UBC Theses and Dissertations

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UBC Theses and Dissertations

Aeroelastic instability of a structural angle section Slater, Jonathan Ernest 1969

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AEROELASTIC INSTABILITY OF A STRUCTURAL ANGLE SECTION by JONATHAN ERNEST SLATER B . A . S c , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA M a r c h , 1969 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 C olumbia, I agre e 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 Study. 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 o r p u b l i c a t i o n of 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 o f M.EcUAui':*'- s~h/c:/^^B.lZW<s The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada ABSTRACT A n g l e s e c t i o n members, u s e d i n open e n g i n e e r i n g s t r u c -t u r e s , have b e e n known t o e x p e r i e n c e l a r g e a m p l i t u d e o s c i l l a t i o n s when e x p o s e d t o n o r m a l a t m o s p h e r i c w i n d s , and i n a few i n s t a n c e s f a i l u r e has been r e p o r t e d . The b l u f f g e o m e t r y t o g e t h e r w i t h low n a t u r a l f r e q u e n c y make t h e s e members s u s c e p t i b l e t o a e r o e l a s t i c v i b r a t i o n s o f a v o r t e x r e s o n a n t o r g a l l o p i n g n a t u r e . The t h e s i s aims a t s t u d y i n g t h e n a t u r e o f t h e a e r o d y n a m i c f o r c e s and t h e r e s u l t i n g i n s t a b i l i t i e s f o r t h e s a f e d e s i g n o f t h e s t r u c t u r e s . I t p r e s e n t s i n f o r m a t i o n on t h e a e r o d y n a m i c s and d y n a m i c s o f t h e a n g l e s e c t i o n d u r i n g s t a t i o n a r y , p l u n g i n g , t o r s i o n a l and combined p l u n g i n g - t o r s i o n a l c o n d i t i o n s . From t h e measurements on s t a t i o n a r y a n g l e m o d e l s , i t i s p o s s i b l e t o p r e d i c t t h e c r i t i c a l v o r t e x r e s o n a n t w i n d s p e e d s f o r v a r i o u s a n g l e s o f a t t a c k . The l a r g e v a r i a t i o n s o f t h e u n s t e a d y a e r o d y n a m i c c o e f f i c i e n t s i n d i c a t e t h e dependence o f t h e r e s o n a n t i n s t a b i l i t y on model o r i e n t a t i o n . I n c o r p o r a t i n g t h e s t a t i o n a r y a e r o d y n a m i c l o a d i n g s , t h e q u a s i - s t e a d y a n a l y s i s i s a b l e t o p r e -d i c t t h e g a l l o p i n g i n s t a b i l i t y and r e s u l t i n g a m p l i t u d e and b u i l d -up t i m e r e s p o n s e . The a b s e n c e o f t o r s i o n a l g a l l o p i n g d u r i n g t h e e x p e r i m e n t i s s u b s t a n t i a t e d by t h e t h e o r y w h i c h shows t h e i n -s t a b i l i t y t o o c c u r o n l y a t h i g h w i n d s p e e d s o r f o r s y s t e m s w i t h v e r y low damping. The d y n a m i c a l s t u d y d e m o n s t r a t e s t h a t s t r u c t u r a l a n g l e s e c t i o n s a r e s u s c e p t i b l e , i n g e n e r a l , t o combined p l u n g i n g and t o r s i o n a l v i b r a t i o n s . The n a t u r e o f t h e i n s t a b i l i t y depends on s u c h s y s t e m p a r a m e t e r s as damping, n a t u r a l f r e q u e n c y , a n g l e o f a t t a c k , s e c t i o n s i z e , e t c . However, due to the e x i s t e n c e of two d i s t i n c t f a m i l i e s of v i r t u a l hinge p o i n t s , i t i s p o s s i b l e to r e p r e s e n t the motion as predominantly p l u n g i n g or t o r s i o n . Furthermore, the frequency of the coupled motion as w e l l as the type and range of the i n s t a b i l i t y are found to be s i m i l a r to those i n the s i n g l e degree of freedom. T h i s makes i t p o s s i b l e to o b t a i n p e r t i n e n t i n f o r m a t i o n by s t u d y i n g , both e x p e r i m e n t a l l y and t h e o r e t i c a l l y , the p l u n g i n g and t o r s i o n a l degrees of freedom, s e p a r a t e l y . During p l u n g i n g resonance, the angle s e c t i o n experiences a v o r t e x capture phenomenon where the shedding frequency i s con-t r o l l e d by the c y l i n d e r motion over a f i n i t e wind speed range. On the other hand, the t o r s i o n a l v i b r a t i o n shows a vortex con-t r o l c o n d i t i o n over a l a r g e v e l o c i t y range where the v o r t e x shedding governs the frequency of o s c i l l a t i o n and f o l l o w s the s t a t i o n a r y model S t r o u h a l curve. Compared to the s t a t i o n a r y and t o r s i o n a l r e s u l t s , the f l u c t u a t i n g p r e s s u r e s on the angle s u r f a c e d u r i n g p l u n g i n g resonance are s u b s t a n t i a l l y l a r g e r i n magnitude with l e s s amplitude modulation and phase v a r i a t i o n . Consequently, the unsteady aerodynamic c o e f f i c i e n t s i n c r e a s e with t h i s i n s t a -b i l i t y . During resonance i n e i t h e r degree of freedom, the vortex v e l o c i t y and l o n g i t u d i n a l s p a c i n g remain e s s e n t i a l l y u n a l t e r e d , however, the wake width experiences s u b s t a n t i a l i n c r e a s e with p l u n g i n g motion. I t appears t h a t the t o r s i o n a l resonance has v i r t u a l l y no e f f e c t on the v o r t e x shedding or wake c h a r a c t e r i s t i c s . TABLE OF CONTENTS Chapter Page 1 I n t r o d u c t i o n 1 1.1 P r e l i m i n a r y Remarks . . . 1 1.2 L i t e r a t u r e Survey 4 1.3 Purpose and Scope of the I n v e s t i g a t i o n . . . 8 2 Aerodynamics of a S t a t i o n a r y Angle S e c t i o n . . . . 12 2.1 P r e l i m i n a r y Remarks 12 2.2 Models, Apparatus, Instrumentation and C a l i b r a t i o n 12 2.2.1 Angle Models 12 2.2.2 F l u c t u a t i n g P ressure Transducer and C a l i b r a t i o n 15 2.2.3 Wake Probe and T r a v e r s i n g Gear . . . . 16 2.3 T e s t Procedures 19 2.3.1 Balance Measurements 19 2.3.2 Vortex Shedding Frequency 19 2.3.3 Mean S t a t i c P ressure on Model Surface 19 2.3.4 F l u c t u a t i n g S t a t i c P ressure on Model Surface 22 2.3.5 Wake Survey 23 2.4 Experimental R e s u l t s and D i s c u s s i o n . . . . . 25 2.4.1 Steady L i f t , Drag and P i t c h i n g Moment D i s t r i b u t i o n s 2 5 2.4.2 Vortex Shedding Frequency and S t r o u h a l Number 2 8 2.4.3 Mean S t a t i c P r e s s u r e D i s t r i b u t i o n s . . 34 2.4.4 F l u c t u a t i n g S t a t i c P ressure D i s t r i b u t i o n s 38 V C h a p t e r Page 2.4.5 F l u c t u a t i n g L i f t , D r a g and Moment C o e f f i c i e n t s 47 2.4.6 Wake Geometry 50 2.5 C o n c l u d i n g Remarks 6 6 3 Dynamics o f an A n g l e S e c t i o n 70 3.1 P r e l i m i n a r y Remarks 70 3.2 E x p e r i m e n t a l A r r a n g e m e n t 72 3.2.1 M o d e l M o u n t i n g S y s t e m 72 3.2.2 I n s t r u m e n t a t i o n and T e s t P r o c e d u r e s 75 3.3 R e s p o n s e o f an A n g l e S e c t i o n w i t h Combined P l u n g i n g and T o r s i o n a l D e g r e e s o f f r e e d o m . . 80 3.4 T h e o r e t i c a l D e v e l o p m e n t 8 8 3.4.1 V o r t e x R esonance 89 3.4.2 G a l l o p i n g I n s t a b i l i t y 90 3.4.2.1 S i n g u l a r i t i e s and S t a b i l i t y i n t h e S m a l l 91 3.4.2.2 L i m i t C y c l e s and B u i l d - u p Time 94 3.5 R e s u l t s and D i s c u s s i o n 95 3.5.1 V o r t e x Resonance 95 3.5.1.1 M o d e l A m p l i t u d e - V e l o c i t y Measurements 9 5 3.5.1.2 S u r f a c e P r e s s u r e and Wake C h a r a c t e r i s t i c s w i t h O s c i l -a t i n g A n g l e M o d e l 105 3.5.1.3 R e s o n a n t T h e o r y P r e d i c t i o n s 117 3.5.2 G a l l o p i n g M o t i o n 118 3.5.2.1 T h e o r e t i c a l P r e d i c t i o n s f o r P l u n g i n g D e g ree o f Freedom 118 v i Chapter Page 3.5.2.2 Pl u n g i n g Amplitude and Bu i l d - u p Time Measurements . 122 3.5.2.3 T h e o r e t i c a l P r e d i c t i o n s f o r T o r s i o n a l Degree of Freedom 127 3.6 C o n c l u d i n g Remarks 132 4 Recommendations f o r Future Work 13 8 B i b l i o g r a p h y 140 Appendices I Geometric P r o p e r t i e s of Angle S e c t i o n Members . . 150 I I Wind Tunnel W a l l C o r r e c t i o n s 15 3 I I I E l e c t r o n i c Instruments 166 IV Theory f o r P l u n g i n g or T o r s i o n a l Degree of Freedom 16 8 LIST OF TABLES Table Page 2-1 Wake Geometry Parameters f o r V a r i o u s Bodies 64 1-1 Geometric Features of Angle S e c t i o n s 152 LIST OP FIGURES F i g u r e Page 2-1 P r e s s u r e tap angle model and numbering of p r e s s u r e h o l e s and contour s i d e s 14 2-2 C a l i b r a t i o n curves f o r B a r o c e l t r a n s d u c e r arrangement 17 2-3 Dimensions and c a l i b r a t i o n d a t a of d i s c probe . . 18 2-4 I n s t r u m e n t a t i o n l a y o u t f o r v o r t e x shedding f r e -quency and f l u c t u a t i n g p r e s s u r e measurements . . . 20 2-5 T y p i c a l f l u c t u a t i n g p r e s s u r e s i g n a l s from (a) wake probe (b) model s u r f a c e 21 2-6 Schematic of i n s t r u m e n t a t i o n f o r wake survey measurements . 24 2-7 D i s t r i b u t i o n of steady l i f t , drag and p i t c h i n g moment c o e f f i c i e n t s f o r balance model B 2 7 2-8 Comparison of aerodynamic c o e f f i c i e n t f o r v a r i o u s s t r u c t u r a l angle s e c t i o n s 29 2-9 V a r i a t i o n o f v o r t e x shedding frequency w i t h wind v e l o c i t y f o r p r e s s u r e tap angle model 30 2-10 V a r i a t i o n of S t r o u h a l number and v o r t e x resonant wind speed w i t h angle of a t t a c k 30 2-11 S t r o u h a l number d i s t r i b u t i o n s f o r (a) d i f f e r e n t s i z e angle models (b) v a r i o u s s t r u c t u r a l angle s e c t i o n s 32 2-12 Midspan d i s t r i b u t i o n s of mean s t a t i c p r e s s u r e c o e f f i c i e n t i -45° £ a 1 35° 35 i i 40° <_ a <_ 135° 36 2-13 Comparison of p r e s s u r e i n t e g r a t e d and balance measured steady aerodynamic c o e f f i c i e n t s 37 2-14 T y p i c a l f l u c t u a t i n g p r e s s u r e s i g n a l s from v a r i o u s model taps i n d i c a t i n g (a) random ampli-tude modulation (b) phase v a r i a t i o n 39 i x F i g u r e Page 2-15 Midspan d i s t r i b u t i o n s of f l u c t u a t i n g s t a t i c p r e s s u r e c o e f f i c i e n t and amplitude modu-l a t i o n r a t i o i -45° 1 a <_ 0° 40 i i 15° <_ a <_ 75° . . . . 41 i i i 90° <_ a <_ 135° 42 2-16 Phase v a r i a t i o n of midspan f l u c t u a t i n g p r e s s u r e i -45° < a < 0° 44 i i 15° <_ a <_ 60° . . . . . . . . . . . . . . 45 i i i 75° <_ a <_ 135° . . . . . . . . . . . . . 46 2-17 Spanwise v a r i a t i o n of f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t and phase . 48 2-18 Comparison of f l u c t u a t i n g and steady aerodynamic c o e f f i c i e n t s • • 49 2-19 L a t e r a l v a r i a t i o n of f l u c t u a t i n g p r e s s u r e amplitude i n the wake of (a) 3 i n . angle model (b) 1 i n . angle model 52 2-20 V a r i a t i o n of peak f l u c t u a t i n g p r e s s u r e w i t h downstream c o o r d i n a t e 53 2-21 L a t e r a l p o s i t i o n o f v o r t e x rows behind 1 i n . and 3 i n . angle models i -45° ± a <_ 30°. 54 i i 45° <_ a <_ 135°. 55 2-22 V a r i a t i o n of l a t e r a l v o r t e x s p a c i n g f o r 1 i n . and 3 i n . angle models 57 2-23 L o n g i t u d i n a l v a r i a t i o n of phase angle i n wake of s t a t i o n a r y 3 i n . angle model 59 2-24 Streamwise v a r i a t i o n o f (a) L o n g i t u d i n a l v o r t e x s p a c i n g (b) v o r t e x v e l o c i t y . . 6 0 2-25 L o n g i t u d i n a l d i s t r i b u t i o n s of wake geometry r a t i o f o r 1 i n . and 3 i n . angle models 61 2-26 D i s t r i b u t i o n s o f the 'near i n f i n i t y 1 values o f the wake survey parameters f o r 1 i n . and 3 i n . angle models 63 2-27 Comparison of S t r o u h a l number,wake geometry and drag c o e f f i c i e n t f o r angel s e c t i o n 65 X F i g u r e Page 3-1 D e t a i l s o f model s u p p o r t s y s t e m w i t h p l u n g i n g and t o r s i o n a l d e g r e e s o f f r e e d o m (a) p l u n g -i n g a r r a n g e m e n t (b) t o r s i o n a l a s s e m b l y 73 3-2 I n s t r u m e n t a t i o n l a y o u t f o r p l u n g i n g and t o r -s i o n a l d i s p l a c e m e n t measurements 76 3-3 D i s p l a c e m e n t measurements f o r a n g l e model a t a Q = -45° w i t h combined p l u n g i n g and t o r s i o n a l d e g r e e s o f f r e e d o m i p l u n g i n g g a l l o p i n g i n i t i a t e d b elow t o r s i o n a l r e s o n a n c e 80 i i p l u n g i n g g a l l o p i n g i n i t i a t e d above t o r s i o n a l r e s o n a n c e 81 3-4 R e s p o n s e o f a n g l e model a t a 0 = -45° w i t h p l u n g i n g d e g r e e o f f r e e d o m o n l y 84 3-5 R e s p o n s e c u r v e f o r a n g l e model a t a Q = -45° w i t h t o r s i o n a l d e g r e e o f f r e e d o m o n l y and r o t a t i o n a l a x i s a t s h e a r c e n t r e 85 3-6 T y p i c a l d i s p l a c e m e n t s i g n a l s f o r a n g l e model e x p e r i e n c i n g v o r t e x e x c i t e d p l u n g i n g o r t o r s i o n a l m o t i o n 95 3-7 • T y p i c a l t o r s i o n a l d i s p l a c e m e n t s i g n a l s f o r a n g l e model e x p e r i e n c i n g v o r t e x e x c i t e d m o t i o n a t v a r i o u s w i n d s p e e d s n e a r r e s o n a n c e 9 7 3-8 S e p a r a t i o n o f v o r t e x r e s o n a n c e and g a l l o p i n g t y p e o f o s c i l l a t i o n by c h o i c e o f damping o r mass p a r a m e t e r f o r a 0 = -45° 98 3-9 P l u n g i n g r e s o n a n t c u r v e s f o r 3 i n . angle' model a t v a r i o u s o r i e n t a t i o n s 9 9 3-10 T o r s i o n a l r e s o n a n t c u r v e s f o r 3 i n . a n g l e model a t a Q = -45° w i t h a x i s o f r o t a t i o n a t s h e a r c e n t r e 100 3-11 T o r s i o n a l r e s o n a n t c u r v e s f o r 3 i n . a n g l e model a t a Q = -45° and 135° w i t h a x i s o f r o t a t i o n a t c e n t r e o f g r a v i t y 101 3-12 S t a b i l i t y d i a g r a m f o r 3 i n . a n g l e model e x p e r -i e n c i n g v o r t e x e x c i t e d p l u n g i n g m o t i o n a t a G = -45° 103 3-13 S t a b i l i t y d i a g r a m f o r 3 i n . a n g l e model e x p e r -i e n c i n g v o r t e x e x c i t e d t o r s i o n a l m o t i o n a t a 0 = -45° w i t h a x i s o f r o t a t i o n a t s h e a r c e n t r e 104 xi F i g u r e Page 3-14 V a r i a t i o n o f c y l i n d e r and v o r t e x shedding f r e -q u e n c i e s , phase, and displacement amplitude near v o r t e x resonance ( a 0 = -45°) . . . . . . . . 107 3-15 V a r i a t i o n o f c y l i n d e r and v o r t e x shedding c h a r a c t e r i s t i c s w i t h wind speed f o r a 0 = -45' i t o r s i o n a l amplitude and frequency r e s u l t s . . . . . . 108 i i frequency, phase and mean t o r s i o n a l amplitude near resonance . . . . . . . . 109 3-16 Midspan and spanwise d i s t r i b u t i o n s o f f l u c t u -a t i n g p r e s s u r e c o e f f i c i e n t , amplitude modula-t i o n r a t i o and phase d u r i n g s t a t i c and dynamic c o n d i t i o n s o f the model . . . . . . . . . . . . . I l l 3-17 Comparison of f l u c t u a t i n g aerodynamic c o e f f i c -i e n t s f o r s t a t i o n a r y and v o r t e x e x c i t e d con-d i t i o n s o f the angle model a t a 0 = -45° . . . . . 112 3-18 V a r i a t i o n o f amplitude and phase o f f l u c t u -a t i n g p r e s s u r e i n wake of angle model ex-p e r i e n c i n g v o r t e x e x c i t e d motion a t o 0 =» -45° (a) p l u n g i n g (b) t o r s i o n . . . . . . . . . . . . 114 3-19 L o n g i t u d i n a l v a r i a t i o n o f wake survey p a r a -. meters f o r p l u n g i n g and t o r s i o n a l c o n d i t i o n s Of model a t a Q = -45° . . . . . . . . . . . . . 115 3-20 Comparison of 'near i n f i n i t y ' v a l u e s o f the wake survey parameters f o r s t a t i o n a r y and v o r t e x e x c i t e d c o n d i t i o n s o f the angle model a t a Q = -45° 116 3-21 P o l y n o m i a l curve f i t o f t y p i c a l l a t e r a l f o r c e c o e f f i c i e n t d a t a 120 3-22 V a r i a t i o n o f g a l l o p i n g p l u n g i n g amplitude w i t h wind v e l o c i t y and model a t t i t u d e as p r e d i c t e d by the q u a s i - s t e a d y theory . . . . . . . . . . . 121 3-23 G a l l o p i n g a m p l i t u d e - v e l o c i t y r e s u l t s f o r angle model a t a 0 = -45° and t h e i r comparison w i t h theory . . 123 3-24 G a l l o p i n g a m p l i t u d e - v e l o c i t y r e s u l t s f o r angle model a t a Q = 90° and t h e i r comparison w i t h •theory 124 x i i F i g u r e Page 3-25 Comparison of experimental and t h e o r e t i c a l amplitude b u i l d - u p time f o r angle model at a 0 = 90° X26 3-26 Moment c o e f f i c i e n t d i s t r i b u t i o n f o r s t a t i o n a r y model a t a Q = 45° 128 3-27 Contour p l o t of t o r s i o n a l moment c o e f f i c i e n t as a f u n c t i o n of e and G f o r a Q = -45° 129 3-2 8 Po l y n o m i a l curve f i t of t o r s i o n a l moment c o e f f i c i e n t data f o r angle model at a Q = -45° . . 130 3-29 V a r i a t i o n o f g a l l o p i n g amplitude and reduced frequency w i t h wind v e l o c i t y f o r angle s e c t i o n a t a c = -45° as p r e d i c t e d by the q u a s i -steady theory 131 1-1 C r o s s - s e c t i o n s of angle member (a) ' i d e a l i z e d ' angle s e c t i o n (b) t y p i c a l s t r u c t u r a l angle . . . 151 I I - l Percentage c o r r e c t i o n a p p l i c a b l e t o 3 i n . angle s e c t i o n t e s t e d i n departmental wind t u n n e l 156 II-2 V a r i a t i o n of S t r o u h a l number wit h blockage f o r v a r i o u s o r i e n t a t i o n s of the angle members . . 164 ACKNOWLEDGEMENT The author wishes t o express h i s s i n c e r e thanks and a p p r e c i a t i o n t o Dr. V . J . Modi f o r the guidance and a s s i s t a n c e g i v e n throughout the r e s e a r c h programme and d u r i n g the pre p a r a -t i o n of the t h e s i s . H is h e l p and encouragement have been i n v a l u a b l e . The co n s t a n t i n t e r e s t and encouragement shown by Dr. G.V. Par k i n s o n d u r i n g the exp e r i m e n t a l work, and h e l p f u l comments on the t h e o r e t i c a l aspects o f t h i s study r e q u i r e s p e c i a l a p p r e c i a t i o n . Thanks are a l s o due t o the Department of Mechanical E n g i n e e r i n g f o r the use of t h e i r f a c i l i t i e s , and t o the t e c h -n i c a l s t a f f members f o r t h e i r v a l u a b l e h e l p d u r i n g the c o n s t r u c -t i o n o f the wind t u n n e l models and support systems, and the assemblying o f e l e c t r o n i c i n s t r u m e n t a t i o n . Reduction o f some of the experimental data and computer output, and t y p i n g o f the o r i g i n a l t h e s i s manuscript were per-formed by my w i f e . Her hours o f a s s i s t a n c e are g r e a t f u l l y a p p r e c i a t e d . F i n a n c i a l a s s i s t a n c e was r e c e i v e d from the N a t i o n a l Research C o u n c i l of Canada i n the form of d i r e c t postgraduate s c h o l a r s h i p and o p e r a t i n g grant (No. A-2181). LIST OF SYMBOLS Steady l i f t , drag and pitc h i n g moment c o e f f i c i e n t s , respectively L a t e r a l force c o e f f i c i e n t Pitching moment c o e f f i c i e n t Sectional steady l i f t , drag and pitching moment c o e f f i c i e n t s , respectively Average peak, f l u c t u a t i n g s e c t i o n a l l i f t , drag and pitching moment c o e f f i c i e n t s , respectively 1 2 Mean s t a t i c pressure c o e f f i c i e n t , (p-p 0)/ j pV Average peak f l u c t u a t i n g pressure c o e f f i c i e n t , P 7 j P V 2 Drag, j P V 2 h l C D 1 2 Lat e r a l force for plunging system, pV hlC y Mass moment of i n e r t i a about e l a s t i c axis Mass moment of i n e r t i a about i n e r t i a l axis Reduced frequency parameter for t o r s i o n a l gallop-ing theory L i f t , j P V 2 h l C L 1 2 2 Pitching moment, j pV h l C ^ Pitching moment for t o r s i o n a l system, 1 2 2 l " V h l C M e Degree of polynomial curve f i t Reynolds number, Vh/v Dimensionless representative radius parameter, r r / h Strouhal numbers, f^e/V and f^h/V, respectively Dimensionless f l u i d v e l o c i t y , V/u^h XV Dimensionless c r i t i c a l v e l o c i t y , 2g/na^ Dimensionless resonant wind v e l o c i t y , 1 / ( 2T T S ^ ) F l u i d v e l o c i t y f a r upstream of angle model F l u i d v e l o c i t y r e l a t i v e t o o s c i l l a t i n g c y l i n d e r Resonant wind v e l o c i t y , w hu •* n res Streamwise v o r t e x v e l o c i t y Dimensionless l a t e r a l displacement of o s c i l l a t i n g model, y/h Dimensionless amplitude of l a t e r a l displacement, y / h Dimensionless i n i t i a l amplitude of l a t e r a l displacement L o n g i t u d i n a l s p a c i n g between v o r t i c e s (i=0 ,1,2 ,3 , • • • • • ,N) c o e f f i c i e n t s o f polynomial curve f i t L a t e r a l s p a c i n g between v o r t i c e s cos X P r o j e c t e d width of angle model Frequency of c y l i n d e r o s c i l l a t i o n s N a t u r a l frequency i n p l u n g i n g , con /2TT y N a t u r a l frequency i n t o r s i o n , GO /2TT e Frequency of v o r t e x shedding Maximum width of angle model P l u n g i n g and t o r s i o n a l s t i f f n e s s e s of the system, r e s p e c t i v e l y Length of angle model Mass of o s c i l l a t i n g system 2 Dimensionless mass parameter, ph 1/(2m) Dimensionless mass moment of i n e r t i a parameter, p h 4 l / ( 2 l ) Mean s t a t i c p r e s s u r e S t a t i c p r e s s u r e f a r upstream o f angle model F l u c t u a t i n g s t a t i c p r e s s u r e Average peak of f l u c t u a t i n g s t a t i c p r e s s u r e R a d i a l d i s t a n c e from e l a s t i c a x i s t o s u r f a c e element P l u n g i n g and t o r s i o n a l v i s c o u s damping c o e f f i -c i e n t s , r e s p e c t i v e l y s i n A Real time V e l o c i t y of v o r t e x r e l a t i v e t o surrounding f l u i d S u r f a c e element v e l o c i t y f o r t o r s i o n a l o s c i l l a t i o n Downstream c o o r d i n a t e from angle s e c t i o n i n e r t i a l a x i s Tranverse c o o r d i n a t e from angle s e c t i o n i n e r t i a l a x i s o r instantaneous l a t e r a l displacement o f o s c i l l a t i n g model, normal t o flow d i r e c t i o n Amplitude of l a t e r a l displacement Coordinate a l o n g e l a s t i c a x i s from model midspan T o r s i o n a l displacement o f o s c i l l a t i n g model, =8 Amplitude of t o r s i o n a l displacement I n i t i a l amplitude o f t o r s i o n a l displacement Phase l a g between model displacement and f l u c t u -a t i n g f o r c e o r moment Vortex frequency r a t i o , ^ ^/^ n System frequency r a t i o , w n^/<») n Instantaneous angle of a t t a c k of o s c i l l a t i n g system or a t t i t u d e o f s t a t i o n a r y model Mean angle o f a t t a c k of o s c i l l a t i n g system Dimensionless damping parameter f o r p l u n g i n g system, r /2mu)n y y Dimensionless damping parameter f o r t o r s i o n a l system, r /2Iu X V I I Angle between r e l a t i v e and approaching f l u i d v e l o c i t i e s f o r o s c i l l a t i n g system Increment i n angle of a t t a c k of o s c i l l a t i n g t o r -s i o n a l system, a-a Q Coordinate along c o n t o u r l i n e of C^ , p a s s i n g through o r i g i n 6 Angle between s u r f a c e element v e l o c i t y and approach-i n g f l u i d v e l o c i t y f o r o s c i l l a t i n g t o r s i o n a l system Value of n r a t a = 0 o T o r s i o n a l displacement of o s c i l l a t i n g system Angle between C and 8/U a x i s Kinematic v i s c o s i t y of the f l u i d C o ordinate p e r p e n d i c u l a r t o C a x i s D e n s i t y of the f l u i d D i s t a n c e between i n e r t i a l and e l a s t i c axes~ Dimensionless time f o r o s c i l l a t i n g system, u n t Dimensionless time f o r v o r t e x shedding, u v t Reduced d i m e n s i o n l e s s time f o r o s c i l l a t i n g system, S T Phase angle Average phase angle between f l u c t u a t i n g p r e ssure s i g n a l s C i r c u l a r frequency of c y l i n d e r o s c i l l a t i o n s N a t u r a l c i r c u l a r frequency of o s c i l l a t i n g system 1/2 N a t u r a l c i r c u l a r frequency i n p l u n g i n g , (k^/m) N a t u r a l c i r c u l a r frequency i n t o r s i o n , ( k . / I ) 1 / / 2 C i r c u l a r frequency of v o r t e x shedding x v i i i . S u b s c r i p t s F Value of the parameter under f r e e stream c o n d i t i o n s e Parameter based on p r o j e c t e d width e m Pres s u r e on model s u r f a c e max maximum r Average v a l u e o f the parameter f o r t o r s i o n a l o s c i l l a t i o n s o f angle s e c t i o n s Value o f the parameter f o r s t a t i o n a r y model w Value o f the parameter i n wake y Value o f the parameter i n p l u n g i n g 6 Value o f the parameter i n t o r s i o n 0 0 Constant v a l u e o f the parameter a t a l a r g e d i s t a n c e downstream S u p e r s c r i p t s (•) D e r i v a t i v e w i t h r e s p e c t t o dime n s i o n l e s s time T (o ) D e r i v a t i v e w i t h r e s p e c t t o r e a l time t * Reduced v a l u e of the displacement o r v e l o c i t y , parameter/U 0 1 INTRODUCTION 1.1 P r e l i m i n a r y Remarks The o s c i l l a t i o n s o f aero d y n a m i c a l l y b l u f f b o d i e s , when exposed t o a f l u i d stream, have been a s u b j e c t o f c o n s i d e r a b l e study. To en g i n e e r s , the a e r o e l a s t i c v i b r a t i o n s o f smoke s t a c k s , t r a n s m i s s i o n l i n e s , suspension b r i d g e s , b u i l d i n g s , e t c . , are of i n t e r e s t . In g e n e r a l , the nature of the wind l o a d i n g , v o r t e x shedding frequency and wake geometry form three important p a r a -meters i n an a e r o e l a s t i c i n s t a b i l i t y study. The d e t e r m i n a t i o n of the c o r r e s p o n d i n g i n f o r m a t i o n a s s o c i a t e d w i t h a s t r u c t u r a l angle, d u r i n g s t a t i c and dynamic c o n d i t i o n s , forms the s u b j e c t o f t h i s t h e s i s . S t r u c t u r a l angles are f r e q u e n t l y used i n the c o n s t r u c t i o n o f open c i v i l e n g i n e e r i n g s t r u c t u r e s , such as h i g h v o l t a g e t r a n s -m i s s i o n towers, antenna masts, and b r i d g e s . I n c o r p o r a t e d as secondary members, these s t r u c t u r a l s e c t i o n s may be long and f l e x -i b l e . Furthermore, r e c e n t advances i n the m e t a l l u r g i c a l s c i e n c e through e x p e r i m e n t a l r e s e a r c h and i n e n g i n e e r i n g d e s i g n with the a i d o f computers have encouraged the use o f l i g h t e r and r e l a t i v e l y more f l e x i b l e i n d i v i d u a l components. B l u f f geometry to g e t h e r w i t h low n a t u r a l frequency make these members p a r t i c -u l a r l y s u s c e p t i b l e t o aero d y n a m i c a l l y induced v i b r a t i o n s . Some long s l e n d e r angle members i n t r a n s m i s s i o n towers have been known to experience l a r g e amplitude o s c i l l a t i o n s when exposed to normal atmospheric winds, and i n a few i n s t a n c e s f a i l u r e has been r e p o r t e d . I t i s , t h e r e f o r e , d e s i r a b l e to understand the nature of 2 the unsteady f o r c e s and the r e s u l t i n g i n s t a b i l i t i e s f o r the s a f e d e s i g n of these s t r u c t u r e s . Depending on the nature of the aerodynamic e x c i t a t i o n , a f l e x i b l e s t r u c t u r a l member may e x h i b i t v a r i o u s forms of v i b r a t i o n , e.g., v o r t e x resonance, geometric-aerodynamic i n s t a b i l i t y c a l l e d g a l l o p i n g , c l a s s i c a l o r s t a l l f l u t t e r , o r random motion e x c i t e d by the t u r b u l e n c e . However, i n g e n e r a l , the aerodynamically induced o s c i l l a t i o n s o f b l u f f c y l i n d e r s are of the v o r t e x resonant or g a l l o p i n g type w i t h the v i b r a t i o n s o c c u r r i n g , predominately, i n one of two degrees of freedom; f l e x u r e t r a n s v e r s e t o the flow d i r e c t i o n , o r t o r s i o n about the l o n g i t u d i n a l e l a s t i c a x i s of the beam. Furthermore, s i n c e the e l a s t i c and i n e r t i a l axes of angle members are not c o i n c i d e n t , coupled t o r s i o n a l - f l e x u r a l v i b r a t i o n s may o c c u r . N e v e r t h e l e s s , as determined by Kosko^ and observed by 2 Wardlaw, the coupled t o r s i o n a l - f l e x u r a l o s c i l l a t i o n s can be c o n s i d e r e d as r o t a t i o n a l motion about v i r t u a l hinge p o i n t s which produces two d i s t i n c t l y d i f f e r e n t f a m i l i e s of v i b r a t i o n . At the lower modes, one f a m i l y appears predominantly f l e x u r a l and the o t h e r t o r s i o n a l . The n a t u r a l t u r b u l e n c e of atmospheric winds t o which the s t r u c t u r a l members are exposed i s not, i n g e n e r a l , comparable to t h a t i n the steady a i r stream o f c o n v e n t i o n a l wind t u n n e l s . The atmospheric winds have f l u c t u a t i n g v e l o c i t y components which may be a l a r g e f r a c t i o n o f the mean wind speed. However, s i n c e the peak of the power spectrum of the v e l o c i t y f l u c t u a t i o n s g e n e r a l l y occurs a t a frequency, which i s much lower than the n a t u r a l f r e -3 q uencies of t y p i c a l angle s e c t i o n beams, the atmospheric t u r -bulence cannot be expected to cause s e r i o u s resonant v i b r a t i o n 3 of i n d i v i d u a l s t r u c t u r a l members. On the other hand, b u f f e t t i n g by h i g h l y t u r b u l e n t wakes from o t h e r b l u f f s t r u c t u r e s i s more l i k e l y t o cause e x c i t a t i o n of members l y i n g downstream s i n c e the t u r b u l e n t energy spectrum can have peak(s) near the n a t u r a l f r e -q u e n c i e s . N e v e r t h e l e s s , a v a i l a b l e l i t e r a t u r e r e p o r t s t h a t t u r b u -lence reduces the spanwise c o r r e l a t i o n of the v o r t e x shedding phenomenon and thereby a i d s i n s u p p r e s s i n g the resonant v i b r a t i o n s . For the g a l l o p i n g i n s t a b i l i t y , i t should be mentioned t h a t the i n f l u e n c e o f t u r b u l e n c e i s not y e t f u l l y understood and may be s i g n i f i c a n t under some circumstances. The d i s t i n c t c h a r a c t e r o f vortex e x c i t e d and g a l l o p i n g o s c i l l a t i o n s should be emphasized. The former i s e s s e n t i a l l y a resonance phenomenon where the vortex formation frequency, and hence the frequency of the f o r c i n g f u n c t i o n , c o i n c i d e s w i t h the n a t u r a l frequency of the system under c o n s i d e r a t i o n . T h i s type of o s c i l l a t i o n i s referred t o as a f o r c e d v i b r a t i o n s i n c e the s u s t a i n -i n g a l t e r n a t i n g f o r c e e x i s t s independent o f the motion and p e r -s i s t s even when the motion i s stopped. Although any b l u f f member of a r b i t r a r y c r o s s - s e c t i o n , when s u i t a b l y mounted, would e x h i b i t v o r t e x e x c i t e d o s c i l l a t i o n , the a v a i l a b l e l i t e r a t u r e i s l a r g e l y c o n f i n e d t o such s t u d i e s on c i r c u l a r c y l i n d e r s because of the geometric s i m p l i c i t y as w e l l as the p r a c t i c a l importance of the s e c t i o n . The second form of i n s t a b i l i t y , r e f e r r e d to as g a l l o p i n g , r e p r e s e n t s an important type of s e l f - e x c i t e d v i b r a t i o n . The f l u i d f o r c e s which c r e a t e a c o n d i t i o n of i n s t a b i l i t y are generated by the f a c t t h a t the c r o s s - s e c t i o n of the body i s aerodynamically 4 u n s t a b l e t o s m a l l d i s t u r b a n c e s . These f o r c e s r e s u l t i n o s c i l l a -t i o n s which grow i n amplitude u n t i l the energy e x t r a c t e d from the f l u i d stream balances t h a t d i s s i p a t e d through v a r i o u s forms of damping. G a l l o p i n g o s c i l l a t i o n s are r e f e r r e d t o as " s e l f -e x c i t e d " because the f l u i d f o r c e s t h a t s u s t a i n the motion are c r e a t e d and c o n t r o l l e d by the motion i t s e l f , and i f the motion s t o p s , the unsteady f o r c e s d i s a p p e a r . T h i s i s i n c o n t r a s t t o v o r t e x resonance. The main f e a t u r e s o f g a l l o p i n g are t h a t the v i b r a t i o n can occur i n a s i n g l e degree o f freedom and the steady-s t a t e amplitude tends t o i n c r e a s e w i t h i n c r e a s i n g wind v e l o c i t y . O f t e n , as i s the case w i t h a s t r u c t u r a l angle beam, the i n s t a b i l i t y a t a g i v e n f l u i d stream v e l o c i t y and angle of a t t a c k may be the combined e f f e c t of both v o r t e x resonance and g a l l o p i n g . To p e r m i t the study of the i n d i v i d u a l forms o f e x c i t a t i o n , the j u d i c i o u s c h o i c e of e i t h e r damping, angle s e c t i o n s i z e o r n a t u r a l frequency i s r e q u i r e d t o separate the two phenomena. 