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Aeroelastic behavior of square section prisms in uniform flow Wawzonek, Mitchell A. 1979

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AEROELASTIC BEHAVIOR OF SQUARE SECTION PRISMS IN UNIFORM FLOW by Mitchell A. Wawzonek B.A.Sc, University of Waterloo, 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1979 © Mitchell Anthony Wawzonek, 1979 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the 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 agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department 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 or 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 . Department n 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 Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date June 28, 1979 i i ABSTRACT The work p r e s e n t e d i n t h i s t h e s i s c o n c e r n s t h e 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 and a n a l y s i s o f t h e o s c i l l a t i o n b e h a v i o r o f e l a s t i c a l l y s u p p o r t e d s q u a r e p r i s m s i n a u n i f o r m f l o w o f a i r . T h i s i s a s t e p i n u n d e r s t a n d i n g t h e b e h a v i o r o f s t r u c t u r e s i n f l o w s i t u a t i o n s s u c h as i n t he a tmosphere o r i n w a t e r . The p r i n c i p a l modes o f e x c i t a t i o n s t u d i e d a r e t h o s e due t o v o r t e x s h e d d i n g , and a e r o d y n a m i c i n s t a b i l i t y known as g a l l o p i n g . The t e s t s showed t h a t when t h e n o n - d i m e n s i o n a l w i n d speed U was l e s s t h a n about 2 . 5 , v o r t e x s h e d d i n g i n f l u e n c e d t h e b e h a v i o r s i g n i f i c a n t l y , so t h a t t h e q u a s i - s t e a d y t h e o r y d e v e l o p e d f o r g a l l o p i n g c o u l d n o t p r e d i c t t h e a m p l i t u d e r e s p o n s e . Some o f t h e d a t a i n t h e l i t e r a t u r e showed t h a t i n t h i s w i n d s p e e d r a n g e , measurements o f a e r o d y n a m i c f o r c e d u r i n g f o r c e d o s c i l l a t i o n c o u l d be u s e d t o more a c c u r a t e l y p r e d i c t t h e a e r o e l a s t i c b e h a v i o r . The q u a s i - s t e a d y a n a l y s i s was v e r i f i e d t o be v a l i d f o r U g r e a t e r t h a n a b o u t 2 .5 i n t h e p r e s e n t t e s t s , b u t t h e measured a e r o d y n a m i c f o r c e b e h a v i o r o n w h i c h t h e a n a l y s i s depends i s i n t u r n dependen t on R e y n o l d s number and end c o n d i t i o n s . The v e l o c i t y a t w h i c h g a l l o p i n g i s p r e d i c t e d t o o c c u r , U 0 , i s p a r t i c u l a r l y s e n s i t i v e t o t h e s e e f f e c t s , s i n c e t h e v a l u e o f t h e a e r o d y n a m i c p a r a m e t e r A t h a t d e f i n e s U 0 i s n o t n e c e s s a r i l y c o n s t a n t . T a b l e o f C o n t e n t s page A b s t r a c t i i L i s t o f T a b l e s i v L i s t o f F i g u r e s v Acknowledgement v i i N o m e n c l a t u r e v i i i CHAPTER ONE 1 1.1 I n t r o d u c t i o n 1 1.2 B r i e f R e v i e w o f t h e Q u a s i - S t e a d y T h e o r y 2 1.3 Rev i ew o f t h e L i t e r a t u r e 3 1 .3 .1 E f f e c t s o f v o r t e x s h e d d i n g 3 1 . 3 . 2 G a l l o p i n g f rom R e s t 4 CHAPTER TWO 6 2 . 1 S t a t i c Measurements and A n a l y s i s 6 2 . 1 . 1 Measurement o f C y ( a ) 6 2 . 1 . 2 A p p l i c a t i o n t o t h e Q u a s i - S t e a d y A n a l y s i s 7 2 . 1 . 3 C o m p a r i s o n t o Measurements f rom F o r c e d O s c i l l a t i o n 8 2 . 2 Dynamic B e h a v i o r E x p e r i m e n t s 9 2 . 2 . 1 T e s t s 9 2 . 2 . 2 R e s u l t s 11 2 . 2 . 3 D i s c u s s i o n 12 CHAPTER THREE 17 - C o n c l u s i o n s 17 B i b l i o g r a p h y 18 A p p e n d i x A 19 A p p e n d i x B 20 A p p e n d i x C 21 A p p e n d i x D 22 i v L i s t o f T a b l e s T a b l e I page 21 V L i s t of Figures page 1(a) Normal force v a r i a t i o n , square section 22 (b) Relative angle of attack f o r moving section 22 (c) Predicted galloping response f o r square section 22 2(a) Measured phase angle from dynamic measurements ( f i g . 18 of (3)) 23 (b) Measured force c o e f f i c i e n t from dynamic measurements 23 ( f i g . 12 of (4)) 3 Measured l i f t c o e f f i c i e n t s 24 4 Measured drag c o e f f i c i e n t s 25 5 Normal force c o e f f i c i e n t s 26 6 Normal force c o e f f i c i e n t s near a = 0 27 7 Equation (2.1) in graphic form 28 8 Predicted galloping response of square section 29 9 Schematic: galloping and damping measurements 30 10(a) Damping c a l i b r a t i o n 31 (b) V a r i a t i o n of damping with o s c i l l a t i o n amplitude 32 11 Typical test model 33 12 Galloping t e s t s , 3.3 cm model, Uo=0.44 34 13 Galloping t e s t s , 3.3 cm model, U0=1.38 35 14 Galloping t e s t s , 3.3 cm model, U0=1.67 36 15 Galloping t e s t s , 3.3 cm model, U0=1.77 37 16 Galloping t e s t s , 3.3 cm model, U0=1.87 38 17 Galloping t e s t s , 3.3 cm model, Uo=2.07 39 18 Galloping t e s t s , 3.3 cm model, UQ=2.27 40 19 Galloping t e s t s , 3.3 cm model, UQ=2.29 41 20 Galloping t e s t s , 3.3 cm model, U0=2.56 42 v i page 21 Galloping tests, 3.3 cm model, U0=2.77 43 22 Galloping tests, 3.3 cm model, U0=3.28 44 23 Galloping tests, 3.3 cm model, Uo=4.07 45 24 Galloping tests, 3.3 cm model, U0=4.24 46 25 Galloping tests, 3.3 cm model, U0=4.76 47 26 Galloping tests, 3.3 cm model, UD=5.67 48 27 Galloping tests, 3.3 cm model, UD=8.91 49 28 Galloping tests, 5.1 cm model, Uo=0.175 50 29 Galloping tests, 5.1 cm model, UQ=1.25 51 30 Galloping tests, 5.1 cm model, U0=1.75 52 31 Galloping tests, 5.1 cm model, U0=1.76 53 32 Galloping tests, 5.1 cm model, U0=2.54 54 33 Galloping tests, 5.1 cm model, Uo=3.06 55 34 Comparison between dynamic measurements and quasi-steady theory 56 35 Collapsed data from galloping tests 57 V l l Acknowledgement I w i s h t o thank Dr. G.V. P a r k i n s o n f o r h i s g u i d a n c e d u r i n g t h e r e s e a r c h . Nomenclature c 2mo> L % p V 2 M Vh v f v h streamwise section dimension viscous damping c o e f f i c i e n t measured force c o e f f i c i e n t , from (4) natural frequency (Hz) wake frequency (Hz) force i n y - d i r e c t i o n cross-stream section dimension model length o s c i l l a t i n g mass time flow v e l o c i t y displacement o s c i l l a t i o n amplitude angle of attack a i r density a i r v i s c o s i t y (kinematic) natural frequency (rad/s) £^y „ . . „ . . a=0 galloping c r i t e r i o n f r a c t i o n of c r i t i c a l damping l i f t c o e f f i c i e n t Ppy'^Yii drag c o e f f i c i e n t - — — normal force c o e f f i c i e n t % p V 2 h £ p h 2 £ 2m mass parameter Reynolds number Strouhal number IX U = ^-r dimensionless wind speed 2B ^ ° = n A c r i t i c a l galloping speed U _ = ^rrr c r i t i c a l resonance speed r 2TTS R dy . , . y = section v e l o c i t y y Y = dimensionless displacement y Y = ^ dimensionless amplitude • dY . , . , . Y = dimensionless section v e l o c i t y Y = d 2Y dimensionless section acceleration dT 2 x = cot dimensionless time CHAPTER ONE 1 1.1 I n t r o d u c t i o n F l o w i n d u c e d s t r u c t u r a l o s c i l l a t i o n s have been o b s e r v e d f o r some t i m e , and have been a p r o b l e m i n t h e d e s i g n o f t o w e r s , l o n g suspended c a b l e s , b r i d g e s , and o t h e r s t r u c t u r e s . The mechanisms o f t h e s e phenomena have been s t u d i e d e x t e n s i v e l y b u t due t o t h e c o m p l e x i t y o f t h e s e p a r a t e d f l o w f i e l d abou t b l u f f s e c t i o n s , many o f t h e d e t a i l s a r e n o t y e t f u l l y u n d e r s t o o d . I n t h e f l o w abou t two d i m e