n l i n e w i t h t h e l o w e r s p e c i m e n s u p p o r t . The l o w e r column mount i s shown i n F i g . I l l - 3 . The u p p e r mount was s i m i l a r e x c e p t t h a t i t was r i g i d l y a t t a c h e d t o t h e frame and i t d i d n o t have t h e a c c e l e r o m e t e r - 25 -attachment. The mounts were intended t o provide clamped end c o n d i t i o n s . As shown i n F i g . I I I - 3 , spacers were forced i n on each side between the column and mount. The top i n s i d e edges of the wedges were b e v e l l e d , and the specimen length f o r f l e x u r a l o s c i l l a t i o n s was the distance between the e x t r e -m i t i e s of the bevels on the upper and lower mounts. To secure 3 the specimen i n p l a c e , a t i g h t f i t t i n g i n c h diameter m i l d s t e e l p i n was i n s e r t e d through the mount, wedges and specimen. The experiments were performed on a s t e e l specimen of re c t a n g u l a r cross s e c t i o n f a b r i c a t e d from hot r o l l e d carbon s t e e l , f l a t stock ( F i g . I I I - 6 ) . F i g . I I I - 6 . Schematic View of Column Specimen The column specimen has a length of 11.625" between f i l l e t s , and a 0.375" by 0.125" re c t a n g u l a r cross s e c t i o n . The dimen-sions of the specimen were chosen t o produce a measurable s t r a i n l e v e l with the force a v a i l a b l e , and t o provide the f i r s t coupled l o n g i t u d i n a l resonant v i b r a t i o n i n the 8 Khz region. The f i r s t E u l e r b u c k l i n g load of the specimen f o r clamped end c o n d i t i o n s was 534 pounds. The surfaces were l i g h t l y ground t o s i z e on a surface g r i n d e r , which gave the column a s l i g h t i n i t i a l curvature, taken care of i n the t h e o r e t i c a l model. The specimen had a wider s e c t i o n on each - 26 -end to accomodate a hole for the pin and to allow a closer approximation to clamped ends. The column specimen was mounted i n the upper portion of the heavy frame, as shown i n F i g . I I I - 4 . The frame was bolted to a s t e e l bed. On top of the bed was a block of wood on which the shaker was supported. Below the bed was a Roots blower which c i r c u l a t e d a i r through the shaker to prevent overheating. The blower and i t s driving motor were i s o l a t e d from the test bed and frame to avoid any unnecessary transmission of extra-neous vibration to the specimen. The concrete base on which the t e s t bed was mounted was i s o l a t e d from the building to minimize external vibration sources. Fig. I I I - 7 i s a photo-graph of the specimen, loading frame, shaker and test bed. F i g . I I I - 7 . P h o t o g r a p h o f S p e c i m e n L o a d i n g Frame - 27 -D e t a i l s of Measuring System Apart from the accelerometer, the use of two types of transducers was attempted to monitor the behavior of the specimen. The f i r s t of these was a non-contacting d i s p l a c e -ment transducer, c a l l e d a Fotonic Sensor, manufactured by Mechanical Technology, L i m i t e d . The Fotonic Sensor i s a s o l i d s t a t e e l e c t r o n i c instrument with a probe c o n s i s t i n g of a packed bundle of s p e c i a l l y constructed glass f i b r e s arranged i n a random transmit-and-receive c o n f i g u r a t i o n . The t r a n s m i t t i n g f i b r e s c a r r y l i g h t to the t a r g e t ; the r e f l e c t e d l i g h t i s returned through the r e c e i v i n g f i b r e s t o i l l u m i n a t e a l i g h t s e n s i t i v e diode. Though the instrument o f f e r s many d e s i r a b l e f e a t u r e s , i t was not found very s u c c e s s f u l i n t h i s work. The instrument probe must be mounted very r i g i d l y so that i t undergoes no movements. With the apparatus used, i t was not found p o s s i b l e t o provide a mount s u f f i c i e n t l y i s o -l a t e d from a l l sources of v i b r a t i o n . The second type of transducer, the s t r a i n gauge, was more s u c c e s s f u l . Four BLH E l e c t r o n i c s SR-4 type FAP-12-12 f o i l gauges were attached to the middle of the column specimen; two side by side a l l i g n e d a x i a l l y on each of the wider faces of the column specimen. To measure bending s t r a i n l e v e l s , the arrangement shown s c h e m a t i c a l l y i n F i g . I I I - 8 was used, where t h e specimen was l a t e r a l l y d i s p l a c e d to produce decreased compression i n two gauges on one side and increased compression i n t h e two gauges on the other s i d e . A four arm bridge of the - 28 -F i g . I I I - 8 Schematic S t r a i n Gauge Arrangement f o r F l e x u r a l S t r a i n Measurement. type shown was used f o r s e v e r a l reasons. I t was the most s e n s i t i v e bridge c o n f i g u r a t i o n , and was completely tempera-ture compensating. This bridge was i n s e n s i t i v e to any u n i -form l o n g i t u d i n a l s t r a i n s i n the specimen. Since small s t r a i n l e v e l s were encountered, a f u r t h e r increase i n s e n s i t i v i t y was obtained by i n t r o d u c i n g as much a d d i t i o n a l voltage as p o s s i b l e i n s e r i e s w i t h the i n t e r n a l e x c i t a t i o n of the bridge a m p l i f i e r . The E l l i s A ssociates BAM 1 Bridge A m p l i f i e r and Meter used was capable of measuring dynamic s i g n a l s over the range encountered. I t s frequency response i s such t h a t the a t t e n u a t i o n at 10 Khz i s approximately 3%. L o n g i t u d i n a l s t r a i n was measured using the bridge con-f i g u r a t i o n shown i n F i g . I I I - 9 . L o n g i t u d i n a l s t r a i n cannot be measured using f o u r a c t i v e arms, so two a c t i v e arms (one on each side of the specimen) and two dummy gauges p r o v i d i n g - 29 -F i g . I I I - 9 . S c h e m a t i c S t r a i n Gauge A r r a n g e m e n t f o r L o n g i t u d i n a l S t r a i n Measurement i n c r e a s e d s e n s i t i v i t y and t e m p e r a t u r e c o m p e n s a t i o n were u s e d i n a f o u r arm b r i d g e . T h i s c o n f i g u r a t i o n was a l s o s e n s i t i v e t o t o r s i o n a l r o t a t i o n s o f t h e m i d d l e c r o s s - s e c t i o n , s i n c e t o r s i o n p r o d u c e s e q u a l s t r a i n i n b o t h s t r a i n g a u g e s . F o r t h i s a r r a n g e m e n t , b e n d i n g s t r a i n s e n s i t i v i t y i s s e v e r a l o r d e r s o f m a g n i t u d e l o w e r t h a n l o n g i t u d i n a l o r t o r s i o n a l s t r a i n s e n s i t i v i t y . The s t r a i n gauge s i g n a l was t a k e n f r o m t h e BAM t o an o s c i l l o s c o p e . The s i g n a l was a l s o a m p l i f i e d and r e c o r d e d on t h e l e v e l r e c o r d e r , and s p e c t r a l a n a l y s e s were a l s o r e c o r d e d a t f r e q u e n c i e s o f i n t e r e s t . The F r e q u e n c y a n a l y z e r u s e d was a B r u e l and K j a e r Type 2107, w h i c h s e r v e d as a v o l t a g e a m p l i f i e r and s p e c t r u m a n a l y -z e r . The l e v e l r e c o r d e r u s e d i n c o n j u n c t i o n w i t h t h e F r e -q u e n c y A n a l y z e r was a B r u e l and K j a e r Type 2 305, o f f e r i n g a - 30 -range of paper and w r i t i n g speeds. Recordings were made by means o f an ink pen on 100 mm. l o g a r i t h m i c frequency c a l i -b r a t e d paper. RMS, DC or peak val u e s c o u l d be p l o t t e d ; . t h e RMS f u n c t i o n was chosen t o minimize the i n f l u e n c e o f sudden extraneous s i g n a l s . F o r most t e s t s , the r e c o r d i n g paper was c a l i b r a t e d by means o f an event marker and a d i g i t a l frequency counter. A General Radio Company D i g i t a l Time and Frequency Meter, Type 1151-A, measured the average frequency of the e x c i t a t i o n t o a t l h*z. accuracy. S p e c t r a l a n a l y s i s of wave-forms of i n t e r e s t were done u s i n g frequency c a l i b r a t e d paper and a mechanical frequency l i n k a g e between the frequency a n a l -y z e r and l e v e l r e c o r d e r . The waveforms were d i s p l a y e d on a T e k t r o n i x 565 o s c i l l o -scope, where they c o u l d be photographed d i r e c t l y . These photographs were taken with a Pentax camera using Kodak P l u s -X f i l m . ) C H A P T E R 4 Test Piro< - 31 -CHAPTER I V TEST PROCEDURE C a l i b r a t i o n The p e r f o r m a n c e and c a l i b r a t i o n o f t h e e x p e r i m e n t a l i n s t r u m e n t s was c h e c k e d . The f r e q u e n c y s c a l e o f t h e c o n t r o l g e n e r a t o r was c a l i b r a t e d by means o f a b u i l t - i n c i r c u i t w h i c h c h e c k s two p o i n t s on t h e f r e q u e n c y s c a l e a g a i n s t a w e l l d e f i n e d f r e q u e n c y . S i n c e f r e q u e n c i e s c a n n o t be r e a d a c c u r a -t e l y a t t h e u p p e r end o f t h e f r e q u e n c y s c a l e on t h e c o n t r o l g e n e r a t o r , a d i g i t a l f r e q u e n c y c o u n t e r was u s e d . Agreement between t h e c o u n t e r and t h e f r e q u e n c y s c a l e was good a t a l l f r e q u e n c i e s . A f a c t o r y c a l i b r a t e d a c c e l e r o m e t e r was u s e d . S c h n e i d e r [9] f o u n d t h a t f o r c e d c o o l i n g d i d n o t a f f e c t t h e a c c e l e r o m e t e r o u t p u t , so no p r o v i s i o n was made t o c o o l t h e a c c e l e r o m e t e r . The a c c e l e r a t i o n l e v e l i n d i c a t e d by t h e a c c e l e r o m e t e r a g r e e d e x a c t l y w i t h t h e a c c e l e r a t i o n s e t on t h e c o n t r o l g e n e r a t o r . S e v e r a l BAMs were e v a l u a t e d , and t h e one w i t h t h e most f a v o u r a b l e n o i s e c h a r a c t e r i s t i c s was u s e d . The f r e q u e n c y r e s p o n s e o f t h e BAM was w e l l w i t h i n t h e m a n u f a c t u r e r ' s s p e c i f i c a t i o n s o v e r t h e f r e q u e n c y r a n g e o f t h e e x p e r i m e n t s . The f r e q u e n c y r e s p o n s e o f t h e f r e q u e n c y a n a l y z e r was c o m p l e -t e l y f l a t o v e r t h e r a n g e i n v o l v e d . F r e q u e n c y s y n c h r o n i z a t i o n between t h e f r e q u e n c y a n a l y z e r and t h e l e v e l r e c o r d e r f o r s p e c t r a l