1.2 L i t e r a t u r e Survey 4 S t r o u h a l was the f i r s t t o c o r r e l a t e the p e r i o d i c i t y of the v o r t e x shedding w i t h the diameter of the c i r c u l a r c y l i n d e r and v e l o c i t y of the f l u i d stream. T h i s was f o l l o w e d by numerous 5 experiments on wake geometry by Benard, the c l a s s i c a l study of 6 7 s t a b i l i t y by Von Karman, and wake a n a l y s i s by Heisenberg. Ever s i n c e , i n t e r e s t i n the v o r t e x shedding phenomenon has r e -s u l t e d i n many t h e o r e t i c a l and 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 s by g Roshko, Kovasnay, Rosenhead, E s k i n a z i and o t h e r s . M a r r i s has p r e s e n t e d an e x c e l l e n t review of t h i s l i t e r a t u r e . The mechanism of g a l l o p i n g e x c i t a t i o n of b l u f f c y l i n d e r s g was p r o b a b l y f i r s t d e s c r i b e d by Den Hartog t o g e t h e r w i t h the d e t e r m i n a t i o n of h i s p l u n g i n g s t a b i l i t y c r i t e r i o n . O r i g i n a l l y , however, i t was Lord R a y l e i g h ^ 0 who i n d i c a t e d the inadequacy of l i n e a r theory and proposed a n o n l i n e a r e q u a t i o n to e x p l a i n the " s u s t a i n e d " o s c i l l a t i o n s . Van der P o l ' s development of h i s c l a s s i c a l n o n l i n e a r e q u a t i o n i n 1920 l e d to a f l u r r y o f r e s e a r c h a c t i v i t y on t h i s s u b j e c t by Appleton, Greaves and o t h e r s . E x c e l l e n t reviews of these developments are summarized by van der 11 12 P o l and Le C o r b e i l l e r , w i t h many r e c e n t r e f e r e n c e s found i n 13 2 Minorsky. Wardlaw extended Den Hartog"s a n a l y s i s by d e v e l o p i n g a g e n e r a l i z e d s t a b i l i t y c r i t e r i o n f o r coupled t o r s i o n a l - f l e x u r a l motion. S i s t o , ^ S c r u t o n , ^ P a r k i n s o n , e t a l , ^ I i > " ^ 20 21 D i c k e r , Novak, e t c . , a p p l i e d the q u a s i - s t e a d y approach t o s o l v e the n o n l i n e a r v i b r a t i o n problems f o r bodies o f v a r i o u s geo-m e t r i c shapes w i t h p l u n g i n g and/or t o r s i o n a l degrees of freedom. The g a l l o p i n g o s c i l l a t i o n s of e x i s t i n g s t r u c t u r e s , mainly of t r a n s m i s s i o n conductor l i n e s , were observed and s t u d i e d by Scruton, 23 24 25 Richardson, e t a l , Cheers, Dryden and H i l l and o t h e r s . 26 Davenport suggested the p o s s i b i l i t y of g a l l o p i n g i n s t a b i l i t y of t a l l t h i n b u i l d i n g s which may be c o n s t r u c t e d i n the near f u t u r e . 27 A paper by P a r k i n s o n d i s c u s s e s the a e r o e l a s t i c behaviour of b l u f f c y l i n d e r s and p r o v i d e s a good survey of the l i t e r a t u r e . S c r u t o n , as w e l l as Davenport and a s s o c i a t e s have i n v e s i t -gated i n d e t a i l the wind l o a d i n g and dynamics of c e r t a i n complete s t r u c t u r e s such as s t a c k s , towers, masts and b u i l d i n g s . On the 2 8 o t h e r hand, Dale, e t a l , have c o n c e n t r a t e d t h e i r study to the 6 dynamic behaviour of hydrophone cables. Intensive investigations into the aerodynamic i n s t a b i l i t y of suspension bridges with s p e c i a l reference to the o r i g i n a l Tacoma Narrows Bridge have 29 30 been conducted by Farquharson, et a l , Kelley, etc. In addition to t h i s , the unsteady forces and wake geo-metry associated with two-dimensional b l u f f cylinders have been 31 32 in t e n s i v e l y studied. McGregor and Gerrard have conducted experimental investigations of the fl u c t u a t i n g pressures on 33 stationary c i r c u l a r cylinders. More recently, Ferguson made wake survey as well as f l u c t u a t i n g surface pressure measurements on the same section. The corresponding r e s u l t s for stationary square, rectangular, and e l l i p t i c a l cylinders were presented by 34 35 36 Modi and Heine, and Wiland. S i m i l a r l y , Grove, et a l , 37 38 39 Bishop and Hassan, Humphreys, and Fung measured the f l u c t u -ating forces on stationary c i r c u l a r cylinders over d i f f e r e n t 3 ranges of Reynolds number. In addition, Wardlaw and Davenport, 40 and Vickery conducted some f l u c t u a t i n g force measurements on d i f f e r e n t shapes i n laminar as well as i n turbulent flow. On the other hand, invest i g a t i o n of wake and surface conditions on o s c i l l a t i n g b l u f f cylinders i s less complete. The study of f l u c t u a t i n g l i f t and drag forces by Bishop and 41 Hassan showed that the vortex frequency i s controlled over a range of c i r c u l a r cylinder frequencies; while Ferguson 42 and Parkinson measured f l u c t u a t i n g pressures and wake geo-metry related to a c i r c u l a r c y l i n d e r experiencing vortex i n -43 duced o s c i l l a t i o n s . Molyneux developed techniques for measur-ing the aerodynamic forces on o s c i l l a t i n g a i r f o i l s . The three-dimensional structure of the wake and c o r r e l a t i o n along a c i r c -ular cylinder during s t a t i c or dynamic conditions have been i n v e s t i -44 45 46 47 gated by G e r r a r d , Toebes, Prendergast, Feng and o t h e r s . 40 V i c k e r y measured the c o r r e l a t i o n of l i f t a long the s u r f a c e of a s t a t i o n a r y square c y l i n d e r . Dynamic amplitude measurements of two-dimensional b l u f f c y l i n d e r s of v a r i o u s c r o s s s e c t i o n s have been i n v e s t i g a t e d by 4 8 49 50 Brooks. Both Smith and Santosham c o n c e n t r a t e d t h e i r s t u d i e s to a e r o e l a s t i c g a l l o p i n g of r e c t a n g u l a r c y l i n d e r s i n a p l u n g i n g 51 52 degree of freedom; w h i l e Chuan and O t s u k i p r e s e n t i n v e s t i -g a t i o n s on t o r s i o n a l o s c i l l a t i o n s o f a i r f o i l s and p r i s m a t i c b a r s , 53 r e s p e c t i v e l y . On the o t h e r hand, Toebes and E a g l e s o n , and 54 E a g l e s o n , e t a l , d e a l t w i t h the h y d r o e l a s t i c a l l y s e l f - e x c i t e d v i b r a t i o n s o f f l a t p l a t e s as r e l a t e d to the t r a i l i n g edge geometry. The i n f l u e n c e of wake-body i n t e r a c t i o n on a e r o e l a s t i c i n s t a b i l i t y are r e p o r t e d f o r bodies o f v a r i o u s c r o s s - s e c t i o n a l geometry. But i t s h o u l d be emphasized t h a t the bulk of the l i t e r a t u r e i s devoted t o the c i r c u l a r s e c t i o n . T h i s i s i n d i c a t e d by the f a c t t h a t i n v e s t i g a t i o n s i n t o a e r o d y n a m 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 o f s t r u c t u r a l angles were not i n i t i a t e d u n t i l 1962. 5 5 Thornton conducted experiments on the v i b r a t i o n of s e v e r a l s i n g l e and double angle s e c t i o n members i n steady flow and was a b l e to suppress the motion by the a d d i t i o n of f l a t p l a t e s p o i l e r s . 2 More r e c e n t l y , Wardlaw has r e p o r t e d the o s c i l l a t i o n s of a 3x3x3/16 i n . aluminum s t r u c t u r a l angle beam t o g e t h e r w i t h d i s -t r i b u t i o n s of s e c t i o n a l l i f t , drag and p i t c h i n g moment. From a s t r u c t u r a l c o n s i d e r a t i o n , the coupled f l e x u r a l - t o r s i o n a l v i b r a t i o n s i n s t i l l a i r of e q u a l - l e g g e d and unsymmetrical angle s e c t i o n beams were examined t h e o r e t i c a l l y and e x p e r i m e n t a l l y by Kosko. 1 8 1.3 Purpose and Scope of the I n v e s t i g a t i o n The problem of a e r o e l a s t i c i n s t a b i l i t y of two-dimensional b l u f f c y l i n d e r s has been a c t i v e l y s t u d i e d , both e x p e r i m e n t a l l y and t h e o r e t i c a l l y , i n t h i s department s i n c e 1958. A review of 56 57 the p r o g r e s s i s r e p o r t e d i n two survey papers. ' These system-a t i c i n v e s t i g a t i o n s have c o n t r i b u t e d p e r t i n e n t data to the gen-e r a l study of 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 . The i n v e s t i g a t i o n d e s c r i b e d here forms a p a r t of t h i s c o n t i n u i n g programme. I t p r e s e n t s i n f o r m a t i o n on the aerodynamics of the angle s e c t i o n , the wind speed and angle of a t t a c k ranges of i n s t a b i l i t y , and the e f f e c t s of the model dynamics on the impor-t a n t aerodynamic parameters. T h i s i s intended to p r o v i d e , even-t u a l l y , a source of i n f o r m a t i o n f o r the s a f e design of open s t r u c t u r e s composed of t h i s simple s e c t i o n . In p a r t i c u l a r , the i n v e s t i g a t i o n s examined the aero-dynamics and dynamics of equal-legged, angle s e c t i o n models i n a c o n v e n t i o n a l low t u r b u l e n c e , r e t u r n - t y p e wind t u n n e l w i t h a t e s t s e c t i o n of 36 i n . x 27 i n . x 8 2/3 f t . The experimental models had l x l i n . and 3x3 i n . c r o s s - s e c t i o n a l dimensions with uniform l e g t h i c k n e s s e s of 1/6 i n . and 1/2 i n . , r e s p e c t i v e l y . The e x t e r i o r s u r f a c e s were smooth wi t h sharp contour edges. Being r i g i d i n themselves, the models were mounted normal to the flow on s t a t i o n a r y or s p r i n g s u p p o r t i n g systems with s t r u c t u r a l and mean aerodynamic c o n d i t i o n s being e s s e n t i a l l y two-dimensional. A comparison of the geometric f e a t u r e s of the ' i d e a l i z e d ' angle models and commercially a v a i l a b l e angle s e c t i o n s i s presented i n Appendix I. The r e s e a r c h programme was d i v i d e d i n t o f o u r stages i n v e s -t i g a t i n g the angle s e c t i o n d u r i n g s t a t i o n a r y , p l u n g i n g , t o r s i o n , and combined p l u n g i n g - t o r s i o n c o n d i t i o n s . For the s t a t i o n a r y model t e s t s , the t h e s i s p r e s e n t s e x p e r i m e n t a l r e s u l t s on: (i) steady l i f t , drag and p i t c h i n g moment c o e f f i c i e n t s ; ( i i ) v a r i a t i o n o f S t r o u h a l number wit h Reynolds number; ( i i i ) mean and f l u c t u a t i n g s t a t i c p r e s s u r e d i s t r i b u t i o n s ; (iv) f l u c t u a t i n g l i f t , drag and p i t c h i n g moment c o e f f i c i e n t s ; (v) wake geometry as f u n c t i o n s of angle of a t t a c k . In most tests: the Reynolds 4 5 number was c o n f i n e d to the range 2x10 t o 10 . For comparison, s t a t i c f o r c e and v o r t e x shedding frequency measurements were con-ducted on commercially a v a i l a b l e aluminum and s t e e l angle members. The r e s u l t s of the remaining three stages are combined i n one chapter o f the t h e s i s t o c o r r e l a t e the important f e a t u r e s of the model dynamics and a s s o c i a t e d aerodynamics. The e x i s t i n g l a t e r a l s u p p o r t i n g equipment was s u i t a b l e f o r the study i n the p l u n g i n g degree of freedom, but a new s p r i n g mounting system and a u x i l i a r y measuring i n s t r u m e n t a t i o n were designed f o r the t o r s i o n a l i n v e s t i g a t i o n s . Both the v o r t e x induced and g a l l o p i n g i n s t a b i l -i t i e s were examined a t v a r i o u s angles of a t t a c k and damping l e v e l s f o r each of the three o s c i l l a t o r y modes of v i b r a t i o n . For the angle s e c t i o n e x p e r i e n c i n g v o r t e x e x c i t e d resonance, the e f f e c t s of the model motion on such parameters as: (i) v o r t e x shedding frequency and phase; ( i i ) f l u c t u a t i n g s t a t i c p r e s s u r e ; ( i i i ) wake geometry were determined. Peak v o r t e x resonant displacement amplitudes were p r e d i c t e d from resonant theory.by i n c o r p o r a t i n g the f l u c t u -a t i n g f o r c e d a t a . In a d d i t i o n , u s i n g the e x p e r i m e n t a l l y o b t a i n e d s t a t i c aerodynamic l o a d i n g s , the q u a s i - s t e a d y a n a l y s i s was a p p l i e d to p r o v i d e t h e o r e t i c a l p r e d i c t i o n s of the g a l l o p i n g model dynamics. Si n c e the i n f l u e n c e s of the wind t u n n e l w a l l s on the measured data are not w e l l e s t a b l i s h e d , the r e s u l t s p resented, u n l e s s otherwise s t a t e d , are u n c o r r e c t e d f o r these e f f e c t s . A number o f experiments were conducted to e s t a b l i s h trends of the w a l l i n t e r f e r e n c e , and t h i s data t o g e t h e r w i t h a summary of e x i s t i n g t h e o r i e s on w a l l confinement c o r r e c t i o n s are presented in-Appendix I I . A comment con c e r n i n g the a p p l i c a b i l i t y of t h i s i n v e s t i g a -t i o n t o a p r a c t i c a l s i t u a t i o n i s p e r t i n e n t here. The r e s u l t s p r e - : sented i n t h i s t h e s i s r e l a t e t o a r i g i d , two-dimensional element of a long angle s e c t i o n beam, w i t h the s t i f f n e s s lumped as s p r i n g s and damping a p p l i e d e x t e r n a l l y . However, f o r an a c t u a l beam, the p o t e n t i a l energy i s due t o the s t r a i n s and damping i s a s s o c i a t e d w i t h the i n t e r n a l f r i c t i o n of the m a t e r i a l . N e v e r t h e l e s s , 21 Novak has r e p o r t e d an i n v e s t i g a t i o n showing c o m p a t a b i l i t y between v i b r a t i o n s of r i g i d s e c t i o n s and continuous systems. A procedure f o r model s i m u l a t i o n o f p h y s i c a l s t r u c t u r e s has been pr e s e n t e d by Whitbread. For s e c t i o n a l models, c e r t a i n s i m i l a r i t y requirements have to be s a t i s f i e d . When geometric s i m i l i t u d e e x i s t s , the s t a t i c aerodynamic r e s u l t s are d i r e c t l y a p p l i c a b l e , even at o t h e r Reynolds numbers w i t h i n a c e r t a i n range, i f flow 11 s e p a r a t i o n i s f i x e d by edge c o n d i t i o n . However, f o r dynamical s t u d i e s , the requirements i n v o l v e correspondence of s t r u c t u r a l parameters, such as s t i f f n e s s , i n e r t i a and damping. They are re p r e s e n t e d i n nondimensional form as n and.0, r e s p e c t i v e l y . In these experiments, the s p r i n g s t i f f n e s s e s have been chosen t o p r o v i d e a frequency r a t i o u>Q/u) = 3 which i s t y p i c a l 1 2 ^ of angle s e c t i o n beams. ' The i n e r t i a c h a r a c t e r i s t i c s of the 1 i n . and 3 i n . angle models (Appendix I) make the former s u i t -a ble f o r p l u n g i n g s t u d i e s , w h i l e the l a t t e r f o r t o r s i o n a l i n v e s -t i g a t i o n s . For the 3 i n . models i n the p l u n g i n g case, the n^ val u e s are approximately 3 times h i g h e r than those o f the p r o t o t y p e . T h i s was s e l e c t e d f o r convenience, but does not l i m i t the a p p l i c a b i l i t y of the r e s u l t s o b t a i n e d . S e v e r a l i n v e s -t i g a t o r s ^ ' 6 0 ' e t a ^ have shown t h a t the i n t e r n a l f r i c t i o n of s o l i d s can be approximated by v i s c o u s r e s i s t a n c e or other energy d i s s i p a t i v e models so t h a t the decay of the v i b r a t i o n i s l o g a r -61 62 i t h m i c . However, ot h e r i n v e s t i g a t i o n s ' i n d i c a t e t h a t the l o g a r i t h m i c decrement i s dependent on amplitude as w e l l , with damping i n c r e a s i n g w i t h displacement. In any case, i t i s w e l l known t h a t the i n t e r n a l f r i c t i o n i s a f u n c t i o n o f the m a t e r i a l and i t s p a r t i c u l a r chemical and p h y s i c a l p r o p e r t i e s . T y p i c a l v a l u e s f o r aluminum would be of the or d e r o f 0.001 < 8 < 0.01, wh i l e f o r s t e e l 0.0G08 < £ < 0.006. With t h i s u n c e r t a i n t y i n the v a l u e o f i n t e r n a l damping, i t was thought a d v i s a b l e to con-duct the t e s t s w i t h damping l e v e l s i n and below the range l i k e l y to occur i n the f u l l - s c a l e s t r u c t u r e . 2 AERODYNAMICS OF A STATIONARY ANGLE SECTION 2.1 P r e l i m i n a r y Remarks In an a e r o e l a s t i c i n s t a b i l i t y study, s t a t i o n a r y model i n v e s t i g a t i o n s p r o v i d e v i t a l i n f o r m a t i o n from which c r i t i c a l o s c i l l a t o r y c o n d i t i o n s such as wind speed and model o r i e n t a t i o n can be p r e d i c t e d . A s e t of angle s e c t i o n models were su b j e c t e d to e x t e n s i v e wind tunnel t e s t i n g t o o b t a i n i n f o r m a t i o n concern-i n g mean wind l o a d i n g , S t r o u h a l number, f l u c t u a t i n g s t a t i c p r e s s u r e and wake geometry. T h i s chapter d e s c r i b e s model c o n s t r u c t i o n , necessary apparatus, i n s t r u m e n t a t i o n , and experimental procedures which a l s o form the b a s i s f o r the dynamical study i n the f o l l o w i n g chapter. The t e s t r e s u l t s are d i s c u s s e d and c o n c l u s i o n s p r e -sented. A l i s t o f the e l e c t r o n i c instruments used i n the e x p e r i -mental programme i s g i v e n i n Appendix I I I . 2.2 Models, Apparatus, Instrumentation and C a l i b r a t i o n 2.2.1 Angle Models Depending on the type of experimental t e s t proposed, the angle s e c t i o n models are c a t e g o r i z e d as f o l l o w s : (i) p r e s s u r e tap model; ( i i ) dynamic model; ( i i i ) balance model. The designed angle models had sharp contour edges and r e l a t i v e l y smooth faces to f a c i l i t a t e c o n s t r u c t i o n and p r o v i d e u n i f o r m i t y of s u r f a c e c o n d i t i o n s . However, commercially a v a i l a b l e angles u s u a l l y have rounded edges and rough s u r f a c e s . To determine the 13 i n f l u e n c e o f the d i s p a r i t i e s between the models and prototypes on c e r t a i n aerodynamic c h a r a c t e r i s t i c s , three balance models were made from s t e e l and aluminum s t r u c t u r a l angle members. To examine the mean and f l u c t u a t i n g s t a t i c p ressures on the s u r f a c e of an angle s e c t i o n d u r i n g s t a t i c and dynamic con-d i t i o n s the p r e s s u r e tap model was designed. The 3x3x1/2 i n . hollow angle model, 26 3/4 i n . l o n g , was c o n s t r u c t e d from 0.020 i n . aluminum sheet bonded t o a c r y l i c bulkheads and 1/4 i n . t h i c k end f a s t e n i n g tab p l a t e s (Figure 2-1). A 39 hole p r e s s u r e r i n g of 0.025 i n . diameter taps, l o c a t e d at the midspan of the model, p r o v i d e d a means of examining the pressure d i s t r i b u t i o n s around the contour of the angle. Two taps were p r o v i d e d i n the span-wise d i r e c t i o n , a t d i s t a n c e s of 4 1/2 i n . and 9 i n . from the mid-s e c t i o n , a t the same contour p o s i t i o n as tap number 5. The s u r f a c e p r e s s u r e s i g n a l s were t r a n s m i t t e d from the pressure taps to e x t e r n a l l y l o c a t e d t r a n s d u c e r s through 5 f t . long, 0.066 i n . diameter "Intramedic" p o l y e t h y l e n e t u b i n g . The 1 i n . and 3 i n . dynamic models, which d i d not have pr e s s u r e taps, were of i d e n t i c a l geometry. S i x d i f f e r e n t angle s e c t i o n models (Appendix I) were designed and b u i l t f o r mounting on the Aerolab s i x component, strain - g a u g e balance. Models A, B and C had the same nominal l e g width and t h i c k n e s s , while model D and E had h a l f the l e g t h i c k n e s s . Models A and B d i f f e r e d o n l y by the 1/4 i n . t h i c k end p l a t e s mounted on the l a t t e r model. T h i s was intended to determine the e f f e c t s of the same end p l a t e s mounted on the pr e s s u r e tap and dynamic angle models. Models C and D were s t e e l s t r u c t u r a l angles with rounded corners and rough s u r -Figure 2-1 P r e s s u r e t a p a n g l e m o d e l a n d n u m b e r i n g o f p r e s s u r e h o l e s a n d c o n t o u r s i d e s 15 f a c e s . A t y p i c a l c o m m e r c i a l aluminum a n g l e s e c t i o n w i t h r e l a -t i v e l y smooth s u r f a c e and s m a l l c o r n e r r a d i u s was r e p r e s e n t e d by m odel E . M o d e l F was g e o m e t r i c a l l y s i m i l a r t o model B w i t h c r o s s - s e c t i o n a l d i m e n s i o n s o f 2x2x1/3 i n . F o r a l l b a l a n c e m o d e l s , t h e e f f e c t i v e l e n g t h w h i c h was e x p o s e d t o t h e w i n d s t r e a m was n o m i n a l l y 2 7 i n . 2.2.2 F l u c t u a t i n g P r e s s u r e T r a n s d u c e r and C a l i b r a t i o n D a t a m e t r i c s I n c . o f Waltham, M a s s a c h u s e t t s , has d e v e l o p e d a new p r e s s u r e t r a n s d u c e r c a l l e d B a r o c e l M o d u l a r P r e s s u r e T r a n s -d u c i n g S y s t e m c o n s i s t i n g o f a p r e s s u r e s e n s o r , s i g n a l c o n d i t i o n e r and power s u p p l y . The B a r o c e l i s a h i g h p r e c i s i o n , s e n s i t i v e i n s t r u m e n t w h i c h p r o v e d t o be s u i t a b l e f o r t h e i n t e n d e d f l u c t u a t -i n g p r e s s u r e measurements. The p r e s s u r e s e n s o r c o n s i s t s b a s i c -a l l y o f a c a p a c i t i v e v o l t a g e d i v i d e r w i t h a s t a i n l e s s s t e e l d i a p h r a g m s e p a r a t i n g t h e two p r e s s u r e chambers. The s i g n a l c o n -d i t i o n e r p r o v i d e s 8 s e n s i t i v i t y r a n g e s f r o m 0-10 mm. o f m e r c u r y on t h e l e a s t s e n s i t i v e t o 0-0.001 mm. o f m e r c u r y on t h e most s e n s i t i v e s c a l e when c o u p l e d t o a p r e s s u r e s e n s o r h e a d . The o u t p u t v o l t a g e i s 0-5 v o l t s d . c . f u l l s c a l e on any o f t h e 8 s e n s i t i v i t y r a n g e s w i t h a l i n e a r i t y o f ± 0 . 1 % and s t a b i l i t y o f ± 0 . 1 % f o r ±15°F a m b i e n t t e m p e r a t u r e c h a n g e s ; The p r e s t r e s s e d d i a p h r a g m has a n a t u r a l f r e q u e n c y o f 2500 cps and t h e t r a n s i e n t r e s p o n s e o f t h e p r e s s u r e s e n s o r head i s l e s s t h a n 2 ms t o a s t e p p r e s s u r e i n p u t . From e x p e r i m e n t t h e H e l m h o l t z r e s o n a t o r f r e q u e n c y o f t h e c a v i t y and c o n n e c t i o n on one s i d e o f t h e d i a p h r a g m was f o u n d t o be a p p r o x i m a t e l y 210 c p s . The Barocel i s accurately calibrated for steady pressures. However, for fluc t u a t i n g pressures transmitted through r e l a t i v e l y long, small diameter tubes considerable attenuation occurred. Therefore, the output e l e c t r i c a l signal requires c a l i b r a t i o n against known input fl u c t u a t i n g pressure at the model surface. This was achieved using the c a l i b r a t i o n system developed by Wiland. The e f f e c t of amplitude and frequency of the source pres-sure on the output signal i s shown i n Figure 2-2(a). These curves indicate the l i n e a r i t y of the system. For convenience the c a l i b r a t i o n curves i n Figure 2-2(a) were replotted i n Figure 2-2(b) as a r a t i o of output to input. This eliminates frequency i n t e r p o l a t i o n . 2.2.3 Wake Probe and Traversing Gear The wake geometry survey was performed using a disc probe 33 6 3 constructed by Ferguson and described i n d e t a i l by Bryer et a l . I t was mounted on a 1 i n . hypodermic needle which i n turn was connected to a 14 i n . long, 1/4 i n . diameter sting. S t a t i c pressure c a l i b r a t i o n data were obtained for this p a r t i c u l a r probe from wind tunnel tests (Figure 2-3). The measurements indicate the probe to be r e l a t i v e l y i n s e n s i t i v e to a pitch of ±5° and yaw of ±20°. To enable the wake probe to be positioned i n the test section of the tunnel with control of movement i n a l a t e r a l and longitudinal d i r e c t i o n , the wake traversing gear designed by 3 3 Ferguson was used. The accuracy i n positioning the probe was approximately 0.02 i n . 17 Input, volt 3 a. 3 - o Frequency , cps F i g u r e 2-2 C a l i b r a t i o n curves f o r B a r o c e l t r a n s d u c e r arrangement F i g u r e 2-3 Dimensions and c a l i b r a t i o n d ata.of d i s c probe 2.3 T e s t Procedures 2.3.1 Balance Measurements A f t e r s e t t i n g the angle of a t t a c k and i n c r e a s i n g the wind speed to a p r e s e l e c t e d v a l u e , the l i f t , drag and p i t c h i n g moment on a balance model were re c o r d e d . The wind speed was then i n c r e a s e d to two f u r t h e r s e t t i n g s and c o r r e s p o n d i n g data o b t a i n e d . T h i s procedure was repeated f o r the f i v e balance models (A to E) over the e n t i r e range of angle of a t t a c k . 2.3.2 Vortex Shedding Frequency Measurement of the S t r o u h a l frequency was accomplished u s i n g the i n s t r u m e n t a t i o n l a y o u t shown i n F i g u r e 2-4. The wake probe was l o c a t e d a t an a p p r o p r i a t e p o s i t i o n i n the mid-plane where the f l u c t u a t i n g p r e s s u r e s i g n a l was r e l a t i v e l y c l e a r . However, to improve the q u a l i t y of the s i g n a l , a band pass f i l t e r was i n c o r p o r a t e d . I l l u s t r a t e d i n F i g u r e 2-5(a), are t y p i c a l u n f i l t e r e d and f i l t e r e d f l u c t u a t i n g p r e s s u r e s i g n a l s from the probe. Using t h i s system, the 1 i n . and 3 i n . angle models and the balance models (B,C,D,E, and F) were t e s t e d f o r v o r t e x shedding frequency v a r i a t i o n with angle of a t t a c k and Reynolds number. 