n s i o n a l b l u f f b o d i e s , e x c e p t a t v e r y l ow R e y n o l d s numbers , v o r t i c i t y i n t h e s e p a r a t e d s h e a r l a y e r s forms a Karman v o r t e x s t r e e t , i m p o s i n g an u n s t e a d y , p e r i o d i c f o r c e on t h e b o d y . When t h i s o c c u r s n e a r a n a t u r a l f r e q u e n c y o f t h e b o d y , o s c i l l a t i o n s known as v o r t e x r e s o n a n c e c a n o c c u r . T h i s b e a r s some s i m i l a r i t y t o a s i m p l e f o r c e d o s c i l l a t i o n , b u t t h i s e x p l a n a t i o n i s l i m i t e d b y s t r o n g c o u p l i n g be tween t h e a m p l i t u d e o f o s c i l l a t i o n and t h e f l u i d f o r c e s . S t u d i e s o f t h i s t y p e o f o s c i l l a t i o n have i n c l u d e d o b s e r v a t i o n o f t h e e f f e c t s o f v a r i o u s p a r a m e t e r s s u c h as damping o r t u r b u l e n c e on a m p l i t u d e . Pu re m a t h e m a t i c a l m o d e l l i n g o f v o r t e x r e s o n a n c e has met w i t h o n l y l i m i t e d s u c c e s s due t o t h e complex n a t u r e o f t h e f l o w f i e l d , and e m p i r i c i s m must be employed t o o b t a i n r e a s o n a b l e r e s u l t s . N u m e r i c a l m o d e l s , w h i c h t a k e more d e t a i l s o f t h e f l o w f i e l d b e h a v i o r i n t o a c c o u n t , can be u s e d bu t a r e a t t h e p r e s e n t t i m e q u i t e c o m p l i c a t e d and r e q u i r e l a r g e c a p a c i t y c o m p u t i n g f a c i l i t i e s . A n o t h e r t y p e o f i n s t a b i l i t y , n o t n e c e s s a r i l y r e l a t e d t o v o r t e x r e s o n a n c e , i s g a l l o p i n g . The name o f t h i s i s b e l i e v e d t o s tem f rom f i r s t o b s e r v a t i o n s o f e l e c t r i c a l c o n d u c t o r s t h a t ' g a l l o p e d ' d u r i n g h i g h w i n d s a f t e r 2 b e i n g c o a t e d w i t h s l e e t d u r i n g w i n t e r s t o r m s . T h i s i s now known t o be c a u s e d by e f f e c t s o f t h e p r o x i m i t y o f t h e s e p a r a t i o n s h e a r l a y e r s t o t h e b o d y , r e s u l t i n g i n an a e r o d y n a m i c i n s t a b i l i t y s i m i l a r « t o w i n g f l u t t e r t h a t has been a p r o b l e m w i t h a i r c r a f t d e s i g n . C e r t a i n b l u f f c r o s s s e c t i o n s , when e l a s t i c a l l y s u p p o r t e d , w i l l g a l l o p when t h e w i n d v e l o c i t y r e a c h e s a c r i t i c a l v a l u e t h a t depends on t h e s y s t e m ' s s t r u c t u r a l damping and p a r t i c u l a r shape o f t h e b o d y . The d e t a i l s o f g a l l o p i n g b e h a v i o r a r e b e t t e r u n d e r s t o o d t h a n t h o s e o f v o r t e x r e s o n a n c e , m a i n l y due t o b e i n g a b l e t o a p p l y a q u a s i - s t e a d y a p p r o a c h , w i t h l i m i t a t i o n s , t o t h e e q u a t i o n o f m o t i o n o f t h e s y s t e m . The p r i n c i p a l l i m i t a t i o n i s t h a t v o r t e x f o r m a t i o n c a n r e n d e r t h e q u a s i - s t e a d y a p p r o a c h i n v a l i d f o r a c e r t a i n r a n g e o f f l o w v e l o c i t i e s . 1.2 B r i e f r e v i e w o f t h e Q u a s i - S t e a d y T h e o r y The a n a l y s i s r e q u i r e s knowledge o f t h e n o r m a l f o r c e c o e f f i c i e n t C y v s . a n g l e o f a t t a c k a , as f o r example i n f i g u r e 1 ( a ) . The k e y a s s u m p t i o n i s t h a t i n t h e f l o w a b o u t an o s c i l l a t i n g b o d y , t h e i n s t a n t a n e o u s a e r o d y n a m i c f o r c e i s t h e same as t h a t on a s t a t i o n a r y body a t t h e same r e l a t i v e a n g l e o f a t t a c k ( f i g u r e 1 ( b ) ) . A p p l y i n g t h e q u a s i - s t e a d y a s s u m p t i o n t o t h e e q u a t i o n o f m o t i o n , one o b t a i n s t h e d i m e n s i o n l e s s f o r m : Y + Y = n U 2 C y ( a r c t a n -gj -2BY ( 1 . 1 ) T h i s n o n - 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 c a n be s o l v e d by a p p r o x i m a t e means, t h e method o f K r y l o v and B o g o l i u b o v b e i n g a f r e q u e n t l y u s e d a p p r o a c h when t h e r i g h t hand s i d e o f t h e e q u a t i o n i s n e a r z e r o , as i t i s f o r o s c i l l a t i o n s i n a i r . I n t h i s c a s e , t h e m o t i o n i s n e a r l y s i n u s o i d a l , so a s s u m i n g t h a t 3 Y = Y s i n tot, where Y i s a s t a t i o n a r y a m p l i t u d e , and e q u a t i n g t h e work done by damping and a e r o d y n a m i c f o r c e s o v e r one c y c l e , one o b t a i n s t h e i n t e g r a l e q u a t i o n : / Cy ( a r c t a n (^j COST)} COST d T = TT ^p- ( 1 - 2 ) I f C y ( a ) can be e x p r e s s e d as a p o l y n o m i a l , t h e e q u a t i o n c a n be i n t e g r a t e d d i r e c t l y t o g i v e a n o t h e r e q u a t i o n i n n , U , B , and Y . T h i s i s shown i n ( 2 ) , and the r e s u l t i s t h a t g a l l o p i n g o c c u r s f rom r e s t f o r U> Uo , where U Q = 2 B / n A , and A i s p o s i t i v e . I n ( 2 ) , t h e r e s u l t i n g e q u a t i o n i s o f t h e f o r m a - b Y 2 + c Y ' * - d Y 6 = 0, so t h a t t h e s t a t i o n a r y a m p l i t u d e s a r e f o u n d f rom t h e r e a l r o o t s Y . I t i s a l s o shown t h a t t h e s o l u t i o n s c o l l a p s e o n t o a s i n g l e c u r v e o f Y / U 0 v s . U / U 0 . The t h e o r e t i c a l g a l l o p i n g r e s p o n s e o f a s q u a r e s e c t i o n i n u n i f o r m f l o w , b a s e d on a 7 t h d e g r e e o d d - o r d e r p o l y n o m i a l f i t o f S m i t h ' s C y ( a ) d a t a (1) i s shown i n f i g u r e 1 ( c ) . Wind t u n n e l g a l l o p i n g e x p e r i m e n t s p e r f o r m e d by S m i t h CI) a g r e e d w e l l w i t h t h e t h e o r y , so l o n g as t h e o n s e t v e l o c i t y o f g a l l o p i n g U Q was w e l l above t h e v o r t e x r e s o n a n c e v e l o c i t y U r . The i n t e g r a l e q u a t i o n 1.2 c a n a l s o be s o l v e d n u m e r i c a l l y , g i v e n a n u m e r i c a l r e p r e s e n t a t i o n o f CyC°9, u s i n g q u a d r a t u r e s i n an i t e r a t i o n scheme t o f i n d t h e s t a t i o n a r y a m p l i t u d e s . 1.3 R e v i e w o f t h e L i t e r a t u r e 1 . 3 . 1 E f f e c t s o f v o r t e x s h e d d i n g I t has been o b s e r v e d t h a t i n c a s e s where t h e t h e o r e t i c a l o n s e t v e l o c i t y o f g a l l o p i n g U 0 was l e s s t h a n t h e v o r t e x r e s o n a n c e v e l o c i t y U r , no o s c i l l a t i o n s o c c u r r e d u n t i l t h e f l o w speed a p p r o a c h e d U r . S e v e r a l i n v e s t i g a t o r s C 3 , 4 , 5 ) have r e p o r t e d measurements o f t h e 4 i n s t a n t a n e o u s a e r o d y n a m i c f o r c e s on o s c i l l a t i n g s q u a r e and r e c t a n g u l a r c r o s s s e c t i o n s i n w i n d t u n n e l s . By u s e o f s t r a i n gauges a t t a c h e d t o s u i t a b l e mode l s f o r c e d t o o s c i l l a t e s i n u s o i d a l l y i n t h e w i n d , u s e f u l i n f o r m a t i o n was o b t a i n e d . I t was n e c e s s a r y t o a p p l y F o u r i e r a n a l y s i s t o t h e p e r i o d i c f o r c e d a t a i n o r d e r t o d e t e r m i n e t h e m a g n i t u d e and p h a s e a n g l e ( r e l a t i v e t o d i s p l a c e m e n t ) o f t h e f o r c e s a t t h e f r e q u e n c y o f o s c i l l a t i o n . By c o n v e r t i n g t o s u i t a b l e l i f t c o e f f i c i e n t s , i t was p o s s i b l e t o compare t h e r e s u l t s t o t h e q u a s i - s t e a d y t h e o r y p r e d i c t i o n . These showed t h e t h e o r y t o