a n a l y s i s t e s t s was e x c e l l e n t p r o v i d e d t h e y were - 32 -a c c u r a t e l y s y n c h r o n i z e d m a n u a l l y b e f o r e h a n d . The s e n s i t i v i t y o f t h e f r e q u e n c y a n a l y z e r was c a l i b r a t e d u s i n g an a c c u r a t e r e f e r e n c e v o l t a g e p r o d u c e d by an i n t e r n a l z e n e r - d i o d e . The l e v e l r e c o r d e r was f o u n d c a p a b l e o f p r o v i d i n g a c c u r a t e r e c o r d s w i t h h i g h r e s p o n s e s p e e d s o f m e a s u r e d s i g n a l l e v e l s . I n t h e p r e s e n t e x p e r i m e n t s , t h e w r i t i n g s p e e d was r e d u c e d t o some e x t e n t t o m i n i m i z e t h e i n f l u e n c e o f n o i s e and e x t r a n e o u s s i g n a l s on t h e s t r a i n r e c o r d . The o s c i l l o s c o p e was c a l i b r a t e d b e f o r e and d u r i n g t h e e x p e r i m e n t . The p e r f o r m a n c e o f one o f t h e o s c i l l o s c o p e p l u g - i n u n i t s became u n a c c e p t a b l e d u r i n g t h e c o u r s e o f t h e e x p e r i m e n t s , so i t was r e p l a c e d . N o i s y s i g n a l s o f t e n o c c u r r e d , and e f f o r t s t o m i n i m i z e n o i s e were n e c e s s a r y . F r e s h d r y c e l l s were al w a y s u s e d i n t h e BAM. A l l c a b l e s and l e a d s u s e d were s h i e l d e d . E l e c t r o -m a g n e t i c p i c k up i n t h e s t r a i n gauge s i g n a l s was m i n i m i z e d by t w i s t i n g t h e l e a d s t o g e t h e r and k e e p i n g t h e gauges and l e a d s o u t o f s t r o n g e l e c t r o m a g n e t i c f i e l d s . Any s t r a i n gauges w h i c h d e v e l o p e d n o i s y s i g n a l s were p r o m p t l y r e p l a c e d . N o i s e a r i s i n g f r o m g r o u n d l o o p s was e l i m i n a t e d by h o o k i n g most o f t h e i n s t r u m e n t s up w i t h a f l o a t i n g g r o u n d and g r o u n d i n g them t h r o u g h t h e o s c i l l o s c o p e . T e s t i n g The s p e c i m e n w i t h s t r a i n gauges a t t a c h e d was p u t i n p l a c e i n t h e u p p e r and l o w e r mounts ( F i g s . I I I - 3 and I I I - 5 ) and t h e b e v e l l e d s p a c e r s d r i v e n i n on e a c h s i d e . The - 33 -s p e c i m e n was removed o n l y when a s t r a i n gauge h a d t o be r e p l a c e d . Any m i s a l i g n m e n t o f t h e s h a k e r c o u l d be d e t e c t e d by t h e o u t p u t waveform f r o m t h e power a m p l i f i e r . When c o r r e c t l y a l i g n e d , t h e power waveform became s h a r p and c l e a n and t h e power r e q u i r e d t o a c h i e v e a g i v e n a c c e l e r a t i o n l e v e l was a minimum. The s p r i n g s on e i t h e r s i d e o f t h e s p e c i m e n were a t t a c h e d and t i g h t e n e d t o p r o v i d e a c o n s t a n t c o m p r e s s i v e f o r c e o f 64 pounds on t h e column. The a c c e l e r o m e t e r was i n s t a l l e d and c o n n e c t e d t o t h e a c c e l e r o m e t e r p r e a m p l i f i e r . The p r e a m p l i f i e r was a d j u s t e d a c c o r d i n g t o t h e combined c a p a c i t a n c e o f t h e a c c e l e r o m e t e r and i t s l e a d . The s t r a i n gauge l e a d s were c o n n e c t e d t o t h e BAM i n an a p p r o p r i a t e , c o n f i g u r a t i o n i n c l u d i n g t h e e x t e r n a l e x c i t a -t i o n , as d e s c r i b e d i n C h a p t e r I I I . The l e a d s e x t e n d e d f r o m t h e gauges i n a l o o s e c o i l w h i c h m i n i m i z e d t h e e f f e c t o f t h e l e a d s on t h e f l e x u r a l r i g i d i t y o f t h e s p e c i m e n . The gauges were c h e c k e d f o r t h e i r r e s i s t a n c e and f o r t h e p o s s i b i l i t y o f any l e a k a g e t o g r o u n d . The s h i e l d s o f t h e l e a d w i r e s and t h e s p e c i m e n were c o n n e c t e d t o t h e o s c i l l o s c o p e g r o u n d . The BAM b a t t e r i e s were c h e c k e d and t h e gauge c i r c u i t e x c i t e d . By means o f v a r i a b l e r e s i s t a n c e s w i t h i n t h e BAM, t h e s t r a i n gauge b r i d g e was b a l a n c e d . A t t h i s p o i n t a l l i n s t r u m e n t s were p r o p e r l y c o n n e c t e d and t u r n e d t o t h e s t a n d b y p o s i t i o n f o r a t l e a s t two h o u r s b e f o r e t e s t i n g . A t i m e s w i t c h was c o n n e c t e d t o t u r n t h e i n s t r u m e n t a t i o n on a u t o m a t i c a l l y a few h o u r s i n a dvance o f t h e a n t i c i p a t e d t e s t i n g t i m e . - 34 -P r e l i m i n a r y t e s t s were n e c e s s a r y t o a s c e r t a i n what a c c e l e r a t i o n l e v e l c o u l d be u s e d o v e r w h i c h f r e q u e n c y r a n g e f o r t h e p a r t i c u l a r s y s t e m s t u d i e d . The v i b r a t i o n e x c i t e r c o n t r o l c o u l d n o t p r o v i d e c o n s t a n t d i s p l a c e m e n t o r v e l o c i t y l e v e l s o v e r t h e e n t i r e f r e q u e n c y r a n g e i n v e s t i g a t e d . The n e x t c h o i c e w o u l d be a c o n s t a n t l e v e l s i n u s o i d a l a c c e l e r a t i o n e x c i t a t i o n . T h i s , however, was a l s o u n s u i t a b l e s i n c e t h e a c c e l e r a t i o n l e v e l u s e d w o u l d h a v e t o be so s m a l l t h a t t h e s t r a i n r e c o r d o v e r most o f t h e f r e q u e n c y r a n g e w o u l d be i n -d i s t i n g u i s h a b l e f r o m i n h e r e n t e l e c t r o n i c n o i s e . The n e x t b e s t p r o c e d u r e w o u l d have been a c o n s t a n t power p l o t j b u t t h e e x p e r i m e n t a l s y s t e m i s n o t c a p a b l e o f t h i s t y p e o f c o n -t r o l . A compromise, c a l l e d an ' a p p r o x i m a t e d c o n s t a n t power s p e c t r a ' , was u s e d . The a p p r o x i m a t e d c o n s t a n t power s p e c t r a i s t h e d i v i s i o n o f t h e f r e q u e n c y r a n g e i n t o a number o f p a r t s o v e r e a c h o f w h i c h t h e a c c e l e r a t i o n l e v e l i s c o n s t a n t . D i f f e r e n t a c c e l e r a -t i o n l e v e l s a r e u s e d o v e r d i f f e r e n t f r e q u e n c y segments s u c h t h a t t h e power i n p u t t o t h e s p e c i m e n r e a c h e s a maximum a t some p o i n t i n t h e f r e q u e n c y segment. The number and s i z e o f t h e s e segments i s an o p t i m i z a t i o n p r o c e s s c h o s e n so t h a t t h e a v e r a g e power d e l i v e r e d t o t h e s p e c i m e n i s maximum w h i l e a t t h e same t i m e a v o i d i n g t h e d i v i s i o n o f t h e f r e q u e n c y r a n g e i n t o t o o many seg m e n t s . T a b l e IV-1 i n d i c a t e s t h e f r e q u e n c y segments and c o r r e s p o n d i n g a c c e l e r a t i o n l e v e l s u s e d f o r t h e column s t u d i e d . I n t h i s way a f i r s t a p p r o x i m a t i o n t o a c o n -s t a n t , maximum power p l o t was a c h i e v e d w h i l e m a k i n g use o f - 35 -the experimental system c a p a b i l i t y f o r constant a c c e l e r a t i o n c o n t r o l . The amplitude o f v i b r a t i o n was thus kept as hi g h as p o s s i b l e so t h a t the s t r a i n s were more r e a d i l y measured. FREQUENCY RANGE ACCELERATION LEVEL 250 t o 750 Hz 7.0 750 850 4.0 850 1200 20.0 1200 2500 15.0 2500 5100 40.0 5100 8000 20.0 8000 9000 55.0 9000 10200 20 .0 Table IV-1 P e r m i s s i b l e A c c e l e r a t i o n L e v e l s Over Various Frequency Ranges f o r Column Specimen The t e s t i n g was now ready t o begin. The f o l l o w i n g steps were adhered t o : (1) the generator c o n t r o l was programmed to d e l i v e r the d e s i r e d a c c e l e r a t i o n l e v e l . (2) the frequency scanning speed on the c o n t r o l g enerator and corresponding paper speed on the l e v e l r e c o r d e r were chosen (the lowest scanning speed was u s u a l l y used t o approximate steady-s t a t e c o n d i t i o n s ) . - 36 -(3) t h e c o m p r e s s o r s p e e d was c h o s e n t o p r o v i d e s t a b i l i t y i n t h e f e e d b a c k . c i r c u i t (4) t h e f r e q u e n c y i n d i c a t e d by t h e l e v e l r e c o r d e r was s y n c h r o n i z e d w i t h t h e c o n t r o l u n i t (5) t h e c o n t r o l u n i t was p u t i n i t s e x c i t a t i o n mode (6) t h e p r o p e r a m p l i f i c a t i o n i n t h e f r e q u e n c y a n a l y z e r was s e t (7) t h e a t t e n u a t i o n and w r i t i n g s p e e d o f t h e l e v e l r e c o r d e r were s e t (8) t h e Roots b l o w e r was a c t i v a t e d t o c o o l t h e s h a k e r (9) t h e CRO waa a d j u s t e d t o a p p r o p r i a t e l y d i s p l a y t h e s i g n a l s o f i n t e r e s t (10) t h e s c a n n i n g mechanism was a c t i v a t e d . T h e s e s t e p s were r e p e a t e d o v e r e a c h segment o f t h e f r e q u e n c y s p e c t r a . No r e c o r d was made o f t h e a c t u a l change from one a c c e l e r a t i o n l e v e l t o a n o t h e r due t o l a r g e t r a n s i e n t v o l t a g e s . The t e s t s were p e r f o r m e d so t h a t t h e f r e q u e n c y segments e x t e n d e d a few H e r t z o v e r t h e i r e nd p o i n t s t o a v o i d t h e p o s s i b i l i t y o f l o s i n g i n f o r m a t i o n a t t h e a c c e l e r a t i o n c h a n g e o v e r f r e q u e n c y . A t p a r t i c u l a r f r e q u e n c i e s o f i n t e r e s t , s p e c t r a l a n a l y s e s were made. The p r o c e d u r e f o l l o w e d i n o b t a i n i n g a s p e c t r a l a n a l y s i s was much t h e same as t h a t o u t l i n e d above, e x c e p t t h a t t h e f r e q u e n c y s c a n n i n g mechanism was n o t a c t i v a t e d , t h e f r e q u e n c y a n a l y z e r was p u t i n t h e a n a l y s i s mode, and a m e c h a n i c a l f r e q u e n c y s y n c h r o n i z a t i o n was p r o v i d e d between t h e l e v e l r e c o r d e r and f r e q u e n c y a n a l y z e r . The e x c i t a t i o n f r e -q u e n c y o f i n t e r e s t was s e t on t h e c o n t r o l g e n e r a t o r and t h e t e s t was c a r r i e d o u t . P h o t o g r a p h y P h o t o g r a p h s o f s t r a i n , a c c e l e r a t i o n and power waveforms were o b t a i n e d o v e