2.3.3 Mean S t a t i c Pressure on Model Surface The mean p r e s s u r e d i s t r i b u t i o n was o b t a i n e d u s i n g a Lambrecht manometer wit h e t h y l a l c o h o l . The o s c i l l a t i o n of the a l c o h o l column caused by the f l u c t u a t i n g component of the s t a t i c p r e s s u r e was reduced u s i n g a r e s t r i c t i o n i n the p r e s s u r e l i n e . To f a c i l i t a t e r e d u c t i o n of the data, the p r e s s u r e on the model 2 0 Probe Polyethylene tubes K l = 5 ' ; d | = 0.066" » Damping bottle Barocel Signal conditioner Visicorder g u r e 2 - 4 I n s t r u m e n t a t i o n l a y o u t f o r v o r t e x s h e d d i n g f r e -q u e n c y a n d f l u c t u a t i n g p r e s s u r e m e a s u r e m e n t s Figure 2-5 Typical fluctuating pressure signals from (a) wake probe (b) model surface 22 was measured r e l a t i v e to the t o t a l head i n the s e t t l i n g s e c t i o n of the wind t u n n e l . T h i s e l i m i n a t e d the e f f e c t of atmospheric p r e s s u r e changes d u r i n g the t e s t s and produced a p r e s s u r e d i f f e r e n t i a l which was always p o s i t i v e . The r e s u l t s were o b t a i n -ed f o r a range of wind speed and angle of a t t a c k . 2.3.4 F l u c t u a t i n g S t a t i c Pressure on Model Surface I n v e s t i g a t i o n of the f l u c t u a t i n g p r e s s u r e on the s u r f a c e of the angle model was separated i n t o amplitude and phase measurements. For s t u d y i n g the amplitude of the p r e s s u r e f l u c t u -a t i o n s , the i n s t r u m e n t a t i o n shown i n F i g u r e 2-4 was used except t h a t the s i g n a l was taken from the model taps and recorded on a 33 35 V i s i c o r d e r . As d i s c o v e r e d by other i n v e s t i g a t o r s , ' the f l u c t u a t i n g p r e s s u r e s i g n a l s on the model s u r f a c e were at the frequency of the v o r t e x shedding, and had seemingly random ampli-tude modulation. F i g u r e 2-5(b) i l l u s t r a t e s t y p i c a l u n f i l t e r e d , f i l t e r e d and random amplitude modulated p r e s s u r e s i g n a l s from the model s u r f a c e . For p r e s e n t a t i o n of the f l u c t u a t i n g pressure amplitude, the average peak value over approximately 150 to 250 c y c l e s was determined from the c h a r t r e c o r d s . In a d d i t i o n , to e v a l u a t e the s i g n a l modulation, the r a t i o of maximum amplitude to average peak was o b t a i n e d . The phase d i f f e r e n c e between the f l u c t u a t i n g pressure s i g n a l s from d i f f e r e n t model taps was obtained by i n c o r p o r a t i n g a second B a r o c e l transducer system and comparing the s i g n a l s on V i s i c o r d e r c h a r t s . To e l i m i n a t e the e f f e c t of phase s h i f t i n t r o -duced by the i n s t r u m e n t a t i o n , the f l u c t u a t i n g pressure from one 23 model tap was a r b i t r a r i l y s e l e c t e d as a r e f e r e n c e . The r e l a t i v e phase angle was averaged over 10 t o 20 c y c l e s . The amplitude and phase measurements f o r the 41 p r e s s u r e taps on the model were performed a t th r e e wind speeds and v a r i o u s angles of a t t a c k . The band pass f i l t e r a n t e n u a t i o n was determined f o r each wind speed and f i l t e r c u t - o f f frequency s e t t i n g s by u s i n g a s i n u s o i d a l s i g n a l from a low frequency f u n c t i o n g e n e r a t o r . The V i s i c o r d e r c a l i b r a t i o n and impedance a t t e n u a t i o n were determined u s i n g the same procedure. 2.3.5 Wake Survey Wake survey measurements were accomplished by examining the f l u c t u a t i n g p r e s s u r e f i e l d a s s o c i a t e d w i t h the v o r t i c e s shed from the model u s i n g the i n s t r u m e n t a t i o n setup shown i n F i g u r e 2-6. S i n c e the wake r e s u l t s were found t o be s u b s t a n t i a l l y independent o f Reynolds number, the measurements were c o n f i n e d to o n l y one wind speed f o r v a r i o u s angles o f a t t a c k . T r a v e r s i n g the d i s c probe l a t e r a l l y a t v a r i o u s x - s t a t i o n s and r e c o r d i n g the average peaks o f the f l u c t u a t i n g p r e s s u r e s i g n a l s p r o v i d e d a s e t of curves each having two maxima near the v o r t e x c e n t r e l i n e s . The y - d i s t a n c e between these maxima a t each x - s t a t i o n was taken t o be a measure of the l a t e r a l s p a c i n g between the two rows of v o r t i c e s shed from the model. I t was convenient t o p l o t the r e s u l t s o f the l a t e r a l t r a v e r s e as a r a t i o of probe t o model tap average f l u c t u a t i n g p r e s s u r e s . The model tap s e l e c t e d f o r the probe r a t i o was somewhat a r b i t r a r y , but re p r e s e n t e d a p o s i t i o n on the model contour having a near maximum f l u c t u a t i n g p r e s s u r e v a l u e . A t r u e rms v o l t m e t e r was used f o r Damping bottle Signal conditioner Oscilloscope Visicorder Filter Damping bottle Power supply R M S voltmeter Signal conditioner Filter R-C damping unit VTVM Figure 2-6 Schematic of instrumentation for wake survey measurements 25 averaging o f the f l u c t u a t i n g p r e s s u r e s i g n a l s . The v a r i a t i o n of the peak v a l u e s w i t h x c o o r d i n a t e gave an i n d i c a t i o n of the decay of the v o r t i c e s i n the downstream d i r e c t i o n . L o n g i t u d i n a l s p a c i n g , a, between c o n s e c u t i v e v o r t i c e s i n one row of the v o r t e x s t r e e t was o b t a i n e d i n d i r e c t l y from l o n g i -t u d i n a l phase measurements. The s p a c i n g d i s t a n c e corresponds to a 360° phase d i f f e r e n c e between the f l u c t u a t i n g p r e s s u r e s i g n a l s a s s o c i a t e d w i t h c o n s e c u t i v e v o r t i c e s i n the same row. T r a v e r s i n g the wake probe a l o n g the c e n t r e l i n e o f a v o r t e x row, the f l u c t u a t i n g p r e s s u r e s from the probe and a r e f e r e n c e model tap were r e c o r d e d s i m u l t a n e o u s l y on a V i s i c o r d e r . The phase angle data, averaged over 10 t o 20 c y c l e s , was p l o t t e d as a f u n c t i o n o f the downstream c o o r d i n a t e from which the v o r t e x s p a c i n g , a, was i n t e r p r e t e d . In a d d i t i o n , the v o r t e x streamwise v e l o c i t y was determined from the l o n g i t u d i n a l s p a c i n g u s i n g the r e l a t i o n (2.1) 2.4 Ex p e r i m e n t a l R e s u l t s and D i s c u s s i o n 2.4.1 Steady L i f t , Drag and P i t c h i n g Moment D i s t r i b u t i o n s The s t a t i c l i f t , drag and p i t c h i n g moment experienced by v a r i o u s angle s e c t i o n s over the Reynolds number range of 4 4 4x10 t o 11x10 are pres e n t e d i n the f o l l o w i n g i l l u s t r a t i o n s . The p i t c h i n g moment has been measured r e l a t i v e t o an a x i s c o i n c i d i n g w i t h the c e n t r o i d o f an angle s e c t i o n . I t was observed t h a t over the wind speed range i n v e s t i g a t e d , the f o r c e and moment 26 c o e f f i c i e n t s were independent of the Reynolds number. F i g u r e 2-7, pr e s e n t s the v a r i a t i o n o f the aerodynamic c o e f f i c i e n t s w i t h angle of a t t a c k , o , f o r the balance model B which i s g e o m e t r i c a l l y s i m i l a r t o the p r e s s u r e tap and dynamic angle models. I t i s apparent t h a t the angle s e c t i o n experiences the maximum n e g a t i v e l i f t and moment a t approximately a = 10° and c o r r e s p o n d i n g maximum p o s i t i v e v a l u e s near a = 40°. On the othe r hand, the drag f o r c e i s maximum a t the symmetric angle a = -45° where the model angle s e c t i o n i s open upstream. The v a r i a t i o n o f the drag c o e f f i c i e n t w i t h angle o f a t t a c k can be reduced by b a s i n g the c o e f f i c i e n t on the p r o j e c t e d f r o n t a l area. However, the drag c o e f f i c i e n t does not become completely indepen-dant o f model o r i e n t a t i o n . A comparison o f t h i s data w i t h the t e s t r e s u l t s o f model A i n d i c a t e d no s i g n i f i c a n t i n f l u e n c e of the 1/4 i n . t h i c k end p l a t e s on the o v e r a l l aerodynamic c h a r a c t e r i s t i c s . The abrupt changes i n the l i f t , drag and moment d i s t r i -b u t i o n s a t c e r t a i n o r i e n t a t i o n such as a = 10°, 40° and 104° are a t t r i b u t e d t o major changes i n the flow f i e l d over the angle contour. T h i s was s u b s t a n t i a t e d by a q u a l i t a t i v e study u s i n g a smoke t u n n e l . With the i n c r e a s e of angle o f a t t a c k i n the ranges 5° t o 25° and 97° t o 125°, the separated flow r e a t t a c h e d on one s i d e forming a s e p a r a t i o n bubble which decreased i n length and f i n a l l y d i sappeared producing a t t a c h e d flow over the e n t i r e s i d e . In the v i c i n i t y of a = 40°, the forward s t a g n a t i o n p o i n t s h i f t e d from the j u n c t i o n o f s i d e s 3 and 4 to approximately tap Figure 2-7 D i s t r i b u t i o n of steady l i f t , drag and pitching moment c o e f f i c i e n t s for balance model B t o 28 number 21 ca u s i n g a change i n the p o s i t i o n o f the separated shear l a y e r s downstream. F i g u r e 2-8 i l l u s t r a t e s some of the e f f e c t s of edge r a d i u s and s u r f a c e roughness a s s o c i a t e d w i t h commercially a v a i l -a ble angle members. The r e s u l t s show t h a t angle model B and s t r u c t u r a l angle members C, D and E have reasonable aerodynamic s i m i l a r i t y even though t h e r e are minor d i f f e r e n c e s i n geometric f e a t u r e s . In F i g u r e 2-8(c) s l i g h t d e v i a t i o n s i n the p i t c h i n g moment d i s t r i b u t i o n s are apparent over the range of -45° to 15° due t o d i f f e r e n c e s i n l e g t h i c k n e s s and c o r n e r c o n d i t i o n s . A comparison o f the r e s u l t s suggests t h a t a r e d u c t i o n i n l e g t h i c k n e s s o r co r n e r r a d i u s decreases the magnitude of the p i t c h -i n g moment. T h i s v a r i a t i o n i s apparent from the mean s t a t i c p r e s s u r e d i s t r i b u t i o n s g i v e n l a t e r i n F i g u r e 2-12. The p r e s e n t l i f t and drag d i s t r i b u t i o n s compare w e l l with 2 Wardlaw's measurements and the few drag r e s u l t s f o r s t r u c t u r a l 64 angles quoted by Hoerner. For the angle models a t a = -45°, the c o r r e c t e d drag c o e f f i c i e n t v a l u e o f approximately 2.0 i s 64 s i m i l a r t o t h a t o f a normal f l a t p l a t e or 90° wedge model. Using the measured base p r e s s u r e v a l u e f o r the p r e s s u r e tap angle s e c t i o n a t a = 135°, Roshko's 6^ notched hodograph s o l u t i o n f o r a 90° wedge, p r e d i c t s a va l u e o f C D = 1.8 which compares w e l l with the ex p e r i m e n t a l drag c o e f f i c i e n t of 1.7 f o r balance model B. 2.4.2 Vortex Shedding Frequency and S t r o u h a l Number The dependence of the v o r t e x shedding frequency on wind speed i s shown i n F i g u r e 2-9 f o r v a r i o u s angles of a t t a c k of the p r e s s u r e tap angle model. The l i n e a r i t y of the p l o t s i n -29 F i g u r e 2-8 Comparison of aerodynamic c o e f f i c i e n t s f o r v a r i o u s s t r u c t u r a l angle s e c t i o n s F i g u r e 2-9 V a r i a t i o n of v o r t e x shedding frequency with wind v e l o c i t y f o r p r e s s u r e tap angle model F i g u r e 2-10 V a r i a t i o n of S t r o u h a l number and v o r t e x resonant wind speed with angle of a t t a c k f o r 3 i n . angle model d i c a t e s t h a t the S t r o u h a l number i s independent of the Reynolds 4 4 number over the range of 10 t o 15x10 i n v e s t i g a t e d . Using t h i s and s i m i l a r data, the S t r o u h a l number v a r i a -t i o n w i t h angle of a t t a c k f o r the p r e s s u r e tap model was obtained (Figure 2-10). Basing the dime n s i o n l e s s frequency parameter on the p r o j e c t e d model width e, r a t h e r than the constant dimension h, reduces the v a r i a t i o n o f S t r o u h a l number but does not make i t completely independent of o r i e n t a t i o n . The sudden i n c r e a s e i n the S t r o u h a l number near a = 1 5 ° and 105° i s a t t r i b u t e d to the reattachment of the flow as e x p l a i n e d i n the d i s c u s s i o n o f the balance measurements. T h i s reduces the wake width which, i n g e n e r a l , i s accompanied by a simultaneous i n c r e a s e i n the S t r o u h a l frequency. T h e r e f o r e , i f the wake width i s used as the c h a r a c t e r i s t i c l e n g t h , the S t r o u h a l number may have l e s s 66 dependency on model o r i e n t a t i o n . Roshko's concept of a U n i v e r s a l S t r o u h a l Number, S A, based on wake width as w e l l as shear l a y e r s e p a r a t i o n v e l o c i t y , has c o n s i d e r a b l e m e r i t . Examin-a t i o n of the mean base p r e s s u r e , wake geometry and shedding frequency data i n d i c a t e d t h a t the angle s e c t i o n w i l l have values of S* f o r v a r i o u s angles of a t t a c k approximately equal to the accepted r e s u l t o f 0.164. A comparison of the S t r o u h a l number d i s t r i b u t i o n s f o r the 1, 2 and 3 i n . angle models t e s t e d i n the same wind t u n n e l (Figure 2-11(a)) i n d i c a t e s the presence of t u n n e l w a l l i n t e r -f e r e n c e e f f e c t s . The t r e n d i s f o r S t r o u h a l number to i n c r e a s e w i t h blockage. For comparison, the estimated curve f o r f r e e stream c o n d i t i o n i s a l s o i n c l u d e d (Appendix I I ) . 32 Figure 2-11 Strouhal number d i s t r i b u t i o n s for (a) d i f f e r -ent size angle models (b) various s t r u c t u r a l angle sections The c o r r e s p o n d i n g S t r o u h a l number data f o r the balance models (B,C,D, and E) are shown.in F i g u r e 2-11(b). Aerodynamic s i m i l a r i t y between the sharp-edged model B and the commercial angle s e c t i o n s i s apparent. Since a l l the angle members are of the same nominal s i z e , 3x3 i n . , the W a l l confinement e f f e c t s would be s i m i l a r . For angle s e c t i o n s of t h i s width, a c o r r e c t i o n curve i s p r e s e n t e d i n Appendix I I , F i g u r e I I - l . Comparison of the p r e s e n t S t r o u h a l number data with the 2 few r e s u l t s p u b l i s h e d by Wardlaw shows good agreement. F u r t h e r -more, the angle s e c t i o n r e s u l t s f o l l o w the t r e n d e s t a b l i s h e d by c y l i n d e r s of d i f f e r e n t geometric form. The S t r o u h a l numbers f o r s e v e r a l r e p r e s e n t a t i v e s e c t i o n s are l i s t e d below: S _ f F (i) angle s e c t i o n a t a - -45° 0.135 ( i i ) f l a t p l a t e normal to flow 0.145 ( i i i ) 90° wedge and angle s e c t i o n a t a = 135° 0.18 (iv) c i r c u l a r c y l i n d e r 0.20 (v) f l a t p l a t e a t a = 40° 0.23 The r e s u l t s suggest a tendency f o r the S t r o u h a l number to i n c r e a s e i n magnitude w i t h a decrease i n b l u f f n e s s . F u r t h e r comparison 64 w i t h o t h e r p u b l i s h e d data p r o v i d e d c o n f i r m a t i o n as to the r e -l i a b i l i t y of the measurements. From the S t r o u h a l number r e s u l t s , important i n f o r m a t i o n r e g a r d i n g the resonant wind speed (V ), a t which v o r t e x induced a e r o e l a s t i c i n s t a b i l i t y may occur, can be o b t a i n e d . v r e s r e p r e s e n t s the c r i t i c a l wind speed a t which the v o r t e x shedding frequency c o i n c i d e s w i t h the n a t u r a l frequency of the system. Expressed i n nondimensional form, the resonant wind speed can be o b t a i n e d from the S t r o u h a l number u s i n g the e x p r e s s i o n r e s 2 U S U (2.2) n Shown i n F i g u r e 2-10 i s the v a r i a t i o n of U with a f o r the 3 res 3 i n . angle s e c t i o n . I t i s apparent t h a t an angle member i s s u s c e p t i b l e t o v o r t e x resonance a t the lowest v e l o c i t y near a = 30° . 2.4.3 Mean S t a t i c P ressure D i s t r i b u t i o n s The mean s t a t i c p r e s s u r e on the s u r f a c e of the angle model was found to be independent of the Reynolds number over 4 4 the range 10 to 12x10 . The p r e s s u r e d i s t r i b u t i o n around the model midspan s e c t i o n i s shown i n F i g u r e 2-12, f o r v a r i o u s angles of a t t a c k . The r e s u l t s i n d i c a t e the l o c a t i o n of the s t a g n a t i o n and s e p a r a t i o n p o i n t s . In a d d i t i o n , the curves provide data on the base p r e s s u r e c o e f f i c i e n t which i s u s e f u l i n the e v a l u a t i o n of the separated shear l a y e r v e l o c i t y and wind t u n n e l w a l l c o r r e c t i o n . For the same o r i e n t a t i o n s , i n v e s t i g a t i o n of the spanwise C d i s t r i b u t i o n showed i t to be r e l a t i v e l y c o n s t a n t , thus P s u b s t a n t i a t i n g t w o - d i m e n s i o n a l i t y of the flow. A comparison of the pressure i n t e g r a t e d and balance measured aerodynamic c o e f f i c i e n t s (Figure 2-13) confirms the above o b s e r v a t i o n . As shown, both l i f t and drag c o e f f i c i e n t s are i n good agreement over the e n t i r e range of angle of a t t a c k . The p i t c h i n g moment measurements show s i m i l a r t r e n d except f o r the 35 F i g u r e 2 - 1 2 - i l Midspan d i s t r i b u t i o n s of mean s t a t i c p r e s -sure c o e f f i c i e n t (40° < a < 135°) T r Balance D ( model B Pressure tap angle model 1-0.4 o - 4 5 45° 90° F i g u r e 2-13 Comparison of pressure i n t e g r a t e d and balance measured steady aero-dynamic c o e f f i c i e n t s 38 d i s c r e p a n c y i n magnitude over the range a = ± 25°. 2.4.4 F l u c t u a t i n g S t a t i c Pressure D i s t r i b u t i o n s O s c i l l o s c o p e t r a c e s of t y p i c a l f l u c t u a t i n g p r e s s u r e s i g n a l s from v a r i o u s model taps are i l l u s t r a t e d i n F i g u r e 2-14. The photographs i n (a) show the amplitude modulation of p r e s s u r e s from two midspan and two spanwise t a p s / The t y p i c a l phase s i g n a l s shown i n (b) were o b t a i n e d from v a r i o u s p r e s s u r e taps as f o l l o w s : (i) n e i g h b o u r i n g midspan taps from the same s i d e o f the model; ( i i ) midspan taps from o p p o s i t e s i d e s of the model; ( i i i ) spanwise taps from the same s i d e o f the model. 3 3 Over the range of 26x10 t o 63x10 r i t was observed t h a t the Reynolds number d i d not have any s i g n i f i c a n t e f f e c t on the average f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t or phase angle. F o r v a r i o u s angles of a t t a c k , the midspan d i s t r i b u t i o n s of the average f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t and modulation r a t i o are i l l u s t r a t e d i n F i g u r e 2-15. I t i s apparent t h a t the amplitude of the f l u c t u a t i n g p r e s s u r e can reach a magnitude comparable to the mean s t a t i c p r e s s u r e l e v e l . The maximum C-, v a l u e s are of the o r d e r of 0.8 i n the v i c i n i t y o f a = -45° and 75° w i t h c o r r e s p o n d i n g modulation r a t i o s between 1.5 and 2.5. A t angles of a t t a c k of 45° and 135°, the average c o e f f i c i e n t d i m i n i s h e s to a l e v e l below 0.2 but the modulation r a t i o remains at approximately 2.0. T h i s v a r i a t i o n of f l u c t u a t i n g p r e s s u r e w i t h model o r i e n t a t i o n was observed q u a l i t a t i v e l y as w e l l d u r i n g the flow v i s u a l i z a t i o n experiment by n o t i n g the change i n the s t r e n g t h of the v o r t e x wake system. Tap no. 10 11,111.1 I i l l ! L y i l U U L l l L i a i l B i i l 41 •» • i r (a) Figure 2-14 Typical fluctuating pressure signals from various model taps i n d i c a -ting (a) random amplitude modulation (b) phase va r i a t i o n 40 F i g u r e 2 - 1 5 - i M i d s p a n d i s t r i b u t i o n s o f f l u c t u a t i n g s t a t i c p r e s s u r e c o e f f i c i e n t and a m p l i t u d e modu-l a t i o n r a t i o (-45° < a < 0°) 41 P' 'max F 3 4 5 Model contour sides Figure 2 - 1 5 - i i Midspan d i s t r i b u t i o n s of fl u c t u a t i n g s t a t i c pressure c o e f f i c i e n t and amplitude modula-tio n r a t i o (15° < a < 75°) 42 F i g u r e 2 - X 5 - i i i Midspan d i s t r i b u t i o n s of f l u c t u a t i n g s t a t i c p r e s s u r e c o e f f i c i e n t and amplitude modula-t i o n r a t i o (90° < a < 135°) 43 For the p o r t i o n of the body i n the wake, the magnitude of the f l u c t u a t i n g p r e s s u r e i s c o m p a r a t i v e l y l a r g e r than t h a t near the forward s t a g n a t i o n p o i n t , S T. A s i m i l a r r e d u c t i o n i n L i C-, occurs i n the v i c i n i t y o f the r e a r " s t a g n a t i o n " r e g i o n marked S T. I t seems reasonable t h a t t h i s tendency of v a n i s h i n g C- ( near the two s t a g n a t i o n areas i s due to the c a n c e l l a t i o n of the pressure f l u c t u a t i o n s , from the two s i d e s of the wake system, which are 180° out of phase. However, t h i s e f f e c t i s l e s s complete a t the r e a r of the body because of i r r e g u l a r i t i e s i n the wake. Quan-t i t a t i v e o b s e r v a t i o n s i n d i c a t e d the second harmonic component of the p r e s s u r e f l u c t u a t i o n s t o be l e s s than 20% of the fundamental frequency v a l u e . The seemingly random amplitude modulation of the pressure f l u c t u a t i o n s i s a t t r i b u t e d t o the g e n e r a l i n s t a b i l i t y o f the s e p a r a t e d shear l a y e r s and a s s o c i a t e d v o r t i c e s . Much of the , v o r t i c i t y generated a t the body i s d i s s i p a t e d immediately behind the c y l i n d e r through t u r b u l e n c e . Even the v o r t i c e s , forming the Karman v o r t e x s t r e e t , are of d i f f e r e n t s t r e n g t h s and tend to l o s e t h e i r i n d i v i d u a l i t y as the d i s t a n c e behind the body i n c r e a s e s . From the sample phase data i n F i g u r e 2-14, i t i s observed t h a t the p r e s s u r e f l u c t u a t i o n s on o p p o s i t e s i d e s of the model contour e x h i b i t the f a m i l i a r 180° phase d i f f e r e n c e . On the other hand, p r e s s u r e s i g n a l s from neighbouring taps on the same s i d e are not n e c e s s a r i l y i n phase as f r e q u e n t l y r e p o r t e d i n l i t e r a t u r e , but may have phase d i f f e r e n c e s as l a r g e as 50° to 100° (Figure 2-16). These phase v a r i a t i o n s are a t t r i b u t e d to the adjustment of the flow f i e l d around the model due to the shedding of a vortex downstream and formation of the next v o r t e x core. Recent measure-44 F i g u r e 2-16-i Phase v a r i a t i o n of midspan f l u c t u a t i n g p r e s s u r e (-45° < a < 0°) F i g u r e 2 - 1 6 - i i Phase v a r i a t i o n of midspan f l u c t u a t i n g p r e s s u r e (-15° <_ a < 60°) ments by W i l a n d on e l l i p t i c c y l i n d e r s i n d i c a t e d a s i m i l a r p h a s e phenomenon. The s p a n w i s e v a r i a t i o n s o f t h e f l u c t u a t i n g p r e s s u r e , shown i n F i g u r e 2-17, s u g g e s t r e a s o n a b l y u n i f o r m p r e s s u r e c o -e f f i c i e n t a l o n g t h e model l e n g t h . However, t h e s c a t t e r and i n c o n s i s t e n c y i n ph a s e d a t a , p a r t i c u l a r l y a t some a n g l e s o f a t t a c k , a r e t o o l a r g e t o e s t a b l i s h any d e f i n i t e t r e n d . I t seems r e a s o n a b l e t o assume on t h e b a s i s o f t h e work r e p o r t e d by 40 46 47 V i c k e r y , P r e n d e r g a s t and F e n g t h a t s p a n w i s e c o r r e l a t i o n o f t h e f l o w e x i s t s o v e r o n l y a f i n i t e l e n g t h o f t h e mod e l . F o r s t a t i o n a r y c i r c u l a r c y l i n d e r s t h e c o r r e l a t i o n l e n g t h i s t y p i c a l l y o f t h e o r d e r o f two t o t h r e e c y l i n d e r d i a m e t e r s w i t h i n c r e a s e d t w o - d i m e n s i o n a l i t y d u r i n g c y l i n d e r m o t i o n . As r e p o r t e d by V i c k e r y , t h e c o r r e l a t i o n l e n g t h i m p r o v e s f o r s h a r p - e d g e d b o d i e s and i s o f t h e o r d e r o f f i v e t o s i x d i a m e t e r s f o r , a s t a t i o n a r y s q u a r e member i n a smooth s t r e a m . However, u n d e r t u r b u l e n t f l o w c o n d i t i o n s , t h e s p a n w i s e c o r r e l a t i o n s u f f e r s s u b s t a n t i a l r e d u c t i o n . As i l l u s -t r a t e d i n F i g u r e 2 - 1 4 ( a ) , t h e a m p l i t u d e m o d u l a t i o n s o f t h e p r e s -s u r e f l u c t u a t i o n s on t h e s t a t i o n a r y a n g l e model a r e i n phase a r o u n d t h e c o n t o u r and a l o n g t h e l e n g t h o f t h e m o d e l . S i m i l a r 31 33 35 o b s e r v a t i o n s have b e e n r e p o r t e d by o t h e r i n v e s t i g a t o r s . ' ' 2.4.5 F l u c t u a t i n g L i f t , D r a g and Moment C o e f f i c i e n t s The m i d s p a n f l u c t u a t i n g p r e s s u r e d i s t r i b u t i o n was i n t e -g r a t e d n u m e r i c a l l y t o o b t a i n t h e f l u c t u a t i n g l i f t , d r a g and p i t c h -i n g moment e x p e r i e n c e d by t h e s t a t i o n a r y a n g l e s e c t i o n ( F i g u r e 2 - 1 8 ) . The p i t c h i n g moment i s a b o u t t h e e.g. a x i s . Two d a t a p o i n t s a r e p l o t t e d f o r e a c h c o e f f i c i e n t t o i n d i c a t e t h e e f f e c t o f t h e m e a s u r e d p h a s e d i f f e r e n c e a t t h e m i d s p a n t a p s . B o t h c a l c u -l a t i o n s , however, t a k e i n t o a c c o u n t t h e 180° phase between t h e <x = - 4 5 ° • * $ 1 ; • i 1 -15° ( 1 75° I -( i i 105° < 1 -30" 50 50 15° « I » o ' ' 30 o 40 41 5 Spanwise tap numbers 40 100 50 0 4 5 ° ' ( r 60° • 1 • i r 50 50 1 90° > ( 1 1-50 50 •50 • V 135° , • T 120°. 41 F i g u r e 2-17 Spanwise v a r i a t i o n of f l u c t u a t i n g p r essure c o e f f i c i e n t and phase F i g u r e 2 - 1 8 C o m p a r i s o n o f f l u c t u a t i n g a n d s t e a d y a e r o -d y n a m i c c o e f f i c i e n t s f l u c t u a t i n g p r e s s u r e s from the two s i d e s of the model. Since the r e s e a r c h programme was not aimed at s t u d y i n g the spanwise v a r i -a t i o n s , t h i s e f f e c t i s not i n c l u d e d i n the above r e s u l t s . For comparison, the d i s t r i b u t i o n s of the s t a t i c s e c t i o n a l c o e f f i c i e n t s from F i g u r e 2-13 are i n c l u d e d . Examination of the f l u c t u a t i n g f o r c e s and moment c o e f f i c -i e n t s i n d i c a t e s t h a t : (i) C j , i s of the same o r d e r as C^; s ( i i ) C^, i s approximately 1/10 of C^; s ( i i i ) C-, i s about 1/2 of C . m m s The f l u c t u a t i n g aerodynamic c o e f f i c i e n t s show l a r g e v a r i a t i o n w i t h model o r i e n t a t i o n r e a c h i n g minimum values near a = 4 5 ° and 135°. T h i s t r e n d was a l s o suggested by the f l u c t u a t i n g p r e s s u r e d i s t r i b u t i o n s . In g e n e r a l , the e f f e c t o f the phase i s to change the above c o e f f i c i e n t s by l e s s than ±10%. Thus, a c c u r a t e d e t e r m i n a t i o n of the phase angle along the s i d e s of the model contour i s l e s s s i g n i f i c a n t i n the f i n a l e v a l u a t i o n of the c o e f f i c i e n t s . Study of the f l u c t u a t i n g p r e s s u r e amplitude and phase d i s t r i b u t i o n s a t each angle of a t t a c k r e v e a l s t h a t the l a r g e phase d i f f e r e n c e s occur over r e g i o n s w i t h near minimum C-, v a l u e s , thus c o n t r i b u t i n g l i t t l e t o the f i n a l summation. 2 . 4 . 6 Wake Geometry Using the techniques d e s c r i b e d i n s e c t i o n 2.3.5, wake geometry data was o b t a i n e d f o r the 1 i n . dynamic and 3 i n . p r e s s u r e tap models. A l l t e s t s were performed a t N R = 59,500 except f o r one a d d i t i o n a l examination of the l a t e r a l v o r t e x 51 s p a c i n g f o r the p r e s s u r e tap angle model at N R =26,800 with a = 135°. W i t h i n the range of Reynolds number i n v e s t i g a t e d , no s i g n i f i c a n t change i n wake geometry was observed. F i g u r e 2-19 shows the l a t e r a l v a r i a t i o n s o f the f l u c t u -a t i n g p r e s s u r e amplitude at v a r i o u s x - s t a t i o n s f o r two t y p i c a l angles of a t t a c k . As expected, the f l u c t u a t i n g p r e s s u r e d i s t r i -b u t i o n s are s i m i l a r on both s i d e s of the wake f o r sy m m e t r i c a l l y o r i e n t e d models. The wake c e n t r e l i n e i s then c o i n c i d e n t with the x - a x i s . However, f o r the models at o t h e r angles of a t t a c k the wake i s unsymmetrical w i t h the peaks of the p r e s s u r e curves h i g h e r on the s i d e f o r which the p o i n t of s e p a r a t i o n i s most rearward. 