be v a l i d , as e x p e c t e d , a t h i g h e r n o n - d i m e n s i o n a l f l o w v e l o c i t i e s , and h e l p e d t o e x p l a i n t h e b e h a v i o r f o r U 0 < U r . I n f i g u r e 2 ( a ) , t h e phase a n g l e be tween f o r c e and d i s p l a c e m e n t i s shown t o be n e g a t i v e f o r U < U r . T h i s i n d i c a t e s a n e t t r a n s f e r o f e n e r g y t o t h e f l u i d f rom t h e o s c i l l a t i n g p r i s m , so t h a t i n an e l a s t i c a l l y s u p p o r t e d s y s t e m , s t a t i o n a r y o s c i l l a t i o n w o u l d n o t o c c u r and any m o t i o n w o u l d be damped o u t . A n o t h e r d e s c r i p t i o n o f t h e b e h a v i o r i s s een i n f i g u r e 2 ( b ) , where t h e m a g n i t u d e o f t h e f o r c e component i n p h a s e w i t h t h e v e l o c i t y i s shown. A t low v e l o c i t i e s , t h e f o r c e i s s een t o be n e g a t i v e , r e t a r d i n g t h e m o t i o n . The q u a s i - s t e a d y t h e o r y w o u l d a l s o p r e d i c t t h e f o r c e t o be n e g a t i v e a t low v e l o c i t i e s f o r f o r c e d o s c i l l a t i o n , b u t t h e v e l o c i t y where t h e f o r c e changes s i g n i s a c t u a l l y s t r o n g l y l i n k e d t o U r , a c h a r a c t e r i s t i c n o t p r e d i c t e d by t h e t h e o r y . 1 . 3 . 2 G a l l o p i n g f rom R e s t As m e n t i o n e d , f o r a l l U > U 0 > > U r , g a l l o p i n g s h o u l d commence f r o m r e s t so l o n g as 2 B / n A i s p o s i t i v e . The c o n s t a n t A , d e t e r m i n e d f r o m Cy measuremen t s , was f o u n d t o have a w i d e r a n g e o f v a l u e s i n t h e l i t e r a t u r e , d e p e n d i n g on t h e measurement c o n d i t i o n s and t e c h n i q u e s . T a b l e { shows some o f t h e v a l u e s f o r d i f f e r e n t f l o w s and g e o m e t r i e s . 5 I n u n i f o r m low t u r b u l e n c e f l o w s t h e v a l u e s f o r t h e s q u a r e s e c t i o n v a r y c o n s i d e r a b l y , s i n c e a p p a r e n t l y t h e end and s u r f a c e c o n d i t i o n s and R e y n o l d s number c a n have l a r g e e f f e c t s . I n t h e u n i f o r m , h i g h e r t u r b u l e n c e f l o w s , t h e v a l u e s a r e l e s s s c a t t e r e d . One w o u l d e x p e c t t h e smooth f l o w r e s u l t s o f B r o o k s (6) t o be u n a f f e c t e d b y end c o n d i t i o n s , s i n c e t h e v a l u e s came f rom i n t e g r a t i o n o f t h e p r e s s u r e d i s t r i b u t i o n measured a t m i d - s p a n o f t h e m o d e l . F o r c e measurements by Kwok (8) on t h e o t h e r hand were p e r f o r m e d w i t h a c a n t i l e v e r model h a v i n g t h e f r e e end l o c a t e d i n abou t t h e c e n t e r o f t h e w i n d t u n n e l t e s t s e c t i o n . I t i s i n t e r e s t i n g t o n o t e ( i n t h e r e s u l t s o f Kwok) t h e e f f e c t o f s m a l l s c a l e t u r b u l e n c e f rom a s m a l l d i a m e t e r r o d p l a c e d u p s t r e a m o f t h e m o d e l , i n c r e a s i n g t h e v a l u e o f A by a f a c t o r o f a l m o s t f o u r . The r e s u l t s p r e s e n t e d by Nakamura (11) were o b t a i n e d i n d i r e c t l y , u s i n g t h e o b s e r v e d o n s e t v e l o c i t y f o r g a l l o p i n g and t h e n f i n d i n g A u s i n g t h e known v a l u e s o f B and n , t h a t i s , A = 2 B / n U Q . P r e l i m i n a r y e x p e r i m e n t s i n t h e p r e s e n t r e s e a r c h s u g g e s t e d t h a t A = 2 . 6 9 , t h e v a l u e u s e d i n ( 2 ) , was somewhat l o w , so t h i s p o s s i b i l i t y was f u r t h e r i n v e s t i g a t e d . CHAPTER TWO 6 2 . 1 S t a t i c Measurements and A n a l y s i s 2 . 1 . 1 Measurement o f C v ( a ) I n o r d e r t o d e t e r m i n e C y ( a ) i t was n e c e s s a r y t o measure t h e l i f t and d r a g on a s t a t i o n a r y m o d e l . The 3 . 3 cm„ s q u a r e s e c t i o n mode l u s e d f o r t h e g a l l o p i n g t e s t s ( f i g u r e 11) was c l amped a t t h e base t o a w i n d t u n n e l f o r c e b a l a n c e . The mode l c o u l d be r o t a t e d i n 0 . 1 ° i n c r e m e n t s , and t h e l i f t and d r a g f o r c e s c o u l d be d e t e r m i n e d f rom t h e c a l i b r a t e d v o l t a g e o u t p u t o f t h e s t r a i n gauge c e l l s and a m p l i f i e r s . The w i n d t u n n e l u s e d i n t h i s s t u d y was a v e r y low t u r b u l e n c e , r e t u r n f l o w t y p e w i t h a t e s t s e c t i o n 0 .69 m by 0 .91 m. The measu red s t r e a m w i s e t u r b u l e n c e i n t e n s i t y o f - t h e f l o w was l e s s t h a n 0 .1%. The l i f t and d r a g c o e f f i c i e n t s a t t h r e e R e y n o l d s numbers a r e p l o t t e d i n f i g u r e s 3 and 4 . C o m p a r i n g t h e h i g h R e y n o l d s number r e s u l t s t o t h e two l o w e r o n e s , t h e peak i n t h e C^ c u r v e and t r o u g h i n t h e Cn move t o t h e l e f t as R e y n o l d s number d e c r e a s e s , and t h e s c a t t e r i n t h e C n measurements i n c r e a s e s . An i n c r e a s e i n t h e s l o p e o f C ^ f a ) a t t h e o r i g i n w i t h R e y n o l d s number d e c r e a s i n g i s a l s o e v i d e n t . F o r c o m p a r i s o n , t h e Cn v a l u e s r e p o r t e d i n (15) show r e a s o n a b l e ag reemen t . N o t e t h a t t h e d a t a o f t h e p r e s e n t measurements a r e n o t c o r r e c t e d f o r b l o c k a g e e f f e c t s , so t h e C Q v a l u e s a r e t h e r e f o r e abou t 6% h i g h . I n f i g u r e 5 , C v ( a ) i s p l o t t e d f o r t h e two h i g h e r R e y n o l d s numbers , d e t e r m i n e d f rom t h e l i f t and d r a g c o e f f i c i e n t s by u s i n g Cy = " ( C t + C n t a n a ) s e c a The c u r v e u s e d t o a p p r o x i m a t e C y ( a ) f o r t h e n u m e r i c a l c a l c u l a t i o n s i s shown, and i s r e p r e s e n t a t i v e o f t h e f o r c e b e h a v i o r a t a R e y n o l d s number o f abou t 7 1 2 , 0 0 0 . The p o l y n o m i a l f i t o f S m i t h ' s d a t a f rom (2) i s shown f o r c o m p a r i s o n . One c a n see t h a t as R e y n o l d s number d e c r e a s e s , t h e v a l u e o f A i n c r e a s e s . T h i s i s more e v i d e n t i n f i g u r e 6, an expanded s c a l e p l o t o f t h e C y d a t a n e a r a =0, w i t h l i n e a r l e a s t s q u a r e s f i t o f t h e d a t a f o r - . 9 ° < a < l . l ? The v a l u e o f A a t t h e h i g h e s t R e y n o l d s number i s c o m p a r a b l e t o t h e one q u o t e d i n ( 2 ) , so t h a t f a i r l y s t r o n g R e y n o l d s n u m b e r - e f f e c t i s a p p a r e n t . T h i s i s somewhat u n e x p e c t e d i n s u c h a s e p a r a t e d f l o w , w i t h f i x e d s e p a r a t i o n l i n e s a t t h e edges o f t h e m o d e l . 2 . 1 . 2 A p p l i c a t i o n t o t h e Q u a s i - S t e a d y A n a l y s i s U s i n g the C y(a) c u r v e o f f i g u r e 5 , one can s e t up an e q u a t i o n t o s o l v e f o r t h e s t a b l e a m p l i t u d e s . B l o c k a g e c o r r e c t i o n s a r e n o t a p p l i e d t o t h e C y d a t a s i n c e t h e n u m e r i c a l r e s u l t w i l l be compared t o u n c o r r e c t e d dynamic t e s t r e s u l t s , and b o t h t y p e s o f measurements were p e r f o r m e d w i t h t h e same w i n d t u n n e l . R e c a l l e q u a t i o n 1 .2 : ( C v ( a r c t a n ( 7 7 COST"))COST'di = TT . ( 1 - 2 ) oJ v v U^ " n ( j 2 S i n c e C y(a) i s odd i n a, f o r t h i s s i t u a t i o n , t h e u p p e r l i m i t c a n be r e d u c e d t o TT/2 t o make t h e n u m e r i c a l i n t e g r a t i o n s h o r t e r . The t e r m on t h e r i g h t hand s i d e must t h e n be d i v i d e d b y 4 , and a f t e r some r e - a r r a n g i n g one o b t a i n s _ = 4 ^ / ^ ( a r c t a n (I COST)) COST dT - ( 2 . 