r most o f t h e f r e q u e n c y r a n g e . T h o s e o f p a r t i c u l a r i n t e r e s t a r e shown i n C h a p t e r IV. P h o t o g r a p h s o f waveforms were t a k e n d i r e c t l y f r o m t h e o s c i l l o s c o p e i n t h e n o r m a l t r i g g e r i n g mode u s i n g a P e n t a x S p o t m a t i c Camera w i t h an f / s t o p o f 2.8, an a p e r a t u r e s p e e d o f s e c o n d , w i t h a 55 mm. l e n s and a no. 3 c l o s e - u p l e n s . Oscosslooi ©f Result; - 38 -CHAPTER V RESULTS AND DISCUSSION I n t e r p r e t a t i o n o f F r e q u e n c y S p e c t r a The s t r a i n v e r s u s f r e q u e n c y p l o t s were o b t a i n e d on a c h a r t p r o d u c e d by t h e l e v e l r e c o r d e r (see F i g . V-2 and F i g . V - 3 ) . To c o r r e l a t e t h e r e c o r d w i t h t h e m e a s u r e d s t r a i n , t h e f o l l o w i n g p a r a m e t e r s must be known: (a) t h e gauge f a c t o r and b r i d g e a r r a n g e m e n t (b) t h e s t r a i n gauge b r i d g e e x c i t a t i o n v o l t a g e u s e d (c) t h e m a g n i f i c a t i o n o f t h e BAM and v o l t a g e a m p l i f i e r (d) t h e a t t e n u a t i o n o f t h e l e v e l r e c o r d e r The b r i d g e e x c i t a t i o n v o l t a g e u s e d i s s l i g h t l y t i m e d e p e n d e n t . The m a g n i f i c a t i o n o f t h e BAM i s b o t h t i m e and f r e q u e n c y de-p e n d e n t . I t i s a p p a r e n t t h a t t h e c o n v e r s i o n o f t h e m e a s u r e d v o l t a g e l e v e l t o a t r u e s t r a i n r e c o r d i s t e d i o u s . F o r t h i s r e a s o n , and b e c a u s e t h e p r i m a r y i n t e n t o f t h e e x p e r i m e n t s i s t o s t u d y t h e e x i s t e n c e o f p r e d i c t e d v i b r a t i o n s , s u c h a c o n -v e r s i o n was n o t c a r r i e d o u t . A p p r o x i m a t e c a l c u l a t i o n s were c a r r i e d o u t a t r e s o n a n t f r e q u e n c i e s o f i n t e r e s t t o s t u d y t h e s i g n i f i c a n c e o f t h e s e v i b r a t i o n s . C a r e must be e x e r c i s e d i n i n t e r p r e t a t i o n o f t h e s t r a i n r e c o r d s , s i n c e t h e s i g n a l i n -c l u d e s a c o n t r i b u t i o n f r o m e l e c t r o m a g n e t i c i n d u c t i o n . However, t h i s e f f e c t was m i n i m i z e d and t h e i n f l u e n c e on t h e r e s o n a n t p e a k s was s m a l l ( l e s s t h a t 1 d b ) . - 39 -The f r e q u e n c i e s at which peaks o c c u r r e d on the s t r a i n records were s t u d i e d , and the s t r a i n waveforms at these f r e -quencies were analyzed. The experimental r e s u l t s are pre-sented and d i s c u s s e d i n t h i s chapter. I d e n t i f i c a t i o n of S t r a i n Peaks The s t r a i n records o b t a i n e d have reproduceable peaks at s e v e r a l d i s c r e t e f r e q u e n c i e s . Some of these peaks correspond to resonant o s c i l l a t i o n s of the specimen. I t i s t h e r e f o r e d e s i r a b l e t o i d e n t i f y the peaks i n order t o d i s t i n g u i s h between ' l i n e a r 1 resonant peaks and n o n l i n e a r peaks. Resonant f l e x u r a l o s c i l l a t i o n s are co n s i d e r e d f i r s t . By l i n e a r theory (Appendix C), the approximate resonant t r a n s -v e r s e f r e q u e n c i e s can be c a l c u l a t e d . A knowledge of the end c o n d i t i o n s i s necessary i n order t o determine the resonant f r e q u e n c i e s a n a l y t i c a l l y . Although the experimental setup was d i r e c t e d towards o b t a i n i n g clamped end c o n d i t i o n s , t h i s o b j e c t i v e was not r e a l i z e d . The end c o n d i t i o n s were some-where between clamped and pinned. F i g . V - l i n d i c a t e s the resonant f r e q u e n c i e s of v a r i o u s modes f o r pinned and clamped ends as c a l c u l a t e d from l i n e a r theory accounting f o r the con-s t a n t compressive end l o a d . A l s o p l o t t e d on the graph are the fr e q u e n c i e s of s t r a i n peaks b e l i e v e d t o correspond t o resonant t r a n s v e r s e f r e q u e n c i e s of the a c t u a l specimen. The e x p e r i -mental p o i n t s are obt a i n e d from the f l e x u r a l s t r a i n versus f r e -quency r e c o r d . The experimental curve i s seen t o f a l l between the f r e q u e n c i e s o f clamped end o s c i l l a t i o n s and those o f pinned end v i b r a t i o n s . The n a t u r a l f r e q u e n c i e s of t r a n s v e r s e v i b r a -- 40 -F i g . V - l . F o r c e d R e s o n a n t F r e q u e n c i e s P l o t t e d V e r s u s Mode Number f o r V a r i o u s End C o n d i t i o n s - 41 -t i o n as t h e y a r e p l o t t e d i n F i g . V - l s e r v e as a q u a l i t a t i v e t e s t o f t h e b o u n d a r y c o n d i t i o n s . R e s o n a n t l o n g i t u d i n a l o s c i l l a t i o n s a r e c o n s i d e r e d i n t h e f o l l o w i n g . U s i n g l i n e a r t h e o r y , and a c c o u n t i n g f o r t h e end l o a d i n g , t h e f u n d a m e n t a l l o n g i t u d i n a l r e s o n a n c e i s p r e -d i c t e d a t 8789 Hz ( A p p e n d i x C ) . A s p i k e on t h e l o n g i t u d i n a l s t r a i n r e c o r d o c c u r s a t 8 700 Hz. F u r t h e r , t h e power r e q u i r e d t o s u s t a i n a c o n s t a n t l e v e l o f a c c e l e r a t i o n i n c r e a s e d a t 8700 Hz. T h i s w o u l d i n d i c a t e a p r o b a b l e r e s o n a n t c o n d i t i o n s i n c e t h e n a t u r a l d e s i r e o f t h e column t o o s c i l l a t e a t t h e f u n d a -m e n t a l l o n g i t u d i n a l f r e q u e n c y w o u l d i n c r e a s e t h e e x t e r n a l l o a d i n g r e q u i r e d t o l i m i t t h e a c c e l e r a t i o n t o t h e p r e s e t l e v e l . F i n a l l y , t o r s i o n a l r e s o n a n c e s a r e c o n s i d e r e d . A g a i n , u s i n g l i n e a r t h e o r y , c o n s i d e r i n g t h e d e c r e a s e i n t o r s i o n a l s t i f f n e s s and t h e c o m p r e s s i v e end l o a d , t h e p r e d i c t e d r e s o n a n t f r e q u e n c i e s a r e c a l c u l a t e d ( A p p e n d i x C ) . The computed f u n d a -m e n t a l t o r s i o n a l r e s o n a n c e o c c u r s a t 3380 Hz; t h e f i r s t and s e c o n d t o r s i o n a l h a r m o n i c s a r e c a l c u l a t e d a t 6 760 Hz and 10140 Hz r e s p e c t i v e l y . No s i g n i f i c a n t s t r a i n peak i s o b s e r v e d on t h e l o n g i t u d i n a l s t r a i n r e c o r d i n t h e n e i g h b o u r h o o d o f 3380 Hz, so i t a p p e a r s t h a t t h e f u n d a m e n t a l t o r s i o n a l r e s o -n a n c e i s n o t e x c i t e d . Peaks w i t h s i m i l a r c h a r a c t e r i s t i c s a p p e a r on t h e e x p e r i m e n t a l s t r a i n r e c o r d a t 6920 Hz and a g a i n a t 10000 Hz. T h e s e w i l l be d i s c u s s e d i n c o n n e c t i o n w i t h t o r s i o n a l v i b r a t i o n s i n a l a t e r s e c t i o n o f t h i s c h a p t e r . - 42 -R e s u l t s o f F l e x u r a l S t r a i n R e c o r d F i g . V-2 shows a f l e x u r a l s t r a i n v e r s u s f r e q u e n c y r e c o r d f o r t h e m i d p o i n t o f t h e column. Most o f t h e s t r a i n peaks a r e i d e n t i f i e d a c c o r d i n g t o t h e d e d u c t i o n s made i n t h e p r e v i o u s s e c t i o n c o n c e r n i n g f l e x u r a l r e s o n a n c e s . I t i s n o t e w o r t h y t h a t t r a n s v e r s e r e s o n a n c e s w i t h e v e n mode numbers a p p e a r as s m a l l e r s t r a i n peaks t h a n t h o s e h a v i n g odd mode numbers. T h i s i s e x p e c t e d a t t h e column m i d p o i n t . • -t < < i i i ( ( ( i if f f -c —f—r —r —r—r—r —r —r —r —r—r — r —r—r—r—r—r —r—r—r—r—r—r—r—i' —r —r —r —t —r — i —i — r —r —r—r--Fifth Flexural Mode E,iein.,„ ed Flexural Mode1 Second Coupled Flexural Mode= Ninths Fl exural Mode 1 Tenth-ElexyraLModp^^ m i n l a r l F l A v u m l M o d e ^ -First Coupled lexural qde EXCITATION ERECIUENCY^ k H z 9 10 F i g . V-2 F l e x u r a l S t r a i n R e c o r d O b t a i n e d a t M i d p o i n t o f Column As shown s c h e m a t i c a l l y i n F i g . V-3, t h e m i d p o i n t o f t h e s p e c i m e n i s a t o r n e a r a node f o r an e v e n mode number. The column m i d p o i n t e x h i b i t s r e l a t i v e l y l i t t l e c u r v a t u r e , and t h e f l e x u r a l s t r a i n i s c o r r e s p o n d i n g l y low. F o r an odd mode number, t h e column m i d p o i n t i s n e a r an a n t i n o d e h a v i n g a l a r g e r c u r v a t u r e , and an a t t e n d a n t l a r g e r a m p l i t u d e s t r a i n s p i k e w i l l r e s u l t . - 43 -M o d e S h a p e a t M i n i m u m E x t e n s i o n M o d e S h a p e a t M a x i m u m E x t e n s i o n ^ M i d p o i n t F l e x u r a l ,' D i s p l a c e m e n t PQ + R c o s r t O D D M O D E N U M B E R M i d p o i n t F l e x u r a l D i s p l a c e m e n t P D + P c o s T t E V E N M O D E N U M B E R F i g . V - 3 . P o s s i b l e F l e x u r a l Mode Shapes F o r A x i a l l y E x c i t e d Column Two s i g n i f i c a n t f l e x u r a l s t r a i n s p i k e s a p p e a r w h i c h a r e n o t y e t a c c o u n t e d f o r . The f i r s t o f t h e s e o c c u r s a t an e x -c i t a t i o n f r e q u e n c y o f a b o u t 2900 Hz; t h e s e c o n d a t an e x c i t a -t i o n f r e q u e n c y o f a p p r o x i m a t e l y 4 350 Hz as i n d i c a t e d by t h e d i g i t a l c o u n t e r . I n t h e n e i g h b o u r h o o d o f t h e s e two f r e q u e n c i e s , t h e v i b r a t i o n s y s t e m has a t e n d e n c y t o become u n s t a b l e . E x c e p t a t low e x c i t a t i o n l e v e l s , t h e v i b r a t i o n s y s t e m b e g i n s ' f l u t t e r i n g ' r a p i d l y . The r e s p o n s e waveforms f l u t t e r on t h e o s c i l l o s c o p e ; t h e f e e d b a c k c i r c u i t i s n o t c a p a b l e o f s t a b i l i -- 44 -z i n g t h e o s c i l l a t i o n . I f t h e a c c e l e r a t i o n l e v e l . o f t h e ex-c i t a t i o n i s l o w e r e d , however, t h e s y s t e m becomes s t a b l e and t h e r e s p o n s e waveforms a t 2900 Hz and 4350. Hz can be s t u d i e d . I n c r e a s i n g the a c c e l e r