6 7 The e x p e r i m e n t a l r e s u l t s by Fage and Johansen showed t h a t v o r t i c i t y i s shed from the upper and lower s u r f a c e s of an asymmetric model a t the same r a t e . T h e r e f o r e , the presence of the h i g h e r p r e s s u r e peaks on one s i d e of the wake may be e x p l a i n e d by the f a c t t h a t the c o r r e s p o n d i n g v o r t e x t r a v e l s r e l a t i v e l y s h o r t e r d i s t a n c e and hence s u f f e r s l e s s d i s s i p a t i o n and d i s p e r s i o n . From l a t e r a l p r e s s u r e d i s t r i b u t i o n r e s u l t s , s i m i l a r t o those i n F i g u r e 2-19, the decay of the peak pr e s s u r e amplitude i n the downstream d i r e c t i o n was o b t a i n e d as shown i n F i g u r e 2-20. The curves i n d i c a t e an approximate i n v e r s e p r o p o r t i o n a l i t y between p r e s s u r e and downstream d i s t a n c e which agrees with the 6 8 a n a l y t i c a l p r e d i c t i o n g i v e n by Schaefer and E s k i n a z i 1 s v o r t e x s t r e e t model. Ex p e r i m e n t a l measurements wi t h c i r c u l a r and e l l i p t i c 33 35 c y l i n d e r s by Ferguson and Wiland, r e s p e c t i v e l y , i n d i c a t e s i m i l a r p r e s s u r e decay c u r v e s . Taking the peaks of the l a t e r a l p r e s s u r e d i s t r i b u t i o n s as the p o s i t i o n s of the two v o r t e x rows, F i g u r e 2-21 shows the stream-5 2 1 F i g u r e 2-20 V a r i a t i o n of peak f l u c t u a t i n g pressure with downstream c o o r d i n a t e F i g u r e 2 - 2 1 - i i L a t e r a l p o s i t i o n s of vortex rows behind 1 i n . and 3 i n . angle models (45° <_ a < 135°) wise v a r i a t i o n of the wake data f o r both the 1 i n . and 3 i n . angle models a t v a r i o u s angles o f a t t a c k . However, as p o i n t e d 69 out by Hooker the maximum v e l o c i t y f l u c t u a t i o n s and, t h e r e f o r e , p r e s s u r e f l u c t u a t i o n s , do not occur along the path o f the v o r t e x c e n t r e s as some experimenters have a s s e r t e d but r a t h e r develop i n the neighbourhood of the edge of the core f a r t h e s t from the s t r e e t c e n t r e l i n e . Hence, the a c t u a l p o s i t i o n s o f the v o r t e x c e n t r e s l i e inward o f the p r e s s u r e boundaries by an amount equal t o the r a d i u s of the v o r t e x c o r e s . Using the mathematical model of an i s o l a t e d v i s c o u s v o r t e x , Schaefer and E s k i n a z i were able t o a r r i v e a t an e x p r e s s i o n f o r c o r r e c t i n g the exp e r i m e n t a l measurements. For the angle models, t h i s g i v e s a c o r r e c t i o n which i n c r e a s e s w i t h x from v i r t u a l l y zero a t the model t o l e s s than 4% a t a d i s t a n c e o f x/h = 1 0 . I t i s apparent, however, t h a t a t l a r g e x/h t h i s c o r r e c t i o n i s not adequate as i t does not account f o r tu r b u l e n c e o r wake i n s t a b i l i t y which may i n f l u e n c e the expansion of the v o r t e x cores or the p o s i t i o n o f t h e i r c e n t r e s . T h i s 70 7 o b s e r v a t i o n i s i n agreement w i t h the experimental measurements ' which show t h a t the v o r t i c e s do not flow downstream i n d e f i n i t e l y i n p a r a l l e l rows but always move away from the c e n t r e l i n e w i t h i n c r e a s i n g x even when an i n t e r m e d i a t e sequence of v o r t i c e s have some u n i f o r m i t y of c o n f i g u r a t i o n . From F i g u r e 2-21, the d i s t r i b u t i o n of the t r a n s v e r s e s e p a r a t i o n o f the v o r t e x rows i s p l o t t e d i n F i g u r e 2-22 f o r the 1 i n . and 3 i n . models. The i n c r e a s e of the l a t e r a l s p a c i n g i n the downstream d i r e c t i o n , t e n d i n g t o some uniform v a l u e , i s i n -d i c a t e d . I t i s shown t h a t the parameter b/h i s a l s o dependent on model o r i e n t a t i o n . F i g u r e 2-22 V a r i a t i o n o f l a t e r a l v o r t e x s p a c i n g f o r 1 i n . and 3 i n . a n g l e models 58 From the r e s u l t s , i t i s apparent t h a t wind t u n n e l i n t e r -f e r e n c e e f f e c t s do e x i s t , as i n d i c a t e d by the r e l a t i v e c o n f i n e -ment of the wake f o r the l a r g e r model. A d i s c u s s i o n of w a l l i n f l u e n c e on wake geometry i s presented i n Appendix I I . R e s u l t s on phase angle d i s t r i b u t i o n i n the wake f o r the p r e s s u r e tap model are p r e s e n t e d i n F i g u r e 2 - 2 3 . The l o n g i t u d i n a l phase v a r i a t i o n appears to become l i n e a r f a r downstream i n d i c a t i n g t h a t the v o r t i c e s have reached a uniform streaming v e l o c i t y and, thereby, c o n s t a n t l o n g i t u d i n a l s p a c i n g . The l i m i t e d l e n g t h of the t u n n e l t e s t s e c t i o n and c l a r i t y of the f l u c t u a t i n g p r e s s u r e s i g n a l s r e s t r i c t e d the l o n g i t u d i n a l measurements t o x/h = 1 3 . U s i n g the f a c t t h a t a phase d i f f e r e n c e o f 3 6 0 ° e x i s t s between the f l u c t u a t i n g p r e s s u r e s i g n a l s a s s o c i a t e d w i t h c o n s e c u t i v e v o r t i c e s i n one row, the c o r r e s p o n d i n g l o n g i t u d i n a l s p a c i n g was o b t a i n e d as shown i n F i g u r e 2 - 2 4 ( a ) . The curves i n d i c a t e t h a t the s p a c i n g of the v o r t i c e s i n c r e a s e s r a p i d l y behind the body and approaches a c o n s t a n t v a l u e a t l a r g e x/h. I t i s apparent t h a t the parameter a/h i s a l s o a f u n c t i o n of angle of a t t a c k . The d i m e n s i o n l e s s v o r t e x v e l o c i t y curves shown i n F i g u r e 2 - 2 4(b) e x h i b i t a s i m i l a r t r e n d as the l o n g i t u d i n a l s p a c i n g s i n c e the two v a r i a b l e s are d i r e c t l y r e l a t e d by the S t r o u h a l number [ e q u a t i o n ( 2 . 1 ) ] . Combining F i g u r e s 2 -22 and 2 - 2 4 ( a ) , d i s t r i b u t i o n of the c l a s s i c a l wake geometry parameter, b/a, f o r both models at v a r i o u s angles of a t t a c k , i s summarized i n F i g u r e 2 - 2 5 . With i n c r e a s i n g x, the g e n e r a l t r e n d i s f o r b/a to reach a.maximum i n the v i c i n i t y of x/h = 2 . 5 and then decrease q u i t e r a p i d l y ^ ^ 6 to a t t a i n some l i m i t i n g v a l u e . A c c o r d i n g to Karman, b/a should F i g u r e 2 - 2 3 L o n g i t u d i n a l v a r i a t i o n o f p h a s e a n g l e i n w a k e o f s t a t i o n a r y 3 i n . a n g l e m o d e l F i g u r e 2 - 2 5 L o n g i t u d i n a l d i s t r i b u t i o n s o f w a k e g e o m e t r y r a t i o f o r 1 i n . a n d 3 i n . a n g l e m o d e l s be approximately equal t o 0.36 behind the o b s t a c l e and then dimin-i s h w i t h downstream d i s t a n c e to the c l a s s i c a l value of 0.2 81. However, f o r some o r i e n t a t i o n s of the angle model an i n c r e a s e i n 5 b/a w i t h x was observed. Benard as w e l l has presented data i n d i c a t i n g a s i m i l a r r e v e r s a l of the t r e n d . The c o n s t a n t v a l u e s of the wake geometry v a r i a b l e s at l a r g e d i s t a n c e s downstream of the model, r e f e r r e d to as "near i n f i n i t y , " are p l o t t e d i n F i g u r e 2-26. The graphs c l e a r l y i n d i c a t e the de-pendence o f wake geometry on model o r i e n t a t i o n . Two curves f o r each of the l a t e r a l and l o n g i t u d i n a l spacings are shown wit h e i t h e r the maximum or p r o j e c t e d h e i g h t of the model as the non-dimensional-i z i n g l e n g t h . Note t h a t the use o f e does not s i g n i f i c a n t l y reduce the v a r i a t i o n of the wake parameter w i t h angle of a t t a c k . I t i s of i n t e r e s t to observe t h a t the d i s t r i b u t i o n of (b/a) i s i n the v i c i n i t y o f Karman's s t a b i l i t y v a lue of 0.281. Wake geometry data f o r an angle s e c t i o n have not been r e p o r t e d i n l i t e r a t u r e . However, numerous p u b l i c a t i o n s on wake c h a r a c t e r i s t i c s behind o b s t a c l e s of s i m p l e r shape are a v a i l a b l e and have been l i s t e d i n Table 2-1. The r e s u l t s f o r the angle model at a = -45° have reasonable s i m i l a r i t y w i t h the f l a t p l a t e c h a r a c t e r i s t i c s , and the data a t a = 135° compares with t h a t f o r a c i r c u l a r c y l i n d e r . However, such s i m i l a r i t y between the bodies cannot be g e n e r a l i z e d due to the complex geometry of the angle s e c t i o n . As a f i n a l summary of the experimental r e s u l t s f o r the s t a t i o n a r y angle s e c t i o n , a comparison of the S t r o u h a l number, wake geometry and drag c o e f f i c i e n t v a r i a t i o n s w i t h model o r i e n -t a t i o n are pr e s e n t e d i n F i g u r e 2-27. I t i s i n t e r e s t i n g to note Figure 2-26 Dis t r i b u t i o n s of the 'near i n f i n i t y ' values of the wake survey parameters for 1 i n . and 3 i n . angle models TABLE 2-1 Wake Geometry Parameters f o r V a r i o u s Bodies body i n v e s t i g a t o r (a/e)„ (b/e)m ( b / a ) m (u/v)„ ( v v ) » c i r c u l a r c y l i n d e r Benard (E) 4.68-6.43 0.15-0.49 average=0.32 Karman (E) 4.3 1.2 0.28 0.14 Fage and Johansen (E) 4.27 0.23 0.80 Rosenhead and Schwabe (E) =4.8 = 1.6 = 0.32 0.225 Schaefer and E s k i n a z i (E) 1.3-1.8 0.24-0.28 Ferguson (E) 0.32 0.82 normal f l a t p l a t e Karman (E) Heisenberg (T) Fage and Johansen (E) 5.50 5.45 5.25 1-7 1.54 0.31 0.283 0.20 0.229 0.77 angle p r e s e n t data (E) a = -45 c 5.58 1.77 0.32 0.77 s e c t i o n a = 135° 4.64 1.36 0.29 0.79 (E) e x p e r i m e n t (T) theory <7\ 66 the s i m i l a r i t y o f the d i s t r i b u t i o n s . A l l curves tend to peak near a = 2 5 ° , d i m i n i s h t o almost constant l e v e l s between a = 60° and 90°, and i n c r e a s e r a p i d l y a t a = 100° to a t t a i n near u n i f o r m v a l u e s over the remaining range. T h i s suggests s i g n i f i c a n t r e l a t i o n s h i p between the wake geometry and body aerodynamic c h a r a c t e r i c s . I n t r o d u c t i o n of the U n i v e r s a l S t r o u h a l Number, and r e p r e s e n t a t i o n of S t r o u h a l number as a f u n c t i o n of drag c o e f f i c i e n t by s e v e r a l i n v e s t i g a t o r s may be a t t r i b u t e d t o s i m i l a r o b s e r v a t i o n s o f the wake-body i n t e r a c t i o n . 2.5 C o n c l u d i n g Remarks Based on the experimental r e s u l t s the f o l l o w i n g g e n e r a l remarks can be made con c e r n i n g the aerodynamics of s t a t i o n a r y angle s e c t i o n s : (i) The d i s t r i b u t i o n s of the steady l i f t , drag and p i t c h i n g moment c o e f f i c i e n t s o b t a i n e d from balance measurements f o r the sharp-edged angle models and the commercial s t r u c t u r a l angles are i n good agreement. S i g n i f i c a n t v a r i a t i o n s i n the aerodynamic f o r c e s and moment with model o r i e n t a t i o n suggest the p o s s i b i l i t y of g a l l o p i n g a e r o e l a s t i c i n s t a b i l i t y . The p r e s s u r e i n t e g r a t e d , steady aerodynamic c o e f f i c i e n t s compare w e l l w i t h the balance r e s u l t s , thus i n d i c a t i n g e s s e n t i a l l y two-dimensional c o n d i t i o n o f the model and mean flow, ( i i ) The sharp-edged and s t r u c t u r a l angle models e x h i b i t comparable S t r o u h a l number d i s t r i b u t i o n s , . The S t r o u h a l 67 number v a r i e s s i g n i f i c a n t l y with model o r i e n t a t i o n but i s e s s e n t i a l l y independent of Reynolds number over the 4 4 range 10 to 15x10 . Basing the S t r o u h a l number on p r o j e c t e d model width reduces i t s dependency on angle of a t t a c k o n l y s l i g h t l y . The nondimensional resonant wind speed, being i n v e r s e l y r e l a t e d t o the S t r o u h a l number, i s a f u n c t i o n of model a t t i t u d e having a minimum near a = 30°. ( i i i ) The p r e s s u r e f l u c t u a t i o n s on the s u r f a c e of the model change wi t h angle of a t t a c k but are independent of 3 3 Reynolds number over the range 26x10 to 63x10 i n -v e s t i g a t e d . The v a r i a t i o n of the pressure c o e f f i c i e n t around the contour of the s e c t i o n i s s i g n i f i c a n t with a minimum value near the forward s t a g n a t i o n p o i n t and peak l e v e l c l o s e t o the p o i n t s of s e p a r a t i o n . The random modulation of the f l u c t u a t i n g p r essure s i g n a l can be as l a r g e as 2.5 i n d i c a t i n g s u b s t a n t i a l peak p r e s s u r e s . The f l u c t u a t i n g p r e s s u r e data confirms the f a m i l i a r 180° phase d i f f e r e n c e between the two apparent s i d e s of the model contour. In a d d i t i o n , however, phase d i f f e r e n c e s as l a r g e as 50° to 100° may e x i s t between neighbouring taps on each s i d e of the angle s e c t i o n . F o r t u n a t e l y , the e f f e c t of these l a r g e phase angles on the i n t e g r a t e d unsteady f o r c e and moment c o e f f i c i e n t s i s l e s s than 10%. (iv) The spanwise d i s t r i b u t i o n of the f l u c t u a t i n g p r essure c o e f f i c i e n t i s reasonably uniform over the l e n g t h of 68 the model examined. However, v a r i a t i o n i n phase i s p r e s e n t which would a f f e c t the c o r r e l a t i o n of the f l u c t u a t i n g f o r c e s along the model. T h i s l a c k of two-d i m e n s i o n a l i t y w i l l a i d i n r e d u c i n g p o s s i b l e v o r t e x induced v i b r a t i o n s , (v) The l a t e r a l and l o n g i t u d i n a l s p a c i n g s , together w i t h the v o r t e x v e l o c i t y i n c r e a s e r a p i d l y behind the body and g r a d u a l l y a t t a i n c o n s t a n t v a l u e s a t a l a r g e d i s t a n c e downstream (x/h>6). As expected, each parameter i s a f u n c t i o n of the a t t i t u d e Of the angle s e c t i o n . However, i t i s independent of Reynolds number i n the range 3 3 26x10 t o 60x10 i n v e s t i g a t e d . The dependence of the 'near i n f i n i t y ' v a l u e s of the wake geometry parameters on the o r i e n t a t i o n of the angle s e c t i o n i s s i m i l a r t o t h a t of the S t r o u h a l number and drag c o e f f i c i e n t . T h i s v a r i a t i o n a l s i m i l a r i t y s u b s t a n t i a t e s the e x i s t e n c e of a r e l a t i o n s h i p between the wake geometry and aerodynamic c h a r a c t e r i s t i c s . The wake survey r e s u l t s f o r the angle s e c t i o n mounted at a = -45° and 135° are comparable to those f o r a normal f l a t p l a t e and a c i r c u l a r c y l i n d e r , r e s p e c t i v e l y . The d i s t r i b u t i o n of the v o r t e x s p a c i n g r a t i o around Karman's s t a b i l i t y v a l u e of 0.281 i s of i n t e r e s t , (vi) Any d e s i g n c r i t e r i o n t o minimize v o r t e x induced v i b r a -t i o n s of angle s e c t i o n beams should c o n s i d e r the nature of the f o r c i n g f u n c t i o n , n a t u r a l frequency and i n h e r e n t s t r u c t u r a l damping. From the s t a t i o n a r y angle model i n v e s t i g a t i o n of the S t r o u h a l number and aerodynamic c h a r a c t e r i s t i c s , i t i s p o s s i b l e t o suggest guide l i n e s f o r the f i r s t two c o n s i d e r a t i o n s . Any o r i e n t a t i o n s i n the ranges -45° <_ a <_ 10° and 50° <_ a <_ 100° should be avoided s i n c e the magnitude of the unsteady f o r c e s (Cr,. ,, C-,) are near maximum. On the other hand, 1 d m s s s the resonant wind speed curve i n d i c a t e s t h a t angles of at t a c k near 30° should not be used i n order t o avoid the onset of v o r t e x v i b r a t i o n s . Hence, the use of angle beams i n open e n g i n e e r i n g s t r u c t u r e s should i n v o l v e s u f f i c i e n t damping or hig h n a t u r a l frequency t o prevent p o s s i b l e s t r u c t u r a l f a i l u r e due to vo r t e x resonance. Moreover, the angle s e c t i o n as a b l u f f c y l i n d e r may experience o s c i l l a t i o n s of a g a l l o p i n g nature. T h i s i s i n d i c a t e d by the d i s t r i b u t i o n s o f the steady l i f t , drag and p i t c h i n g moment which e x h i b i t l a r g e v a r i a t i o n s over the complete range of model o r i e n t a t i o n . In the f o l l o w -i n g chapter the qua s i - s t e a d y theory i n c o r p o r a t i n g the steady aerodynamic data i s employed t o p r e d i c t the occur-rence of the g a l l o p i n g i n s t a b i l i t y . 3. DYNAMICS OF AN ANGLE SECTION 3.1 P r e l i m i n a r y Remarks As i n d i c a t e d i n the i n t r o d u c t i o n , b l u f f c y l i n d e r s are s u s c e p t i b l e t o two d i s t i n c t forms of i n s t a b i l i t y , v o r t e x resonance and g a l l o p i n g , w i t h the v i b r a t i o n s o c c u r r i n g , p r e -dominately, i n one degree of freedom. The e x t e n t of the c o u p l i n g depends on the r e l a t i v e p o s i t i o n of e l a s t i c and i n e r t i a l axes. T h e r e f o r e , t o study the dynamics and a s s o c i a t e d aerodynamics of an angle s e c t i o n , the wind t u n n e l models were mounted on a support system p r o v i d i n g p l u n g i n g and/or t o r s i o n a l degree Is) of freedom. Of p a r t i c u l a r i n t e r e s t were the wind speed and angle of a t t a c k ranges conducive t o model v i b r a t i o n , and the e f f e c t of the r e s -onant motion on the important aerodynamic and wake parameters. In a d d i t i o n , i t was d e s i r a b l e t o i n t r o d u c e c e r t a i n a p p r o p r i a t e a n a l y t i c a l models t o p r e d i c t the dynamics of the system. E x p e r i m e n t a l l y , the frequency of the coupled motion, and the types and ranges of i n s t a b i l i t i e s were found to be s i m i l a r t o those i n the s i n g l e degree of freedom. Furthermore, the phase phenomenon between the p l u n g i n g and t o r s i o n a l modes confirmed the concept of two d i s t i n c t f a m i l i e s of coupled v i b r a t i o n s , one predominantly p l u n g i n g and the other t o r s i o n , 1 2 as r e p o r t e d by Kosko and Wardlaw. T h e r e f o r e , i t was p o s s i b l e t o o b t a i n p e r t i n e n t i n f o r m a t i o n about c r i t i c a l o r i e n t a t i o n s , wind speeds and the c h a r a c t e r i s t i c f e a t u r e s of the i n s t a b i l i t i e s by s t u d y i n g , both e x p e r i m e n t a l l y and t h e o r e t i c a l l y , the plung-i n g and t o r s i o n a l degrees of freedom, s e p a r a t e l y . In g e n e r a l , the i n s t a b i l i t y may be a combined e f f e c t of both v o r t e x resonance and g a l l o p i n g e x c i t a t i o n . T h i s s i t u a t i o n was observed f o r the p l u n g i n g degree of freedom a t s e v e r a l o r i e n t a t i o n s and damping l e v e l s . A j u d i c i o u s c h o i c e of damping or model s i z e was r e q u i r e d i n such i n s t a n c e s to i s o l a t e the two phenomena, thus p e r m i t t i n g separate study of each form of i n s t a b i l i t y . Based on t h i s c o n s i d e r a t i o n , the experimental i n v e s -t i g a t i o n was d i v i d e d i n t o two p o r t i o n s ; v o r t e x resonance, and g a l l o p i n g o s c i l l a t i o n s . However, f o r the t o r s i o n a l degree of freedom, no g a l l o p i n g was observed over the wide ranges of wind speed and model o r i e n t a t i o n i n v e s t i g a t e d , thus s u g g e s t i n g t h a t a s t r u c t u r a l angle s e c t i o n i s not l i k e l y t o e x h i b i t t h i s form of i n s t a b i l i t y i n t o r s i o n u n l e s s very low damping or r e l a t i v e l y h i g h wind speed i s encountered. T h i s o b s e r v a t i o n i s substan-t i a t e d by a t h e o r e t i c a l a n a l y s i s based on the n o n l i n e a r , q u a s i -steady approach. Of g r e a t e r importance i s the v i o l e n t , v o r t e x resonant type of i n s t a b i l i t y which was observed and i n v e s t i g a t e d w i t h the angle models a t v a r i o u s damping l e v e l s and model o r i e n t a t i o n s . Since q u a s i - s t e a d y g a l l o p i n g theory f o r the p l u n g i n g 17 case i s w e l l e s t a b l i s h e d , more a t t e n t i o n i s d i r e c t e d towards the development o f an analogous theory f o r the t o r s i o n a l mode. The problem i s somewhat complicated due to the f a c t t h a t the non-l i n e a r f o r c i n g f u n c t i o n (moment) depends on the instantaneous 14 19 angular p o s i t i o n as w e l l as the v e l o c i t y . S i s t o and I i , a n a l y z i n g the problem of s t a l l - f l u t t e r , e l i m i n a t e d t h i s compli-c a t i o n by assuming a mean moment d i s t r i b u t i o n as a f u n c t i o n of the i n s t a n t a n e o u s angle of a t t a c k a which i n t u r n was chosen to s a t i s f y the downwash a t the t h r e e - q u a r t e r chord p o i n t . T h i s e m p i r i c a l approach p a r a l l e l s the c l a s s i c a l f l u t t e r theory. How-ever, s i n c e the flow f i e l d s d u r i n g s t a l l and c l a s s i c a l f l u t t e r are q u i t e d i f f e r e n t , t h e r e i s some doubt as to the s i g n i f i c a n c e of the t h r e e - q u a r t e r chord p o i n t i n s t a l l f l u t t e r a n a l y s i s . A m o d i f i e d theory f o r the g a l l o p i n g i n s t a b i l i t y of b l u f f c y l i n d e r s i s p r e s e n t e d . T h i s chapter o u t l i n e s the important aspects o f the v o r t e x resonant and q u a s i - s t e a d y g a l l o p i n g t h e o r i e s w i t h a p p l i c a t i o n t o the s p e c i f i c c o n f i g u r a t i o n under i n v e s t i g a t i o n . A d e s c r i p t i o n of the a d d i t i o n a l i n s t r u m e n t a t i o n and experimental procedure i s p r e s e n t e d , f o l l o w e d by a comparison and d i s c u s s i o n of the a n a l y t i c a l and exp e r i m e n t a l r e s u l t s . A c o n c l u d i n g s e c t i o n summarizes the u s e f u l i n f o r m a t i o n concerning angle s e c t i o n s under dynamical c o n d i t i o n . 3.2 Ex p e r i m e n t a l Arrangement 3.2.1 Model Mounting System The i n v e s t i g a t i o n s i n t o the dynamic i n s t a b i l i t y of an angle s e c t i o n were conducted by mounting the angle models, pr e -v i o u s l y d e s c r i b e d , on a s u i t a b l y designed support system shown s c h e m a t i c a l l y i n F i g u r e 3-1. The mounting arrangement p r o v i d i n g p l u n g i n g and t o r s i o n a l degrees of freedom was e s s e n t i a l l y 4 9 composed of a l a t e r a l a i r b e a r i n g system and c r o s s - s p r i n g p i v o t s . The system of j o u r n a l a i r - b e a r i n g s l o c a t e d on a frame e n c i r c l i n g the t u n n e l t e s t s e c t i o n gave the model a p l u n g i n g degree of freedom normal t o the wind d i r e c t i o n . The c r o s s e d Lateral displacement transducer ^101 Torsional pivot unit o ° o Air bearing frame Wind tunnel Lateral shaft, • Model Scale. Lateral spring Torsional damper \ ] Lateral damper Torsional damper Counter weight Air bearing frame — Air bearing and shaft •i JTT ^]—*(TO BAM-I) Angular displacement transducer (b) F i a u r e 3-1 D e t a i l s of model support system with p l u n g i n g and t o r s i o n a l degrees of freedom (a) p l u n g i n g arrangement (b) t o r s i o n a l assembly 74 beam p i v o t u n i t s , p r o v i d i n g r o t a t i o n a l s t i f f n e s s f o r the t o r -s i o n a l mode, were f i x e d t o the l a t e r a l a i r b e a r i n g s h a f t s by s u i t a b l e b r a c k e t s and h e l d t r a n s v e r s e l y by the l a t e r a l c o i l s p r i n g s to make,the t w o . v i b r a t i o n a l systems independent. The model was f a s t e n e d to the f r e e ends of the p i v o t u n i t s by two L-shaped f i n g e r s . L a t e r a l and angular displacements of the model were 'detected by a i r - c o r e transformer and strain-gauge b r i d g e t r a n s d u c e r s , r e s p e c t i v e l y . V a r i a b l e damping was i n t r o -duced independently i n t o each degree of freedom system through e l e c t r o m a g n e t i c eddy-current dampers.. To permit i n v e s t i g a t i o n of the model dynamics i n the i n d i v i d u a l degrees of freedom, clamps were p r o v i d e d to e l i m i n a t e the unwanted mode. D e t a i l s of the l a t e r a l a i r b e a r i n g system and a u x i l i a r y i n s t r u m e n t a t i o n 49 50 designed by Smith have been r e p o r t e d i n the l i t e r a t u r e . ' However, s i n c e the c r o s s - s p r i n g p i v o t u n i t s were designed p a r t i c u l a r l y f o r the p r e s e n t r e s e a r c h programme a b r i e f summary of the p e r t i n e n t c o n s t r u c t i o n a l d e t a i l s i s p r o v i d e d . B a s i c a l l y , the p i v o t s c o n s i s t of two p a i r s of uniform, c r o s s e d t h i n - m e t a l s t r i p s r i g i d l y fixed.:, a t t h e i r ends to the two p a r t s between which r e l a t i v e angular movement i s r e q u i r e d . I t i s s i g n i f i c a n t t h a t t h i s d e v i c e i n t r o d u c e s a minimum of i n h e r e n t damping, and f o r s m a l l movement (<15°), the a x i s of r o t a t i o n remains e s s e n t i a l l y s t a t i o n a r y at the i n t e r s e c t i o n of the unde-72 f l e e t e d c r o s s e d s t r i p s . T h e o r e t i c a l i n v e s t i g a t i o n s i n the 73 d e s ign of c r o s s - s p r i n g p i v o t s have been presented by Eastman, 74 75 76 77 Haringx, W i t t r i c k ' and o t h e r s . Young conducted an e xperimental i n v e s t i g a t i o n and a r r i v e d at s e v e r a l e m p i r i c a l 75 relationships for the design c h a r a c t e r i s t i c s . Based on the 7 6 t h e o r e t i c a l development by Wittrick and the experimental information given by Young, a workable assembly was arrived at for supporting the angle models with a t o r s i o n a l degree of free-dom. The two beams, connected to suitable end brackets, crossed orthogonally at 87.3% ( i . e . , 1/2(1+/5/3)) of the distance from the moving end. This cross over position was selected because of i t s superior performance c h a r a c t e r i s t i c s as pointed out by Wittrick. The thin metal beams, made from blue tempered spring s t e e l , have a free working length of 2.80 i n . The p a r t i c u l a r cross-sectional dimensions, 0.030x0.500 i n . , were selected from consideration of r i g i d i t y and s t i f f n e s s . A counter weight (Figure 3-1(b)) was added to produce a pivot assembly for which the o v e r a l l centre of gravity of the o s c i l l a t i n g components of the pivot coincided with the i n e r t i a l axis of the model. 