1 ) n u 2 U 2 TTY 0 Y U T h i s r e p r e s e n t s a b a l a n c e be tween t h e v i s c o u s s t r u c t u r a l damping c o e f f i c i e n t on t h e l e f t , and t h e a v e r a g e a e r o d y n a m i c f o r c e c o e f f i c i e n t on t h e r i g h t . T h i s i s d e p i c t e d g r a p h i c a l l y i n f i g u r e 7. The damping c o e f f i c i e n t t e r m i s shown as a f u n c t i o n o f B / n and U . The a e r o d y n a m i c t e r m on t h e r i g h t was i n t e g r a t e d f o r d i f f e r e n t v a l u e s o f Y and U t o ge t t h e i n d i c a t e d c u r v e s . I n t e r s e c t i o n p o i n t s r e p r e s e n t s t a b l e a m p l i t u d e s Y a t t h e c o r r e s p o n d i n g v a l u e s o f U and B / n . F o r example t h e c u r v e (2B/n )=6 i n t e r s e c t s Y=0 a t U = 1 . 5 , w h i c h i s t h u s t h e v a l u e o f U Q . F o r a l l U g r e a t e r t h a n U Q , Y i s p o s i t i v e , i . e . o s c i l l a t i o n s o c c u r . The c o m p l e t e s e t o f i n t e r s e c t i o n p o i n t s was d e t e r m i n e d n u m e r i c a l l y , and t h e r e s u l t i n g p l o t shown i n f i g u r e 8. Shown f o r c o m p a r i s o n i s t h e r e s u l t f rom ( 2 ) , i n d i c a t i n g d i f f e r e n c e s i n p r e d i c t e d h y s t e r e s i s b e h a v i o r due t o t h e d i f f e r e n t C y ( a ) u s e d . I t i s i n t e r e s t i n g t o see how t h e shape o f C y ( a ) a f f e c t s t h e p r e d i c t e d g a l l o p i n g r e s p o n s e . On f i g u r e 8, t h e l i n e s r a d i a t i n g f rom t h e o r i g i n a r e l i n e s o f c o n s t a n t Y / U , and t h e y can r e p r e s e n t t h e maximum a n g l e o f a t t a c k r e a c h e d d u r i n g one c y c l e o f s t a t i o n a r y o s c i l l a t i o n , s i n c e t a n a = Y / U . The c r i t i c a l p o i n t s o f t h e h y s t e r e s i s l o o p c a n be shown t o c o r r e s p o n d t o p o i n t s on t h e Cy c u r v e . F o r e x a m p l e , t h e o n s e t o f g a l l o p i n g , ( U / U 0 ) = l , i s o f c o u r s e dependen t on ^ C y da , and s i m i l a r l y , t h e jump t o t h e h i g h e r l i m i t c y c l e n e a r a=0 Q J / U 0 ) = 2 . 1 seems t o c o r r e s p o n d t o a p o i n t on C y ( a ) n e a r a = 9 . 5 ° w i t h a p p r o x i m a t e l y t h e same s l o p e as a t a = 0. The jump down t o t h e l o w e r l i m i t c y c l e n e a r U / U 0 = 1.6 c o r r e s p o n d s t o t h e Cy peak n e a r a = 1 2 . 6 ° . T h i s s o r t o f a n a l y s i s , h o w e v e r , o n l y g i v e s i n f o r m a t i o n on Y / U , and i s n o t enough t o s p e c i f y p a i r s o f ( U , Y ) , f o r w h i c h more q u a n t i t a t i v e methods a r e r e q u i r e d . 2 . 1 . 3 C o m p a r i s o n t o Measurements f rom F o r c e d O s c i l l a t i o n I n t h e c a s e o f f o r c e d o s c i l l a t i o n s , one c a n compare some o f t h e measurements o f f i g u r e 2 t o t h e q u a s i - s t e a d y p r e d i c t i o n . The t h e o r y p r e d i c t s Cor assumes) t h e p h a s e a n g l e t o be p l u s o r minus 9 0 ° , d e p e n d i n g on t h e s i g n o f C y ( ° 0 , w h i c h depends on a = a r c t a n ( Y / U ) . Tha t i s , when Y / U i s l a r g e , t h e q u a s i - s t e a d y t h e o r y a p p l i e d t o f o r c e d o s c i l l a t i o n p r e d i c t s t h e a n g l e t o be - 9 0 ° . C o n v e r s e l y , when Y / U i s s m a l l , t h e p r e d i c t e d p h a s e a n g l e i s + 9 0 ° . The change i n a v e r a g e phase a n g l e s i g n o c c u r s when t h e a e r o d y n a m i c f o r c e t e r m o f f i g u r e 7 changes s i g n , and t h i s c a n o c c u r o v e r a w i d e r a n g e o f w i n d s p e e d s . 9 The o b s e r v e d phase a n g l e s i n f i g u r e 2 ( a ) do n o t change s i g n f o r U < U r , h o w e v e r , so t h a t f o r U < U r , v o r t e x s h e d d i n g e f f e c t s d o m i n a t e . The o b s e r v e d m a g n i t u d e s d i f f e r f rom t h e p r e d i c t i o n as w e l l , g o i n g f rom - 1 8 0 ° a t low speed t o be tween + 2 0 ° and + 8 0 ° a t h i g h e r s p e e d s . F o r low v e l o c i t y o s c i l l a t i o n s an a n a l o g y c a n be d rawn t o t h e e f f e c t s o f added mass t h a t u n s t e a d y p o t e n t i a l f l o w t h e o r y s u g g e s t s . In t h e c a s e o f an o s c i l l a t i n g t h i n ( f l a t p l a t e ) a i r f o i l , t h e mass o f f l u i d o s c i l l a t i n g w i t h t h e a i r f o i l r e s u l t s i n an i n e r t i a l f o r c e e f f e c t w h i c h d o m i n a t e s a t low f l o w v e l o c i t i e s . I n t h e l i m i t , as v e l o c i t y a p p r o a c h e s z e r o , t h e p r e d i c t e d phase a n g l e a p p r o a c h e s - 1 8 0 ° . T h i s i s c l o s e t o t h e o b s e r v e d v a l u e s f o r t h e s q u a r e s e c t i o n n e a r U = 0 . 7 . A r o u n d U = 1 . 0 , t h e phase a n g l e i s s c a t t e r e d about - 9 0 ° , i n a p p r o x i m a t e agreement w i t h t h e q u a s i - s t e a d y t h e o r y . A t h i g h e r v e l o c i t y , t h e o b s e r v e d phase a n g l e s a r e be tween + 2 0 ° and + 8 0 ° , compared t o t h e p r e d i c t e d v a l u e o f + 9 0 ° . 2 . 2 Dynamic B e h a v i o r E x p e r i m e n t s 2 . 2 . 1 T e s t s The b a s i c s y s t e m d e s i g n e d and u s e d by S m i t h (1) w a s u s e d i n t h i s s t a g e o f t h e s t u d y . I t i s b a s e d on a i r f i l m j o u r n a l b e a r i n g s t h a t a r e u s e d t o s u p p o r t t h e m o d e l , w h i c h i s e l a s t i c a l l y c o n s t r a i n e d i n one d e g r e e o f f r e e d o m . F i g u r e 9 i s a s c h e m a t i c o f t h e a p p a r a t u s . I n s t r u m e n t a t i o n and mode l d e s i g n have been r e f i n e d c o n s i d e r a b l y s i n c e S m i t h ' s e x p e r i m e n t s t h o u g h , and t h u s i t was p o s s i b l e t o make a c c u r a t e measurements a t l o w e r o s c i l l a t i o n a m p l i t u d e s , and w i t h more p r e c i s e c o n t r o l o f t h e r e l e v a n t p a r a m e t e r s . The v a r i a b l e v i s c o u s - t y p e r e s i s t a n c e a f f o r d e d by t h e D . C . eddy c u r r e n t dampers was measured w i t h t h e w i n d o f f , u s i n g a t h i n C-32 x 3 . 5 x 68 cm,) 10 s t r e a m l i n e d a luminum b a r i n p l a c e o f t h e m o d e l . U s i n g a B r i i e l and K j a e r t y p e 2305 l e v e l r e c o r d e r w i t h l o g a r i t h m i c p o t e n t i o m e t e r ZR0005 , ( l o g ) a m p l i t u d e v s . t i m e r e c o r d s c o u l d be o b t a i n e d . F o r a model g i v e n an i n i t i a l d i s p l a c e m e n t and a l l o w e d t o o s c i l l a t e w i t h no e x t e r n a l e x c i t a t i o n , t h e s l o p e o f t h e ( l o g ) a m p l i t u d e t r a c 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 v i s c o u s damping c o e f f i c i e n t . From r e c o r d s o f a m p l i t u d e d e c a y f o r v a r i o u s v a l u e s o f damper c u r r e n t , a c a l i b r a t i o n o f c v s . I was o b t a i n e d . The c u r v e ( f i g u r e 10(a) ) a g r e e s w i t h t h o s e o b t a i n e d b y S m i t h ( 1 ) , and Santosham ( 1 4 ) , b u t i t was f o u n d t h a t a t low v a l u e s o f I , damping depended p a r t l y on o t h e r p a r a m e t e r s s u c h as s y s t e m mass , and o s c i l l a t i o n f r e q u e n c y and a m p l i t u d e . T y p i c a l a m p l i t u d e e f f e c t s a r e shown i n f i g u r e 10(b) where c / c ^ i s t h e r a t i o o f damping t o a r e f