a t i o n l e v e l from a c o n d i t i o n o f s y s t e m s t a b i l i t y , on t h e o t h e r hand, r e s u l t s i n a sudden jump t o s y s t e m i n s t a b i l i t y as i n d i c a t e d by t h e f l u t t e r i n g waveforms d e s c r i b e d e a r l i e r . The f r e q u e n c y o f t h e u n s t a b l e r e s p o n s e i s s t i l l t h e same as t h e e x c i t a t i o n f r e q u e n c y , so t h i s jump phenomena i s n o t a s n a p - t h r o u g h t y p e o f o s c i l l a t i o n . I f s n a p - t h r o u g h were o c c u r r i n g , t h e f r e q u e n c y o f t h e f l e x u r a l s t r a i n waveform w o u l d be o n e - h a l f t h a t o f t h e e x c i t a t i o n . F i n a l l y , when t h e e x c i t a t i o n f r e q u e n c y p a s s e s t h r o u g h t h e f r e q u e n c y f o r t h e f u n d a m e n t a l l o n g i t u d i n a l v i b r a t i o n a t a p p r o x i m a t e l y 8700 Hz a s m a l l n o n l i n e a r s t r a i n s p i k e a p p e a r s on t h e f l e x u r a l s t r a i n r e c o r d . The h i s t o g r a m shown i n F i g . V-4 i n d i c a t e s t h o s e e x c i t a -t i o n f r e q u e n c i e s a t w h i c h s t r a i n s p i k e s c o r r e s p o n d i n g t o r e s o n a n t f r e q u e n c i e s a r e o b s e r v e d on t h e f l e x u r a l s t r a i n r e c o r d . The s o l i d b a r s c o r r e s p o n d t o f l e - x u r a l r e s o n a n c e s as p r e d i c t e d by o r d i n a r y l i n e a r t h e o r y , w h i l e t h e c r o s s - h a t c h e d b a r s i n d i c a t e e x c i t a t i o n f r e q u e n c i e s a s s o c i a t e d w i t h non-l i n e a r r e s o n a n t p e a k s . N o n l i n e a r f l e x u r a l s t r a i n p e aks a r e p r e s e n t when t h e e x c i t a t i o n f r e q u e n c y c o i n c i d e s . w i t h t h e f u n d a m e n t a l l o n g i t u d i n a l f r e q u e n c y a t 8700 Hz, when t h e e x c i t a t i o n f r e q u e n c y i s one h a l f o f t h e f u n d a m e n t a l l o n g i t u d i -n a l f r e q u e n c y a t 4 350 Hz, and when t h e e x c i t a t i o n f r e q u e n c y - 45 -i s one t h i r d o f t h e f u n d a m e n t a l l o n g i t u d i n a l f r e q u e n c y a t 2900 Hz. EXCITATION FREQUENCY kHz F i g . V-4. F r e q u e n c i e s o f E x p e r i m e n t a l l y O b s e r v e d F l e x u r a l R e s o n a n t S t r a i n Peaks The r e s o n a n t t r a n s v e r s e c o n d i t i o n o c c u r r i n g a t t h e s e f r e -q u e n c i e s a r e r e f e r r e d t o as t h e f i r s t , s e c o n d , and t h i r d c o u p l e d f l e x u r a l modes r e s p e c t i v e l y . Arrows below t h e h i s t o g r a m i n F i g . V-4 i n d i c a t e t h o s e e x c i t a t i o n f r e q u e n c i e s a t w h i c h c o u p l e d f l e x u r a l modes a r e e x p e c t e d from 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 . Shown i n F i g . V-5 a r e p h o t o g r a p h s o f s e v e r a l waveforms c o r r e s p o n d i n g t o o s c i l l a t i o n s o f t h e c o u p l e d f l e x u r a l modes. The u p p e r waveform i n e a c h p i c t u r e i s t h e e x c i t a t i o n v o l t a g e - 4 6 -• F i r s t C o u p l e d F l e x u r a l Mode t ( E x c i t a t i o n Frequency=8700 Hz O r d e r o f S i g n a l s ( t o p t o b o t t o m ) E x c i t a t i o n V o l t a g e F l e x u r a l S t r a i n A c c e l e r a t i o n S e c o n d C o u p l e d F l e x u r a l Mode E x c i t a t i o n F r e q u e n c y = 4 3 5 0 Hz O r d e r o f S i g n a l s ( t o p t o b o t t o m ) E x c i t a t i o n V o l t a g e F l e x u r a l S t r a i n A c c e l e r a t i o n T h i r d C o u p l e d F l e x u r a l Mode E x c i t a t i o n Frequency=2900 Hz O r d e r o f S i g n a l s ( t o p t o bottom) E x c i t a t i o n V o l t a g e F l e x u r a l S t r a i n A c c e l e r a t i o n F i g . V-5 Waveforms C o r r e s p o n d i n g t o C o u p l e d F l e x u r a l Modes supplied to the shaker, the middle waveform is the flexural strain level at the midpoint of the column, and the lower waveform is the acceleration monitored at the lower end mount of the column. The time base is the same for a l l the wave-forms in each picture. The strain at the column midpoint corresponding to the second and third flexural modes are comparable in magnitude to the strains associated with l inear flexural resonances, while the strain level at the f i r s t coupled flexural mode is somewhat less. The nonlinearity at an excitation frequency of approxi-mately 8700 Hz may be the result of longitudinal inertia forces, which can influence the dynamic behavior of a column when the frequency of the^external force is near the longi-tudinal natural frequency of the column; that i s , when the longitudinal vibrations have a resonance character. In other-words, this nonlinearity may represent the parametric influenc of resonant longitudinal vibrations which give rise to a flexural vibration as indicated in the theory by the term (u x w x ) x . The second and third coupled flexural modes may be parametrically excited in a similar manner. Recalling the theoretical predictions, the transverse response was seen to- be,";appT5S?mately sinusiodal on the f i r s t substitution (equation 2-12) . This apparently describes the. transverse motion quite well except when the fundamental longitudinal frequency is excited. On the third substitution the transverse motion included a sinusoidal term with twice t h e e x c i t a t i o n f r e q u e n c y ( e q u a t i o n 2-19) . T h i s b e t t e r d e s c r i b e s t h e r e s p o n s e when t h e e x c i t a t i o n f r e q u e n c y e q u a l s t h e f u n d a m e n t a l l o n g i t u d i n a l f r e q u e n c y . The s e c o n d and t h i r d c o u p l e d f l e x u r a l r e s o n a n c e s e x h i b i t no c o n t r i b u t i o n from t h e s i n u s o i d a l t e r m h a v i n g t w i c e t h e e x c i t a t i o n f r e -q u e n c y . The a p p e a r a n c e o f t h e t h i r d c o u p l e d f l e x u r a l v i b r a t i o n i n d i c a t e s t h a t h i g h e r c o u p l e d v i b r a t i o n s t h a n t h e s e c o n d do o c c u r . T h e o r e t i c a l l y , o n l y t h e f i r s t and s e c o n d c o u p l e d f l e x u r a l v i b r a t i o n s were a n t i c i p a t e d a t f r e q u e n c i e s i n d i c a t e d by t h e a r r o w s i n F i g . V-4. S m a l l n o n l i n e a r i t i e s i n t h e f l e x u r a l s t r a i n were n o t e d a t l o w e r f r e q u e n c i e s , p a r t i c u l a r l y below 1000 Hz. T h e s e a r e n o t d i s c u s s e d , s i n c e t h e a m p l i t u d e o f t h e s e i r r e g u l a r i t i e s i s s m a l l compared t o t h e amplitude, o f t h e p r i m a r y waveform h a v i n g t h e f r e q u e n c y o f t h e e x c i t a t i o n . F u r t h e r , t h e t e s t s were f o c u s e d on h i g h f r e q u e n c y r e s p o n s e . No e x p l a n a t i o n i s p r o v i d e d f o r one r e m a i n i n g f l e x u r a l s t r a i n peak w h i c h i s n o t a c c o u n t e d f o r a t 900 Hz. R e s u l t s o f A x i a l S t r a i n R e c o r d F i g . V-6 i s an a x i a l s t r a i n r e c o r d f o r t h e m i d p o i n t o f o f t h e column. I n a d d i t i o n t o t h e a x i a l s t r a i n , t h e r e c o r d a l s o c o n t a i n s a s m a l l i n f l u e n c e due t o t h e f l e x u r a l s t r a i n . However, f o r t h e b r i d g e c o n f i g u r a t i o n u s e d , t h e m easured v o l t a g e i s d i r e c t l y p r o p o r t i o n a l t o t h e a x i a l s t r a i n and t h e - 49 -F i g . V-6. A x i a l S t r a i n R e c o r d O b t a i n e d a t M i d p o i n t o f Column s q u a r e o f t h e f l e x u r a l s t r a i n . S i n c e t h e maximum f l e x u r a l s t r a i n and maximum a x i a l s t r a i n a r e c o m p a r a b l e and a r e o f t h e o r d e r o f 200 u i n / i n , i t i s a p p a r e n t t h a t t h e c o n t r i b u t i o n t o t h e r e c o r d by t h e f l e x u r a l s t r a i n i s n e g l i g i b l e . T h r e e p r o m i n e n t s p i k e s a p p e a r on t h e a x i a l s t r a i n r e c o r d . The f i r s t i s a t an e x c i t a t i o n f r e q u e n c y o f a p p r o x i -m a t e l y 8700 Hz, t h e s e c o n d when t h e e x c i t a t i o n f r e q u e n c y i s abo u t 4350 Hz, and t h e t h i r d o c c u r s when t h e e x c i t a t i o n f r e q u e n c y i s 2900 Hz. The r e s o n a n t c o n d i t i o n s c o r r e s p o n d i n g t o t h e s e s t r a i n s p i k e s a r e r e f e r r e d t o as t h e f i r s t , s e c o n d and t h i r d c o u p l e d a x i a l modes r e s p e c t i v e l y , as i n d i c a t e d i n F i g . V-6. The e x c i t a t i o n f r e q u e n c i e s a s s o c i a t e d w i t h t h e s e t h r e e r e s o n a n t s t r a i n s p i k e s c o i n c i d e w i t h t h e e x c i t a t i o n f r e q u e n c i e s o f c o u p l e d f l e x u r a l r e s o n a n c e s d i s c u s s e d e a r l i e r . - 50 -In g e n e r a l , r e s o n a n t a x i a l s t r a i n peaks b u i l d up more q u i c k l y t h a n r e s o n a n t f l e x u r a l s t r a i n p e a k s . The s t r a i n p e a k s f o r r e s o n a n t f l e x u r a l and a x i a l o s c i l l a t i o n s have a m p l i t u d e s i n t h e same o r d e r o f m a g n i t u d e . I n v e r t e d peaks a p p e a r on t h e a x i a l s t r a i n r e c o r d i n F i g . V-6 a t two f r e q u e n c i e s o f e x c i t a t i o n . The f i r s t o f t h e s e i s a t 6920 Hz, t h e s e c o n d i s a t 10000 Hz. The shape o f t h e s t r a i n r e s p o n s e as t h e v i b r a t i o n s y s t e m sweeps t h r o u g h t h e s e f r e q u e n c i e s i n d i c a t e s p a r a m e t r i c e x c i t a t i o n o f t o r s i o n a l modes. F o r m a t e r i a l s s u c h as s t e e l , t w i s t i n g p u t s t h e f i b r e s away f r o m t h e c e n t r a l p l a n e f i b r e o f t h e column i n t e n s i o n , i f i t i s o r i g i n a l l y u n l o a d e d . Remembering t h a t t h e column i s u n d e r c o m p r e s s i v e a x i a l l o a d i n g , and r e f e r r i n g t o F i g . V-7 i t c a n be s e e n t h a t t w i s t i n g o f t h e column w i l l r e d u c e t h e s t r a i n i n t h o s e f i b r e s away from t h e c e n t r a l f i b r e , s u c h as AB 1 F i b e r AB must e l o n g a t e t o AB' when t h e column i s t w i s t e d t h r o u g h a n g l e © . Thus, when t o r s i o n a l o s c i l l a t i o n s a r e p a r a -m e t r i c a l l y e x c i t e d , t h e a x i a l s t r a i n l e v e l w i l l be r e d u c e d . S u ch i s t h e c a s e i n F i g . V-6 a t e x c i t a t i o n f r e q u e n c i e s o f 6920 Hz and 10000 Hz. The h i s t o g r a m shown i n F i g . V-8 i n d i c a t e s t h o s e e x c i t a t i o n f r e q u e n c i e s a t w h i c h s t r a i n s p i k e s c o r r e s p o n d i n g t o r e s o n a n t f r e q u e n c i e s a r e o b s e r v e d on t h e a x i a l s t r a i n r e c o r d . The c r o s s -h a t c h e d b a r s c o r r e s p o n d t o c o u p l e d a x i a l modes w h i l e t h e s o l i d b a r s i n d i c a t e e x c i t a t i o n f r e q u e n c i e s a s s o c i a t e d w i t h c o u p l e d t o r s i o n a l r e s o n a n c e s . The arrows below t h e h i s t o g r a m i n d i c a t e - 51 -F i g . V-7. A P o r t i o n o f A Column U n d e r g o i n g T w i s t i n g IA 8? 