3.2.2 Instrumentation and Test Procedures The l a t e r a l and angular displacements of the angle models were recorded using the instrumentation layout shown i n Figure 3-2. The top a i r bearing shaft was of s u f f i c i e n t length 49 to project into the l a t e r a l displacement transducer. With a 10 kc sinusoidal s i g n a l , the interference of the aluminum shaft with the magnetic coupling between the co-axial c y l i n d r i c a l c o i l s gave r i s e to an amplitude modulated output proportional to the model displacement. A f u l l wave r e c t i f i e r and RC f i l t e r c i r c u i t was used for demodulating the high-frequency c a r r i e r . The t o r s i o n a l displacement of the o s c i l l a t i n g model was measured using a strain-gauge type of transducer incorporated in the 7 6 Lateral displacement transducer. Shaft Function generator Figure 3-2 Instrumentation layout for plunging and to r s i o n a l displacement measurements 77 lower c r o s s - s p r i n g p i v o t u n i t . M e t a l l i c f o i l s t r a i n gauges, bonded to the beams and connected to form a 4-arm Wheatstone b r i d g e c i r c u i t i n c o n j u n c t i o n w i t h an E l l i s B r i d g e - A m p l i f i e r -Meter, produced an e l e c t r i c a l s i g n a l c o r r e s p o n d i n g to the angular displacement. The model displacement s i g n a l s were re c o r d e d e i t h e r on a storage o s c i l l o s c o p e or V i s i c o r d e r . When amplitude modulation of the response s i g n a l s o c c u r r e d , maximum and minimum as w e l l as mean (using the rms v o l t m e t e r c i r c u i t ) d isplacements were re c o r d e d . C o n t r o l l e d damping, i n a d d i t i o n t o t h a t i n h e r e n t with the i n d i v i d u a l o s c i l l a t i n g support systems, was i n t r o d u c e d by means of e l e c t r o m a g n e t i c dampers, which d i s s i p a t e d energy through eddy c u r r e n t s . The amount of magnetic damping was pro-p o r t i o n a l t o the i n p u t d.c. c u r r e n t . For the l a t e r a l system, 49 the dampers designed by Smith were employed with the aluminum mounting s h a f t s b e i n g used as the d i s s i p a t i v e medium. Analogous, horseshoe shaped dampers wi t h t h i n copper s t r i p s i n t e r s e c t i n g the magnetic f i e l d i n the gaps were c o n s t r u c t e d f o r the t o r s i o n a l assembly. A v a r i a b l e a.c. c u r r e n t source was i n c o r p o r a t e d i n each damper arrangement t o e r a s e the u n d e s i r a b l e r e s i d u a l magnet-ism induced i n the i r o n cores o f the electromagnets. Using t h e s t a n d a r d technique o f l o g a r i t h m i c d e c r e m e n t , the e l e c t r o m a g n e t i c dampers were c a l i b r a t e d o v e r a r a n g e o f d.c. c u r r e n t u s i n g s u i t -50 a b l e s t r e a m l i n e d models i n p l a c e o f the angle s e c t i o n . The v i s c o u s n a t u r e o f t h e m a g n e t i c damping was i n d i c a t e d by the l i n e a r i t y o f the l o g a r i t h m i c decrement c u r v e s f r o m w h i c h the v a l u e s o f the damping c o e f f i c i e n t s were d e t e r m i n e d . To i n t r o d u c e 78 f u r t h e r damping d u r i n g the t o r s i o n experiments, the horseshoe electromagnets were r e p l a c e d by permanent magnets (Cinaudagraph Corp., Type 6.30540) w i t h approximate f i e l d s t r e n g t h of 5225 Gauss each. The magnitude o f the damping was v a r i e d by p l a c i n g aluminum s t r i p s of v a r i o u s t h i c k n e s s e s i n t o the gap between the p o l e s . From the S t r o u h a l number da t a g i v e n i n F i g u r e 2-11 and u s i n g e q u a t i o n (2*1), the approximate wind speeds f o r v o r t e x resonance of the 1 i n . and 3 i n . angle models were c a l c u l a t e d . The model o r i e n t a t i o n , system n a t u r a l frequency and model s i z e were the parameters a f f e c t i n g the resonant v e l o c i t i e s . Exten-s i v e p l u n g i n g and t o r s i o n a l amplitude measurements were conducted around the fundamental resonant c o n d i t i o n s w i t h i n c r e a s i n g and d e c r e a s i n g wind speeds f o r v a r i o u s model a t t i t u d e s and damping l e v e l s . No d e t e c t a b l e sub- or h i g h e r - harmonic resonance was observed d u r i n g any of the amplitude t e s t s . The wake and aerodynamic c h a r a c t e r i s t i c s a s s o c i a t e d with an angle s e c t i o n e x p e r i e n c i n g v o r t e x e x c i t e d motion were obtained u s i n g the procedure d e s c r i b e d i n Chapter 2 ( s e c t i o n 2.3). A combination of the instrument l a y o u t s shown i n F i g u r e s 2-4 and 3-2 was used f o r measuring the f r e q u e n c i e s of v o r t e x shedding and c y l i n d e r o s c i l l a t i o n . Corresponding phase data between the p r e s s u r e and displacement s i g n a l s was o b t a i n e d over the wind speed ranges f o r which the c y l i n d e r and v o r t e x shedding f r e q u e n c i e s c o i n c i d e d . The s i g n a l s were averaged on V i s i c o r d e r c h a r t s . Inves-t i g a t i o n s of the wake geometry, as w e l l as the amplitude and phase of the f l u c t u a t i n g s u r f a c e p r e s s u r e s f o r the o s c i l l a t i n g model were conducted a t wind speeds corresponding to peak reson-ance. Due t o the dynamical c o n d i t i o n of the angle s e c t i o n , the v o r t e x v e l o c i t y was c a l c u l a t e d u s i n g the m o d i f i e d formula JV _ / T"v / fn \ r ,a\ (3.1) The f i n a l r e s u l t s were compared wi t h the c o r r e s p o n d i n g valmes f o r the s t a t i o n a r y model. For the g a l l o p i n g i n s t a b i l i t y , i n v e s t i g a t i o n of the model dynamics a t v a r i o u s angles of a t t a c k and damping l e v e l s was con-ducted over the wind speed range 0-70 f t / s e c . I f o s c i l l a t i o n s o c c u r r e d , f u r t h e r i n t e n s i v e study was conducted f o r i n c r e a s i n g and d e c r e a s i n g wind speeds w i t h the model s t a r t i n g from r e s t or from the e x i s t i n g displacement. Depending on the o s c i l l a t o r c h a r a c t e r i s t i c s a t some o r i e n t a t i o n s , the model r e q u i r e d an i n i t i a l d isplacement to s t a r t o s c i l l a t i n g a f t e r which i t would a t t a i n a l a r g e r , s t a b l e amplitude. The time taken f o r g a l l o p i n g o s c i l l a t i o n s t o b u i l d - u p from a predetermined i n i t i a l displacement to 95% of the f i n a l s t a b l e amplitude was o b t a i n e d by d i s p l a y i n g the s i g n a l on a storage o s c i l l o s c o p e . T h i s gave an i n d i c a t i o n of the s e v e r i t y of the i n s t a b i l i t y . 3.3 Response of an Angle S e c t i o n w i t h Combined P l u n g i n g and T o r s i o n a l Degrees of Freedom The model amplitude data (Figure 3-3) was o b t a i n e d f o r the 3 i n . dynamic angle model mounted a t a D = -45° and with the a x i s of r o t a t i o n of the t o r s i o n a l system c o i n c i d i n g w i t h shear c e n t r e of the angle s e c t i o n . The p l u n g i n g and t o r s i o n a l s p r i n g s t i f f n e s s e s were s e l e c t e d t o p r o v i d e a frequency r a t i o 80 0.151 1 1 1 1 n = 0.01952 0 P0= 0.00513 Sl = 2.92 0.10-Motion with 0 0.05-0 1 2 3 4 5 u y F i g u r e 3 - 3 - i Displacement measurements f o r angle model at a 0 = - 4 5 ° with combined p l u n g i n g and t o r -s i o n a l degrees of freedom (plunging g a l l o p i n g i n i t i a t e d below t o r s i o n a l resonance) 81 i-ll-t 0 0.15 0.10 0 0.05 / M A X I I 1 1 1 i T ne= 0.01952 j i i j Motion with Pe= 0.00513 2.92 i i 1 Torsional 1 j ' s ! ' '"e resonance j I A maximum A i 1 ? • mean A 1 |A T minimum _ A 1 i f ' A , A S • A V y^ V A X A A j ^ - T ^ A • Jkv*-^, T * ^ « ^ A 1 u. Figure 3 - 3 - i i Displacement measurements for angle model at a G = -45° with combined plunging and to r s i o n a l degrees of freedom (plunging galloping i n i t i a t e d about t o r s i o n a l resonance) 82 r e p r e s e n t a t i v e of t y p i c a l s t r u c t u r a l angle beams, (O J _ / O J ^ = 2.92). n 6 n y Two clamping l e v e l s f o r the p l u n g i n g system were c o n s i d e r e d so as t o i n i t i a t e l a t e r a l g a l l o p i n g o s c i l l a t i o n s a t wind speeds below and above the t o r s i o n a l resonant c o n d i t i o n . T h i s p e r m i t t e d a study of the g a l l o p i n g motion e f f e c t s on the t o r s i o n a l reson-ance. The maximum damping o b t a i n a b l e w i t h the t o r s i o n a l e l e c -tromagnetic dampers was not s u f f i c i e n t t o r e s t r i c t the v i o l e n t , t o r s i o n a l v o r t e x resonance to the l i m i t o f the s u p p o r t i n g system (dotted l i n e a t & = 0.14 i n F i g u r e 3 - 3 - i i ) . Both the induced and n a t u r a l frequency displacement components f o r the p l u n g i n g and t o r s i o n a l o s c i l l a t i o n s are p l o t t e d . The presence of ampli-tude modulation, e x h i b i t e d a t p a r t i c u l a r wind speeds, i s i n d i c a t e d by the maximum, mean and minimum displacement v a l u e s . I t i s of i n t e r e s t t o note t h a t the response of the angle s e c t i o n over d i s c r e t e wind speed ranges can be c a t e g o r i z e d by the type of i n s t a b i l i t y and predominant degree of freedom as f o l l o w s : ( i ) v o r t e x e x c i t e d p l u n g i n g ; ( i i ) g a l l o p i n g induced p l u n g i n g ; ( i i i ) v o r t e x e x c i t e d t o r s i o n . No occurrence of n a t u r a l g a l l o p i n g i n s t a b i l i t y i n the t o r s i o n a l mode of v i b r a t i o n was observed over the wind speed range i n v e s t i -gated. The s e p a r a t i o n of the p l u n g i n g and t o r s i o n a l resonant c o n d i t i o n s , a r e s u l t of the d i f f e r e n c e i n the n a t u r a l frequency v a l u e s , i s s i m i l a r t o the v i b r a t i o n a l c h a r a c t e r i s t i c s of c o n t i n -2 uous angle beams except i n t h a t case many d i s c r e t e ranges of wind speed may be o b t a i n e d because of the i n f i n i t e number of p r i n c i p a l f r e q u e n c i e s . To b e t t e r understand the nature of the 83 i n s t a b i l i t y o f the r i g i d angle s e c t i o n , the amplitudes i n the i n d i v i d u a l modes of v i b r a t i o n were measured. The t e s t s were per-formed by clamping the t o r s i o n a l and l a t e r a l systems c o n s e c u t i v e -l y , thus o b t a i n i n g p l u n g i n g and t o r s i o n a l amplitude data as shown i n F i g u r e s 3 -4 and 3-5. The l a t e r a l and t o r s i o n a l damping l e v e l s were i d e n t i c a l to the v a l u e s used f o r the two degree of freedom i n v e s t i g a t i o n . The r e s u l t s i n d i c a t e t h a t , although the e l a s t i c and i n e r t i a l axes f o r the s e c t i o n do not c o i n c i d e , the coupled motion i s a t the n a t u r a l f r e q u e n c i e s f„ and f_, of the s i n g l e degrees n v "e J 2 — of freedom. T h i s i s due to the f a c t t h a t the r a t i o ma / I Q i s c o m p a r a t i v e l y s m a l l f o r an angle s e c t i o n , thus r e d u c i n g the coup-l i n g e f f e c t . Comparison of the response data r e v e a l s t h a t the coupled system e x h i b i t s p l u n g i n g motion which i s almost i d e n t i c a l to t h a t f o r the s i n g l e degree of freedom. On the other hand, the t o r s i o n a l amplitude depends on the r e l a t i v e p o s i t i o n of the p l u n g i n g g a l l o p i n g i n s t a b i l i t y . For the h i g h e r damping l e v e l (Figure 3 - 3 - i i ) , the t o r s i o n a l resonant displacements are s i m i l a r to those i n F i g u r e 3-5, c h a r a c t e r i z e d by a sharp peak and s i g n i f -i c a n t amplitude modulation. However, at the lower damping (Figure 3 - 3 - i ) , the t o r s i o n a l r e s u l t s do not resemble the normal resonant curve due to the presence of the p l u n g i n g , g a l l o p i n g motion. The resonant peak i s s u b s t a n t i a l l y reduced i n magnitude and the modulation c h a r a c t e r i s t i c i s v i r t u a l l y n o n - e x i s t e n t . The induced 6 curves near U = 1 are a r e s u l t of the l a t e r a l y resonant c o n d i t i o n . For h i g h e r wind speeds, the l a t e r a l g a l l o p -F i g u r e 3-4 Response o f a n g l e model a t a 0 = -45° w i t h p l u n g i n g d e g r e e o f f r e e -dom o n l y F i g u r e 3-5 Response curve f o r angle model at a Q = -45° with t o r s i o n a l degree of freedom only and r o t a t i o n a l a x i s at shear c e n t r e CD LT1 86 i n g i n s t a b i l i t y produces a t o r s i o n a l motion of frequency f n y which i n c r e a s e s i n magnitude p r o p o r t i o n a l to the p l u n g i n g displacement. However, f o r the lower damping case, the induced amplitude i s c o n t r o l l e d by the n a t u r a l t o r s i o n a l frequency u n t i l the range of v o r t e x resonance i s exceeded. U n f o r t u n a t e l y , the pl u n g i n g o s c i l l a t i o n rose to the maximum a l l o w a b l e displacement of the a i r - b e a r i n g support system d u r i n g the range of t o r s i o n a l resonance. T h e r e f o r e , the t o r s i o n a l curves f o r U y > 2.5 r e p r e -sent o n l y the a n t i c i p a t e d values f o r u n r e s t r i c t e d g a l l o p i n g mode. S i m i l a r induced p l u n g i n g motion of frequency f n i s observed at 6 the h i g h e r damping l e v e l d u r i n g the t o r s i o n a l resonance. A comment concerning the phase of the coupled motion i s a p p r o p r i a t e here. I t was observed t h a t the l a t e r a l and t o r s i o n a l displacements o f the same frequency were i n phase f o r the plung-i n g resonant and g a l l o p i n g o s c i l l a t i o n s , w h i le at. t o r s i o n a l r e s -onance the modes were 180° out of phase. T h i s suggests the presence of v i r t u a l c e n t r e s of r o t a t i o n , denoted as hinge p o i n t s . For the former case, the r e s u l t s i n F i g u r e 3-3-i i n d i c a t e t h a t the v i r t u a l c e n t r e of r o t a t i o n i s l o c a t e d f a r downstream at approximately x/h = 3 0 . On the other hand, the l a t t e r case (Figure 3 - 3 - i i ) corresponds t o a hinge p o i n t l o c a t e d j u s t forward of the i n e r t i a l a x i s . Thereby, depending on the predominant mode, the coupled o s c i l l a t i o n s can be co n s i d e r e d as r o t a t i o n a l motion about two d i f f e r e n t c e n t r e s . The f i r s t corresponds to e s s e n t i a l l y plung-i n g motion, while the other r e p r e s e n t s t o r s i o n . Thus, the present measurements c o n f i r m the e x i s t e n c e of the two f a m i l i e s of v i r t u a l 1 2 hinge p o i n t s p r e d i c t e d by Kosko and observed by Wardlaw f o r an 87 7 8 angle s e c t i o n beam. Garland has pres e n t e d s i m i l a r t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s f o r a c a n t i l e v e r , extended-channel beam. I t i s of i n t e r e s t t o note, t h a t the experimental r e s u l t s r e p o r t e d by Wardlaw on the dynamics of an angle beam are com-p a t i b l e , i n g e n e r a l , w i t h the trends o b t a i n e d from the presen t i n v e s t i g a t i o n s . The resonant v i b r a t i o n s can be p r e d i c t e d from the S t r o u h a l d a t a , w i t h o n l y a few e x c e p t i o n s . In a d d i t i o n , Wardlaw's measurements r e v e a l t h a t the o s c i l l a t i o n s a t many angles o f a t t a c k are predominantly resonant i n nature r a t h e r than g a l l o p i n g . N e v e r t h e l e s s , a t c e r t a i n o r i e n t a t i o n s , the o s c i l l a -t i o n s over d i s c r e t e wind v e l o c i t y ranges appeared t o e x h i b i t g a l l o p i n g c h a r a c t e r i s t i c s . In summary, the response of an angle s e c t i o n w i t h combined p l u n g i n g and t o r s i o n i n d i c a t e s t h a t the o s c i l l a t i o n s occur essen-t i a l l y i n one of the two degrees of freedom. The measurements of frequency and phase f u r t h e r s u b s t a n t i a t e t h i s o b s e r v a t i o n . Thereby, the type of i n s t a b i l i t y , r e l a t i v e amplitude and the co r r e s p o n d i n g c r i t i c a l wind speed can be determined by s t u d y i n g the system dynamics i n the i n d i v i d u a l degrees o f freedom. In what f o l l o w s , t h i s approach i s adopted t o understand the nature of the v o r t e x resonant and g a l l o p i n g i n s t a b i l i t i e s . 3.4 T h e o r e t i c a l Development D e r i v a t i o n of the governing p l u n g i n g and t o r s i o n a l equations o f motion and t h e i r s o l u t i o n s are given i n Appendix IV with a summary of the b a s i c r e l a t i o n s p r o v i d e d here. Assuming l i n e a r s p r i n g and v i s c o u s damping and e x p r e s s i n g the a p p l i e d 88 aerodynamic f o r c e F or moment M. i n c o e f f i c i e n t form, the y o equations of motion can be w r i t t e n as m V + r y y + k y y = i ^ V ' h l C ^ C y . y - t ) 0 . 2 ) i e • c e + k.e » i ^ t c ^ e ^ i ) 0.3) E v a l u a t i o n of these equations d u r i n g p a r t i c u l a r types of aero-dynamic e x c i t a t i o n s forms the s u b j e c t o f t h i s a n a l y s i s . 3.4.1 V o r t e x Resonance Since the p l u n g i n g and t o r s i o n a l equations are s i m i l a r i n form, o n l y the p l u n g i n g a n a l y s i s i s p r e s e n t e d . Assuming a 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 , e q u a t i o n (3.2) , i n nondimensional form, becomes + Y + Y = n . U a C p s i n t (3.4) g i v i n g the s t e a d y - s t a t e s o l u t i o n as v r U 2 = ^ c ? / ( i - n ; r + ( A ^ / i / < 3 - 5 ) E v a l u a t i o n of (3.5) g i v e s the f a m i l i a r resonance curves with damping c o e f f i c i e n t as the d i s t r i b u t i o n parameter. For the peak amplitude (ft v = 1.0), e q u a t i o n (3.5) reduces to 79 P a r k i n s o n e t a l c o n s i d e r e d a s l i g h t l y d i f f e r e n t approach to o b t a i n a s o l u t i o n of the form 89 Yma* = ^ i V C r S ' n * A F < 3 ' 7 ) which incorporates the experimentally observed vortex shedding phenomenon through the variables and w /u> * AF n c y y 3.4.2 Galloping I n s t a b i l i t y 17 Using the quasi-steady approach, which assumes no aero-dynamic hysteresis e f f e c t s i n the force and moment characteris-t i c s and the vortex shedding frequency to,be far removed from the cylinder values, the governing d i f f e r e n t i a l equations of motion (3.2) and (3.3) can be put i n the nondimensional form Y + Y = f , ( Y ) (3.8) ® + ® s /^ V®'®' (3'9) where the variables y y and uQ are modified mass and moment of i n e r t i a parameters, respectively. The function f y ( Y ) , incorpor-ating the aerodynamic and viscous forces, i s represented as a polynomial i n Y using the steady l i f t and drag r e s u l t s . Simi-l a r l y , f (0, 0) i s a polynomial related to the steady pitching moment with an assumption that the contourlines of C M (0, 0) 8 are l i n e a r and p a r a l l e l . For a system i n a i r , both y y and p Q << 1 making equations (3.8) and (3.9) amenable to quasi-linear solution techniques. 90 3.4.2.1 S i n g u l a r i t i e s and S t a b i l i t y i n the Small A n a l y t i c a l l y , the c o n d i t i o n of s t a b i l i t y of the motion i n 80 the s m a l l can be determined by i n v e s t i g a t i n g the s i n g u l a r i t i e s i n the phase p l a n e . For the p l u n g i n g system, equation (3.8) can be reduced to v = z (3.10) z = -Y + /.Jiz) g i v i n g az - Y + M / U Z ) — = S (3.11) <JY Z I t i s apparent t h a t the o n l y s i n g u l a r i t y f o r the system i s l o c a t e d at the o r i g i n of the phase pl a n e . S i m i l a r l y , the t o r s i o n a l e q u a t i o n (3.9) becomes S - X (3.12) g i v i n g X = - 0 + yu e f e (©,x) de x The s i n g u l a r i t i e s of (3.13) are l o c a t e d on the 0 a x i s at the p o i n t s g i v e n by the r o o t s of the p o l y n o m i a l equation ®n eU*l-(a , C * Lf) + Q C 1 e - a c 3 e , + . . . . + (.,)-aNcN®-H ) = 0 (3.14) 9 ' Again the o r i g i n r e p r e s e n t s one of the s i n g u l a r i t i e s . The remain-i n g r o o t s of the e q u a t i o n , b e i n g r e l a t i v e l y l a r g e , do not a f f e c t the c o n d i t i o n of s t a b i l i t y u n l e s s the t o r s i o n a l d i s t u r b a n c e s are severe. 91 The nature of the s i n g u l a r p o i n t s and the phase t r a j e c -t o r i e s i n t h e i r v i c i n i t y can be s t u d i e d through a l i n e a r a n a l y s i s of the system u s i n g the c h a r a c t e r i s t i c r o o t s (M,» = ^ ( U - U o ) ± / C ^ ( U - U „ ) ] 2 - 1 (3.15) f o r the p l u n g i n g system (3.9), and ( X J = M U s - U 0 ) i / [ £ ( U s - U e ) ] 2 - ( » + y u e U 2 c ; (3.16) ' 2 f o r the t o r s i o n a l r e l a t i o n (3.11). Si n c e and p g << 1, the r o o t s are complex conjugates i n d i c a t i n g the s i n g u l a r i t y a t the o r i g i n to be e i t h e r a c e n t r e or a focus depending on the magnitude of the r e a l p a r t . For the p a r t i c u l a r case when U = U 0 or U Q/s, the s i n g u l a r p o i n t i s a c e n t r e . These c r i t i c a l v a l u e s r e p r e s e n t the i n i t i a l wind v e l o c i t i e s f o r the o n s e t of the i n s t a b i l i t y . A l l o t h e r v a l u e s of U g i v e r i s e to a f o c u s , whose s t a b i l i t y , based on the s i g n of the r e a l p a r t , i s determined by the c r i t e r i o n s t a b l e (3.17) u n s t a b l e s t a b l e (3.18) u n s t a b l e These c o n d i t i o n s are based on the assumption t h a t U 0 and U 0/s are p o s i t i v e . However, i t i s p o s s i b l e f o r them to be n e g a t i v e as governed by the s i g n of the c o e f f i c i e n t a^ , which d e s i g n a t e s the s l o p e of the aerodynamic f o r c e or moment curve at the o r i g i n , and on the s i g n of the t o r s i o n a l parameter s. The u < u0 p l u n g i n g : u > ue U < U 0/5 t o r s i o n ; , . U > L L / * system i s c a l l e d a " s o f t " o s c i l l a t o r i f a^ or a^*s i s p o s i t i v e because f o r s u f f i c i e n t l y l a r g e value of U ( g r e a t e r than the c r i t i c a l v e l o c i t y ) , the model w i l l o s c i l l a t e from r e s t i n p r e -sence of an a r b i t r a r i l y s m a l l d i s t u r b a n c e . On the other hand, when i t i s n e g a t i v e , the system i s denoted as a "hard" o s c i l l a t o r s i n c e no v i b r a t i o n w i l l commence from r e s t . However, s u s t a i n e d o s c i l l a t i o n may occur i f an i n i t i a l d i s t u r b a n c e exceeding some r e q u i r e d minimum value i s p r o v i d e d . T h i s c r i t e r i o n i s analogous to the concept of n e g a t i v e and p o s i t i v e aerodynamic damping used by many i n v e s t i g a t o r s . Thus, from the s t a b i l i t y a n a l y s i s i n the s m a l l , p e r t i n e n t i n f o r m a t i o n about the c r i t i c a l angles of a t t a c k and wind speeds can be o b t a i n e d by examining the s t a t i o n a r y aerodynamic f o r c e and moment d i s t r i b u t i o n s g i v e n i n Chapter 2, F i g u r e 2-8. For the angle s e c t i o n i n p l u n g i n g , s t a b i l i t y can be s t u d i e d u s i n g Den 9 Hartog's c r i t e r i o n > 0 s t a b l e (3.19) < 0 u n s t a b l e which i n c o r p o r a t e s the steady l i f t and drag c o e f f i c i e n t s . I t i s apparent t h a t , i f the l i f t curve has s u f f i c i e n t l y l a r g e n e g a t i v e s l o p e , the angle s e c t i o n may be s e l f - e x c i t e d . For the t o r s i o n a l motion, such a simple r e l a t i o n i s not a v a i l a b l e , hence d i r e c t study of the system parameters i s r e q u i r e d . In the range -45° <_ aof_ 45°, s i s p o s i t i v e but becomes n e g a t i v e f o r ot0 > 45°. T h e r e f o r e , the s i g n of a^•s depends on-a^. As C M i s p r o p o r t i o n a l 6 to -C M, the system w i l l behave as a s o f t o s c i l l a t o r i n the range -45° <_ a0<_ 45° i f the s l o p e of C M p l o t i s n e g a t i v e . L i k e w i s e , f o r a Q> 45°, an angle s e c t i o n i s a s o f t o s c i l l a t o r i f the slope of the moment curve i s p o s i t i v e . Thus, as i l l u s t r a t e d i n F i g u r e 2 - 8 ( c ) , a s t r u c t u r a l angle member may o s c i l l a t e from r e s t i n a t o r s i o n a l mode f o r -45° <_ a 0 <_ 10° and 40° <_ a Q <_ 60°. 3.4.2.2 L i m i t C y c l e s and B u i l d - u p Time The g a l l o p i n g motion and i t s s t a b i l i t y i n the l a r g e can be s t u d i e d u s i n g the method of V a r i a t i o n of Parameters. For the p l u n g i n g system, the s o l u t i o n can be w r i t t e n as 7 = -YSJ9) (3.20) (p - 0 where 6y(Y) i s a pol y n o m i a l r e l a t e d t o f y ( Y ) . For the t o r s i o n a l mode, however, 0 does not reduce t o zero as i n the p l u n g i n g case but becomes a f u n c t i o n o f 0 and U g i v i n g the s o l u t i o n as Q * -©4(©) (3.21) where 6.(0) and K. are polynomials r e l a t e d t o f (0,0). The term D O 0 (1-K.) r e p r e s e n t s a reduced frequency parameter. 6 The amplitude of the s u s t a i n e d motion i s e v a l u a t e d by s e t t i n g Y = 0 = 0, and o b t a i n i n g the r e a l p o s i t i v e r o o t s , Y_. or 0j , of the a l g e b r a i c equations i^(Y) = 0 <3-22> SB(e) = 0 (3.23) The nature o f the response i s determined by Lyapunov's S t a b i l i t y c r i t e r i o n 94 m <0 p l u n g i n g : T 5 _ - 1 M 'Y = Yj t > O t o r s i o n : 3 ® 4© o o o r , i n t h e l i g h t o f (3.20) and (3.21) f > o p l u n g i n g : t o r s i o n : Y = Y; J 40 i: < o o o s t a b l e u n s t a b l e s t a b l e u n s t a b l e s t a b l e u n s t a b l e s t a b l e u n s t a b l e (3.24) (3.25) (3.26) (3.27) The t i m e o f a m p l i t u d e b u i l d - u p i s r e a d i l y o b t a i n e d f r o m (3.20) and (3.21) by e v a l u a t i n g t h e i n t e g r a l s A p l u n g i n g : (3.28) A t o r s i o n : id (3.29) where Y 0 , e 0 r e p r e s e n t s m a l l i n i t i a l d i s p l a c e m e n t s and Y.., Q.. a r e some p r e s c r i b e d f r a c t i o n s ( say 95%) o f t h e Y_. , 0.. v a l u e s , r e s p e c t i v e l y . 3.5 R e s u l t s and D i s c u s s i o n 3.5.1 V o r t e x R e s o n a n c e 3.5.1.1 M o d e l A m p l i t u d e - V e l o c i t y Measurements Shown i n F i g u r e 3-6 a r e t y p i c a l l a t e r a l and t o r s i o n a l d i s p l a c e m e n t s i g n a l s o b t a i n e d f o r w i n d s p e e d s c o r r e s p o n d i n g t o peak v o r t e x r e s o n a n c e . The s i g n a l s i n d i c a t e t h e s i n u s o i d a l wave-( a ; (b) K / U r e s = 0 ' 9 8 3 ) (C) 8 I M -• • • M B ! m wm ( d ) • . H i l i l l i llllfltllllAI ,1 (e) (4,/Ures= ° - 9 8 0 ) • (f) Figure 3-6 Typical displacement signals for angle model experiencing vortex excited plunging or torsional motion form and con s t a n t frequency o f the disp l a c e m e n t s . The p l u n g i n g s t e a d y - s t a t e amplitude i s uniform at peak resonance with some s l i g h t v a r i a t i o n s o c c u r r i n g o n l y a t wind speeds above and below resonance. On the oth e r hand, the t o r s i o n a l motion e x h i b i t s s u b s t a n t i a l random, amplitude modulation (Figure 3 - 6 ( f ) ) , which may a t t a i n a d e f i n i t e frequency a t o t h e r wind speeds as i l l u s -t r a t e d i n F i g u r e 3-7(a) and (b). When f u r t h e r above the r e s o -nant wind v e l o c i t y (Figure 3 - 7 ( c ) ) , the t o r s i o n a l displacement again shows c o n s i d e r a b l e random modulation. Thus, i n p r e s e n t i n g r e s u l t s f o r the t o r s i o n a l mode, i t i s important t o r e c o r d the maximum as w e l l as the mean amplitudes of o s c i l l a t i o n , s i n c e the peak displacement may be many times the average l e v e l . V o r tex resonance amplitude r e s u l t s f o r the 1 i n . and 3 i n . angle models a t v a r i o u s o r i e n t a t i o n s , damping l e v e l s and support c o n d i t i o n s are summarized i n F i g u r e s 3-8 through 3-11. The h o r i z o n t a l d o t t e d l i n e s a t Y = 0.70 and 0 = 0.14 i n d i c a t e the l i m i t s o f the mounting systems. F i g u r e 3-8 i l l u s t r a t e s the p l u n g i n g a m p l i t u d e - v e l o c i t y r e s u l t s a t the symmetrical angle of a t t a c k o f -45°. The p l o t s i n d i c a t e the presence of combined v o r t e x resonance and g a l l o p i n g e x c i t a t i o n , and the e f f e c t of damping on the~two types o f i n s t a b i l i t y . An i n c r e a s e i n damp-i n g or decrease i n model s i z e ( i . e . , mass parameter) e f f e c t -i v e l y s h i f t s the e n t i r e g a l l o p i n g curve t o the r i g h t on the v e l o c i t y a x i s t o g e t h e r w i t h a r e d u c t i o n i n v o r t e x resonant amplitude. For the lower damping l e v e l s , the amplitude became very l a r g e exceeding the l i m i t i n g displacement a t t a i n a b l e with the l a t e r a l system. Of course, the angle s e c t i o n e x h i b i t s 97 I PII min r « ii III ii n i l mill iiin'iiiii iii in mini nm uiMI Him1 II II in m minim llllilIIIIM u r n v v 'r t. H ^ ^ H l ^ ^ ^ n HHHre wwWf9 in/Ww* inrwlNI WR^PI WHHHIWIi^^H ii Ii •? _ i l J U U I I K ( a ) ( U 9 / U r « = 0 - 8 4 ° ) •m I f f l l t f f p i t i i p T t m TI • ' | H l i j u L I I 4 ,1 i l l MUli J J i J ld l J III lill (b) •r •i nun IH PT I'li.ilit i uniJidM.ifiu^y i iL » H 'T if"1111 n1 (c) ( U e / U r e s = ' « ) F i g u r e 3 - 7 T y p i c a l t o r s i o n a l displacement s i g n a l s f o r angle model e x p e r i e n c i n g v o r t e x e x c i t e d motion a t v a r i o u s wind speeds near resonance 98 0.81 0.4 —i r - 4 5 ° ->—id n y = 0.00648 P = 0.00618 F i g u r e 3-8 S e p a r a t i o n of v o r t e x resonance and g a l l o p i n g type of o s c i l l a t i o n by c h o i c e of damping or mass parameter f o r a Q = -45° 99 Figure 3-9 Plunging resonant curves for 3 i n . angle model at various orientations 100 0.15 CX= -45 o nQ= 0.01952 pQ= 0.002055 (a) A maximum o mean v minimum 0.5 1.0 , 1.5 U /U 6 res F i g u r e 3-10 T o r s i o n a l resonant curves f o r 3 i n . angle model at a 0 = centre Q = -45° with a x i s of r o t a t i o n at shear 101 F i g u r e 3-11 T o r s i o n a l r e s o n a n t c u r v e s f o r 3 i n . a n g l e model a t a G = -45° and 135° w i t h a x i s o f r o t a t i o n a t c e n t r e o f g r a v i t y 102 v o r t e x resonance a t o t h e r o r i e n t a t i o n s as w e l l (Figure 3-9) but the r e s u l t s show the peak displacements i n these cases may be s m a l l even a t low damping. Being d i r e c t l y r e l a t e d to the f o r c i n g f u n c t i o n , the peak amplitudes vary w i t h a t t i t u d e i n a manner s i m i l a r t o the unsteady aerodynamic c h a r a c t e r i s t i c s f o r the s t a t i o n a r y model (F i g u r e 2-18(a)). Besides the modulation c h a r a c t e r i s t i c , the t o r s i o n a l a m p l i t u d e - v e l o c i t y r e s u l t s i n F i g u r e s 3-10 and 3-11 i n d i c a t e the s e v e r i t y of the resonant c o n d i t i o n a t a 0 = -45° w i t h the p r e s -ence of a s i g n i f i c a n t l y sharp peak even at moderate damping l e v e l s . Furthermore, the displacement curves may be spread over a r e l a -t i v e l y l a r g e v e l o c i t y range as compared t o the p l u n g i n g case, s u g g e s t i n g a p o s s i b i l i t y f o r the t o r s i o n a l resonance to be c r i t i c a l . . From the d a t a g i v e n i n F i g u r e 3-10, i t i s apparent t h a t the t o r s i o n a l displacement decreases w i t h i n c r e a s e d damping, which i s t y p i c a l o f a resonant c o n d i t i o n . T r a n s f e r of the a x i s of r o t a t i o n from the shear c e n t e r (Figure 3-10) to the c e n t r e of g r a v i t y ( Figure 3-11(a)) has c o n s i d e r a b l e i n f l u e n c e on the dynam-i c s of the system. The s u b s t a n t i a l decrease i n amplitude i s a d i r e c t consequence of a r e d u c t i o n i n the f l u c t u a t i n g moment c o e f f i c i e n t accompanying such a t r a n s f e r . The occurrence of the s m a l l amplitude o s c i l l a t i o n s a t a c = 135° (Figure 3-11 (b)) i s a s s o c i a t e d w i t h a s i m i l a r v a r i a t i o n of the unsteady f o r c i n g f u n c t i o n w i t h angle of a t t a c k as observed d u r i n g the s t a t i o n a r y model measurements (Figure 2-18 ( c ) ) . The i n f l u e n c e of the mass and damping parameters on the v o r t e x resonance can be summarized i n the form of a s t a b i l i t y 2 2 diagram. F i g u r e s 3-12 and 3-13 i l l u s t r a t e such p l o t s f o r the 103 F i g u r e 3-12 S t a b i l i t y diagram f o r 3 i n . angle model exper-i e n c i n g v o r t e x e x c i t e d p l u n g i n g motion at a c = -45° 104 F i q u r e 3-13 S t a b i l i t y d i a g r a m f o r 3 i n . a n g l e model e x p e r -i e n c i n g v o r t e x e x c i t e d t o r s i o n a l m o t i o n a t a = -45° w i t h a x i s o f r o t a t i o n a t s h e a r c e n t r e 105 3 i n . a n g l e s e c t i o n a t a Q = -45° f o r the p l u n g i n g and t o r s i o n a l degrees of freedom, r e s p e c t i v e l y . The graphs show the v a r i a t i o n of the peak amplitude and c o r r e s p o n d i n g wind speed, as w e l l as the upper and lower v e l o c i t y bounds f o r some l i m i t i n g , permis-s i b l e displacement. Reduction of the i n s t a b i l i t y domain with i n c r e a s i n g damping or d e c r e a s i n g mass parameter i s apparent. As the mass parameter i s d i r e c t l y p r o p o r t i o n a l t o the model s i z e , a s m a l l e r angle s e c t i o n w i l l e f f e c t i v e l y reduce the resonant v i b r a t i o n s . The d o t t e d p o r t i o n s of the curves i n F i g u r e 3-13 2 extended t o 2g./n„U = 20 were ob t a i n e d by p r o j e c t i n g the mean 9' 6 res ^ peak displacement to 8 = 0.0045 r e p r e s e n t i n g the value f o r which the v e l o c i t y bounds were determined. Furthermore, the upper and lower l i m i t s s hould meet a t the resonant peak v e l o c i t y . Hence, 2 f o r v a l u e s of 2 3 Q/n QU > 20, t o r s i o n a l resonance w i l l be y o res v i r t u a l l y n o n - e x i s t e n t . I t should be noted, however, t h a t a sub-s t a n t i a l amount of damping i s r e q u i r e d t o reach t h i s c o n d i t i o n . For the g i v e n s e t of data, a 3.value g r e a t e r than 8% of the c r i t i c a l damping was necessary w i t h an i n e r t i a parameter of 0.01 which i s r e p r e s e n t a t i v e of 3 i n . s t r u c t u r a l aluminum ang l e s . Thus, the i n v e s t i g a t i o n suggests t h a t the t o r s i o n a l mode i s l i k e l y to be c r i t i c a l and may l e a d t o f a i l u r e o f the s t r u c t u r e . 