e r e n c e v a l u e (c @ 2y = 0 .1 c m ) . Thus w i t h no c u r r e n t , t h e damping i s s een t o i n c r e a s e by abou t 75% be tween d o u b l e a m p l i t u d e s 0 .1 cm and 10 . cm, bu t a t I = 300 mA«, t h e change i s o n l y about 8%. The e f f e c t o f v i s c o u s d r a g on t h e s t r e a m l i n e d b a r was e s t i m a t e d t o be a p p r o x i m a t e l y 3% o f t h e t o t a l damping a t z e r o c u r r e n t , b a s e d on t h e t h e o r e t i c a l v e l o c i t y p r o f i l e o f a v i s c o u s f l u i d n e a r a s i n u s o i d a l l y o s c i l l a t i n g i n f i n i t e p l a n e w i t h t h e f l u i d b e i n g a t r e s t a t i n f i n i t y . T h u s , t h e measured damping w i t h t h e b a r i n p l a c e i s n e a r l y a l l due t o t h e s u p p o r t s y s t e m . S t a t i c l o a d v a r i a t i o n (as c a u s e d b y w i n d b e i n g t u r n e d o n ) , and s u p p l y p r e s s u r e t o t h e a i r b e a r i n g s were a l s o f o u n d t o have some e f f e c t on t h e d a m p i n g , bu t t h e s e were m i n i m i z e d t h r o u g h e x p e r i m e n t a l t e c h n i q u e . C a l i b r a t i o n o f t h e l i n e a r d i s p l a c e m e n t t r a n s d u c e r was a c c o m p l i s h e d b y o b s e r v i n g o s c i l l a t i o n a m p l i t u d e w i t h a s t r o b o s c o p e and s c a l e , and c o m p a r i n g t h i s t o t h e RMS v o l t a g e o u t p u t o f t h e t r a n s d u c e r s y s t e m t o g e t a c a l i b r a t i o n c o n s t a n t . To i n v e s t i g a t e t h e e f f e c t s o f v o r t e x s h e d d i n g on g a l l o p i n g b e h a v i o r i n a i r , mode l s were c o n s t r u c t e d so t h a t U Q c o u l d be much l e s s t h a n U r . (In t e rms 11 o f t h e d i m e n s i o n a l sy s t em p a r a m e t e r s , U 0 = 2 c / ( p £ A c o h 2 ) . The l i g h t w e i g h t b a l s a and a luminum mode l s were 3 .3 and 5 .1 cm s q u a r e , and 6 8 . 1 cm i n l e n g t h . E n d p l a t e s were f i t t e d t o m i n i m i z e end e f f e c t s , s i n c e t h e w i n d t u n n e l w a l l s l o t s n e c e s s a r y to a l l o w model m o t i o n were f o u n d t o be a d m i t t i n g o u t s i d e a i r i n t o t h e r e g i o n n e a r t h e m o d e l , a f f e c t i n g t h e o s c i l l a t i o n b e h a v i o r . A t y p i c a l mode l i s shown i n f i g u r e 1 1 . 2 . 2 . 2 R e s u l t s The g a l l o p i n g r e s p o n s e o f t he two mode l s was i n v e s t i g a t e d , by m e a s u r i n g t h e o s c i l l a t i o n a m p l i t u d e as a f u n c t i o n o f w i n d s p e e d , d a m p i n g , and o s c i l l a t i o n f r e q u e n c y . U s i n g s e v e r a l c o m b i n a t i o n s o f t e n s i o n s p r i n g s , t h e n a t u r a l f r e q u e n c y o f t h e s y s t e m c o u l d be c h a n g e d . The damping was a d j u s t a b l e by v a r y i n g t h e c u r r e n t t h r o u g h t h e dampers . F o r g i v e n v a l u e s o f damping and f r e q u e n c y , t h e a m p l i t u d e was a l l o w e d t o s t a b i l i z e a t e a c h v a l u e o f w i n d s p e e d , and t h e a v e r a g e v a l u e s o r r a n g e o f a m p l i t u d e s were r e c o r d e d . The r e s u l t s a r e p r e s e n t e d i n f i g u r e s 12 t h r o u g h 3 3 , t o g e t h e r 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 b a s e d on n u m e r i c a l r e s u l t s , as i n f i g u r e 8. S i n c e t h e b e h a v i o r was somet imes d i f f e r e n t f o r t h e two m o d e l s , even when t h e n o n - d i m e n s i o n a l p a r a m e t e r s were m a t c h e d , f i g u r e s 12 t o 27 a r e f o r t h e 3 . 3 cm m o d e l , and 28 t o 33 a r e f o r t h e o t h e r . I n f i g u r e 12 , t he e x p e r i m e n t a l a m p l i t u d e b e h a v i o r f o r U Q = 0 . 4 4 i s shown. H e r e , U 0 i s l e s s t h a n U r and no o s c i l l a t i o n s o c c u r u n t i l r e s o n a n c e i s a p p r o a c h e d . T h e n , n e a r U r , t h e o s c i l l a t i o n s a r e b e a t m o d u l a t e d , as shown by t h e i n s e t a m p l i t u d e o s c i l l o g r a p h . As t h e w i n d s p e e d i s i n c r e a s e d , t h e a m p l i t u d e i n c r e a s e s and n e a r U=1.5 shows a s a w - t o o t h b e h a v i o r w i t h r a t h e r s l o w b u i l d u p and r a p i d d r o p - o f f t o t he l o w e r a m p l i t u d e . F u r t h e r i n c r e a s e i n f l o w speed i n c r e a s e s t h e a m p l i t u d e and r e d u c e s t h e v a r i a t i o n u n t i l e v e n t u a l l y the t h e o r e t i c a l a m p l i t u d e i s a p p r o a c h e d n e a r U = 2 . 5 . No te a l s o t h e v a r i a t i o n 12 abou t l i n e a r i n c r e a s e a t h i g h e r w i n d s p e e d . F i g u r e s 13 t o 15 show b e h a v i o r f o r U Q s l i g h t l y g r e a t e r t h a n U r , where t h e e x i s t e n c e o f two s t a b l e l i m i t c y c l e s o v e r a r a n g e o f w i n d speeds i s somet imes a p p a r e n t . In f i g u r e 15 f o r e x a m p l e , n e a r U = 1 . 5 , t h e model i s s t a b l e b o t h n e a r Y=0 and Y = 0 . 2 . T h i s i s i n a d d i t i o n t o t h e g a l l o p i n g t y p e h y s t e r e s i s o b s e r v e d n e a r U = 2 . 2 . I n f i g u r e s 16 t o 2 5 , U 0 becomes p r o g r e s s i v e l y g r e a t e r t h a n U r , and t h e s e p a r a t i o n o f r e s o n a n c e and g a l l o p i n g b e h a v i o r i s e v i d e n t . The amount o f o b s e r v e d h y s t e r e s i s o v e r l a p v a r i e d somewhat , w i t h f i g u r e s 19 and 20 s h o w i n g t h e s o r t o f v a r i a t i o n . F i g u r e s 23 t o 27 do n o t show t h e u p p e r l i m i t c y c l e s o f g a l l o p i n g b e h a v i o r due t o t h e p h y s i c a l l i m i t a t i o n on mode l o s c i l l a t i o n a m p l i t u d e , bu t do i n d i c a t e t h e o n s e t o f g a l l o p i n g i n c o m p a r i s o n t o t h e t h e o r y . As p r e v i o u s l y m e n t i o n e d , f i g u r e 28 t o 33 show t h e o s c i l l a t i o n b e h a v i o r o f t h e 5 .1 cm, m o d e l . F i g u r e 28 r e p r e s e n t s b e h a v i o r f o r t h e l o w e s t v a l u e o f U Q o b t a i n e d . A g a i n , as f o r a l l e x p e r i m e n t s w i t h U 0 < U r , o s c i l l a t i o n b e g i n s n e a r U r and a p p r o a c h e s t h e p r e d i c t e d a m p l i t u d e o n l y a t h i g h e r w i n d s p e e d s . F i g u r e 29 shows t h e e x i s t e n c e o f two s t a b l e l i m i t c y c l e s n e a r U = 1 . 5 . Over t h i s r a n g e o f w i n d s p e e d , t h e mode l w o u l d n o t r e a c h t h e u p p e r a m p l i t u d e s i f r e l e a s e d f rom t h e n e u t r a l p o s i t i o n a t r e s t . M a n u a l p e r t u r b a t i o n above t h e u n s t a b l e a m p l i t u d e s i n d i c a t e d , r e s u l t e d i n a m p l i t u d e g r o w t h and e v e n t u a l c o n v e r g e n c e on t h e h i g h e r l i m i t c y c l e . F i g u r e s 30 and 31 show more p r o n o u n c e d b e h a v i o r o f t h i s s o r t , w i t h a l a r g e s t a b l e - u n s t a b l e l o o p f o r 1 . 4 < U < 2 . F u r t h e r i n c r e a s e i n U 0 r e s u l t e d i n s e p a r a t i o n o f r e s o n a n c e and g a l l o p i n g as o b s e r v e d f o r t h e 3 . 3 cm. m o d e l . 2 . 2 . 