8 EXCITATION FREQUENCY kHz F r e q u e n F i g . V-8. c i e s o f E x p e r i m e n t a l l y O b s e r v e d A x i a l S t r a i n Peaks t h o s e e x c i t a t i o n f r e q u e n c i e s a t w h i c h p a r a m e t r i c a l l y e x c i t e d modes.were t h e o r e t i c a l l y a n t i c i p a t e d on t h e a x i a l s t r a i n r e c o r d . Two d i s c r e p a n c i e s between t h e t h e o r e t i c a l p r e d i c -t i o n s and e x p e r i m e n t a l r e s u l t s a r e a p p a r e n t . The t h i r d c o u p l e d a x i a l mode i s s i g n i f i c a n t t h o u g h i t i s n o t p r e s e n t i n t h e 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 . F u r t h e r , t h e s e c o n d c o u p l e d t o r s i o n a l mode, i f i t were t o be e x c i t e d , was e x p e c -t e d when t h e e x c i t a t i o n f r e q u e n c y was i n t h e n e i g h b o u r h o o d o f t h e f u n d a m e n t a l l o n g i t u d i n a l f r e q u e n c y a t 8700 Hz. However, i t a p p e a r s t o be p a r a m e t r i c a l l y e x c i t e d a t t h e f r e q u e n c y c o r r e s p o n d i n g t o t h e t h i r d t o r s i o n a l mode. F i r s t C o u p l e d A x i a l Mode ( F u n d a m e n t a l L o n g i t u d i n a l Resonance) E x c i t a t i o n Frequency=8700 O r d e r o f S i g n a l s ( t o p t o bottom) E x c i t a t i o n V o l t a g e A x i a l S t r a i n A c c e l e r a t i o n F i g . V - 9 . Waveforms a t F u n d a m e n t a l L o n g i t u d i n a l Resonance F i g . V-9 i s a p i c t u r e o f t h e waveforms c o r r e s p o n d i n g t o an e x c i t a t i o n f r e q u e n c y o f 8700 Hz. S i m i l a r waveforms f o r e x c i t a t i o n f r e q u e n c i e s i n t h e n e i g h b o u r h o o d o f 4 350 Hz and - 53 -2900 Hz a r e shown i n F i g . V-10 and V - l l r e s p e c t i v e l y . A s p e c t r a l a n a l y s i s , o f t h e s t r a i n s i g n a l a t t h e s e e x c i t a t i o n f r e q u e n c i e s was p e r f o r m e d . The f o l l o w i n g was r e v e a l e d : a t an e x c i t a t i o n f r e q u e n c y o f 3700 Hz, a r e s o n a n t l o n g i t u d i n a l v i b r a t i o n o f f r e q u e n c y 8700 e x i s t s ; a t an e x c i t a -t i o n f r e q u e n c y o f 4350 Hz a l o n g i t u d i n a l v i b r a t i o n o f f r e -q u e n c y 8700 Hz e x i s t s ; and a t an e x c i t a t i o n f r e q u e n c y o f 2900 Hz a l o n g i t u d i n a l v i b r a t i o n o f f r e q u e n c y 8700 Hz e x i s t s . The c o u p l e d t e r m s i n e q u a t i o n (2-8) a p p e a r t o be s i g n i f i c a n t as p r e d i c t e d t h e o r e t i c a l l y a t e x c i t a t i o n f r e q u e n c i e s o f 8700 Hz and 4350 Hz. The t h e o r e t i c a l development, s u g g e s t e d t h a t t h e l o n g i t u d i n a l r e s p o n s e was t h e sum o f a s i n u s o i d a l t e r m h a v i n g t h e e x c i t a t i o n f r e q u e n c y and a s i n u s o i d a l t e r m h a v i n g t w i c e t h e e x c i t a t i o n f r e q u e n c y . The f u n d a m e n t a l l o n g i t u d i n a l mode i s t h e r e f o r e e x p e c t e d t o be e x c i t e d when t h e e x c i t a t i o n r'.'i • freq.uency i s one h a l f o f "the f u n d a m e n t a l l o n g i t u d i n a l f r e q u e n c y . * O b s e r v a t i o n o f t h e s t r a i n waveforms i n F i g . V - l l f o r e x c i t a t i o n f r e q u e n c i e s f r e q u e n c i e s i n t h e n e i g h b o u r h o o d o f 4350 Hz r e v e a l s a c o n t i n u o u s phase change between t h e s i n u -s o i d a l t e r m h a v i n g t w i c e t h e e x c i t a t i o n f r e q u e n c y and t h e s i n u s o i d a l t e r m h a v i n g t h e e x c i t a t i o n f r e q u e n c y as t h e v i b r a t i o n s y s t e m moves t h r o u g h t h i s f r e q u e n c y r a n g e . T h i s phase r e l a t i o n s h i p i s i l l u s t r a t e d i n F i g . V-12. The fundamen-t a l l o n g i t u d i n a l f r e q u e n c y i s f i r s t e x c i t e d i n F i g . V-12(a) when t h e two s i n u s o i d a l terms a r e ' i n p h a s e ' as i n d i c a t e d by p o i n t s A and A'. As t h e f r e q u e n c y r a n g e i s s c a n n e d , t h e two <9 - 54 -! E x c i t a t i o n F r e q u e n c y = 4320 Hz I , O r d e r o f S i g n a l s ( t o p t o bottom) E x c i t a t i o n V o l t a g e A x i a l S t r a i n A c c e l e r a t i o i . E x c i t a t i o n F r e q u e n c y = 4345 Hz O r d e r o f S i g n a l s ( t o p t o bottom) E x c i t a t i o n V o l t a g e A x i a l S t r a i n A c c e l e r a t i o n E x c i t a t i o n F r e q u e n c y = 4360 Hz O r d e r o f S i g n a l s ( t o p t o bottom) E x c i t a t i o n V o l t a g e A x i a l S t r a i n A c c e l e r a t i o n F i g . V-10. Waveforms C o r r e s p o n d i n g t o S e c o n d C o u p l e d A x i a l Mode - 55 -E x c i t a t i o n F r e q u e n c y = 2890 Hz O r d e r o f S i g n a l s ( t o p t o bottom) E x c i t a t i o n V o l t a g e A x i a l S t r a i n A c c e l e r a t i o n E x c i t a t i o n F r e q u e n c y = 2905 Hz O r d e r o f S i g n a l s ( t o p t o bottom) » E x c i t a t i o n V o l t a g e A x i a l S t r a i n A c c e l e r a t i o n E x c i t a t i o n F r e q u e n c y = 2910 Hz O r d e r o f S i g n a l s ( t o p t o bottom) E x c i t a t i o n V o l t a g e A x i a l S t r a i n A c c e l e r a t i o n F i g . V - l l . Waveforms C o r r e s p o n d i n g t o T h i r d C o u p l e d A x i a l Mode Increasing Excitation Frequency Y (N lb) (c) F i g . V-12. I l l u s t r a t i o n Showing C h a n g i n g Phase R e l a t i o n s Between The Two V i b r a t i o n s C o m p r i s i n g t h e Second C o u p l e d A x i a l Mode waveforms move 'out o f p h a s e ' as shown by t h e movement o f p o i n t A i n F i g . V - 1 2 ( b ) . F i n a l l y , t h e phase between t h e two waveforms has c h a n g e d by one h a l f o f a c o m p l e t e c y c l e o f t h e e x c i t a t i o n , shown i n F i g . V - 1 2 ( c ) , and h e r e t h e ex-c i t a t i o n o f t h e f u n d a m e n t a l l o n g i t u d i n a l r e s o n a n t c o n d i t i o n d i s c o n t i n u e s . . -S7 F i r s t C o u p l e d T o r s i o n a l Mode E x c i t a t i o n F r e q u e n c y = 6920 Hz O r d e r o f S i g n a l s ( t o p t o bottom) E x c i t a t i o n V o l t a g e A x i a l S t r a i n A c c e l e r a t i o n S e c o n d C o u p l e d T o r s i o n a l Mode E x c i t a t i o n F r e q u e n c y = 10000 Hz O r d e r o f S i g n a l s ( t o p t o bottom) E x c i t a t i o n V o l t a g e A x i a l S t r a i n A c c e l e r a t i o n Waveforms C o r r e s p o n d i n g t o F i r s t and Se c o n d C o u p l e d T o r s i o n a l Modes The waveforms c o r r e s p o n d i n g t o t h e f i r s t and s e c o n d c o u p l e d t o r s i o n a l modes a t e x c i t a t i o n f r e q u e n c i e s o f 6920 Hz and 10000 Hz r e s p e c t i v e l y a r e shown i n F i g . V-13. As expec-t e d , t h e s t r a i n waveform r e m a i n s s i n u s o i d a l when t h e f i r s t c o u p l e d t o r s i o n a l mode i s e x c i t e d , s i n c e t h e r e s u l t a n t a x i a l m o t i o n has t h e same f r e q u e n c y as t h e a p p l i e d l o a d . The r e s -ponse when t h e s e c o n d c o u p l e d t o r s i o n a l mode i s e x c i t e d , - 58 -however, i s somewhat n o n l i n e a r * A n o n l i n e a r i t y i s r e a s o n -a b l e h e r e , s i n c e t h e f r e q u e n c y o f t h e v a r i a t i o n i n t h e a x i a l s t r a i n due t o t h e p a r a m e t r i c e x c i t a t i o n o f a t o r s i o n a l mode, w i l l be d i f f e r e n t t h a n t h e f r e q u e n c y o f t h e v a r i a t i o n o f t h e a x i a l s t r a i n due t o t h e l o n g i t u d i n a l e x c i t a t i o n . P a r a m e t r i c e x c i t a t i o n o f t h e t h i r d t o r s i o n a l mode m i g h t be a c c o u n t e d f o r by o b s e r v i n g t h a t a " l i n e a r 1 f l e x u r a l r e s o n a n c e i s n e a r b y on t h e f r e q u e n c y s c a l e . AjeuuuuoiQ - 59 -CHAPTER VI  SUMMARY, & CONCLUSIONS Summary An 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 was made o f t h e dynamic b e h a v i o r o f a column s u b j e c t e d t o p e r i o d i c a x i a l l o a d i n g . O b s e r v e d r e s o n a n t f o r c e d v i b r a t i o n s and n o n l i n e a r c o u p l e d v i b r a t i o n s a r e compared w i t h t h o s e p r e d i c t e d by 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 . To a c c o m p l i s h t h i s s t u d y , c o u p l e d n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s o f m o t i o n were d e r i v e d f o r t h e column t o d e s c r i b e t h e r e l a t i o n s h i p between a x i a l and f l e x u r a l o s c i l l a t i o n s and between a x i a l and t o r s i o n a l v i b r a t i o n s . Of p a r t i c u l a r i m p o r t a n c e i s t h a t i n i t i a l s t r a i n s i n t h e c e n t r a l p l a n e o f t h e column a r e assumed i n f o r m u l a t i n g t h e e q u a t i o n s r e l a t i n g t o a x i a l and f l e x u r a l m o t i o n . T h e s e s t r a i n s a r i s e f r o m c o n s i d e r a t i o n s r e g a r d i n g i n i t i a l c r o o k e d n e s s and l o n g i -t u d i n a l i n e r t i a . By s u i t a b l y m a n i p u