3.5.1.2 Surface P r e s s u r e and Wake C h a r a c t e r i s t i c s A s s o c i a t e d w i t h O s c i l l a t i n g Angle Model To understand the i n f l u e n c e of the resonant motion on the fundamental parameters, such as the shedding p r o p e r t i e s of the v o r t i c e s , r e s u l t i n g unsteady s u r f a c e p r e s s u r e s and wake geometry, measurements were conducted w i t h the p r e s s u r e tap 106 angle modellat a Q = -45° over the wind speed range conducive to t h i s form of i n s t a b i l i t y . F i g u r e s 3-14 and 3-15 p r e s e n t these r e s u l t s f o r the p l u n g i n g and t o r s i o n a l degrees of freedom, r e s p e c t i v e l y . For the p l u n g i n g case, the graph i l l u s t r a t e s the phenomenon of v o r t e x capture where the frequency of the shedding v o r t i c e s i s c o n t r o l l e d by the c y l i n d e r o s c i l l a t i o n s over a f i n i t e range of wind v e l o c i t y around the resonant v a l u e . Above and below t h i s capture r e g i o n the v o r t e x frequency tends to f o l l o w the S t r o u h a l value obtained from s t a t i o n a r y model t e s t s . The c y l i n d e r frequency g e n e r a l l y remains c l o s e to the n a t u r a l frequency o f the undamped system. For the t o r s i o n a l mode, however, the v o r t e x f o r m a t i o n dominates the c y l i n d e r frequency over a l a r g e v e l o c i t y range w i t h the f r e q u e n c i e s c l o s e l y f o l l o w i n g the S t r o u h a l curve. T h i s t o r -s i o n a l phenomenon, i n c o n t r a s t to v o r t e x c a p t u r e , i s denoted as v o r t e x c o n t r o l . However, f o r u / u r e s > 1-2, there develops a d i v i s i o n of the c y l i n d e r frequency i n t o two branches each having d i s t i n c t l y d i f f e r e n t p e r i o d s of o s c i l l a t i o n . Branch 1, f o l l o w i n g the v o r t e x shedding frequency, corresponds to s m a l l amplitude displacements o c c u r r i n g d u r i n g motion b u i l d - u p or decay. For the l a r g e amplitudes, the frequency of o s c i l l a t i o n s approaches the t o r s i o n a l n a t u r a l frequency, f_ , thus momentarily d e s t r o y -"e i n g the phenomenon of v o r t e x c o n t r o l . I t appears t h a t v o r t e x shedding i n i t i a t e s the t o r s i o n a l v i b r a t i o n s which b u i l d up with the i n p u t of energy, but l a r g e r amplitudes cause l o s s of v o r t e x c o n t r o l l e a d i n g f i n a l l y t o the d i m i n i s h i n g of the o s c i l l a t i o n s . T h i s e x p l a i n s the h i g h l y modulated displacement s i g n a l s o b t a i n e d 107 F i g u r e 3-14 V a r i a t i o n o f c y l i n d e r and v o r t e x s h e d d i n g f r e -q u e n c i e s , p h a s e , and d i s p l a c e m e n t a m p l i t u d e n e a r r e s o n a n c e ( a D = -45°) 108 F i g u r e 3-15-i V a r i a t i o n of c y l i n d e r and v o r t e x shedding c h a r a c t e r i s t i c s with wind speed f o r a Q = -45° ( t o r s i o n a l amplitude, and frequency r e s u l t s ) 109 F i g u r e 3 - 1 5 - i i V a r i a t i o n of c y l i n d e r and v o r t e x shedding c h a r a c t e r i s t i c s w i t h wind speed f o r a 0 = - 4 5 ° (frequency, phase and mean t o r s i o n a l ampli-tude near resonance) 110 f o r wind speeds w e l l above U r e s as shown by the sample t r a c e i n F i g u r e 3-7 ( c ) . The d i s t r i b u t i o n of the phase l a g between the motion and the f o r c i n g f u n c t i o n v a r y i n g from n e a r l y in-phase at the b e g i n n i n g of resonance and approaching 180° when above u " r e g , i s t y p i c a l of a l i n e a r , damped, o s c i l l a t o r e x p e r i e n c i n g f o r c e d harmonic v i b r a t i o n . The c o n t r o l o f the t o r s i o n a l c y l i n d e r frequency by the v o r t e x e x c i t a t i o n i s c o n s i s t e n t w i t h the math-e m a t i c a l model, but the theory f a i l s t o p r e d i c t the p l u n g i n g v o r t e x capture phenomenon. The e f f e c t of model dynamics on the f l u c t u a t i n g s u r f a c e p r e s s u r e s was s t u d i e d a t the wind v e l o c i t y c o r r e s p o n d i n g to peak amplitude f o r the angle s e c t i o n a t a G = -45°. F i g u r e 3-16 summarizes the midspan and spanwise d i s t r i b u t i o n s of the mean f l u c t u a t i n g p r e s s u r e amplitude, modulation r a t i o and phase v a r i a t i o n s ; and i n c l u d e , f o r comparison, the s t a t i o n a r y model r e s u l t s . In both, the p l u n g i n g and t o r s i o n a l degrees of freedom, the p r e s s u r e c o e f f i c i e n t i s comparatively l a r g e r over the e n t i r e s u r f a c e of the model. For the p l u n g i n g case, the modulation r a t i o and phase v a r i a t i o n are c o n s i d e r a b l y l e s s ; w h i l e f o r the t o r s i o n a l mode, the r e s u l t s show onl y s l i g h t r e d u c t i o n . The span-wise v a r i a t i o n s of phase and amplitude are s t i l l p r e s e n t and have tre a d s s i m i l a r t o the s t a t i o n a r y model curves. A f i n a l summary showing the i n f l u e n c e of v o r t e x resonance on the magnitude of the unsteady aerodynamic c o e f f i c i e n t s i s g i v e n i n F i g u r e 3-17. For the t o r s i o n a l case, the unsteady l i f t and moment a t the c e n t r e of g r a v i t y decrease from the l a r g e p l u n g i n g v a l u e s t o almost the s t a t i o n a r y model r e s u l t s . On the o t h e r hand, the f l u c t u a t i n g moment about the shear c e n t r e shows I l l Model contour sides Spanwise tap numbers F i g u r e 3-16 M i d s p a n and s p a n w i s e d i s t r i b u t i o n s o f f l u c t u -a t i n g p r e s s u r e c o e f f i c i e n t , a m p l i t u d e m o d u l a t i o n r a t i o and p h a s e d u r i n g s t a t i c and dynamic c o n d i t i o n s o f t h e model 1 1 2 1.2 A C T Wll ith <(> • O O without (j) oiQ = - 4 5 C 0.8 C7 Cn7 0.4 about e.g. position Stationary model Plunging motion at U y/U r e s=0.983 Torsional motion at Ue / U res = 0 - 9 8 0 F i g u r e 3 - 1 7 C o m p a r i s o n o f f l u c t u a t i n g a e r o d y n a m i c c o e f f i c -i e n t s f o r s t a t i o n a r y a n d v o r t e x e x c i t e d c o n -d i t i o n s o f t h e a n g l e m o d e l a t a G = - 4 5 ° 113 a s u b s t a n t i a l i n c r e a s e . T h i s v a r i a t i o n o f p i t c h i n g moment w i t h a x i s p o s i t i o n s u b s t a n t i a t e s t h e o b s e r v e d d i f f e r e n c e i n t h e peak a m p l i t u d e s r e p o r t e d i n F i g u r e s 3-10 (a) and 3-11 (a) . A t a 0 = -45°, t h e f u n d a m e n t a l component o f t h e f l u c t u a t i n g d r a g c o e f f i c i e n t i s z e r o f o r a l l c a s e s . The e f f e c t o f t h e m i d s p a n p r e s s u r e p h a s e on t h e a e r o d y n a m i c c o e f f i c i e n t s compares w i t h t h e 10% d i f f e r e n c e o b t a i n e d f o r t h e s t a t i o n a r y model ( F i g u r e 2 - 1 8 ) . I n v e s t i g a t i o n o f t h e wake geo m e t r y d u r i n g peak p l u n g i n g and t o r s i o n a l r e s o n a n t c o n d i t i o n s was p e r f o r m e d u s i n g t h e p r e s -s u r e t a p model o r i e n t e d a t a Q = - 4 5 ° . The f l u c t u a t i n g p r e s s u r e d i s t r i b u t i o n s i n t h e l a t e r a l and l o n g i t u d i n a l d i r e c t i o n s ( F i g u r e 3-18) a r e s i m i l a r i n f o r m t o t h e s t a t i o n a r y model measurements p r e s e n t e d i n C h a p t e r 2. As a r e s u l t , t h e p l o t s o f v o r t e x v e l o c -i t y , and wake g e o m e t r y w i t h downstream c o o r d i n a t e ( F i g u r e 3-19) a l s o show c o m p a r a b l e t r e n d s a l t h o u g h t h e l i m i t i n g v a l u e s may be d i f f e r e n t i n m a g n i t u d e . A summary o f t h e 'near i n f i n i t y ' v a l u e s f o r t h e a n g l e s e c t i o n d u r i n g s t a t i c and dynamic c o n d i t i o n s i s shown i n F i g u r e 3-20. F o r t h e o s c i l l a t i n g m o d e l , t h e l o n g i t u d -i n a l s p a c i n g and v o r t e x v e l o c i t y r e m a i n e s s e n t i a l l y u n c h a n g e d . On t h e o t h e r hand, i t i s a p p a r e n t t h a t t h e l a t e r a l s p a c i n g o f t h e v o r t e x rows e x p e r i e n c e s s u b s t a n t i a l i n c r e a s e d u r i n g p l u n g i n g r e s o n a n c e . T h e r e i s a s i m u l t a n e o u s i n c r e a s e i n t h e s p a c i n g r a t i o as w e l l . N e v e r t h e l e s s , an i n t e r e s t i n g f e a t u r e becomes a p p a r e n t i f t h e p l u n g i n g d i s p l a c e m e n t i s u s e d i n c o n j u n c t i o n w i t h t h e model f r o n t a l w i d t h when n o n d i m e n s i o n a l i z i n g t h e row s p a c i n g . By t a k i n g t h e e f f e c t i v e b l o c k a g e t o e q u a l (2y + h ) , t h e l a t e r a l s p a c i n g p a r e m e t e r r e d u c e s t o a l m o s t t h e same v a l u e as t h e s t a t i o n a r y o r t o r s i o n a l model r e s u l t , t h u s s u g g e s t i n g 1 1 4 F i g u r e 3 - 1 8 V a r i a t i o n o f a m p l i t u d e a n d p h a s e o f f l u c t u a t i n g p r e s s u r e i n w a k e o f a n g l e m o d e l e x p e r i e n c i n g v o r t e x e x c i t e d m o t i o n a t a Q = - 4 5 ° ( a ) p l u n g i n g ( b ) t o r s i o n F i g u r e 3-19 L o n g i t u d i n a l v a r i a t i o n of wake survey parameters f o r p l u n g i n g and t o r s i o n a l c o n d i t i o n s of model at a D = -45° 116 r ' i g u r e 3 = 2 0 C o m p a r i s o n o f ' n e a r i n f i n i t y ' v a l u e s o f t h e w a k e s u r v e y p a r a m e t e r s f o r s t a t i o n a r y a n d v o r t e x e x c i t e d c o n d i t i o n s o f t h e a n g l e m o d e l a t a0 = - 4 5 ° t h a t the i n c r e a s e i n wake width i s e f f e c t i v e l y caused by the amplitude of the p l u n g i n g o s c i l l a t i o n s . In summary, the wake survey and v o r t e x shedding frequency i n v e s t i g a t i o n s i n d i c a t e t h a t the t o r s i o n a l motion of the angle s e c t i o n has v i r t u a l l y no e f f e c t on the v o r t e x shedding frequency and wake c h a r a c t e r i s t i c s . T h i s i s i n c o n t r a s t to the pronounced enlargement of the wake width and v o r t e x capture phenomenon which occurs w i t h the p l u n g i n g resonant motion. I t should be noted t h a t these l i m i t e d r e s u l t s f o r the f l u c t u a t i n g p r e s s u r e and wake geometry may not be e n t i r e l y r e p r e s e n t a t i v e o f what may 33 occur a t o t h e r wind speeds and damping c o n d i t i o n s . Ferguson, 47 74 Feng and P a r k i n s o n , e t a l , d u r i n g t e s t s on c i r c u l a r and D-s e c t i o n c y l i n d e r s , observed decrease of the p r e s s u r e and wake parameters near the resonant peak. The parameters a l s o depended on the magnitude o f the o s c i l l a t i o n s c o n t r o l l e d by the damping l e v e l . S i m i l a r behaviour i s a n t i c i p a t e d f o r the angle s e c t i o n where, l i k e the D - s e c t i o n , s e p a r a t i o n i s c o n t r o l l e d by the edge geometry. 3.5.1.3 Resonant Theory P r e d i c t i o n s The a p p l i c a b i l i t y of the a n a l y t i c a l models can be t e s t e d by comparing the t h e o r e t i c a l l y p r e d i c t e d and e x p e r i m e n t a l l y measured v a l u e s . However, i t may be p o i n t e d out t h a t s i n c e e x p e r i m e n t a l data i s i n c o r p o r a t e d i n the t h e o r e t i c a l c a l c u l a t i o n s , the f i n a l v a l i d i t y of the a n a l y t i c a l s o l u t i o n s r e s t s p a r t i a l l y on the accuracy of the e x p e r i m e n t a l r e s u l t s . For example, c o n s i d e r the p l u n g i n g motion of the angle s e c t i o n mounted at ct0 = -45° (Figures 3-14 and 3-17). Here 6 y = 0.00414. n y = 0.00505 , 118 f c y / f n y = 1-0.0, U y / U r e s = 0.983, U r e s = ° ' 9 8 6 ' $ A F = 6 7 ° ' a n d C t , = 1.057. T h e r e f o r e , f r o m e q u a t i o n s (3.6) and ( 3 . 7 ) , Y x max = 0.63 and 0.55, r e s p e c t i v e l y . C o m p a r i s o n w i t h t h e e x p e r i m e n t a l v a l u e o f Y = 0 . 3 9 i n d i c a t e s a c l o s e r agreement w i t h t h e max 3 a n a l y t i c a l s o l u t i o n d e v e l o p e d by P a r k i n s o n , e t a l , s i n c e t h e v o r -t e x c a p t u r e e f f e c t s a r e i n c o r p o r a t e d . N o t e , however, 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 a r e l a r g e r t h a n t h e o b s e r v e d a m p l i t u d e s u g g e s t -i n g t h e i n f l u e n c e o f a d d i t i o n a l f a c t o r s n o t i n c l u d e d i n t h e a n a l y t i c a l c o n s i d e r a t i o n s . I t i s a n t i c i p a t e d t h a t w i n d t u n n e l w a l l i n t e r f e r e n c e i s one s o u r c e o f d i s c r e p a n c y , s i n c e t h e d i s -c u s s i o n i n A p p e n d i x I I i n d i c a t e s an i n c r e a s e i n t h e u n s t e a d y f o r c e w i t h f l o w c o n f i n e m e n t . L a c k o f s p a n w i s e c o r r e l a t i o n o f t h e un-s t e a d y a e r o d y n a m i c s w i l l a l s o c o n t r i b u t e t o r e d u c t i o n i n t h e ex-p e r i m e n t a l a m p l i t u d e . F o r t h e t o r s i o n a l mode, c o n s i d e r t h e d a t a g i v e n i n F i g u r e s 3 - 1 5 - i i and 3-17. Here g Q = 0.0235, n Q = 0.01013, fn / f = 1.01, U Q/U = 0.980, U = 0.986, = 66° and c 1 n ' %' r e s r e s T A F C-, = 0.213. T h e r e f o r e , f r o m e q u a t i o n s a n a l o g o u s t o (3.6) and ( 3 . 7 ) , t h e peak d i s p l a c e m e n t s a r e G = 0.045 and 0.039, r e s p e c -t i v e l y . Good a g r e e m e n t w i t h t h e e x p e r i m e n t a l v a l u e o f 0.041 s u b s t a n t i a t e s t h e a n a l y s e s . 3.5.2 G a l l o p i n g M o t i o n 3.5.2.1 T h e o r e t i c a l P r e d i c t i o n s f o r P l u n g i n g Degree o f Freedom The a p p l i c a t i o n o f t h e t h e o r e t i c a l a n a l y s i s t o a p h y s i c a l s y s t e m s u c h as an a n g l e s e c t i o n , f o r p r e d i c t i o n o f t h e g a l l o p i n g o s c i l l a t i o n s , r e q u i r e s a e r o d y n a m i c f o r c e d i s t r i b u t i o n s a t v a r i o u s o r i e n t a t i o n s as the i n p u t data. Knowledge of the model ampli-t u d e - v e l o c i t y curves can be o b t a i n e d from the c a l c u l a t i o n of the s t a r t i n g v e l o c i t y U Q and the a p p r o p r i a t e r o o t s of the 6^ (Y) p o l y n o m i a l . Determination of the model b u i l d - u p time versus v e l o c i t y p r o v i d e s u s e f u l i n f o r m a t i o n concerning the s e v e r i t y of the g a l l o p i n g o s c i l l a t i o n s . U s ing the a c c u r a t e l y measured steady l i f t and drag d i s -t r i b u t i o n s (Figure 2 - 7 ) f t h e f o r c e curves a t v a r i o u s a D (Figure 3-21) were expressed as polynomials from which the amplitude-v e l o c i t y curves were o b t a i n e d . The a n a l y t i c a l data i s summa-r i z e d i n F i g u r e 3-22 i n the form of a t h r e e - d i m e n s i o n a l graph of p l u n g i n g amplitude Y as a f u n c t i o n of U and a 0 f o r a s e t of t y p i c a l mass and damping parameters. The q u a s i - s t e a d y a n a l y s i s p r e d i c t s two l a r g e r e g i o n s of i n s t a b i l i t y w i t h two s m a l l e r areas i n be-tween. For comparison, the v o r t e x resonant v e l o c i t y curve from F i g u r e 2-10 i s a l s o i n c l u d e d . I t i s apparent t h a t there are r e g i o n s i n which the angle s e c t i o n i s s u s c e p t i b l e to pure v e r t e x resonance t o g e t h e r w i t h areas of combined v o r t e x and g a l l o p i n g o s c i l l a t i o n s . T h i s phenomenon of combined e x c i t a t i o n s was con-fir m e d by the e x p e r i m e n t a l r e s u l t s g i v e n i n F i g u r e 3-8. As shown by p r e v i o u s i n v e s t i g a t o r s , " ^ ' " ^ the governing e q u a t i o n of motion can be transformed by i n t r o d u c i n g s t r e t c h e d or reduced v a r i a b l e s Y*> U*, T * to p r o v i d e a u n i v e r s a l s o l u t i o n where the r e s u l t s are independent of the system parameters n y and 3 . The reduced a m p l i t u d e - v e l o c i t y curve i n i t i a t e s from the a b s c i s s a a t U* equal to u n i t y . In a d d i t i o n , v a r i a t i o n of the b u i l d - u p time w i t h v e l o c i t y i s o n l y a f u n c t i o n of the i n i t i a l d isplacement Y Q * when Y . * i s taken to be 95% or some oth e r con-Figure 3-21 Polynomial curve f i t of t y p i c a l l a t e r a l force c o e f f i c i e n t data i — • t o o QJ tr, 122 s t a n t v a l u e of Y j * - Comparison of the t h e o r e t i c a l amplitude-v e l o c i t y and b u i l d - u p time curves w i t h the e x p e r i m e n t a l l y ob-t a i n e d r e s u l t s i s presented i n the f o l l o w i n g s e c t i o n . 3.5.2.2 P l u n g i n g Amplitude and B u i l d - u p Time Measurements F i g u r e s 3-23 and 3-24 summarize the a m p l i t u d e - v e l o c i t y data f o r the 1 i n . angle model o r i e n t e d a t a 0 = -45° and 90° wit h v a r i o u s damping l e v e l s . For comparison, the i n i t i a l g a l l o p -i n g r e s u l t s o b t a i n e d w i t h the 3 i n . model a t a D = -45° (Figure 3-8) are i n c l u d e d . Note t h a t a s m a l l e r model extends the dimen-s i o n l e s s v e l o c i t y and amplitude ranges, b e s i d e s s e p a r a t i n g the v o r t e x and g a l l o p i n g i n s t a b i l i t i e s . The experimental r e s u l t s at both o r i e n t a t i o n s i n d i c a t e the angle s e c t i o n t o be a s o f t o s c i l l a t o r undergoing a d i s c o n t i n u o u s jump phenomenon. In a d d i -t i o n , an i n c r e a s e i n 3 or a r e d u c t i o n of n d e l a y s the i n i t i -y y 1 a t i o n of g a l l o p i n g and has a tendency t o s h i f t the curves toward l a r g e r v e l o c i t y and displacement v a l u e s . Using the a p p r o p r i a t e s t a r t i n g wind v e l o c i t i e s , the r e s u l t s are a l s o p l o t t e d i n the form of reduced parameters. C o l l a p s i n g of the experimental data to approximately the same d i s t r i b u t i o n confirms the qu a s i - s t e a d y theory. A s l i g h t i n c o n s i s t e n c y at low Y* i s due t o the d i f f i -c u l t y i n d e t e r m i n i n g the exact s t a r t i n g v e l o c i t y , and the presence of extraneous i n f l u e n c e s such as t u n n e l v i b r a t i o n , v o r t e x shed-d i n g , and u n c e r t a i n t y of damping a t s m a l l displacement and i t s minor f l u c t u a t i o n s . The d i f f e r e n c e between the p o s i t i o n s of the knee of the t h e o r e t i c a l and measured curves may be a t t r i b u t e d to s l i g h t i n a c c u r a c i e s i n the steady f o r c e d i s t r i b u t i o n near the o r i g i n and the co r r e s p o n d i n g p o l y n o m i a l r e p r e s e n t a t i o n used i n the a n a l y s i s . 123 F i g u r e 3-23 G a l l o p i n g a m p l i t u d e - v e l o c i t y r e s u l t s f o r a n g l e model a t a Q = -45° and t h e i r c o m p a r i s o n w i t h t h e o r y 1 2 4 F i g u r e 3 - 2 4 G a l l o p i n g a m p l i t u d e - v e l o c i t y r e s u l t s f o r a n g l e m o d e l a t " a Q = 9 0 ° a n d t h e i r c o m p a r i s o n w i t h t h e o r y 125 F u r t h e r measurements were conducted a t other angles of a t t a c k which g e n e r a l l y confirmed the presence or absence of the g a l l o p i n g i n s t a b i l i t y as p r e d i c t e d by the theory (Figure 3-22) . One r e g i o n of p a r t i c u l a r i n t e r e s t i s the range -15° <_ a <_ 15° where the theory p r e d i c t s two s m a l l zones of i n s t a b i l i t y . The experiment s u b s t a n t i a t e d the e x i s t e n c e of the g a l l o p i n g motion over approximately the same range of angle of a t t a c k except that the amplitudes were s l i g h t l y l a r g e r i n magnitude and the two zones were connected t o g e t h e r t o form one continuous r e g i o n . The b u i l d - u p time r e s u l t s f o r the 1 i n . angle model a t a 0 = 90° are presented i n F i g u r e 3-25. The measurements were conducted a t the damping l e v e l s used i n the model amplitude i n v e s t i g a t i o n s , and the i n i t i a l d isplacements, Y Q , were s e l e c t e d t o g i v e approximately the same value of the reduced v a r i a b l e Y* . F i g u r e 3-25(a) shows an upward s h i f t o f the t i m e - v e l o c i t y curve w i t h i n c r e a s i n g £ s i m i l a r t o t h a t observed f o r the model y y amplitude c h a r a c t e r i s t i c s . In the reduced t i m e - v e l o c i t y p l o t (Figure 3-25(b)), the data f o r the d i f f e r e n t damping l e v e l s c o l l a p s e very n e a r l y to one curve and compare w e l l with the theor-e t i c a l p r e d i c t i o n . The lower value of the experimental r e s u l t s a t s m a l l U* i s due t o the d i f f e r e n c e i n p o s i t i o n of the theor-e t i c a l and experimental d i s c o n t i n u o u s jumps (Figure 3-24(b)). The s c a t t e r of b u i l d - u p time data may be a t t r i b u t e d t o the s l i g h t d i f f e r e n c e s i n the co r r e s p o n d i n g amplitude r e s u l t s and the f a c t t h a t the Y* v a l u e s are not i d e n t i c a l f o r a l l B Y l e v e l s . The r e s u l t s , i n g e n e r a l , i n d i c a t e a m o n o t o n i c a l l y d e c r e a s i n g b u i l d - u p time with i n c r e a s i n g v e l o c i t y . T h e r e f o r e , g a l l o p i n g o s c i l l a t i o n s are more severe at h i g h e r wind speeds s i n c e the time f o r b u i l d -126 60 40 o i o 20 Y0 =0.01107 n y = 0.000733 (a) 60 40 h 20 0 2 4 6 y 1 1 1 1 1 (b) Py % o 0.000844 2.20 0.00503 ./Theory * 0.001313 3.52 0.00500 - Y*= 0.00500 o ? 0.001452 3.60 0.00459 -° A • 0.00177 4.94 0.00536 -• \ — X \ . o — • Q. ft -i 1 i 1 i 1 1 l • 1.0 1.4 1.8 2.2 2.6 3.0 u F i g u r e 3-25 Comparison of experimental and t h e o r e t i c a l amplitude b u i l d - u p time f o r angle model at a D = 90° up i s s h o r t e r and t h e r e s u l t i n g d i s p l a c e m e n t a m p l i t u d e i s l a r g e r . 