3 D i s c u s s i o n I n f i g u r e s 12 t o 3 3 , t h e ' l o c k i n g i n r e g i o n ' d e s c r i b e d i n (4) i s shown, t h e upper r e g i o n be tween t h e shaded l i n e s n e a r U £ . T h i s r e p r e s e n t s t h e r e g i o n i n w h i c h t h e v o r t e x f r e q u e n c y ' l o c k s i n ' t o t h e o s c i l l a t i o n f r e q u e n c y , and hence becomes i n d e p e n d e n t o f t h e S t r o u h a l f r e q u e n c y . . T h i s r e g i o n was d e f i n e d by power s p e c t r a a n a l y s i s o f f o r c e d o s c i l l a t i o n measurements i n ( 4 ) , and may be u s e f u l i n g a i n i n g i n t u i t i v e u n d e r s t a n d i n g o f v o r t e x s h e d d i n g c h a r a c t e r i s t i c s . I n f i g u r e 1 2 , t h e b e a t m o d u l a t e d a m p l i t u d e b e h a v i o r n e a r U r i s commonly o b s e r v e d i n v o r t e x e x c i t e d o s c i l l a t i o n s , and i s l i k e l y due t o s l i g h t m i s m a t c h be tween v o r t e x s h e d d i n g and o s c i l l a t i o n f r e q u e n c i e s . O n l y a t h i g h e r a m p l i t u d e do t h e f r e q u e n c i e s ' l o c k i n " . The s a w - t o o t h m o d u l a t i o n a p p a r e n t n e a r U=1.5 may i n d i c a t e t h a t v o r t e x f o r c e s s t i l l impose a l i m i t on t h e maximum a m p l i t u d e and t h a t v o r t e x s h e d d i n g i s s t i l l l o c k e d i n t o t h e o s c i l l a t i o n f r e q u e n c y . F u r t h e r i n c r e a s e i n f l o w v e l o c i t y t e n d s t o r e t u r n t h e v o r t e x s h e d d i n g f r e q u e n c y t o t h e s t a t i o n a r y S t r o u h a l f r e q u e n c y , and g a l l o p i n g f o r c e s become dominan t as t h e a m p l i t u d e s a p p r o a c h t h e t h e o r e t i c a l v a l u e s . I t i s p o s s i b l e t o draw c e r t a i n p a r a l l e l s be tween t h e o b s e r v e d o s c i l l a t i o n b e h a v i o r f o r U 0 n e a r U r and l o w e r i n t h e p r e s e n t s t u d y , and t h e f o r c e d o s c i l l a t i o n measurements o f ( 4 ) . As a l r e a d y m e n t i o n e d , t h e n e g a t i v e phase a n g l e f o r U < U r i s c o n s i s t e n t w i t h o b s e r v e d s t a b i l i t y a t r e s t i n c a s e s where U 0 < U r . The e x i s t e n c e o f h i g h e r t h a n e x p e c t e d a m p l i t u d e s , as i n f i g u r e s 13 , 1 5 , 29 , 30 , and 3 1 , f o r U 0 > U r . (bu t c l o s e ) c a n be p a r t i a l l y e x p l a i n e d u s i n g a ' g r a p h i c a l a p p r o a c h as i n f i g u r e 7 d i s c u s s e d p r e v i o u s l y . I n f i g u r e 34 a r e shown t h e d a t a p o i n t s t a k e n d i r e c t l y f rom f i g u r e 13 o f ( 4 ) , t h a t r e p r e s e n t t h e a e r o d y n a m i c w o r k i n p u t o v e r one c y c l e , as a f u n c t i o n o f Y and U . The m a g n i t u d e s n e a r U=1.4 a r e g r e a t e r t h a n t h e maximum t h a t t h e q u a s i - s t e a d y t h e o r y w o u l d p r e d i c t f o r f o r c e d o s c i l l a t i o n , f o r v a l u e s o f A be tween 3 and 4 . T h u s , t h e s e c t i o n w o u l d be l e s s s t a b l e t h a n t h e t h e o r y p r e d i c t s , and h i g h e r t h a n e x p e c t e d a m p l i t u d e s a r e q u i t e p l a u s i b l e . As t h e 14 damping - mass p a r a m e t e r i s i n c r e a s e d , s t a b i l i t y i n c r e a s e s , and t h e q u a s i - s t e a d y t h e o r y i s v a l i d f o r d e t e r m i n i n g g a l l o p i n g i n s t a b i l i t y . The measurements o f f o r c e do n o t seem t o h e l p e x p l a i n t h e l o w e r s t a b l e - u n s t a b l e l o o p somet imes o b s e r v e d f o r t h e same r a n g e o f w i n d s p e e d s . S u c h q u a l i t a t i v e a n a l y s i s does n o t t a k e i n t o a c c o u n t measured f o r c e s i n (4) f o u n d t o be i n phase ( o r 1 8 0 ° ou t o f phase ) w i t h d i s p l a c e m e n t , t h e e f f e c t s o f w h i c h a r e n o t u n d e r s t o o d . Though t h i s c o m p a r i s o n does n o t g i v e d e t a i l s a b o u t t h e f l o w f i e l d , i t a t l e a s t l e n d s c o n f i d e n c e t o t h e v a l i d i t y o f two i n d e p e n d e n t s t u d i e s . I t s h o u l d be m e n t i o n e d t h a t i n m a k i n g t h e i r measu remen t s , t h e a u t h o r s o f (4) t o o k t h e t o t a l a e r o d y n a m i c f o r c e t o be t h e s i g n a l d i f f e r e n c e o f s t r a i n gauges a t t a c h e d t o two mode l s o s c i l l a t i n g i n p a r a l l e l , one i n t h e w i n d and one o u t s i d e t h e w i n d t u n n e l i n s t i l l a i r . T h i s method i n t r o d u c e s e r r o r , due t o n o n - z e r o a e r o d y n a m i c f o r c e on t h e 'dummy' mode l i n s t i l l a i r , i n b o t h phase a n g l e and m a g n i t u d e . A more a p p r o p r i a t e t e c h n i q u e i s t o u s e a s t r e a m l i n e d c o n c e n t r a t e d mass i n p l a c e o f t h e 'dummy' mode l so t h a t o n l y i n e r t i a l f o r c e s w i l l be s u b t r a c t e d f rom t h e a c t i v e mode l f o r c e s i g n a l . ( T h i s was done f o r subsequen t measurements on n o n s q u a r e r e c t a n g u l a r s e c t i o n s i n (5) . ) I t was o b s e r v e d t h a t t h e 5 .1 cm, mode l was more u n s t a b l e t h a n t h e o t h e r mode l i n e x h i b i t i n g t h e h i g h e r t h a n e x p e c t e d a m p l i t u d e s n e a r r e s o n a n c e . Tha t i s , v o r t e x f o r c e s seemed more s i g n i f i c a n t f o r t h e l a r g e r , m o d e l . I t i s c o n j e c t u r e d t h a t t h i s may be due t o b l o c k a g e e f f e c t s , c a u s i n g h i g h e r p e r i o d i c l i f t c o e f f i c i e n t s t h a n f o r t h e l o w e r b l o c k a g e c a s e . Once U 0 i s g r e a t e r t h a n abou t 2 . 5 , v o r t e x s h e d d i n g does n o t have as much e f f e c t on b e h a v i o r , even f o r t h e l a r g e r m o d e l , and t h e q u a s i - s t e a d y t h e o r y i s f a i r l y a c c u r a t e i n p r e d i c t i n g g a l l o p i n g r e s p o n s e . The c o l l a p s e d d a t a f o r a s e l e c t i o n o f t e s t s w i t h U > 2 . 3 i s shown i n f i g u r e 3 5 , w i t h t h e 15 t h e o r e t i c a l c u r v e i n d i c a t e d . F o r h i g h e r v a l u e s o f U 0 , t h e agreement i s f a i r l y good f o r t h e l o w e r l i m i t c y c l e a m p l i t u d e s , b u t n o t as good f o r t h e h i g h e r . A f r e q u e n t l y o b s e r v e d t e n d e n c y i s f o r t h e a m p l i t u d e o f t h e uppe r l i m i t c y c l e a t h i g h e r w i n d speeds t o show a w a v e - l i k e v a r i a t i o n , f o r example i n f i g u r e s 12 and 15 n e a r U=3. R e s u l t s p r e s e n t e d i n (1) show s i m i l a r b e h a v i o r o v e r a w i d e r r a n g e o f w i n d s p e e d s , where more measurements were made o f t h e u p p e r l i m i t c y c l e a m p l i t u d e s . The a m p l i t u d e 'hump' o f f i g u r e 21 n e a r U=5.5 may be an e x t r e m e example o f t h e same e f f e c t . T h i s b e h a v i o r may i n d i c a t e a n o t h e r e f f e c t o f v o r t e x s h e d d i n g , s i n c e t h e m i n i m a o f t h e v a r i a t i o n s seem t o o c c u r n e a r i n t e g e r m u l t i p l e s o f U r . I t i s o f i n t e r e s t t o n o t e some p r o g r e s s a c h i e v e d i n ( 5 ) , u s i n g t h e f o r c e d o s c i l l a t i o n measurements t o p r e d i c t g a l l o p i n g r e s p o n s e . One p r i n c i p a l r e s u l t i s t h a t f o r a r e c t a n g u l a r ( b / h = 2) s e c t i o n w i t h U 0 s l i g h t l y g r e a t e r t h a n U r , t h e s o r t o f b e h a v i o r o f f i g u r e 29 i s p r e d i c t e d , w i t h r e s o n a n c e c o n t i n u i n g on i n t o g a l l o p i n g o s c i l l a t i o n as U i s i n c r e a s e d . (A l o w e r s t a b l e - u n s t a b l e l o o p i s n o t p r e d i c t e d . ) T h i s agreement i n d i c a t e s t h a t measurements u s i n g f o r c e d o s c i l l a t i o n t e c h n i q u e s g i v e s u f f i c i e n t i n f o r m a t i o n t o p r e d i c t f r e e o s c i l l a t i o n b e h a v i o r , and may p r o v e t o be a v a l u a b l e t o o l f o r a n a l y s i s i n c a s e s when U 0 i s n e a r t o o r l e s s t h a n U r , where t h e q u a s i - s t e a d y t h e o r y i s n o t e n t i r e l y v a l i d . I f one compares t h e r e s u l t s o f S m i t h (1) t o t h e p r e s e n t r e s u l t s , t h e most s i g n i f i c a n t d i f f e r e n c e i s t h e v a l u e o f A , w h i c h a f f e c t s U G , t h e w i n d speed where g a l l o p i n g b e g i n s . I n t h e p r e s e n t r e s u l t s , g e n e r a l l y a t l o w e r R e y n o l d s numbers t h a n i n (1)> g a l l o p i n g began a t w i n d speeds abou t t h i r t y p e r c e n t l o w e r ( f o r t h e same B / n ) t h a n one w o u l d have p r e d i c t e d u s i n g A=2 .69 f rom t h e d a t a o f ( 1 )« T h i s becomes a p p a r e n t when one u s e s t h e same v a l u e o f A f o r b o t h S m i t h ' s d a t a and t h e p r e s e n t d a t a , i n r e d u c i n g t o a g r a p h s u c h as 16 i n f i g u r e 3 5 . The two s e t s o f d a t a c o l l a p s e o n t o s e p a r a t e bu t s i m i l a r c u r v e s . P a r t o f t h i s v a r i a t i o n o f A seems due t o R e y n o l d s number e f f e c t , and p a r t due t o end c o n d i t i o n s . The R e y n o l d s number e f f e c t i m p l i e s t h a t s u r f a c e r o u g h n e s s and c o r n e r r a d i u s may a l s o be s i g n i f i c a n t i n d e t e r m i n i n g t h e v a l u e o f A . End e f f e c t s a l s o a p p e a r t o be i m p o r t a n t , s i n c e t h e a d d i t i o n o f e n d p l a t e s t o t h e model e f f e c t i v e l y i n c r e a s e t h e s l o p e o f C y ( a ) @ a^O. I t s h o u l d be m e n t i o n e d t h a t t h e g a l l o p i n g measurements were made i n two s e p a r a t e s e r i e s o f t e s t s , and some d i f f e r e n c e s i n r e s p o n s e a r e n o t i c e a b l e be tween t h e two g r o u p s . The d a t a p o i n t s i n f i g u r e s 18 and 19 show d i f f e r e n t t r e n d s , even t h o u g h U D i s n e a r l y t h e same i n b o t h . The f i r s t s e r i e s r e s u l t s a r e p r e s e n t e d i n f i g u r e s 12 , 15 , 18 , 20 , 24 , 28 , 29 , and 30 . The s e c o n d s e r i e s , p e r f o r m e d s e v e r a l months l a t e r , a r e p r e s e n t e d i n t h e o t h e r f i g u r e s . The p a r a m e t e r most d i f f i c u l t t o c o n t r o l i n s e t t i n g up t h e a p p a r a t u s was t h e d a m p i n g , and o n l y f o r t he s e c o n d s e t o f t e s t s was t h e damping v s . a m p l i t u d e i n v e s t i g a t e d , as i n f i g u r e 10 ( b ) . S i n c e i t i s p o s s i b l e t h a t t h e damping c h a r a c t e r i s t i c s were d i f f e r e n t a t h i g h e r a m p l i t u d e s f o r t h e f i r s t s e t , t h e a u t h o r has more c o n f i d e n c e i n t h e r e s u l t s o f t h e s e c o n d g roup o f t e s t s . The l o w e r a m p l i t u d e r e s u l t s a r e l i k e l y more c o m p a r a b l e be tween t h e two s e t s . Though t h e r e s u l t s p r e s e n t e d h e r e a r e f o r t h e c a s e o f smooth and n e a r l y 2 d i m e n s i o n a l f l o w , and i t w o u l d be somewhat r i s k y t o a p p l y t h e r e s u l t s t o p r e d i c t i n g b e h a v i o r i n t u r b u l e n t , 3 d i m e n s i o n a l f l o w , t h e o b s e r v e d . b e h a v i o r f o r U 0 s l i g h t l y g r e a t e r t h a n U r may be s i g n i f i c a n t . I n t o w e r s and t a l l b u i l d i n g s , s i m i l a r p r o x i m i t y can be a c h i e v e d i n t h e c r i t i c a l w i n d s p e e d s , and i f t h e b e h a v i o r i n f u l l s c a l e i s s i m i l a r t o t h e t e s t r e s u l t s , t h e r e i s t h e r i s k t h a t t h e s t r u c t u r e s may be s u b j e c t e d t o h i g h e r l o a d i n g t h a n e x p e c t e d n e a r r e s o n a n c e , s i n c e t h e o b s e r v e d a m p l i t u d e s c a n ge t q u i t e l a r g e even f o r U < U 0 . V e r i f i c a t i o n o f t h i s w o u l d be w o r t h w h i l e , t h r o u g h t e s t s i n t u r b u l e n t a n d / o r 3 d i m e n s i o n a l f l o w s . C o n c l u s i o n s CHAPTER THREE By e x a m i n i n g t h e a e r o e l a s t i c b e h a v i o r o f a s q u a r e s e c t i o n p r i s m i n u n i f o r m f l o w , i t was d e t e r m i n e d t h a t : 1) As e x p e c t e d , t h e q u a s i - s t e a d y t h e o r y i s v a l i d a t w i n d speeds w e l l above t h e c r i t i c a l r e s o n a n c e w i n d s p e e d . Howeve r , t h e e f f e c t s o f R e y n o l d s number and end c o n d i t i o n s can have s i g n i f i c a n t i n f l u e n c e on the a e r o d y n a m i c c h a r a c t e r i s t i c s o f t h e s e c t i o n , e s p e c i a l l y t h e o n s e t speed o f g a l l o p i n g U o . 2) A t w i n d speeds l e s s t h a n about 2 . 5 , t h e q u a s i - s t e a d y t h e o r y does n o t a c c u r a t e l y p r e d i c t t h e b e h a v i o r due t o t h e i n f l u e n c e o f t h e v o r t e x f o r m a t i o n mechan i sm, a l t h o u g h g a l l o p i n g - l i k e b e h a v i o r i s o b s e r v e d a t w i n d speeds above t h e c r i t i c a l r e s o n a n c e v a l u e . Dynamic f o r c e measurements r e p o r t e d i n the l i t e r a t u r e c a n be u s e d t o e x p l a i n t h e b e h a v i o r i n t h i s r a n g e o f w i n d s p e e d s , b u t t h e d e t a i l s o f t h e f l o w f i e l d b e h a v i o r a r e s t i l l n o t known. F l o w v i s u a l i z a t i o n o f an o s c i l l a t i n g p r i s m may be h e l p f u l i n g a i n i n g u n d e r s t a n d i n g o f v o r t e x e f f e c t s . 18 B i b l i o g r a p h y (1) S m i t h , J . D . , " A n E x p e r i m e n t a l S t u d y o f t h e A e r o e l a s t i c I n s t a b i l i t y o f R e c t a n g u l a r C y l i n d e r s " , M . A . S c . T h e s i s , 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 , 1962 (2) P a r k i n s o n , G . V . , and S m i t h , J . D . , "The S q u a r e P r i s m as an A e r o e l a s t i c N o n - L i n e a r O s c i l l a t o r " , Q u a r t e r l y J o u r n a l o f M e c h a n i c s and A p p l i e d M a t h e m a t i c s , V o l . 17 , p a r t 2 , May 1964 (3) Nakamura , Y . N . , "Some R e s e a r c h on A e r o e l a s t i c I n s t a b i l i t i e s o f B l u f f S t r u c t u r a l S e c t i o n s " , P r o c e e d i n g s o f t h e 4 t h I n t e r n a t i o n a l C o n f e r e n c e on Wind E f f e c t s on B u i l d i n g s and S t r u c t u r e s , 1975 , Hea th row (4) O t s u k i , Y . , W a s h i z u , K . - , Tomizawa , H . , and O h y a , A . , " A n o t e on t h e A e r o e l a s t i c I n s t a b i l i t y o f a P r i s m a t i c Ba r w i t h S q u a r e S e c t i o n " , J o u r n a l o f Sound and V i b r a t i o n , 1974, 3 4 ( 2 ) , 233-248 (5) W a s h i z u , K . , Ohya , A . , O t s u k i , Y . , and F u j i i , K . , " A e r o e l a s t i c I n s t a b i l i t y o f R e c t a n g u l a r C y l i n d e r s i n a H e a v i n g M o d e " , J o u r n a l o f Sound and V i b r a t i o n , 1978 , 5 9 ( 2 ) , 195-210 (6) B r o o k s , P . N . H . , " 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 t h e A e r o e l a s t i c I n s t a b i l i t y o f B l u f f T w o - D i m e n s i o n a l C y l i n d e r s ' , M . A . S c . T h e s i s , 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 , 1960 (7) C o w d r e y , C . F . , and Lawes , J . A . , " F o r c e Measurements on S q u a r e and D o d e c a g o n a l S e c t i o n a l C y l i n d e r s a t H i g h N R " , , N . P . L . - / A e r o / 3 5 1 , 1959 (8) Kwok, K . C . S . , " C r o s s - W i n d Response o f S t r u c t u r e s Due t o D i s p l a c e m e n t Dependent