l a t i n g t h e e q u a t i o n s and i n t e r p r e t i n g some o f t h e c o u p l e d t e r m s , s e v e r a l n o n l i n e a r c o u p l e d r e s o n a n t v i b r a t i o n s were p r e d i c t e d . F i r s t l y , two c o u p l e d l o n g i t u d i n a l v i b r a t i o n s w i t h a f r e q u e n c y r a t i o o f 1:2 were a n t i c i p a t e d . I n a d d i t i o n , two c o u p l e d f l e x u r a l v i b r a t i o n s h a v i n g a f r e q u e n c y r a t i o o f 1:2 were e x p e c t e d . F i n a l l y , t h e s e c o n d and p o s s i b l y t h e t h i r d t o r s i o n a l modes were e x p e c t e d t o be p a r a m e t r i c a l l y e x c i t e d . - 60 -The 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 p r o v i d e d good agreement w i t h t h e t h e o r e t i c a l p r e d i c t i o n s . The l o n g i t u d i n a l r e s p o n s e was f o u n d t o e x h i b i t t h e f i r s t c o u p l e d a x i a l r e s o n a n c e when t h e e x c i t a t i o n f r e q u e n c y c o r r e s p o n d e d t o t h e f u n d a m e n t a l l o n g i t u d i n a l r e s o n a n t f r e q u e n c y . The s e c o n d c o u p l e d a x i a l mode a p p e a r e d when t h e e x c i t a t i o n f r e q u e n c y was one h a l f o f t h e f u n d a m e n t a l l o n g i t u d i n a l f r e q u e n c y , and t h e r e s p o n s e c o n t a i n e d a v i b r a t i o n ' h a v i n g t h e f r e q u e n c y o f t h e f u n d a m e n t a l l o n g i t u d i n a l r e s o n a n c e ; h e n c e a f r e q u e n c y r a t i o o f 1 : 2 . The f l e x u r a l r e s p o n s e a l s o e x h i b i t e d r e s o n a n t v i b r a t i o n s when t h e e x c i t a t i o n f r e q u e n c y c o i n c i d e d w i t h t h e f u n d a m e n t a l l o n g i t u d i n a l r e s o n a n t f r e q u e n c y and one h a l f o f t h e f u n d a -m e n t a l l o n g i t u d i n a l r e s o n a n t f r e q u e n c y . A waveform a n a l y s i s i n d i c a t e d t h a t t h e r e s o n a n t r e s p o n s e a t t h e s e two f r e q u e n c i e s was s i n u s o i d a l w i t h a f r e q u e n c y e q u a l t o t h a t o f t h e e x c i t a -t i o n f r e q u e n c y . T h e s e r e s o n a n t o s c i l l a t i o n s r e p r e s e n t t h e p a r a m e t r i c i n f l u e n c e o f l o n g i t u d i n a l o s c i l l a t i o n s on f l e x u r a l o s c i l l a t i o n s . P a r a m e t r i c a l l y e x c i t e d t o r s i o n a l r e s o n a n c e s were o b s e r -v e d when t h e l o n g i t u d i n a l e x c i t a t i o n f r e q u e n c y was t w i c e t h e f u n d a m e n t a l t o r s i o n a l f r e q u e n c y , and a g a i n when t h e a p p l i e d f r e q u e n c y was e q u a l t o t h e f r e q u e n c y c o r r e s p o n d i n g t o t h e t h i r d t o r s i o n a l mode. F u r t h e r c o u p l e d o s c i l l a t i o n s were o b s e r v e d on t h e a x i a l and f l e x u r a l r e s p o n s e when t h e l o n g i t u d i n a l e x c i t a t i o n f r e -q u e n c y e q u a l e d one t h i r d o f t h e f u n d a m e n t a l l o n g i t u d i n a l - 61 -r e s o n a n t f r e q u e n c y . A waveform a n a l y s i s o f t h e t h i r d c o u p l e d a x i a l r e s o n a n c e i n d i c a t e d t h a t t h e • r e s p o n s e c o n t a i n e d a v i b -r a t i o n e x h i b i t i n g t h e f r e q u e n c y o f t h e f u n d a m e n t a l l o n g i t u d i -n a l r e s o n a n c e . The t h i r d c o u p l e d f l e x u r a l r e s o n a n c e was a s i n u s o i d a l v i b r a t i o n h a v i n g t h e same f r e q u e n c y as t h e a p p l i e d l o a d i n g . B o t h t h e c o u p l e d and ' l i n e a r ' r e s o n a n t s t r a i n peaks on t h e a x i a l d ynamic s t r a i n r e c o r d b u i l t up and d i e d o u t more q u i c k l y t h a n r e s o n a n t peaks on t h e f l e x u r a l d ynamic s t r a i n r e c o r d . C o u p l e d f l e x u r a l modes have a m p l i t u d e s c o m p a r a b l e t o t h o s e o f ' l i n e a r ' f l e x u r a l modes. The s t r a i n l e v e l a s s o c i a t e d w i t h c o u p l e d a x i a l modes i s l e s s t h a n t h e s t r a i n l e v e l a t t h e • f u n d a m e n t a l l o n g i t u d i n a l r e s o n a n t f r e q u e n c y . The two v i b -r a t i o n s o f d i f f e r e n t f r e q u e n c i e s c o m p r i s i n g t h e c o u p l e d a x i a l modes c h a n g e d phase w i t h r e s p e c t t o e a c h o t h e r as t h e c o u p l e d r e s o n a n t f r e q u e n c y was p a s s e d . The i n c i p i e n t waveform ex-h i b i t i n g t h e f u n d a m e n t a l l o n g i t u d i n a l f r e q u e n c y a p p e a r e d 90° 'ahead' o f t h e p r i m a r y waveform h a v i n g t h e e x c i t a t i o n f r e -q u e n c y . The p a r a m e t r i c r e s o n a n t a m p l i t u d e became maximum when t h e two v i b r a t i o n s were ' i n p h a s e 1 a t t h e r e s o n a n t f r e q u e n c y , and a t h i g h e r f r e q u e n c i e s t h e r e s o n a n t c o n d i t i o n d i s c o n t i n u e d w i t h t h e component h a v i n g t h e f u n d a m e n t a l l o n g i -t u d i n a l f r e q u e n c y 90° ' b e h i n d ' t h e p r i m a r y v i b r a t i o n . C o n c l u s i o n s The f o l l o w i n g c o n c l u s i o n s were drawn f r o m t h e f o r e g o i n g t h e o r e t i c a l and e x p e r i m e n t a l a n a l y s i s on t h e f l e x u r a l , l o n g i t u d i n a l , and t o r s i o n a l r e s p o n s e o f a column s u b j e c t e d t o p e r i o d i c a x i a l l o a d i n g : 1. R e s o n a n t c o u p l e d v i b r a t i o n s a p p e a r t o e x i s t as t h e o r e t i c a l l y a n t i c i p a t e d due t o t h e i n t e r a c t i o n between l o n g i t u d i n a l and f l e x u r a l o s c i l l a t i o n s . The f u n d a m e n t a l l o n g i t u d i n a l v i b r a t i o n i s e x c i t e d when t h e f r e q u e n c y o f t h e a p p l i e d l o a d i n g i s one h a l f o f t h e f u n d a m e n t a l l o n g i t u d i n a l r e s o n a n t f r e q u e n c y . A t t h i s same e x c i t a t i o n f r e q u e n c y a r e s o n a n c e a p p e a r s on t h e f l e x u r a l r e s p o n s e h a v i n g t h e same f r e q u e n c y as t h e e x t e r n a l l o a d i n g . 2. A r e s o n a n t c o u p l e d v i b r a t i o n i s o b s e r v e d e x p e r i -m e n t a l l y w h i c h i s n o t a c c o u n t e d f o r by t h e 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 . A waveform a n a l y s i s s u g g e s t s t h a t t h e f u n d a m e n t a l l o n g i t u d i n a l r e s o n a n t v i b r a t i o n i s e x c i t e d when t h e e x c i t a t i o n f r e q u e n c y i s one t h i r d o f t h e f u n d a m e n t a l l o n g i t u d i n a l f r e -q uency . A c o r r e s p o n d i n g c o u p l e d f l e x u r a l r e s o n a n c e e x h i b i t -i n g t h e f r e q u e n c y o f t h e a p p l i e d l o a d a l s o e x i s t s . 3. The a n a l y t i c a l d e v e l o p m e n t i n d i c a t e s t h a t t h e r e s -p o nse o f t h e column i s t h e sum o f two v i b r a t i o n s h a v i n g a f r e q u e n c y r a t i o o f 1:2. T h i s i n t e r p r e t a t i o n seems c o r r e c t , t h o u g h i t may n o t be c o m p l e t e . The e x p e r i m e n t a l o b s e r v a t i o n o f a r e s o n a n t c o n d i t i o n c o n t a i n i n g two v i b r a t i o n s w i t h a f r e q u e n c y r a t i o o f 1:3 s u g g e s t s t h a t t h e a x i a l r e s p o n s e c o n -t a i n s a t h i r d o s c i l l a t i o n e x h i b i t i n g t h r e e t i m e s t h e e x c i t a -t i o n f r e q u e n c y . 4. T h e o r e t i c a l l y , t h e f l e x u r a l r e s p o n s e i s a l s o t h e sum o f two v i b r a t i o n s w i t h a f r e q u e n c y r a t i o o f 1:2. W h i l e t h e p r e d i c t e d c o u p l e d r e s o n a n c e s do o c c u r , t h e e x p e r i m e n t a l r e s u l t s r e v e a l no c o n t r i b u t i o n t o t h e f l e x u r a l r e s p o n s e f r o m t h e v i b r a t i o n h a v i n g t w i c e t h e e x t e r n a l l o a d f r e q u e n c y . T h a t i s , t h e f l e x u r a l r e s p o n s e i s s i n u s o i d a l w i t h t h e e x c i t a t i o n f r e q u e n c y o v e r t h e f r e q u e n c y range i n v e s t i g a t e d . 5. The e x i s t e n c e o f c o u p l e d t o r s i o n a l v i b r a t i o n s was v e r i f i e d e x p e r i m e n t a l l y . When t h e a p p l i e d f r e q u e n c y i s c l o s e t o t w i c e t h e f u n d a m e n t a l t o r s i o n a l f r e q u e n c y , t h e fundamental, t o r s i o n a l mode i s e x c i t e d p a r a m e t r i c a l l y . I t i s t o be e x p e c -t e d t h a t h i g h e r t o r s i o n a l r e s o n a n c e s c o u l d s i m i l a r i l y be p a r a m e t r i c a l l y e x c i t e d . A c o u p l e d t o r s i o n a l mode was n o t e d e x p e r i m e n t a l l y a t an e x c i t a t i o n f r e q u e n c y c o r r e s p o n d i n g t o t h e f r e q u e n c y f o r t h e t h i r d r e s o n a n t t o r s i o n a l mode. T h i s p a r a m e t r i c e x c i t a t i o n m i g h t be a t t r i b u t e d t o t h e f a c t t h a t a ' l i n e a r ' f l e x u r a l r e s o n a n c e was a l s o n e a r b y . 6. The a m p l i t u d e s o f a l l c o u p l e d v i b r a t i o n s a p p e a r q u i t e s i g n i f i c a n t . C o u p l e d f l e x u r a l r e s o n a n t a m p l i t u d e s a r e o f t h e same o r d e r o f m a g n i t u d e as ' l i n e a r ' f l e x u r a l r e s o n a n t a m p l i t u d e s . C o u p l e d a x i a l r e s o n a n t a m p l i t u d e s a r e i n g e n e r a l s l i g h t l y l e s s . C o u p l e d and ' l i n e a r ' a x i a l r e s o n a n t s t r a i n p e a k s b u i l d up and d e c l i n e more s h a r p l y t h a n do f l e x u r a l r e s o n a n c e s . The two v i b r a t i o n s c o m p r i s i n g a c o u p l e d a x i a l r e s o n a n c e change phase by a t o t a l o f 180° r e l a t i v e t o e a c h o t h e r as t h e r e s o n a n t f r e q u e n c y i s p a s s e d . The maximum - 64 - • a m p l i t u d e r e s o n a n t c o n d i t i o n a p p e a r s when t h e two v i b r a t i o n s a r e ' i n ph a s e r . 