3.5.2.3 T h e o r e t i c a l P r e d i c t i o n s f o r T o r s i o n a l Degree o f Freedom As d i s c u s s e d b e f o r e i n s e c t i o n 3.4.2.1, t h e s t a b i l i t y a n a l y s i s o f t h e s i n g u l a r i t y p r e d i c t s s e l f - e x c i t e d o s c i l l a t i o n s i n t h e r a n g e s -45° ± a0 <_ 10° and 40° a D <_ 6 0 ° . However, f o r v a r i o u s o r i e n t a t i o n s o f t h e 1 i n . and 3 i n . a n g l e models w i t h a v a r i e t y o f damping and m o u n t i n g c o n f i g u r a t i o n s , no g a l l o p i n g i n s t a b i l i t y i n t h e t o r s i o n a l d e g r e e o f f r e e d o m c o u l d be i n d u c e d . e i t h e r f r o m r e s t o r w i t h l a r g e i n i t i a l a m p l i t u d e . E x t r a n e o u s v i b r a t i o n o f t h e w i n d t u n n e l and model s u p p o r t s y s t e m a t h i g h w i n d s p e e d s l i m i t e d t h e t e s t s t o a r a n g e b e l o w a p p r o x i m a t e l y 50 f t / s e c . The a b s e n c e o f t h e g a l l o p i n g o s c i l l a t i o n s c a n be e x p l a i n e d u s i n g t h e q u a s i - s t e a d y t h e o r y . F o r example, t h e r e l e v a n t p o r t i o n o f t h e s t e a d y a e r o d y n a m i c moment d i s t r i b u t i o n f r o m t h e s t a t i o n a r y model i n v e s t i g a t i o n ( F i g u r e 2-7) i s r e p l o t t e d i n F i g u r e 3-26 f o r a Q = - 4 5 ° . E v a l u a t i o n o f t h e moment c o e f f i c i e n t u s i n g e q u a t i o n s g i v e n i n A p p e n d i x IV g i v e s a g r a p h o f (0, 0) as shown i n F i g u r e 3-27. The r e p r e s e n t a t i v e c u r v e s a r e l i n e s o f c o n s t a n t C w h i c h a p p e a r n e a r l y l i n e a r and p a r a l l e l , t h u s s u b s t a n t i a t i n g t h e s i m p l i f i c a t i o n i n t r o d u c e d by t a k i n g C.. as a f u n c t i o n o f 5 . The JXLQ r e s u l t i n g d i s t r i b u t i o n o f C.. (£) i s p l o t t e d i n F i g u r e 3-28 and a p p r o x i m a t e d by a 1 7 t h d e g r e e p o l y n o m i a l . O n l y t h e p o s i t i v e v a l u e s o f £ a r e shown s i n c e t h e moment i s s y m m e t r i c a b o u t t h e o r i g i n f o r a c = - 4 5 ° . U s i n g t h e c o e f f i c i e n t s o f t h e C U ) e x p r e s s i o n i n t h e e p o l y n o m i a l s f o r 6„ and K . t h e d i s p l a c e m e n t and r e d u c e d f r e q u e n c y C 8 0 c u r v e s c a n be d e t e r m i n e d . F i g u r e 3-29 summarizes t h e r e s u l t s 128 129 Figure 3-27 Contour p l o t of t o r s i o n a l moment c o e f f i c i e n t as a function of 6 and 0 for a 0 = 45° 130 Figure 3-28 Polynomial curve f i t of t o r s i o n a l moment c o e f f i -cient data for angle model at a 0 = -45° F i g u r e 3-29 V a r i a t i o n o f g a l l o p i n g a m p l i t u d e and r e d u c e d f r e q u e n c y w i t h w i n d v e l o c i t y f o r a n g l e s e c t i o n a t a0 = - 4 5 ° as p r e d i c t e d by t h e q u a s i - s t e a d y t h e o r y i—1 132 f o r the 3 i n . angle model at a Q = -45° w i t h t y p i c a l v a l u e s of the system parameters. The p l o t s i n d i c a t e t h a t the angle s e c t i o n a t t h i s o r i e n t a t i o n i s a s o f t o s c i l l a t o r w i t h a frequency of o s c i l l a t i o n s l i g h t l y h i g h e r than the n a t u r a l frequency. For case 1, the g a l l o p i n g theory p r e d i c t s a s t a r t i n g v e l o c i t y U 0/s = 7.76 w i t h an a m p l i t u d e - v e l o c i t y curve as shown. S i m i l a r c a l c u -l a t i o n f o r the 1 i n . angle model at a 0 = -45° g i v e s an i n i t i a l v e l o c i t y U 0/s = 590. The r e s u l t s i n d i c a t e t h a t g a l l o p i n g w i l l occur o n l y a t very h i g h wind speeds or f o r systems of very low damping, which are o u t s i d e the range of p r a c t i c a l i n t e r e s t . From F i g u r e 3-29, f u r t h e r i n f o r m a t i o n concerning the nature of the g a l l o p i n g i n s t a b i l i t y i n t o r s i o n can be o b t a i n e d . The g a l l o p i n g motion s h i f t s t o h i g h e r v e l o c i t i e s with i n c r e a s e d damping or reduced i n e r t i a parameter. L i k e w i s e , the reduced frequency parameter (1-K Q) i n c r e a s e s with wind v e l o c i t y and i s d i r e c t l y p r o p o r t i o n a l to the i n e r t i a parameter n g as i n d i c a t e d by the e x p r e s s i o n f o r K Q. However, as suggested by the curves or by d i r e c t examination of the e q u a t i o n f o r 6 Q , the t o r s i o n a l g a l l o p i n g system does not c o l l a p s e to a reduced or u n i v e r s a l form as i n the p l u n g i n g case. 3.6 C o n c l u d i n g Remarks Based on the experimental and t h e o r e t i c a l r e s u l t s the f o l l o w i n g g e n e r a l remarks can be made con c e r n i n g the nature of the a e r o e l a s t i c i n s t a b i l i t y of angle s e c t i o n s with p l u n g i n g and/or t o r s i o n a l degree(s) of freedom: (i) Angle s e c t i o n s are s u s c e p t i b l e to v o r t e x induced and g a l l o p i n g types of a e r o e l a s t i c i n s t a b i l i t i e s . The combined p l u n g i n g and t o r s i o n a l motion i n d i c a t e s the 133 e x i s t e n c e of two d i s t i n c t c e n t r e s of r o t a t i o n , thus s u b s t a n t i a t i n g the v i r t u a l hinge concept. During l a t e r a l resonance and g a l l o p i n g , the coupled motion i s predominantly p l u n g i n g . On the oth e r hand, when d i s t i n c t t o r s i o n a l resonance o c c u r s , the v i b r a t i o n i s p r i n c i p a l l y about a hinge p o i n t near the e l a s t i c a x i s . T h e r e f o r e , the o s c i l l a t i o n s may be c a t e g o r i z e d by the type of i n s t a b i l i t y and predominant mode of v i b r a t i o n . Furthermore, the frequency o f the coupled motion i s e s s e n t i a l l y the n a t u r a l f r e q u e n c i e s o f the i n d i v i d u a l modes. In g e n e r a l , the p l u n g i n g - t o r s i o n a l amplitude r e -s u l t s are comparable w i t h the s i n g l e degree o f freedom measurements. T h e r e f o r e , w i t h c o n s i d e r a t i o n o f the v i r t u a l hinge concept, the i n v e s t i g a t i o n of the aero-e l a s t i c v i b r a t i o n o f angle members can be approximated by s t u d y i n g the i n d i v i d u a l degrees of freedom. T h i s p r o v i d e s important i n f o r m a t i o n about the dynamics and aerodynamics of the s t r u c t u r a l s e c t i o n . N e v e r t h e l e s s , when p l u n g i n g , g a l l o p i n g precedes the range of t o r s i o n a l resonance, the r e s u l t i n g angular displacement and a s s o c i a t e d modulation w i l l be s u b s t a n t i a l l y reduced, ( i i ) In p l u n g i n g degree of freedom, angle s e c t i o n s are sus-c e p t i b l e t o both v o r t e x resonance and g a l l o p i n g . There e x i s t ranges of angle of a t t a c k of pure v o r t e x r e s o -nance or combined v o r t e x and g a l l o p i n g e x c i t a t i o n s . E x p e r i m e n t a l measurements a t v a r i o u s o r i e n t a t i o n s confirmed the presence of the g a l l o p i n g i n s t a b i l i t y 134 as p r e d i c t e d by the q u a s i - s t e a d y theory, thus substan-t i a t i n g the a p p l i c a b i l i t y of the a n a l y s i s . The theory i n d i c a t e s two l a r g e r e g i o n s o f g a l l o p i n g i n s t a b i l i t y w i t h two s m a l l e r areas i n between. E x h i b i t i n g e i t h e r a s o f t o r hard o s c i l l a t o r c h a r a c t e r i s t i c , the angle s e c t i o n e xperiences an i n c r e a s e i n g a l l o p i n g amplitude w i t h wind v e l o c i t y , and undergoes a d i s c o n t i n u o u s jump a t s e v e r a l o r i e n t a t i o n s . The time f o r amplitude b u i l d up reduces w i t h i n c r e a s e d wind speed, thus i n c r e a s i n g the s e v e r i t y o f the g a l l o p i n g i n s t a b i l i t y . N e v e r t h e l e s s , a h i g h e r damping o r reduced mass parameter s h i f t s the amplitude and b u i l d - u p time curves toward h i g h e r wind v e l o c i t y , thus d e l a y i n g the onset o f g a l l o p i n g . C o l l a p s -i n g o f the experimental r e s u l t s t o a s i n g l e curve i n the reduced v a r i a b l e plane confirms the concept of a u n i v e r s a l p l o t as p r e d i c t e d by the q u a s i - s t e a d y theory. Angle s e c t i o n s experience v o r t e x e x c i t a t i o n a t a l l o r i e n t a t i o n s but the s e v e r i t y of the resonance depends on the damping l e v e l , mass parameter and unsteady l i f t c o e f f i c i e n t . The s t a b i l i t y diagram shows r e d u c t i o n i n resonant v e l o c i t y range and peak displacement with i n c r e a s e d damping or decreased mass parameter, ( i i i ) In t o r s i o n a l degree of freedom, the v o r t e x induced r e s o -nance may be severe even a t moderate damping l e v e l s . However, the s e c t i o n appears t o be f r e e from g a l l o p i n g type o f a e r o e l a s t i c i n s t a b i l i t y . T h i s i s s u b s t a n t i a t e d by the q u a s i - s t e a d y a n a l y s i s which p r e d i c t s t h a t g a l l o p -i n g w i l l occur o n l y a t r e l a t i v e l y h i g h wind v e l o c i t i e s 135 or f o r systems w i t h very low damping. Furthermore, the g a l l o p i n g theory suggests a v a r i a t i o n i n the frequency of the t o r s i o n a l o s c i l l a t i o n which i s i n c o n t r a s t to the p l u n g i n g case. Although, i t i s not p o s s i b l e to v e r i f y the a p p l i c a b i l i t y of the t o r s i o n a l theory, s i n c e no g a l l o p i n g was observed e x p e r i m e n t a l l y , the q u a s i -steady approach appears promising based on i t s success w i t h the p l u n g i n g a n a l y s i s . The 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 , i n g e n e r a l , ex-h i b i t l a r g e random amplitude modulation over the com-p l e t e range of wind v e l o c i t y conducive to resonant v i b r a t i o n . Only a t c e r t a i n wind speeds above and below U does the modulation a t t a i n d e f i n i t e f r e q u e n c i e s , res ^ The maximum as w e l l as the mean amplitudes should be recorded s i n c e the peak v a l u e s , which may be many times g r e a t e r than the average, may u l t i m a t e l y cause the f a i l u r e of the s t r u c t u r a l members. T h e r e f o r e , the v i o -l e n t , v o r t e x resonance type of i n s t a b i l i t y i s , r e l a t i v e -l y , o f more importance than g a l l o p i n g , (iv) The p l u n g i n g resonance e x h i b i t s the f a m i l i a r v o r tex capture phenomenon where the shedding frequency i s con-t r o l l e d by the c y l i n d e r motion over a f i n i t e wind speed range. On the other hand, the t o r s i o n a l v i b r a t i o n shows a v o r t e x c o n t r o l phenomenon where the v o r t e x shedding governs the frequency of o s c i l l a t i o n . T h i s extends over a l a r g e v e l o c i t y range f o l l o w i n g the S t r o u h a l d i s t r i b u -t i o n o b tained from the s t a t i o n a r y model t e s t s . However, at wind speeds f a r above the resonant v a l u e , the c y l i n -136 der frequency i s a f u n c t i o n o f model amplitude. I t appears t h a t the v o r t e x shedding i n i t i a t e s the o s c i l l a -t i o n s but l o s e s c o n t r o l as the model amplitude b u i l d s up w i t h a r e s u l t i n g drop i n o s c i l l a t i o n frequency t o the n a t u r a l t o r s i o n a l v a l u e . N e v e r t h e l e s s , the d i s p l a c e -ment e v e n t u a l l y d i m i n i s h e s and the c y l i n d e r frequency r e t u r n s t o the S t r o u h a l v a l u e . During p l u n g i n g or t o r s i o n a l motion, the d i s t r i -b u t i o n o f the phase l a g between the displacement and f o r c i n g f u n c t i o n v a r i e s , approximately, from 0°-180° over the r e g i o n o f v o r t e x resonance, (v) Compared t o the s t a t i o n a r y and t o r s i o n a l r e s u l t s , the f l u c t u a t i n g p r e s s u r e s on the p l u n g i n g model are substan-t i a l l y l a r g e r i n magnitude with l e s s amplitude modula-t i o n , and reduced phase v a r i a t i o n between neighbouring midspan t a p s . Consequently the unsteady l i f t , drag and p i t c h i n g c o e f f i c i e n t i n c r e a s e d u r i n g p l u n g i n g resonance. L a r g e r p i t c h i n g moment about the e l a s t i c a x i s e x p l a i n s the g r e a t e r t o r s i o n a l resonance observed w i t h the t r a n s -f e r o f a x i s from the ce n t r e of g r a v i t y t o the shear c e n t e r . S i m i l a r t o the s t a t i o n a r y model r e s u l t s , the l o c a l phase d i f f e r e n c e around the model contour d u r i n g the dynamic c o n d i t i o n s has comparatively l i t t l e e f f e c t (< 10%) on the s e c t i o n a l , unsteady aerodynamic c o e f f i c i e n t . (vi) The wake survey r e s u l t s show s i m i l a r t r e n d w i t h down-stream c o o r d i n a t e f o r both the s t a t i o n a r y and v o r t e x e x c i t e d models. Dur i n g resonance i n e i t h e r degree of 137 freedom, the v o r t e x v e l o c i t y and l o n g i t u d i n a l s p a c i n g remain e s s e n t i a l l y u n a l t e r e d , however, the wake width e x p e r i e n c e s s u b s t a n t i a l i n c r e a s e w i t h p l u n g i n g motion. N e v e r t h e l e s s , i f the l a t e r a l s p a c i n g i s based on the e f f e c t i v e blockage (2y + h ) , the r e s u l t i n g nondimen-s i o n a l i z e d parameter i s almost i d e n t i c a l to the s t a t i o n -ary o r t o r s i o n a l model v a l u e . I t appears t h a t the t o r -s i o n a l resonant motion has v i r t u a l l y no e f f e c t on the v o r t e x shedding or wake c h a r a c t e r i s t i c s , ( v i i ) From the dynamic i n v e s t i g a t i o n s , i t i s shown t h a t angle s e c t i o n beams i n open e n g i n e e r i n g s t r u c t u r e s may e x h i b i t o s c i l l a t i o n s o f a v o r t e x resonance or g a l l o p i n g nature. T h e r e f o r e , t o prevent v i b r a t i o n and p o s s i b l e occurrence of s t r u c t u r a l f a i l u r e of the i n d i v i d u a l members, the d e s i g n should i n v o l v e s u f f i c i e n t damping or h i g h n a t u r a l frequency. T h i s i s i n agreement wi t h the p r e d i c t i o n from the s t a t i o n a r y model study. 4. RECOMMENDATIONS FOR FUTURE WORK P o s s i b l e t h e o r e t i c a l and exp e r i m e n t a l extensions to the pr e s e n t work may be summarized as f o l l o w s : ( i ) A n a l y s i s of the coupled system d u r i n g g a l l o p i n g u s i n g , p r o b a b l y , the method of V a r i a t i o n of Parameters i s c e r t a i n l y d e s i r a b l e . The e x p r e s s i o n s f o r the aerodynamic f o r c e and moment r e q u i r e d i n the a n a l y s i s w i l l be, i n g e n e r a l , q u i t e complex even wi t h the q u a s i - s t e a d y approach. ( i i ) E x p e r i m e n t a l measurements of t o r s i o n a l g a l l o p i n g t o con-f i r m the a p p l i c a b i l i t y of the q u a s i - s t e a d y a n a l y s i s would be a v a l u a b l e study. T h i s may be c a r r i e d out u s i n g an angle s e c t i o n o r some s i m p l e r model such as a r e c t a n g u l a r c y l i n d e r . F o r such a study a c c u r a t e p i t c h -i n g moment da t a i s r e q u i r e d . ( i i i ) D e t e r m i n a t i o n of spanwise v a r i a t i o n s of the steady and unsteady aerodynamics along normal and yawed angle s e c t i o n s would be u s e f u l . T h i s i s important i n e v a l u -a t i n g the o v e r a l l t h r e e - d i m e n s i o n a l nature of the e x c i t a t i o n . (iv) A flow v i s u a l i z a t i o n study should prove u s e f u l i n under-s t a n d i n g the c h a r a c t e r of the flow, such as s e p a r a t i o n , reattachment, wake geometry, e t c . (v) In g e n e r a l , i n f o r m a t i o n c o n c e r n i n g wind t u n n e l w a l l 139 i n t e r f e r e n c e e f f e c t s on the steady and unsteady aero-dynamic c h a r a c t e r i s t i c s and wake geometry f o r b l u f f b odies i s l a c k i n g . T h e r e f o r e , a s y s t e m a t i c study d u r i n g s t a t i c and dynamic c o n d i t i o n s of the model would be of v a l u e . (vi) S ince angle s e c t i o n s , when used i n open e n g i n e e r i n g s t r u c t u r e s , are exposed to atmospheric t u r b u l e n c e , i t i s o f paramount importance to determine i t s e f f e c t on the a e r o e l a s t i c v i b r a t i o n s . The i n v e s t i g a t i o n may be conducted i n the wind t u n n e l under c o n t r o l l e d t u r b u l e n c e to o b t a i n an understanding of the phenomenon, but should be supplemented w i t h f i e l d t e s t s over t e r r a i n exposed t o t y p i c a l t u r b u l e n t winds. 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Brooks, P.N.H., "Experimental I n v e s t i g a t i o n o f the Aero-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 Two-Dimensional C y l i n d e r s , " Univ. o f B r i t i s h Columbia, M.A.Sc. T h e s i s , J u l y 1960. 49. Smith, J.D., "An Experimental Study o f the A e r o e l a s t i c I n s t a b i l i t y of Rectangular C y l i n d e r s , " Univ. of B r i t i s h  Columbia, M.A.Sc. T h e s i s , August 1962. 50. Santosham, T.V., "Force 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 of a Rectangular C y l i n d e r , " Univ. o f B r i t i s h Columbia, M.A.Sc. T h e s i s , February 1966. 51..Chuan, R.L., "Study o f Vortex Shedding as Rel a t e d to S e l f -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 o f an A i r f o i l , " NACA, Tech. Note 2429, September 1951. 145 52. O t s u k i , Y., " T a l l B u i l d i n g s and A e r o e l a s t i c Problems," Conf. Earthquake and Wind E f f e c t s , 1967. 53. Toebes, G.H. and Eagleson, P.S., "Hydroelastic Vibrations of F l a t Plates Related to T r a i l i n g Edge Geometry," J . Basic  Engng. , ASME. Vol.83, Series D, No. 4 , December 196T~, pp. 671-678. 54. Eagleson, P.S., Noutsopoulos, G.K. and Daily, J.W., "The Nature of S e l f - E x c i t a t i o n i n the Flow-Induced Vibration of F l a t Plates," J . Basic Engng., ASME, Series D, No.4, November 1963. 55. Thornton, CP., "Wind Tunnel Tests of the Aerodynamically Induced Vibrations of Some Simple Structural Members," National Research Council of Canada, Mech, Engng. Report MA-245, 1962; also Movie Film, "Aeroelastic I n s t a b i l i t y of Aluminum Angle Section," National Research Council of  Canada, Aeronautical Establishment. 56. Parkinson, G.V., "Aeroelastic Galloping i n One Degree of Freedom," Proc. F i r s t Int. Conf. on Wind E f f e c t s on Bldgs. and Structures, NPL, Teddington, Vol.11, 1965, pp.581-609. 57. Parkinson, G.V. and Modi, V.J., "Recent Research on Wind Ef f e c t s on B l u f f Two-Dimensional Bodies," Int. Research  Seminar: Wind E f f e c t s on Bldgs. and Structures, NRC, Ottawa, September 196 7. 58. Whitbread, R.E., "Model Simulation of Wind Ef f e c t s on Structures," Pro. F i r s t Int. Conf. on Wind E f f e c t s on  Bldgs. and Structures, NPL, Teddington, Vol.1, 1965, pp.283-306. [ " 59. Soroka, W.W., "Note on the Relation between Viscous and Structural Damping C o e f f i c i e n t s , " J . Aeronautical Sciences, Vol.16, 1949, pp.409-410. 60. Kimball, A.L. and L o v e l l , D.E., "Internal F r i c t i o n i n Solids," Physics Review, Vol.30, Series 2, December 1927, p.948. 61. Iokibe, K. and Sakai, S., "The E f f e c t of Temperature on the Modulus of R i g i d i t y and on the V i s c o s i t y of Solid Metals," P h i l . Mag., Vol.42, Series 6, 1921, pp.397-418. 146 62. O c k l e s t o n , A . J . , "The Damping o f the L a t e r a l V i b r a t i o n of a M i l d S t e e l Bar," P h i l . Mag., Vol.26, S e r i e s 7, 1938, pp.705-712. 63. B r y e r , D.W., Walshe, D.E. and Garner, H.C., "Pressure Probes S e l e c t e d f o r Three-Dimensional Flow Measurement," A e r o n a u t i c a l Research C o u n c i l , R. and M. No.3037, 1958. 64. Hoerner, S.F., Fluid-Dynamic Drag, 1965, Chapter 3, pp.6-18 and Chapter 4, p.6. 65. Roshko, A., "A New Hodograph For Fr e e - S t r e a m l i n e Theory," NACA, Tech. Note 3168, J u l y 1954. 66. Roshko, A. "On the Wake and Drag o f B l u f f Bodies," J . A e r o n a u t i c a l S c i e n c e s , Vol.22, No.2, February 1955, pp.124-133. 67. Fage, A. and Johansen, F.C., "The S t r u c t u r e s o f Vortex Sheets," P h i l . Mag., S e r i e s 7, V o l . 5 , No.28, 1928, pp.317-440. 68. Schaefer, J.W. and E s k i n a z i , S., "An A n a l y s i s o f the Vortex S t r e e t Generated i n a Vi s c o u s F l u i d , " J . F l u i d Mech., Vol . 6 , 1959, pp.241-260. 69. Hooker, S.G., "On the A c t i o n of V i s c o s i t y i n I n c r e a s i n g the Spacing R a t i o of a Vortex S t r e e t , " Proc. Roy. Soc. of London, S e r i e s A, Vol.154, 1936, pp.67-89. 70. Fage, A. and Johansen, F.C., "On the Flow of A i r Behind an I n c l i n e d F l a t P l a t e of I n f i n i t e Span," Proc. Roy. Soc. of London, S e r i e s A, Vol.116, 1927, pp.170-197. 71. Rosenhead, L. and Schwabe, M., "An Experimental I n v e s t i -g a t i o n of the Flow Behind C i r c u l a r C y l i n d e r s i n Channels of D i f f e r e n t Breadths," Proc. Roy. Soc. of London, S e r i e s A, Vol.129, 1930, pp.ll5 - I T 5 " : 72. " A p p l i c a t i o n of S p r i n g S t r i p s to Instrument Design," NPL, Notes on Appl. Sc., No.15, 4th ed., 1965, pp.2-5. 147 .73. Eastman, F.S., "The Design of Flexure P i v o t s , " J . Aero-n a u t i c a l Sciences, Vol.5, No.11, November 193 7, pp.16-21. 74. Haringx, J.A., "The Cross-Spring P i v o t as a C o n s t r u c t i o n a l Element," Appl. S c i . Res., V o l . A l , 1947-49, pp.313-332. 75. W i t t r i c k , W.H. "The Theory of Symmetrical Crossed Flexure P i v o t s , " A u s t r a i l i a n J . S c i . Res., A, Vo l . 1 , No.2, 1948, pp.121-134. 76. W i t t r i c k , W.H., "The P r o p e r t i e s of Crossed Flexure P i v o t s , and the Influence of the P o i n t at which the S t r i p s Cross," The A e r o n a u t i c a l Q u a r t e r l y , Vol.11, February 1951, pp.272-292. 77. Young, W.E., "An I n v e s t i g a t i o n of the Cross-Spring P i v o t , " J . Appl. Mech., Vol.11, June 1944, pp.A113-A120. 78. Garland, C.F., "The Normal Modes of V i b r a t i o n s of Beams Having N o n c o l l i n e a r E l a s t i c and Mass Axes," J . Appl. Mech., ASME Trans., Vol.62, 1940, pp.A97-Al05. 79. Parkinson, G.V., Feng, C.C. and Ferguson, N., "Mechanisms of V o r t e x - E x c i t e d O s c i l l a t i o n of B l u f f C y l i n d e r s , " Symposium  on Wind E f f e c t s on Bldgs. and S t r u c t u r e s , NPL/Roy. Aero-n a u t i c a l S o c i e t y , London, March 1968. 80. Cunningham, W.J., I n t r o d u c t i o n to Nonlinear A n a l y s i s , McGraw-Hill, New York, 195 8. 81. Pankhurst, R.C. and Holder, D.W., Wind Tunnel Technique, Pitman and Sons L i m i t e d , London, 1952, Chapter 8. 82. G l a u e r t , M., "The Interference of Wind Channel Walls on the Aerodynamic C h a r a c t e r i s t i c s of an A e r o f o i l , " B r i t i s h A.R.C., R. and M. 867, March 1923; and, B r i t i s h A.R.C., R. and M. 889, 1923-24. 83. Fage, A., "On the Two-Dimensional Flow Past a Body of Symmetrical Cross-Section Mounted i n a Channel of F i n i t e Breadth," B r i t i s h A.R.C., R. and M. 1223, 1928-29. 1 4-J '84. Durand, W.F., Aerodynamic Theory, Durand Reprinting Committee, C a l i f o r n i a , V o l . I l l , Div.I, Part 1, Chapter I I I , "Influence on the Dimensions of the A i r Stream,'' 194 ->, pp.280-319. 85. A l l e n , H.J. and Vincenti, W.G., "Wall Interference i n a Two-Dimensional-Flow Wind Tunnel, with Consideration of the E f f e c t of Compressibility," NACA 13th Annual Report, No. 782, 1944, pp. 155-184. 86. Lock, M., "The Interference of a Wind Tunnel on a Symmet-r i c a l Body," B r i t i s h A.R.C., R. and M. 1275, 1929. 87. Glauert, H., "The Cha r a c t e r i s t i c s of a Karman Vortex Street i n a Channel of F i n i t e Breadth," B r i t i s h A.R.C., R. and M. 1151, 1928-29. 88. Maskell, E.C., "Theory of the Blockage Ef f e c t s on B l u f f Bodies and S t a l l e d Wings i n a Closed Wind Tunnel," B r i t i s h A.R.C., R. and M. 3400, November 1963. . 89. 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APPENDIX I G e o m e t r i c a l P r o p e r t i e s of Angle S e c t i o n Members The c r o s s - s e c t i o n a l f e a t u r e s of the designed models were somewhat ' i d e a l i z e d ' s i n c e t h e i r edges were sharp and the s u r f a c e s smooth. T h i s was necessary to f a c i l i t a t e the c o n s t r u c t i o n of the t h i n - w a l l e d , hollow models. N e v e r t h e l e s s , commercially a v a i l a b l e aluminum angle s e c t i o n s have s i m i l a r geometry and, t h e r e f o r e , are w e l l approximated by the sharp-edged models. On the o t h e r hand, s t r u c t u r a l s t e e l angles do have rounded edges as w e l l as rough s u r f a c e s (Figure 1-1). Table 1-1 compares the geometric f e a t u r e s of the v a r i o u s wind t u n n e l models i n c l u d i n g the commercially a v a i l a b l e s t r u c t u r a l angles t e s t e d d u r i n g the experiment. A l l models were s i m i l a r i n l e n g t h extending across the e f f e c t i v e h e i g h t of the wind t u n n e l . The f o u r angle s e c t i o n s l i s t e d a t the bottom of the t a b l e were not t e s t e d e x p e r i m e n t a l l y but are i n c l u d e d f o r f u r t h e r comparison of the geometric f e a t u r e s of commercially a v a i l a b l e s t r u c t u r a l angles and the wind t u n n e l models. Survey of the t a b u l a t e d data i n d i c a t e s t h a t the pressure tap, as w e l l as 1 i n . and 3 i n . dynamic models compare fav o u r a b l y w i t h the s t r u c t u r a l aluminum angles of s i m i l a r s i z e except f o r the f i l l e t e d i n n e r corner on the commercial members. How-ever, t h i s d i f f e r e n c e i n i n n e r r a d i u s w i l l be of minor aerodynamic importance s i n c e the f l u i d near t h i s area on the s e c t i o n i s almost stagnant. Flow v i s u a l i z a t i o n u s i n g a smoke tu n n e l s u b s t a n t i a t e d t h i s o b s e r v a t i o n . Figure 1-1 Cross-sections of angle member (a) 'idealized' angle section (b) t y p i c a l s t r u c t u r a l angle TABLE 1-1 Geometric Features of Angle S e c t i o n s Model Nominal dimensions e.g. a i n . shear c e n t r e a - i n . I X 4 i n . system m1 lbm/ft system I l b m - i n ? / f t M a t e r i a l d i n . t i n . l e n g t h i n . .R o i n . .R i i n . p r e s sure tap 3 1/2 26 3/4 0 0 0.93 0.96 2.3 0.91 8.2 alumi n u m , a c r y l i c hollow s e c t i o n l a r g e dynamic 3 1/2 26 3/4 0 0 0.92 0.96 2.1 0.71 7.9 a c r y l i c hollow s e c t i o n s m a l l dynamic 1 1/6 26 3/4 0 0 0.31 0.32 0.027 0.70 7.0 aluminum s o l i d s e c t i o n balance A 3 1/2 27 0 0 0.90 0.93 1.95 2.92 7.2 aluminum s o l i d s e c t i o n balance B I d e n t i c a l to model A except f o r 1/4 i n . t h i c k end p l a t e s balance C 3 1/2 27 1/8 5/16 0.93 0.98 2.2 9.4 24.0 s t r u c t u r a l s t e e l angle balance D 3 1/4 27 1/16 5/16 0.84 1.03 1.2 4.9 13.4 s t r u c t u r a l s t e e l angle balance E 3 1/4 27 1/64 7/32 0.84 1.01 1.2 1.71 4.5 s t r u c t u r a l alum-inum angle balance F 2 1/3 27 0 0 0.62 0.64 0.44 1.42 1.6 aluminum s o l i d s e c t i o n - 3 1/2 - 1/64 1/4 0.93 0.96 2.2 3.30 8.3 s t r u c t u r a l alum-inum - 1 3/16 - 1/16 3/16 0.31 0.32 0.030 1.16 0.32 s t r u c t u r a l s t e e l angle - 1 1/8 - 1/16 3/16 0.30 0.33 0.022 0.80 0.24 s t r u c t u r a l s t e e l angle - 1 3/16 - 1/6 4 5/32 0.31 0.32 0.030 0.41 0.11 s t r u c t u r a l alum-inum angle (—1 to APPENDIX I I Wind T u n n e l W a l l C o r r e c t i o n s T h e r e i s a c o n s i d e r a b l e body o f i n f o r m a t i o n 8 1 8 7 e t a l on w i n d t u n n e l i n t e r f e r e n c e f o r s t a t i o n a r y s t r e a m l i n e d models i n s t e a d y f l o w . I n g e n e r a l , t h e t h e o r e t i c a l and s e m i - e m p i r i c a l i n v e s t i g a t i o n s e x p r e s s t h e w a l l e f f e c t s as a s e r i e s i n a s c e n d i n g powers o f t h e model t o t u n n e l c h a r a c t e r i s t i c d i m e n s i o n r a t i o w i t h t h e a n a l y s i s c o n f i n e d t o t h e f i r s t o r s e c o n d o r d e r t e r m s . T h e s e c o r r e c t i o n s have b e e n f o u n d s a t i s f a c t o r y i n most s i t u a t i o n s and c a n be a p p l i e d w i t h some measure o f c o n f i d e n c e . F o r s t a t i o n a r y b l u f f c y l i n d e r s , t h e r e i s some e x p e r i m e n t a l i n f o r m a t i o n on w a l l c o n f i n e m e n t b u t t h e t h e o r e t i c a l a n a l y s i s i s 81 84 r e l a t i v e l y l e s s c o m p l e t e . P a n k h u r s t and H o l d e r , and Durand p r o v i d e t e c h n i q u e s f o r e x t r a p o l a t i n g s t r e a m l i n e body a n a l y s i s t o b l u f f c i r c u l a r c y l i n d e r s b u t t h e i r a p p l i c a b i l i t y i s r a t h e r l i m i t e d . More a p p r o p r i a t e , u n d e r t h i s c o n d i t i o n o f c o m p l e t e l y s e p a r a t e d 5 8 f l o w , i s t h e s i m p l i f i e d e x p r e s s i o n q u o t e d by W h i t b r e a d w h i c h i s 8 8 a p a r t i c u l a r c a s e o f a g e n e r a l a n a l y s i s by M a s k e l l . U s i n g 40 M a s k e l l ' s m a t h e m a t i c a l m o d e l , V i c k e r y o b t a i n e d b l o c k a g e c o r r e c -t i o n s t o S t r o u h a l number and f l u c t u a t i n g l i f t f o r c e and a p p l i e d i t t o t h e f l o w a r o u n d a s q u a r e c y l i n d e r . F o r a c i r c u l a r c y l i n d e r n e a r t h e c r i t i c a l R e y n o l d s number, where t h e d r a g c o e f f i c i e n t d r o p s t o a p p r o x i m a t e l y 0.2 t h u s making t h e s o l i d b l o c k a g e as 8 9 i m p o r t a n t as wake b l o c k a g e , Bearman has s u g g e s t e d t h e a p p l i c a t i o n 8 5 o f t h e method o f A l l e n and V i n c e n t i d e v e l o p e d f o r low d r a g , 8 7 83 s t r e a m l i n e d a i r f o i l s . G l a u e r t and Fage have c o n s i d e r e d , t h e o r e t i c a l l y , t h e t w o - d i m e n s i o n a l f l o w a r o u n d c y l i n d e r s i n 1 5 4 channels of f i n i t e b r eadth. The experimental measurements by Rosenhead and Schwabe,^ and Fage and J o h a n s e n ^ p r o v i d e u s e f u l i n f o r m a t i o n c o n c e r n i n g the i n f l u e n c e of w a l l confinement on flow c h a r a c t e r i s t i c s . T e s t r e s u l t s on s t a t i o n a r y angle models taken d u r i n g the course of t h i s r e s e a r c h programme, gave some i n f o r -mation on w a l l i n t e r f e r e n c e w i t h regards to the departmental wind t u n n e l system. The a n a l y s i s of w a l l i n t e r f e r e n c e on o s c i l l a t i n g models i s c o n s i d e r a b l y more complex, and to date, e f f e c t i v e methods f o r p r e d i c t i n g the c o r r e c t i o n s have not been completely e v o l v e d . N e v e r t h e l e s s , f o r o s c i l l a t i n g a i r f o i l s i n wind t u n n e l s , there 9 0 - 9 3 are v a r i o u s t h e o r e t i c a l and experimental approaches r e p o r t e d e t a l 94 Molyneux has presented a good review of t h i s l i t e r a t u r e , On the o t h e r hand, f o r o s c i l l a t i n g b l u f f b o d i e s , i n f o r m a t i o n on w a l l confinement i s almost completely l a c k i n g . 