E x c i t a t i o n s " , P h . D . T h e s i s , Monash U n i v e r s i t y , V i c t o r i a , A u s t r a l i a , 1977 (9) D u g u n d j i , J . , and Chung , F . K . , Addendum t o ( 1 0 ) , J o u r n a l o f Sound and V i b r a t i o n , 1978 , 5 6 ( 2 ) , 309-311 (10) Mukhopadyay , V . , and D u g u n d j i , J . , " W i n d E x c i t e d V i b r a t i o n o f a S q u a r e S e c t i o n C a n t i l e v e r Beam i n Smooth F l o w " , J o u r n a l o f Sound and V i b r a t i o n , 1976 , 4 5 ( 3 ) , 329 -339 (11) Nakamura , Y . , and T o m o n a r i , Y . , " G a l l o p i n g o f R e c t a n g u l a r P r i s m s i n a Smooth and i n a T u r b u l e n t F l o w " , J o u r n a l o f Sound and V i b r a t i o n , 1977 5 2 ( 2 ) , 233-241 (12) L a n e v i l l e , A . , " E f f e c t s o f T u r b u l e n c e on Wind I n d u c e d V i b r a t i o n s o f B l u f f C y l i n d e r s " , P h . D . T h e s i s , 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 , 1973 (13) S u l l i v a n , P . P . , " A e r o e l a s t i c G a l l o p i n g o f T a l l S t r u c t u r e s i n S i m u l a t e d W i n d s " , M . A . S c . T h e s i s , 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 , 1977 (14) San tosham, T . V . , " F o r c e Measurements on B l u f f C y l i n d e r s and A e r o e l a s t i c G a l l o p i n g o f a R e c t a n g u l a r C y l i n d e r " , M . A . S c . T h e s i s , 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 , 1966 (15) Cowdrey , C . F . , " A Note on t h e Use o f End P l a t e s t o P r e v e n t Th ree -D i m e n s i o n a l F l o w a t t h e Ends o f B l u f f C y l i n d e r s " , N . P . L . / A e r o / 1 0 2 5 , 1962 A p p e n d i x A C y ( a ) d a t a , u s e d f o r n u m e r i c a l c a l c u l a t i o n s a 0 C y 0 0 .5 .035 1.0 .0625 2 . 0 .118 3 . 0 .164 4 . 0 .2005 4 . 5 .214 5 . 0 .2225 6 . 0 .232 7 . 0 .238 8 . 0 . 246 8 .5 . 259 9 . 0 .2793 9 . 5 .3075 1 0 . 0 .341 1 0 . 5 .378 1 1 . 0 . 419 1 1 . 5 .466 1 2 . 0 . 540 1 2 . 2 5 .578 1 2 . 5 . 6045 1 2 . 6 . 608 1 2 . 7 .607 1 2 . 9 .591 13 .25 . 552 1 4 . 0 . 453 1 5 . 0 .31 1 6 . 0 . 17 1 7 . 0 .0375 1 8 . 0 - . 0 9 5 1 9 . 0 - . 2 2 8 5 2 0 . 0 - . 3 6 1 2 1 . 0 - . 4 9 3 A p p e n d i x B_ P r e d i c t e d G a l l o p i n g Response o f Squa re S e c t i o n P r i s m Lower l i m i t c y c l e Upper l i m i t c y c l e u/u 0 Y/U 0 U / U Q Y/U 0 1.0 0. 1. 70 0 .385 1.0125 0 .01 1.70 0 . 3 9 1.05 0 . 0 2 1.713 0 . 4 0 1.10 0. 03 1.825 0 .45 1.175 0 .05 1.963 0 . 5 0 1.35 0 . 1 0 2 . 1 0 0 . 5 5 1.538 0 .15 2 . 2 3 7 0 . 6 0 1.725 0 . 2 0 2 . 5 1 2 0. 70 1.888 0. 25 2 . 794 0 . 8 0 2 . 0 3 8 0. 30 3 .075 0 . 9 0 2 . 0 8 8 0 .325 3 .35 1.00 2 . 0 8 8 0 . 3 5 0 ( n u m e r i c a l r e s u l t s b a s e d on d a t a i n A p p e n d i x A) A p p e n d i x C T a b l e I b / h R e y n o l d s No . A T u r b u l e n c e Type * S o u r c e 1.0 22000 2 . 6 9 < . l % I S m i t h ( 1 ) , p o l y n o m i a l c o e f f . , s m a l l end gaps 27000 2 . 7 2 < . l I S m i t h ( 1 ) , g r a p h , s m a l l end gaps 66000 3 . 0 < . l I B r o o k s ( 6 ) , g r a p h , p r e s s u r e measurements 66000 3 . 3 < . l I B r o o k s ( 6 ) , f rom CL § Cn c u r v e s 270000 1.6 I C7) 1 .13 2 . I Kwok ( 8 ) , c a n t i l e v e r , f r e e end 3000 1.32 I ( 9 ) , c a n t i l e v e r , end gap 4 . 0 .1 - - ( 1 1 ) , o n s e t method 1700 . 6 8 I I ( 1 0 ) , c a n t i l e v e r , end gap , r o u g h s u r f a c e 3100 . 9 4 I I ( 1 0 ) , c a n t i l e v e r , end g a p , rough s u r f a c e 2 . 6 2 13 I I L a n e v i l l e ( 1 2 ) , l a r g e s c a l e t u r b . , s m a l l gaps 3 . 1 9 7-9 I I L a n e v i l l e ( 1 2 ) , s m a l l s c a l e t u r b . , s m a l l gaps 3 . 4 9 1 2 . 5 I I L a n e v i l l e ( 1 2 ) , s m a l l s c a l e t u r b . , s m a l l gaps 4 . 1 9 . , r o d I I Kwok ( 8 ) , c a n t i l e v e r , f r e e end 1.5 b . l . I I Kwok ( 8 ) , c a n t i l e v e r , f r e e end 1.25 b . l . I I S u l l i v a n ( 1 3 ) , c a n t i l e v e r , f r e e end 3 . 7 12 — ( 1 1 ) , o n s e t method 2 . 0 2 . 3 3 <. 1 Santosham ( 1 4 ) , p o l y n o m i a l c o e f f . , s m a l l gaps 20400 4 . 0 • < . l Santosham ( 1 4 ) , f rom CL £ Cn c u r v e s 32500 3 . 2 < . l Santosham ( 1 4 ) , f rom CL 5 Cn c u r v e s 38000 2 . 9 5 < . l Santosham ( 1 4 ) , f rom CL £ Cn c u r v e s 33000 3 . 3 < . l B r o o k s ( 6 ) , g r a p h , p r e s s u r e measurements 22 A p p e n d i x D 0 1 2 3 u Uo Figure 1(c) - Predicted Galloping Response for Square Section -180 A 0 . 025 • 0 . 0 5 0 o 0 . 1 0 0 • 0 . 1 5 0 F i g u r e 2 ( a ) - M e a s u r e d phase a n g l e f rom dynamic measurements ( f i g . 18 o f ( 3 ) ) oA R • D " • 4A y • 2 mm A 5 o 10 A 15 • 20 1.0 Ur 1.5 2.0 h=150mm F i g u r e 2 ( b ) - Measu red f o r c e c o e f f i c i e n t f rom dynamic measurements ( f i g . 12 o f ( 4 ) ) 1.0 a 0.8 Re & ° § 0.6 A 8800 • 12400 o 28800 a ° 0 A • 0.4 a ° g ° { } A 9 o 0.2 -n A I L • i i i . 1. 1 • A 0 4 8 12 16 20 a ( d e g r e e s ) F i g u r e 3 - Measu red l i f t c o e f f i c i e n t s a ( d e g r e e s ) F i g u r e 5 - Norma l f o r c e c o e f f i c i e n t s - 1 0 1 2 a (degrees ) F i g u r e 6 - Normal f o r c e c o e f f i c i e n t s n e a r a = 0 F i g u r e 7 - E q u a t i o n (2.1) i n g r a p h i c form 00 F i g u r e 8 - P r e d i c t e d G a l l o p i n g Response o f S q u a r e S e c t i o n Air Bearing Displacement Transducer Circuitry Tension Spring D.C. Power Supply (Log) Amplitude Chart Recorder l i nn RMS Voltmeter gure 9 - S c h e m a t i c : g a l l o p i n g and damping measurements Figure 10(a) - Damping c a l i b r a t i o n 32 0.10 1.0 10. 2y (cm) Figure 10(b) - Variation of damping with oscillation amplitude V F i g u r e 11 - T y p i c a l t e s t model 34 B=0.00156 f=5.25 Hz A=4.0 F i g u r e 13 - G a l l o p i n g t e s t s , 3 . 3 cm m o d e l , U 0 = 1 . 3 8 36 Y 1 u r 2 3 4 5 U F i g u r e 14 - G a l l o p i n g t e s t s , 3 . 3 cm m o d e l , U 0 = 1 . 6 7 37 Y F i g u r e 15 - G a l l o p i n g t e s t s , 3 . 3 cm m o d e l , U 0 = 1.77 F i g u r e 16 - G a l l o p i n g t e s t s , 3 . 3 cm m o d e l , U 0 = 1 . 8 7 39 F i g u r e 17 - G a l l o p i n g t e s t s , 3 . 3 cm m o d e l , U o = 2 . 0 7 40 F i g u r e 18 - G a l l o p i n g t e s t s , 3 . 3 cm m o d e l , U D = 2 . 2 7 F i g u r e 19 - G a l l o p i n g t e s t s , 3 . 3 cm m o d e l , U D = 2 . 2 9 42 F i g u r e 20 - G a l l o p i n g t e s t s , 3 . 3 cm m o d e l , U 0 = 2 . 5 6 43 Y 1 u r 2 3 4 5 6 U F i g u r e 21 - G a l l o p i n g t e s t s , 3 . 3 cm m o d e l , U 0 = 2 . 7 7 44 45 F i g u r e 23 - G a l l o p i n g t e s t s , 3 . 3 cm m o d e l , U o = 4 . 0 7 46 F i g u r e 24 - G a l l o p i n g t e s t s , 3 . 3 cm m o d e l , U 0 = 4 . 2 4 47 B=0.00540 f=5.25 Hz A=4.0 48 B=0.00644 f=5.25 Hz A=4.0 F i g u r e 26 - G a l l o p i n g t e s t s , 3 . 3 cm m o d e l , U 0 = 5 . 6 7 49 B=0.00855 f=5.25 Hz A=3.5 50 F i g u r e 28 - G a l l o p i n g t e s t s , 5 .1 cm m o d e l , U o = 0 . 1 7 5 51 F i g u r e 29 - G a l l o p i n g t e s t s , 5.1 cm m o d e l , U0=1.25 52 F i g u r e 30 - G a l l o p i n g t e s t s , 5 .1 cm m o d e l , U 0 = 1 . 7 5 53 F i g u r e 31 - G a l l o p i n g t e s t s , 5 .1 cm m o d e l , U 0 = 1 . 7 6 54 F i g u r e 32 - G a l l o p i n g t e s t s , 5 . 1 cm m o d e l , U 0 = 2 . 5 4 B=0.00664 f=5.0 Hz A = 3 . 5 F i g u r e 33 - G a l l o p i n g t e s t s , 5 .1 cm m o d e l , U o = 3 . 0 6 56 q u a s i - s t e a d y t h e o r y f rom (4) Y • 0.013 A 0.033 o 0.067 A 0.100 • 0.133 •10 A • A F i g u r e 34 - C o m p a r i s o n be tween dynamic measurements and q u a s i - s t e a d y t h e o r y F i g u r e 35 - C o l l a p s e d d a t a from g a l l o p i n g t e s t s 

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