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 F i r s t c o n s i d e r a t i o n s i n s u g g e s t i n g f u r t h e r r e s e a r c h w o u l d p r o b a b l y aim a t o v e r c o m i n g t h e l i m i t a t i o n s o f the work done t o d a t e . To t h i s end, a c l o s e d f o r m s o l u t i o n f o r t h e t h e o r e t i c a l e q u a t i o n s ' d e r i v e d i s d e s i r a b l e . A n a l y t i c a l p r e d i c t i o n s c o u l d t h e n be made c o n c e r n i n g t h e a m p l i t u d e s o f c o u p l e d o s c i l l a t i o n s ; t h e r e l a t i o n s h i p between r e s p o n s e and e x c i t a t i o n c o u l d be i n v e s t i g a t e d ; and t h e i n f l u e n c e o f a d d i -t i o n a l v a r i a b l e s s u c h as i n t e r n a l damping and t e m p e r a t u r e c o u l d be c o n s i d e r e d . V e r i f i c a t i o n f o r t h e s e t h e o r e t i c a l r e s u l t s w o u l d be n e c e s s a r y u s i n g an e x p e r i m e n t a l a p p a r a t u s s i m i l a r t o t h e one. u s e d . The s t u d y m i g h t t h e n be e x t e n d e d t o e v a l u a t e means o f c o n t r o l l i n g c o u p l e d and ' l i n e a r ' o s c i l l a t i o n s , p r o b a b l y t h r o u g h t h e a p p l i c a t i o n o f v i s c o e l a s t i c a n d / o r e l a s t i c - v i s c o e l a s t i c damping l a y e r s t o t h e s u r f a c e s o f t h e b a r . O p t i m i z i n g p a r a m e t e r s and i s o l a t i o n p r o p e r t i e s f o r damping l a y e r s would be o f i n t e r e s t i n e v a l u a t i n g t h e e f f e c t i v e n e s s o f v a r i o u s damping m a t e r i a l s . Though i t w o u l d n o t be as f r u i t f u l as an a n a l y t i c a l s o l u t i o n , a n u m e r i c a l s o l u t i o n o b t a i n e d on a computer f o r t h e d i f f e r e n t i a l e q u a t i o n s d e s c r i b i n g t h e column c o u l d com-p l e m e n t e x p e r i m e n t a l r e s u l t s . The s i g n i f i c a n c e o f p a r a m e t r i c -a l l y e x c i t e d o s c i l l a t i o n s and t h e i n f l u e n c e o f v a r i o u s p a r a -m e t e r s on c o u p l e d v i b r a t i o n s :fcould be i n v e s t i g a t e d . The - 65 -m a j o r s h o r t c o m i n g o f a n u m e r i c a l s o l u t i o n i s t h a t i t does n o t s a y a n y t h i n g a b o u t t h e mechanisms w h i c h m i g h t g i v e r i s e t o t h e t y p e o f r e s p o n s e o b s e r v e d . Such a s o l u t i o n , however, i s much more r e a d i l y o b t a i n e d t h a n an a n a l y t i c a l s o l u t i o n . O t h e r l i m i t a t i o n s on t h e s t u d y a r e i n h e r e n t i n t h e a p p a -r a t u s u s e d . The d e s i r e d b o u n d a r y c o n d i t i o n s s h o u l d be a p p r o -x i m a t e d more c l o s e l y . B o u n d a r y c o n d i t i o n s o t h e r t h a n c l a m p e d ends w o u l d a l s o be o f p r a c t i c a l i n t e r e s t . A m u l t i c h a n n e l r e c o r d i n g s y s t e m i s d e s i r a b l e t o m o n i t o r t h e r e s p o n s e o f t h e column s p e c i m e n a t s e v e r a l d i f f e r e n t p o i n t s s i m u l t a n e o u s l y . I n t h i s way, t h e a x i a l and f l e x u r a l b e h a v i o r o f t h e column c o u l d be t r a c e d s i m u l t a n e o u s l y . A v i b r a t i o n c o n t r o l g e n e r a -t o r c a p a b l e o f p r o v i d i n g c o n s t a n t d i s p l a c e m e n t c o n t r o l t o much h i g h e r f r e q u e n c i e s t h a n 10 KHz w o u l d be u s e f u l . Con-s t a n t d i s p l a c e m e n t c o n t r o l w o u l d be b e t t e r t h a n t h e p i e c e -w i s e c o n s t a n t l e v e l a c c e l e r a t i o n u s e d f o r t h i s work. The i m p o r t a n c e o f c o u p l e d o s c i l l a t i o n s a t f r e q u e n c i e s c o r r e s p o n d i n g t o h i g h e r h a r m o n i c l o n g i t u d i n a l v i b r a t i o n s i s unknown. A t t e n t i o n s h o u l d be f o c u s e d on t h e v a r i o u s means o f m o n i t o r i n g t h e column b e h a v i o r . An a m p l i f i e r s u p e r i o r t o t h e BAM 1 u n i t i s e s s e n t i a l i f r e s p o n s e a m p l i t u d e s a r e t o be i n v e s i g a t e d u s i n g s t r a i n g a u g e s . The a m p l i f i e r must have v e r y f a v o u r a b l e n o i s e c h a r a c t e r i s t i c s , s i n c e t h e m easured s i g n a l s a r e s m a l l . A l t e r n a t i v e l y , e l e c t r i c a l n o i s e p r o b l e m s a r e e l i m i n a t e d t h r o u g h use o f t h e F o t o n i c S e n s o r . However, a v e r y s e n s i t i v e F o t o n i c S e n s o r i s r e q u i r e d , and g r e a t c a r e - 6 6 -must be e x e r c i s e d i n . p r o v i d i n g a r i g i d m o u n t i n g s y s t e m f o r t h e S e n s o r p r o b e . A t some expense,' a p r o b e h o l d e r c o u l d be d e v e l o p e d t o a c c u r a t e l y s c a n t h e l e n g t h o f t h e column a t p a r t i c u l a r f r e q u e n c i e s o f i n t e r e s t . O t h e r methods o f m e a s u r i n g t h e column v i b r a t i o n , i n c l u d i n g s o u n d l e v e l meas-urements and o p t i c a l t e c h n i q u e s s h o u l d be c o n s i d e r e d . To f u r t h e r s t u d y p a r a m e t r i c a l l y e x c i t e d t o r s i o n a l o s c i l l a t i o n s , s t r a i n gauges p o s i t i o n e d a t 45°to t h e m a j o r a x i s o f t h e column m i g h t h e l p , s i n c e t h e s e n s i t i v i t y t o t o r -s i o n a l movement w o u l d be i m p r o v e d t h e r e b y . T e s t s c o u l d be. c o n d u c t e d on columns h a v i n g o t h e r c r o s s - s e c t i o n g e o m e t r i e s o f p r a c t i c a l i n t e r e s t . F i n a l l y , t h e e f f e c t o f c o u p l e d o s c i l l a t i o n s on t h e b e h a v i o r o f more c o m p l i c a t e d s t r u c t u r e s s u b j e c t e d t o dynamic e x c i t a t i o n w i l l d e s e r v e a t t e n t i o n . <7 - 67 -BIBLIOGRAPHY 1. L o v e , A.E.H., "A T r e a t i s e on t h e M a t h e m a t i c a l T h e o r y o f E l a s t i c i t y " , 3 r d E d . , C a m b r i d g e , U n i v e r s i t y P r e s s , BT20 2. B e l i a e v , N.M., " S t a b i l i t y o f P r i s m a t i c R o d s . S u b j e c t t o V a r i a b l e L o n g i t u d i n a l F o r c e s " , C o l l e c t i o n o f E n g i n e e r i n g  C o n s t r u c t i o n and S t r u c t u r a l M e c h a n i c s ( I n z h i n e r n y e s o o r z h e i i a i s t r o i t e l ' n a i a m e k h a n i k a ) , L e n i n g r a d , P u t , 1924 3. M e t t l e r , E . , Dynamic B u c k l i n g , "Handbook o f E n g i n e e r i n g M e c h a n i c s , 1 s t Ed.., F l u e g g e , W., e d i t o r , M c G r a w - H i l l Book Company, I n c . 1962 4. S o m e r s e t , J.H., " P a r a m e t r i c I n s t a b i l i t y o f E l a s t i c Columns", S y r a c u s e U n i v e r s i t y R e s e a r c h I n s t i t u t e TR SURI no. 1053 - 7, J a n u a r y , 1963 5. E v a n - I w a n o s k i , R.M., " P a r a m e t r i c (Dynamic) S t a b i l i t y o f E l a s t i c S y s t e m s " , P r o c e e d i n g s o f t h e F i r s t S o u t h e a s t e r n  C o n f e r e n c e on T h e o r e t i c a l and A p p l i e d M e c h a n i c s " , Plenum P r e s s , 1963, pp. I l l - 130 6. B o l o t i n , T.V., "Dynamic S t a b i l i t y o f E l a s t i c S y s t e m s " , ( t r a n s l a t e d f r o m R u s s i a n ) , H o l d e n - D a y , San F r a n s i s c o , C a l i f . , 1964 7. S o m e r s e t , J.H., and E v a n - I w a n o s k i , R.M., " E x p e r i m e n t s on P a r a m e t r i c I n s t a b i l i t y o f Columns", P r o c e e d i n g s o f  t h e S e c o n d S o u t h e a s t e r n C o n f e r e n c e on T h e o r e t i c a l and  A p p l i e d M e c h a n i c s , A t l a n t a , Ga., March, 196 4, pp. 503 -525 8. T s o , W.K., P a r a m e t r i c T o r s i o n a l S t a b i l i t y o f a B a r Under A x i a l E x c i t a t i o n " , J o u r n a l o f A p p l i e d M e c h a n i c s , March 196 8, pp. 13 - 19 9. S c h n e i d e r , B.C., " 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 o f N o n l i n e a r C o u p l e d V i b r a t i o n s o f B a r s and P l a t e s " , M.A.Sc. T h e s i s , 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 , A p r i l , 1969 Append.c< - 68 -APPENDIX A DEVELOPMENT OF STRAIN EXPRESSION FOR A COLUMN ACCOUNTING FOR I N I T I A L CROOKEDNESS The f o l l o w i n g d e r i v a t i o n o r i g i n a l l y f o r m u l a t e d by M e t t l e r [ 3 ] r e s u l t s i n a s t r a i n e x p r e s s i o n f o r a column w h i c h t a k e s i n t o a c c o u n t any i n i t i a l c r o o k e d n e s s o r s t a t i c d i s p l a c e m e n t t h e column may e x h i b i t , p r o v i d e d t h e s e d i s p l a c e m e n t s a r e s m a l l . The L a g r a n g i a n d e f i n i t i o n o f s t r a i n i s used; t h a t i s , dynamic d e f l e c t i o n s a r e measured r e l a t i v e t o an i n i t i a l d e f -l e c t i o n c u r v e . Love [1] d e r i v e s an a n a l o g o u s E u l e r i a n d e s -c r i p t i o n o f s t r a i n . The d i f f e r e n c e between t h e two d e r i v a -t i o n s l i e s i n t h e f a c t t h a t i n t h e d e r i v a t i o n p r e s e n t e d dynamic d e f l e c t i o n s a r e measured r e l a t i v e t o an i n i t i a l d e f l e c t i o n c u r v e , w h i l e L o v e ' s d e d u c t i o n assumes a f i x e d c o o r d i n a t e s y s t e m i n wh i c h t h e i n i t i a l d e f l e c t i o n o f any p o i n t i n t h e n e u t r a l p l a n e i s z e r o . F i g . A - l shows s c h e m a t i c a l l y a f i b e r o f t h e c e n t r a l p l a n e o f a column u n d e r g o i n g p l a n e m o t i o n . The b a r r e d l e t t e r s r e p r e s e n t t h e s t a t i c d i s p l a c e m e n t w h i l e t h e o r d i n a r y l e t t e r s show dynamic d i s p l a c e m e n t s r e l a t i v e t o t h e i n i t i a l d e f l e c t i o n c u r v e . An e l e m e n t o f t h e column f i b e r i n t h e c e n t r a l p l a n e b e f o r e - d e f o r m a t i o n has a l e n g t h ds ( A - l ) - 69 -F i g . A - l . E x t e n s i o n and R o t a t i o n o f C e n t r a l P l a n e P'iber The d e f o r m e d l e n g t h o f t h e e l e m e n t i s ds = V ( 1 + U x ) 2 + <*5 2 The t o t a l s t r a i n o f t h e c e n t r a l f i b e r i s p = ds - ds  x ds (A-2) (A-3) E x p a n d i n g e a c h r a d i c a l i n e q u a t i o n s ( A - l ) and (A-2) by t h e b i n o m i a l t h e o r e m and r e t a i n i n g s e c o n d o r d e r t e r m s , t h e s t r a i n as g i v e n by e q u a t i o n (A-3) becomes 1 2 u + w w + —wv c = x x x 2 x 1 + 4^  2 2 x (A-4) - 70 -Assuming s m a l l i n i t i a l crookedness or s t a t i c t r a n s v e r s e displacements, " x 2 « 1 (A-5) and the s t r a i n e x p r e s s i o n becomes 1 o x x x x 2 x (.A-bj Equation (A-6) i s the most gen e r a l s t r a i n e x p r e s s i o n f o r the s t r a i n s i n the c e n t r a l plane f i b e r of a column f o r s t a t i c or dynamic d e f l e c t i o n s i n two orthogonal d i r e c t i o n s . A d d i t i o n a l s t r a i n s due to bending or t w i s t i n g must be accou-ted f o r i n o b t a i n i n g the s t r a i n at some d i s t a n c e from the c e n t r a l plane f i b e r . - 71 -APPENDIX B DETAILS OF ELECTRONIC INSTRUMENTATION USED FOR VIBRATION CONTROL SYSTEM The f o l l o w i n g p a r a g r a p h s p r o v i d e a d e s c r i p t i o n o f t h e e l e c t r o n i c a p p a r a t u s u s e d i n t h i s work t o c o n t r o l t h e a p p l i e d v i b r a t i o n l e v e l t o t h e column s p e c i m e n . The c o n t r o l g e n e r a t o r i s a B r u e l and K j a e r A u t o m a t i c V i b r a t i o n E x c i t e r C o n t r o l Type 1025 c a p a b l e o f p r o v i d i n g peak t o peak d i s p l a c e m e n t , v e l o c i t y , o r a c c e l e r a t i o n . C o n s t a n t d i s p l a c e m e n t c o n t r o l i s a v a i l a b l e up t o 2 KHz w i t h a maximum a m p l i t u d e o f 0.1 i n c h e s peak t o peak; c o n s t a n t v e l o c i t y c o n -t r o l i s a l s o p o s s i b l e t o 2KHz w i t h a maximum peak v a l u e o f 100 i n c h e s p e r s e c o n d ; and c o n s t a n t a c c e l e r a t i o n c o n t r o l i s a t t a i n a b l e up t o 10 KHz and 1000 g's peak v a l u e , where 1 g i s t h e a c c e l e r a t i o n due t o t h e e a r t h ' s g r a v i t a t i o n a l f o r c e . The c o n t r o l g e n e r a t o r i s e q u i p p e d t o a u t o m a t i c a l l y s c a n t h e e n t i r e f r e q u e n c y s p e c t r u m up t o 10 KHz. I t can s c a n f o r -ward o r r e v e r s e c o n t i n u o u s l y o v e r t h e whole f r e q u e n c y r a n g e , o r o v e r any segments o f t h e r a n g e . The c o n t r o l g e n e r a t o r has s i x f i x e d s c a n n i n g s p e e d s i f u s e d w i t h t h e f r e q u e n c y s y n c h r o n i z e d l e v e l r e c o r d e r , o r 132 o t h e r f i x e d s c a n n i n g s p e e d s . A f i x e d s c a n n i n g s p e e d i m p l i e s t h a t t h e f r e q u e n c y d i a l moves w i t h a f i x e d a n g u l a r v e l o c i t y ; t h e s c a n n e d f r e -q u e n c i e s a c t u a l l y change l o g a r i t h m i c a l l y w i t h t i m e , s i n c e t h e f r e q u e n c y d i a l i s c a l i b r a t e d l o g a r i t h m i c a l l y . The v i b r a t i o n f e e d b a c k s i g n a l f r o m t h e a c c e l e r o m e t e r i s t h e i n p u t t o t h e r e g u l a t i o n o r c o m p r e s s o r c i r c u i t s i n t h e c o n t r o l g e n e r a t o r . The o p e r a t i o n o f t h e s e c i r c u i t s i s s u c h t h a t i f t h e s h a k e r t a b l e v i b r a t i o n becomes t o o g r e a t , a l a r g e r b i a s s i g n a l a p p e a r s i n t h e c o m p r e s s o r c i r c u i t s and t h e g e n e r a t o r o u t p u t c o r r e s p o n d i n g l y d r o p s u n t i l t h e c o r r e c t v i b r a t i o n l e v e l i s o b t a i n e d . The i n t e g r a t i o n t i m e c o n s t a n t i n t h e c o m p r e s s o r c i r c u i t i s v e r y i m p o r t a n t s i n c e i t d e t e r -m i n e s t h e s p e e d w i t h w h i c h a s u d d e n change a t t h e s h a k e r t a b l e i s r e g u l a t e d b a c k t o n o r m a l . A c h o i c e o f c o m p r e s s o r s p e e d s i s a v a i l a b l e . A c o m p r e s s o r s p e e d as s l o w as p o s s i b l e i s c h o s e n i n o r d e r t o a v o i d d i s t o r t i o n o f t h e d e s i r e d v i b r a -t i o n a t low f r e q u e n c i e s , w h i l e s t i l l p r o v i d i n g a d e q u a t e r e g u l a t i o n o f t h e s h a k e r . The a c c e l e r o m e t e r u s e d was a B r u e l and K j a e r Type 43 35 s t u d mounted on t h e l o w e r c o l u m n mount. The B r u e l and K j a e r d u a l c h a n n e l A c c e l e r o m e t e r P r e a m p l i f i e r Type 2622 has a b u i l t - i n s e n s i t i v i t y a t t e n u a t o r , w h i c h , when c o r r e c t l y a d -j u s t e d , p r o v i d e s an o u t p u t s i g n a l on t h e v o l t a g e c h a n n e l o f 10 mv p e r g s e n s e d by t h e a c c e l e r o m e t e r . - 73 -APPENDIX C LINEAR EQUATIONS FOR COLUMN VIBRATIONS Equation of Motion f o r F l e x u r a l V i b r a t i o n s The l i n e a r p a r t i a l d i f f e r e n t i a l equation d e s c r i b i n g the bending v i b r a t i o n s of a uniform, e l a s t i c column s u s t a i n i n g an a x i a l load P0 i s E I x x x x + P o W x x + A < W t t .= 0 { C ~ l ) Proceeding t o a s o l u t i o n of equation ( C - l ) , the column i s assumed to v i b r a t e h a r m o n i c a l l y at a n a t u r a l frequency and i n a normal c o n f i g u r a t i o n . Thus a s o l u t i o n i s assumed having the form w = X(x)(Acospt + B s i n pt) (C-2) where X(x) i s a f u n c t i o n of x and A and B are co n s t a n t s . S u b s t i t u t i n g equation (C-2) i n t o equation (C-l) provides an o r d i n a r y d i f f e r e n t i a l equation d 4X d 2 x 2 EI 7 + Po—TT =-Ad>p X (C-3) d x dx^ The s o l u t i o n s of equation (C-3) s a t i s f y i n g the p r e s c r i b e d end c o n d i t i o n s f u r n i s h the ap p r o p r i a t e normal f u n c t i o n s . The s i m p l e s t case r e s u l t s i f the ends of the bar are simply supported. These c o n d i t i o n s are s a t i s f i e d by x i - S l n "T— (C-4) where i i s an i n t e g e r . The frequency equation r e s u l t s i f equation (C-4) i s s u b s t i t u t e d i n t o equation (C-3), and the r e s u l t a n t resonant f l e x u r a l f r e q u e n c i e s are given by - 74 -(C- 5) 2 EI where c = . The c o m p r e s s i v e a x i a l l o a d P0 i s seen t o d e c r e a s e t h e r e s o n a n t t r a n s v e r s e f r e q u e n c i e s . The c a l c u l a t e d r e s o n a n t t r a n s v e r s e f r e q u e n c i e s f o r t h e column u s e d i n t h i s work f o r p i n n e d ends and clamped ends a r e p l o t t e d i n F i g . V-1 , u s i n g a c o n s t a n t c o m p r e s s i v e end l o a d o f P 0 = 64 pounds p r o v i d e d by t h e s p r i n g s i n t h e e x p e r i m e n t a l s e t - u p . E q u a t i o n o f M o t i o n f o r L o n g i t u d i n a l V i b r a t i o n s The l i n e a r d i f f e r e n t i a l e q u a t i o n o f m o t i o n f o r t h e a x i a l v i b r a t i o n o f a s t r a i g h t , e l a s t i c column i s u = c 2 u ( C " 6 ) t t u x x 2 EI where c = as b e f o r e . A s s u m i n g t h a t t h e column o s c i l l a -t e s i n one o f i t s n a t u r a l modes o f v i b r a t i o n , a s o l u t i o n i s t a k e n o f t h e f o r m u = X ( x ) ( c 3 c o s p t + c 4 s i n pt) (C-7) where X(x) i s a f u n c t i o n o f x and c^ and c^ a r e c o n s t a n t s . The p r o c e d u r e f o l l o w e d i n o b t a i n i n g t h e f r e q u e n c y e q u a t i o n i s t h e same as t h a t o u t l i n e d above f o r f l e x u r a l v i b r a t i o n s . F o r t h e column s t u d i e d , t h e d e f l e c t i o n r e m a i n e d z e r o a t x = 0, and a t x = L t h e a p p l i e d l o a d p r o v i d e d c o n s t a n t l e v e l s i n u s o i d a l a c c e l e r a t i o n . U s i n g t h e s e b o u n d a r y c o n d i -t i o n s , t h e f r e q u e n c y e q u a t i o n becomes: S c o s 1ft = - p z c 5 s i n k c-2 C t - s  & cos p t (C-8) - 75 -where S = a m p l i t u d e o f a p p l i e d a c c e l e r a t i o n . When a n o r m a l mode o c c u r s , t h e e x c i t a t i o n f r e q u e n c y t c o i n c i d e s w i t h t h e r e s o n a n t f r e q u e n c y p. The a x i a l r e s p o n s e t h e n becomes u = " p 2 s L e^( s i n ^ c o s p t ) ( c ' - 9 ) When x = L, t h e l o n g i t u d i n a l d e f l e c t i o n can be a p p r o x i m a t e d by s t a t i c c o n s i d e r a t i o n s . U s i n g H o o k e 1 s Law and an e n d l o a d o f 25 pounds, t h e e x t e n s i o n o f t h e column t e s t e d i s about 200 u i n . S u b s t i t u t i n g t h i s a c c e l e r a t i o n v a l u e f o r u i n t o e q u a -t i o n (C-9) a l o n g w i t h t h e a c c e l e r a t i o n l e v e l S p r o v i d e d i n th e e x p e r i m e n t a l t e s t s ( T a b l e IV-1) an a p p r o x i m a t e l o n g i t u d i -n a l r e s o n a n t f r e q u e n c y o f 8790 Hz r e s u l t s . The c a l c u l a t e d r e s o n a n t f r e q u e n c y o f 8790 Hz does n o t change by more t h a n a few Hz f o r a wide range o f end l o a d i n g f o r c e s and a p p l i e d a c c e l e r a t i o n l e v e l s . E q u a t i o n o f M o t i o n f o r T o r s i o n a l V i b r a t i o n s The l i n e a r d i f f e r e n t i a l e q u a t i o n d e s c r i b i n g t h e t o r s i o n a l o s c i l l a t i o n s o f a u n i f o r m , e l a s t i c column s u s t a i n i n g an a x i a l l o a d P 0 i s ® ~ aZe = 0 (C-10) t t xx where a 2 = ^ - ^ A h ^ + 3 A A s s u m i n g t h a t t h e b a r p e r f o r m s a n a t u r a l mode o f v i b r a t i o n i n t o r s i o n o f f r e q u e n c y w t , . . , ' c .. , . , , ^ 1 — , and u s i n g t h e f o l l o w i n g b o u n d a r y c o n d i t i o n s , - 76 -U = 0; 0 = 0 a t x = 0 U t t = X c o s f t , Q = 0 a t x = L the f r e q u e n c y e q u a t i o n r e s u l t s nrra w l " ~ ( C - l l ) In c a l c u l a t i n g a, t h e f o l l o w i n g v a l u e s a r e used : G = 11.5 x 1 0 6 p s i A = 4.66 x I O " 2 i n . 2 h = 0.125 i n . 1*2 = 1 ( f o r u n i f o r m end l o a d i n g ) ,,.-4 . 4 I . - 5.5 x 10 i n . P = 64 l b s . -4 4 I 0 = 6 . 1 1 x 1 0 i n . From e q u a t i o n ( C - l l ) , t h e r e s o n a n t t o r s i o n a l f r e q u e n c i e s f o r t h e f i r s t t h r e e modes a r e c a l c u l a t e d a s : 3380 Hz, 6760 Hz and 10140 Hz. 

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