81 As i n d i c a t e d by Pankhurst and Holder, the i n t e r f e r e n c e from wind t u n n e l w a l l s d u r i n g steady flow c o n d i t i o n may be d i v i d e d i n t o : (i) s o l i d blockage; ( i i ) wake blockage; ( i i i ) l i f t or c i r c u l a t i o n e f f e c t ; (iv) boundary l a y e r i n t e r f e r e n c e ; and (v) streamwise s t a t i c p r e s s u r e g r a d i e n t i n f l u e n c e . For an o b j e c t s y m m e t r i c a l l y p l a c e d irt the flow f i e l d , ( i i i ) does not e x i s t . For the wind t u n n e l a t the U n i v e r s i t y of B r i t i s h Columbia, the w a l l boundary l a y e r t h i c k n e s s i n the t e s t s e c t i o n i s r e l a t i v e l y s m a l l p a r t l y due to the f i l l e t e d corners which compensate f o r boundary l a y e r growth. Furthermore, the pressure 155 i n t e g r a t e d and balance measured aerodynamic c o e f f i c i e n t s (Figure 2-13) showed good c o r r e l a t i o n . T h e r e f o r e , (iv) appears t o be n e g l i g i b l e . S i m i l a r l y , f o r the t u n n e l (v) was found to be of minor s i g n i f i c a n c e ( l e s s than 1%). T h e r e f o r e , s o l i d and wake blockages r e p r e s e n t the major i n t e r f e r e n c e s on model aerodynamic c h a r a c t e r i s t i c s . A v a i l a b l e c o r r e c t i o n s a p p r o p r i a t e f o r the s t a t i o n a r y angle s e c t i o n measurements are summarized i n the f o l l o w i n g pages. (1) Mean Free Stream V e l o c i t y 81 A c c o r d i n g t o Pankhurst and Holder, the wind speed c o r r e c t i o n i s a sum of the s o l i d and wake blockages given by y F = ( I + cr ) (1) where V p = f r e e stream or c o r r e c t e d v e l o c i t y ; V = measured approaching v e l o c i t y v a l u e ; ° = °s + °w ' c o r r e c ^ o n f a c t o r depending on model and tu n n e l geometry. Another estimate o f the v e l o c i t y c o r r e c t i o n can be ob-t a i n e d from M a s k e l l ' s s i m p l i f i e d r e l a t i o n (equation ( 2 ) ) . I l l u s t r a t e d i n F i g u r e I I - l , i s a comparison between the percen-tage v e l o c i t y c o r r e c t i o n s given by the above two methods when a p p l i e d t o the 3x3 i n . angle models. The d i s c r e p a n c y between the two c o r r e c t i o n s can be e x p l a i n e d by the f a c t t h a t the former i s more a p p l i c a b l e to s t r e a m l i n e d bodies while the l a t t e r i s f o r separated flow c o n d i t i o n . The v e l o c i t y c o r r e c t i o n of A l l e n and V i n c e n t i as s t a t e d by Bearman i s i d e n t i c a l to t h a t of Pankhurst and Holder. (2) F l u c t u a t i n g Free Stream C o n d i t i o n s 9 5 Roberts i n v e s t i g a t e d the flow c o n d i t i o n s ahead of an ~ T 1 1 1 Velocity (Pankhurst and Holder) Velocity ( Maskell ) Strouhal number , S, Lateral vortex spacing \ \ - 4 5 135 F i g u r e I I - l P e r c e n t a g e c o r r e c t i o n a p p l i c a b l e t o 3 i n . a n g l e s e c t i o n t e s t e d i n d e p a r t m e n t a l w i n d t u n n e l 157 o s c i l l a t i n g model by u s i n g the unsteady form of the B e r n o u l l i ' s e q u a t i o n . Two-dimensional c i r c u l a r c y l i n d e r s o s c i l l a t i n g h a r m o n i c a l l y i n the streamwise d i r e c t i o n showed the upstream .flow v a r i a b l e s to be f u n c t i o n s of the frequency and amplitude of the motion. For amplitudes of one diameter, the v e l o c i t y ; and s t a t i c p r e s s u r e v a r i a t i o n s were as l a r g e as 20 and 40 percent, r e s p e c t i v e l y . Thus, one would expect the time-dependent phenom-enon to be s i g n i f i c a n t where l a r g e amplitude, low frequency, model o s c i l l a t i o n s e x i s t . (3) E f f e c t i v e Angle of A t t a c k For t h i n a i r f o i l s e c t i o n s , both Durand, as w e l l as A l l e n and V i n c e n t i p r o v i d e e x p r e s s i o n s f o r w a l l i n t e r f e r e n c e c o r r e c t i o n s to the angle of a t t a c k by c o n s i d e r i n g an e q u i v a l e n t induced v e l o c i t y f i e l d o f a system of images on the base p r o f i l e . How-ever, f o r a b l u f f s e c t i o n no a p p r o p r i a t e a n a l y s i s i s apparently a v a i l a b l e . (4) Mean S t a t i c P r e s s u r e and R e s u l t a n t Steady Forces 8 8 Using the f r e e s t r e a m l i n e model, M a s k e l l has shown t h a t the steady s t a t i c p r e s s u r e and drag on b l u f f s t r u c t u r e s can be c o r r e c t e d as f o l l o w s : ' - c c I = -FT = ~ (2) i - cp cD , c 0 s c c where C_ , = f r e e stream or c o r r e c t e d s t a t i c p r e ssure and drag c o e f f i c i e n t s , r e s p e c t i v e l y ; V °F C , C = measured s t a t i c pressure and drag c o e f f i c i e n t s ; p D 158 2 C = f r e e stream base p r e s s u r e c o e f f i c i e n t , -(k - 1 ) ; % S = model area on which C Q i s based; C = t u n n e l t e s t s e c t i o n area. 40 T h i s i s i d e n t i c a l t o the blockage c o r r e c t i o n s g i v e n by V i c k e r y . The e x p r e s s i o n i s a l s o a p p l i c a b l e t o the c o r r e c t i o n of other f o r c e and moment c o e f f i c i e n t s . 87 G l a u e r t d e r i v e d an e x p r e s s i o n f o r the drag c o e f f i c i e n t f o r a body forming a wake of a l t e r n a t e v o r t i c e s i n a channel of f i n i t e w i dth. The a n a l y s i s , based on Lock's image method and Karman's v o r t e x s t r e e t t h e o r y , was developed f o r a p l a t e normal to the flow but, as s t a t e d by Durand, i s a p p l i c a b l e f o r other bodies o f abrupt or sharp-edged form. The t h e o r e t i c a l e x p r e s s i o n f o r the drag -coefficient i n c o n f i n e d flow i s g i v e n by the eq u a t i o n 0 v V've W e / H (3) where u = W - V y W = surrounding f l u i d v e l o c i t y on which the v o r t e x system i s superimposed (equation (15)); V y = v o r t e x streamwise v e l o c i t y ; b = measured l a t e r a l v o r t e x row s p a c i n g . T h i s e x p r e s s i o n i s s i m i l a r t o t h a t given by Karman f o r the case of a plane u n l i m i t e d flow, C n = ( 5 . 6 5 6 - 2 . 2 4 ttr)^f (4) where 159 and a F = l ° n 9 i t u c l i n a l s p a c i n g between c o n s e c u t i v e v o r t i c e s of s t r e n g t h T . However, to use the e x p r e s s i o n f o r C D i t i s necessary to know. a. p r i o r i , the v a l u e s of u/V and b/e f o r the body i n a c o n s t r a i n e d flow. G l a u e r t gave an approximate t h e o r e t i c a l s o l u t i o n to t h i s problem by c o n s i d e r i n g the r a t e of v o r t i c i t y generated by the body and the amount of v o r t i c i t y p a s s i n g downstream i n the form of d i s c r e t e v o r t i c e s . T h i s study p r o v i d e d an e x p r e s s i o n f o r the drag c o e f f i c i e n t i n c o n s t r a i n e d flow as C = C n + I 32 - * i f c ^ ' - 4 U r / V , 1 0 D' 1 VF i-2u F/v FHv;e/ H (5) where u_/V_ and b„/e correspond to the u n l i m i t e d flow. In p r a c t i c e , F i t the second term i n the p a r e n t h e s i s i s g e n e r a l l y n e g l i g i b l e and, t h e r e f o r e , the drag e x p r e s s i o n reduces to C n = C n + 3 2 / ^ V f e 0 D r V V F e / H ( 6 ) However, the theory i s s t i l l incomplete, s i n c e i t does 7 not p r e d i c t u F / V p and b F / e . An a n a l y s i s by Heisenberg, to complete the theory f o r a f l a t p l a t e , gave the v a l u e s L__F - 0.2295 and 4f]V = 0.3535 ( 7 ) I f Heisenberg's r e s u l t s are adopted, drag c o r r e c t i o n f o r a f l a t p l a t e normal to the flow becomes 160 k- - • - a which can be r e f e r r e d to as Glauert-Heisenberg's formula. I t s p r e d i c t i o n s are i n agreement wi t h experimental r e s u l t s given by 70 Fage and Johansen. I t should be noted t h a t t h i s e x p r e s s i o n i s of the same form as M a s k e l l ' s s i m p l i f i e d e q uation (2) except t h a t -C D/Cp b i s r e p l a c e d by 4/C D < A p p l i c a b i l i t y o f G l a u e r t ' s s i m p l i f i e d equation (6) f o r C D to the measurements on the angle s e c t i o n can be shown by the f o l l o w i n g example. For the 1 i n . angle model mounted at a = -45°, and c o n s i d e r i n g the approximate experimental values of (u/V}^ = 0.201 and (b/e)^ = 1.77, the drag e x p r e s s i o n becomes C D C 0 H which compares w e l l w i t h equation (8) f o r the f l a t p l a t e . Bearman, as w e l l as Pankhurst and Holder suggested t h a t t h e i r v e l o c i t y c o r r e c t i o n s can a l s o be a p p l i e d to s t a t i c p ressures and f o r c e s by ~ C J V s £ d F = SF = _Sv = ! k i - 2 < r ( 1 0 ) (5) F l u c t u a t i n g Forces Caused by Vortex Shedding 40 V i c k e r y has presented a blockage c o r r e c t i o n f o r the 96 f l u c t u a t i n g l i f t based on the a n a l y s i s by Ross, which p r e d i c t s that the r a t i o C][,/CD i s almost independent of the wall confine-ment. Therefore, to the f i r s t approximation, the fluc t u a t i n g forces can be corrected using the steady force expressions. (6) Strouhal Number and Vortex V e l o c i t y Using the same model as employed by Maskell and introduc-6 6 ing Roshko's Universal Strouhal Number (S*), Vickery developed an expression for the correction of the Strouhal number, Q J L _ F = S.w £ F (11) d w where -5— i s the e f f e c t of wall constraint on the wake width; dw F 1/2 and k = (1-C p ) On the other hand, the Strouhal number, which i s a function of the f l u i d v e l o c i t y and vortex shedding frequency, may be affected by t h e i r v a r i a t i o n s . Bearman assumed that the correction for the Strouhal number was a function of the v e l o c i t y correction only, and neglected any e f f e c t of wall interference on the shed-ding frequency. To introduce the correction for the vortex shedding frequency, i t i s appropriate to consider the expression { = v / a d2) and, thereby, investigate the variations of the vortex v e l o c i t y and l o n g i t u d i n a l spacing with wall interference e f f e c t s . The vortex v e l o c i t y i s a function of the absolute v e l o c i t y of the surrounding f l u i d and the r e l a t i v e v e l o c i t y of the vortices i n the wake, and i s given by V v = W-u. The o r e t i c a l l y , for an 162 unconfined flow, W would equal Vp, and u would be given by Karman ' s v o r t e x v e l o c i t y r e l a t i o n . For the flow i n a channel, 87 G l a u e r t has d e r i v e d u s i n g the image method an e x p r e s s i o n f o r the r e d u c t i o n of the v o r t e x v e l o c i t y by the presence of the w a l l s , — = I •- 8 e cosh HP ( 1 3 ) F a where ^ _ 2 TT M " a However, s i n c e e * i s g e n e r a l l y very s m a l l , the i n f l u e n c e of the w a l l s appears t o be n e g l i g i b l e . The frequency c o r r e c t i o n can then be expressed i n the form l v - VJr 5 a ^ - U r / Q i , 1 4 ) Using flow c o n t i n u i t y as suggested by G l a u e r t , the v e l o c i t y W on which the v o r t e x system i s superimposed i s g r e a t e r than the approaching stream v e l o c i t y by the amount equal to the backward flow induced by the v o r t e x s t r e e t and image system. Hence, v e l o c i t y W i s determined by the equation Using Karman's s t a b i l i t y r e s u l t s , W s V + J / T u p i d 6 ) which can be s u b s t i t u t e d i n t o the frequency c o r r e c t i o n e q uation. The f i n a l e x p r e s s i o n f o r the f r e e stream S t r o u h a l number i s of the form 163 (17) Although the S t r o u h a l number forms one of the important parameters i n a e r o e l a s t i c i n s t a b i l i t y s t u d i e s , the a v a i l a b l e l i t e r a t u r e f o r i t s c o r r e c t i o n do not c o r r e l a t e w e l l . T h e r e f o r e , a s e r i e s of e x p e r i m e n t a l measurements were conducted with 1, 2, and 3 i n . angle s e c t i o n s . The r e s u l t s , p l o t t e d i n F i g u r e I I - 2 , show the v a r i a t i o n of the S t r o u h a l number with blockage f o r v a r i o u s angles of a t t a c k . The e x t r a p o l a t i o n of these r e s u l t s g i v e s an e s t i m a t e of the e q u i v a l e n t S t r o u h a l number i n an un-l i m i t e d flow. The percentage c o r r e c t i o n f o r the 3 i n . model as p l o t t e d i n F i g u r e I I - l , though s i m i l a r i n form to the v e l o c i t y c u r v e s , i s g e n e r a l l y l a r g e r over most of the angle of a t t a c k range. (7) Wake Geometry 71 The measurements by Rosenhead and Schwabe i n d i c a t e t h a t both the l o n g i t u d i n a l and l a t e r a l v o r t e x spacings decrease with i n c r e a s i n g blockage a t such a r a t e as to maintain the wake geom-e t r y r a t i o b/a e s s e n t i a l l y c o n s t a n t . T h i s was determined f o r model t o t u n n e l width r a t i o s of up to 1/3 and, t h e r e f o r e , i s a l s o 8 7 l i k e l y t o be r e p r e s e n t a t i v e of the angle model wakes. G l a u e r t , on the o t h e r hand, has developed an e x p r e s s i o n f o r determining the c h a r a c t e r i s t i c s o f the v o r t e x s t r e e t i n a channel, F i g u r e II-2 V a r i a t i o n of S t r o u h a l number with blockage f o r v a r i o u s o r i e n t a t i o n s of the angle models 165 which suggests t h a t the dimensions of the wake i n a c o n s t r a i n e d flow are g r e a t e r than those i n an u n l i m i t e d flow. T h i s i s an o p p o s i t e t r e n d t o the exp e r i m e n t a l measurements by Rosenhead and Schwabe, and c o n t r a d i c t s g e n e r a l p h y s i c a l i n t u i t i o n . 8 8 M a s k e l l has d e r i v e d t h e r e l a t i o n s h i p , ^w F _ , + Co ~ Co r _S (19) aw " (ka-0(k F-i) c i n d i c a t i n g the i n f l u e n c e o f w a l l c o n s t r a i n t on the wake width. However, based on the experimental data f o r the 3 i n . angle model, the second term i n the above eq u a t i o n i s of the order 0.02 and, t h e r e f o r e , M a s k e l l ' s a n a l y s i s suggests t h a t the wake width i s approximately independent o f the w a l l c o n s t r a i n t s . To e s t a b l i s h the c o r r e c t t r e n d f o r w a l l i n t e r f e r e n c e on wake -geometry, a s e r i e s o f wake measurements were conducted with the 1 i n . and 3 i n . angle s e c t i o n s . T h i s s e t of dat a , i n d i c a t e s t h a t the l a t e r a l v o r t e x s p a c i n g i s c o n f i n e d by the presence o f the w a l l s and, thereby, agrees w i t h the t r e n d of the experimental r e s u l t s o f Rosenhead and Schwabe. An approximate estimate of the percentage c o r r e c t i o n f o r the l a t e r a l s p a c i n g i n the w a k e of the 3 i n . model i s p l o t t e d i n F i g u r e I I - l f o r comparison. APPENDIX I I I E l e c t r o n i c Instruments F o l l o w i n g i s a l i s t o f the r e c o r d i n g and a u x i l i a r y -e l e c t r o n i c instruments used i n the c a l i b r a t i o n and experimental t e s t s : P r e s s u r e Transducer: F i l t e r : O s c i l l o s c o p e : C h art Recorder: V o l t m e t e r s : F u n c t i o n Generators: V i b r a t i o n Generator: A m p l i f i e r and Power S u p p l i e s : R-C damping c i r c u i t : B alance: Datametric, B a r o c e l Pressure Sensor, Type 511-10; S i g n a l C o n d i t i o n e r , Type 1015; Power Supply, Type 700. Krohn-Hite, band pass v a r i a b l e f i l t e r 0.02 cps-2Kc, model 330B. T e k t r o n i x , Type 564, d u a l beam storage o s c i l l o s c o p e . Honeywell, 906C V i s i c o r d e r ; Sub-miniature Galvanometers, Type M100-120 and M200-120; Standard, spec. 2, V i s i c o r d e r Record-i n g Paper. Hewlett Packard, HP-3400A t r u e rms v o l t -meter; and HP-412 vacuum tube v o l t m e t e r . Hewlett Packard, low frequency f u n c t i o n g enerator, model 202A; H e a t h k i t , audio frequency g e n e r a t o r , model 1G-72. Goodmans, Type V47. Low frequency, t r a n s i s t o r i z e d power ampli-f i e r w i t h two 12 v o l t d.c. power s u p p l i e s , b u i l t i n the department. [97] R e s i s t o r - c a p a c i t o r system, v a r i a b l e time c o n s t a n t 0 t o 60 seconds, b u i l t i n the department. [35] A e r o l a b , 6 component, pyramidal s t r a i n gauge balance; two co r r e s p o n d i n g 3 channel readout e l e c t r o n i c c a b i n e t s with b u i l t - i n d.c. v o l t m e t e r s . Time-Mark Generator: T e k t r o n i x , Type 184, 2 nanoseconds to 5 seconds c r y s t a l o s c i l l a t o r . 167 L a t e r a l Displacement Transducer: L a t e r a l Transducer Demodulator: Dampers and Power S u p p l i e s : Power Supply: V a r i a c : Strobotachometer: Angular Displacement Transducer: Angular Transducer Power Supply and Output System: A i r - c o r e t r a n s f o r m e r , b u i l t i n the de-partment. [49] F u l l wave r e c t i f i e r and RC f i l t e r , b u i l t i n the department. [49] E l e c t r o m a g n e t i c dampers; v a r i a b l e d.c. and a.c. power s u p p l i e s , b u i l t i n the department. [49] E l e c t r o , v a r i a b l e , f i l t e r e d , d.c. power supply, Model D-612T. General Radio Company, Type W5M, a d j u s t -able ciutotransf ormer. General Radio Company, S t r o b o t a c , Type 1531. 4-arm strai n - g a u g e b r i d g e , b u i l t i n the department. E l l i s , B r i d g e - A m p l i f i e r - M e t e r , Model BAM-1. APPENDIX IV T h e o r y f o r P l u n g i n g o r T o r s i o n a l Degree' o f Freedom 1 G e n e r a l E q u a t i o n s o f M o t i o n / / / / / / The s y s t e m o f mass m ( F i g u r e IV - 1 ), r e s t r a i n e d t o a p l u n g -i n g d e g r e e o f f r e e d o m p e r p e n d i c u l a r t o t h e f l o w d i r e c t i o n , i s s u b j e c t e d t o l i n e a r s p r i n g and v i s c o u s damping f o r c e s and a e r o -d y n a m ic l o a d i n g F . A c o r r e s p o n d i n g s y s t e m w i t h a t o r s i o n a l d e g r e e o f f r e e d o m i s shown i n F i g u r e I V - 2 . I t i s c o n v e n i e n t t o e x p r e s s t h e a e r o d y n a m i c terms i n c o e f f i c i e n t f o r m w h i c h , i n g e n e r a l , may be a f u n c t i o n o f model d i s p l a c e m e n t and v e l o c i t y , and t i m e t . U s i n g t h e L a g r a n g i a n f o r m u l a t i o n , t h e e q u a t i o n o f m o t i o n f o r t h e p l u n g i n g and t o r s i o n a l d e g r e e s o f f r e e d o m become v F i g u r e I V - 1 F i g u r e I V - 2 (1) (2) 169 These equations are analyzed w i t h v o r t e x resonant or g a l l o p i n g e x c i t a t i o n s t o determine the a e r o e l a s t i c i n s t a b i l i t y of angle s e c t i o n s . 2 Vortex Resonance The a n a l y s i s c o n s i d e r s p l u n g i n g degree of freedom only s i n c e the t o r s i o n a l s o l u t i o n i s s i m i l a r . For t r a n s v e r s e e x c i t -a t i o n due to v o r t e x shedding, the f o r c e c o e f f i c i e n t i s approx-imated by a s i n u s o i d a l f u n c t i o n of amplitude C^, and frequency to . Equation (1) then becomes which can be w r i t t e n i n the nondimensional form fi'Y *2^Sl<{ • Y = i^l/Cji *in f w ( 4 ) g i v i n g the s t e a d y - s t a t e amplitude as • 2 Y c „ „ C , U T h e r e f o r e , the resonant amplitude becomes (5) V - 1 J2y r - U 2 ' m a x 2 P y J ' r c , s ( 6 ) For a p h y s i c a l system c o n s i s t i n g of an e l a s t i c a l l y ' m o u n t e d c y l i n d e r i n an a i r flow, i t has been observed t h a t the o s c i l l a t i o n s near the resonance peak do not comply with the p r e d i c t i o n of the simple mathematical model, but r a t h e r e x h i b i t a phenomenon c a l l e d 79 v o r t e x capture. Parkinson,et a l s o l v e d equation (1), approx-i m a t e l y , under the c o n d i t i o n s of vortex capture by assuming 170 S and J (7) The f i n a l e x p r e s s i o n f o r t h e r e s o n a n t d i s p l a c e m e n t now becomes c7 Py 3 G a l l o p i n g o f A e r o d y n a m i c a l l y U n s t a b l e S e c t i o n 3.1 P r e l i m i n a r y Remarks G e o m e t r i c a l l y b l u f f s e c t i o n s may e x h i b i t a g a l l o p i n g t y p e o f o s c i l l a t i o n b e c a u s e o f t h e n a t u r e o f t h e a e r o d y n a m i c f o r c e s o r moment. I f , f r o m t h e mean a n g l e o f a t t a c k a0 , t h e a e r o d y n a m i c l o a d i n g shows an i n c r e a s e w i t h model a t t i t u d e and t h e amount o f r e s u l t i n g e n e r g y i n p u t by t h e f l u i d f o r c e e x c e e d s t h a t d i s s i p a t e d by t h e v i s c o u s damping, t h e n o s c i l l a t i o n s w i l l c o n -t i n u e t o grow u n t i l a n e t e n e r g y b a l a n c e i s e s t a b l i s h e d . The q u a s i - s t e a d y a p p r o a c h i s a d o p t e d so t h a t t h e i n s t a n t a n e o u s f o r c -i n g f u n c t i o n a c t i n g on t h e o s c i l l a t i n g model can be r e p l a c e d by i t s s t e a d y v a l u e a c t i n g on t h e s t a t i o n a r y model o r i e n t e d a t t h e same a p p a r e n t a n g l e o f a t t a c k . T h i s a s s u m p t i o n i m p l i e s t h a t n o a e r o d y n a m i c h y s t e r e s i s e f f e c t s e x i s t i n t h e f o r c e o r moment c h a r a c t e r i s t i c s and 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 i s f a r removed f r o m t h e c y l i n d e r f r e q u e n c y . From t h e n a t u r e o f t h e p r o b l e m , t h e l a t e r a l f o r c e o r t w i s t i n g moment t u r n o u t t o be h i g h l y n o n l i n e a r f u n c t i o n s . How-e v e r , c e r t a i n s i m p l i f i c a t i o n s a r e p o s s i b l e i n t h e a e r o e l a s t i c p r o b l e m s i n c e : 171 (i) the r a t i o of a i r d e n s i t y t o model d e n s i t y i s sm a l l and consequently, the aerodynamic f o r c e i s sm a l l compared with the e l a s t i c and i n e r t i a l f o r c e s of the system; ( i i ) the frequency of o s c i l l a t i o n i s c l o s e to the n a t u r a l frequency; ( i i i ) the e q u i l i b r i u m motion i s n e a r l y s i n u s o i d a l . Under these assumptions the problem becomes q u a s i - l i n e a r which can be s o l v e d by v a r i o u s a n a l y t i c a l techniques. For the p l u n g i n g degree of freedom, the quasi-steady theory has been e s t a b l i s h e d by Scruton"^ and Parkinson, e t a l . ^ 14 19 S i s t o and I i , c o n s i d e r i n g the problem of s t a l l - f l u t t e r , i n -c o r p o r a t e d the downwash c o n d i t i o n a t the t h r e e - q u a r t e r p o i n t to s i m p l i f y the a n a l y s i s . For the t o r s i o n a l g a l l o p i n g of b l u f f c y l i n d e r s a m o d i f i e d theory f o l l o w i n g the p l u n g i n g quasi-steady approach i s presented. However, the t o r s i o n a l a n a l y s i s i s s l i g h t -l y more complicated s i n c e the n o n l i n e a r aerodynamic moment i s a f u n c t i o n of the instantaneous angular p o s i t i o n as w e l l as the v e l o c i t y . A d e r i v a t i o n of the b a s i c e x p r e s s i o n s f o r the plung-i n g and t o r s i o n a l degrees of freedom i s given i n the f o l l o w i n g s e c t i o n s . 3.2 Plung i n g Degree of Freedom F i g u r e IV-3 1 7 2 F o l l o w i n g t h e a n a l y s i s e s t a b l i s h e d b y P a r k i n s o n a n d 1 7 a s s o c i a t e s , 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 ( F i g u r e I V - 3 ) i s r e l a t e d t o t h e s t e a d y l i f t a n d d r a g b y Fy = -(LcosY + DsinV) ( 9 ) w h e r e a n d t h e i n s t a n t a n e o u s a n g l e o f a t t a c k a = a0 + y. N o t i n g t h a t V = V r e l c o s y , t h e f i n a l e x p r e s s i o n f o r t h e f o r c e c o e f f i c i e n t b e c o m e s CF = - ( C + C DtanT) S tc T ( I D 1 T h e c h a n g e i n m o d e l a t t i t u d e i s r e l a t e d t o t h e t r a n s v e r s e v e l o c i t y c o m p o n e n t b y T h e r e f o r e , t h e g o v e r n i n g e q u a t i o n ( 1 ) b e c o m e s 1117 + + Sv = j?v2hlcF(Ci) (i3; I t h a s b e e n c u s t o m a r y t o o b t a i n a n e x p r e s s i o n f o r Cp i n y t h e f o r m o f a p o l y n o m i a l u s i n g C T a n d C e x p e r i m e n t a l d a t a , a n d It u a n a l y z e t h e p a r t i c u l a r c a s e w h e r e t h e m o d e l i s m o u n t e d a t a s y m m e t r i c a l a n g l e o f a t t a c k . I n t h i s c a s e , C i s a n o d d f u n c -y 5 0 t i o n . S a n t o s h a m c o n s i d e r e d t h e p r o b l e m o f a e r o e l a s t i c i n s t a -173 b i l i t y o f r e c t a n g u l a r s e c t i o n s and u s e d odd f u n c t i o n e d C h e b y s h e v p o l y n o m i a l s o f 1 1 t h d e g r e e f o r o b t a i n i n g t h e s o l u t i o n s . However, f o r a n a l y z i n g t h e g a l l o p i n g o s c i l l a t i o n s o f an a n g l e s e c t i o n , t h e t h e o r y has been g e n e r a l i z e d t o c o n s i d e r t h e model o r i e n t e d a t u n s y m m e t r i c a l a n g l e s o f a t t a c k and t h e d e g r e e o f t h e p o l y n o -m i a l c o n t a i n i n g a l l t erms has been e x t e n d e d t o 25. However, t h e a c t u a l d e g r e e o f t h e p o l y n o m i a l u s e d i s 25 o r l e s s b a s e d on a l e a s t s q u a r e e r r o r c r i t e r i o n . T h e r e b y , t h e e x p r e s s i o n f o r C p y c a n be w r i t t e n i n t h e p o l y n o m i a l f o r m (14) where |sl < 25 . N o t e t h a t a Q = 0 b e c a u s e y r e p r e s e n t s t h e d i s p l a c e m e n t f r o m t h e a p p a r e n t z e r o p o s i t i o n g o v e r n e d by t h e s t e a d y l i f t f o r c e . C F i s y o J g e n e r a l l y p l o t t e d as a f u n c t i o n o f t a n y, s i n c e (y/V) and t a n y a r e d i r e c t l y r e l a t e d by ( 1 2 ) . E q u a t i o n (13) c a n be n o n d i m e n s i o n a l i z e d t o ¥ • Y = -20 y Y + lyUX/Y) ,15) o r c o m b i n i n g t h e two terms on t h e r i g h t hand s i d e t o g i v e (16) (17) B a s e d on t h e a f o r e m e n t i o n e d a s s u m p t i o n , y v < < 1; t h e r e f o r e , (16) i s a q u a s i - 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 t h e autonomous t y p e . A c t u a l l y , t h e q u a s i - l i n e a r f o r m r e q u i r e s t h e c o m p l e t e r i g h t hand s i d e o f e q u a t i o n (16) t o r e m a i n s m a l l , t h u s s p e c i f y -i n g , i m p l i c i t y , an u p p e r l i m i t t o t h e v e l o c i t y U. U s i n g t h e V a r i a t i o n o f P a r a m e t e r method, t h e f i r s t o r d e r a p p r o x i m a t e s o l u t i o n o f e q u a t i o n ( 1 6 ) , when y = 0, i s o f t h e f o r m Y = Y s i n ( x • <p) ( 1 8 ) t h e r e f o r e . Y = Y C O S ( T + 0) F o r y < < 1, t h e f i r s t o r d e r s o l u t i o n i s d e t e r m i n e d by r e d u c -i n g (16) t o a s y s t e m o f f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n s Y = z c o n s i d e r i n g Y = Y s i n ( T + Z = Y C O S ( T + 0) (20) + = x + 4> and a s s u m i n g Y and 0 t o be f u n c t i o n s o f t i m e T, e q u a t i o n (19) r e d u c e s t o dx y ' (21) ^ - f [ Ycos^J s in t Y y dY d<6 — S i n c e and a r e p r o p o r t i o n a l t o y , Y and p w i l l be s l o w l y v a r y i n g f u n c t i o n s o f t i m e . Hence, Y and jz$ can be c o n s i d e r e d , a p p r o x i m a t e l y , c o n s t a n t d u r i n g one c y c l e . E q u a -t i o n (21) can then be r e p l a c e d by t h e i r average values ay = Eii f [ Y c o s ^ ] cos*/' (22) By examining $ and r e c a l l i n g the polynomial form of the f u n c t i o n f ^ ( Y ) , the i n t e g r a n d can be r e p l a c e d by a F o u r i e r s i n e s e r i e s . Upon i n t e g r a t i o n each term i n t h i s s e r i e s vanishes and, t h e r e f o r e , (22) can be w r i t t e n as where i l - -Y<5 (Y) dx - * d j > = 0 d t (23) J ( Y ) = - J ^ l C { [ Y c o s + 1 c o s + (24! 0 On s u b s t i t u t i n g (17) i n t o (24), 5^ becomes ^ ) . . ^ ^ Y • I*,* b ^ Y ' " ' } ( 2 5 ) where and k = b, , i r M for M o<dd M - l for W even The displacement amplitudes of the l i m i t c y c l e s are obt a i n e d by determining the r e a l p o s i t i v e r o o t s Y_. of the (Y) 176 p o l y n o m i a l , from which the s t a b i l i t y of the s u s t a i n e d motions can be analyzed. 3 . 3 T o r s i o n a l Degree of Freedom F i g u r e IV-4 For the a n a l y s i s of the t o r s i o n a l degree of freedom, the e x c i t a t i o n i s the instantaneous moment M., which u s i n g the quasi-steady approach can be r e l a t e d to the experimental v a l u e , M, by .2 c = - c (°0 (26) where Q «S ( 2 7 ) I t i s apparent t h a t V r e ^ i s d i f f e r e n t from V i n both magnitude o and d i r e c t i o n because of the angular v e l o c i t y 9. Furthermore, each s u r f a c e element on the contour experiences a d i f f e r e n t r e l a t i v e v e l o c i t y governed by i t s p o s i t i o n with r e s p e c t to the centre of r o t a t i o n . I t i s assumed t h a t an e f f e c t i v e r e l a t i v e 1 7 7 v e l o c i t y can be w r i t t e n as = e n e t TIr ( 2 8 ; r r where i s the e f f e c t i v e r a d i u s of r o t a t i o n and n r i s the d i r e c -t i o n of v . For the angle model suspended about the centre of g r a v i t y a t a Q = 0°, the r e p r e s e n t a t i v e parameter values s u i t a b l y nondimensionalized are R - 0 . 3 0 1 5 ( 2 9 ) For d i f f e r e n t angles of a t t a c k , the r e p r e s e n t a t i v e d i r e c t i o n angle n r can be obtained u s i n g the r e l a t i o n (30) The approach when a p p l i e d to a f l a t p l a t e a i r f o i l gives an e f f e c t i v e downwash v e l o c i t y at the t h r e e - q u a r t e r chord p o i n t . T h i s agrees with the value adopted i n the s t a l l f l u t t e r a n a l y s i s , _. . 14 , _. 19 by S i s t o and I i . For the o s c i l l a t i n g c y l i n d e r , the o v e r a l l v e l o c i t y v e c t o r diagram i s as shown i n F i g u r e IV-4. Hence, the r e l a t i v e v e l o c -i t y can be w r i t t e n as ( I f - , - ill cosX • tyf and the r e p r e s e n t a t i v e angle y r becomes 1 V - e r r c o s T ] r f (31) (32) 178 I n n o n d i m e n s i o n a l f o r m , t h e g o v e r n i n g 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 ( 2 ) becomes © • e . - 2 p e • n 9 u ' c ( » , * ) ( 3 3 ) 9 w i t h a u x i l i a r y e x p r e s s i o n s t r a n s f o r m e d t o (34) Upon e v a l u a t i o n o f as a f u n c t i o n o f 0 and © f r o m e e q u a t i o n s (34), i t was o b s e r v e d t h a t l i n e s o f c o n s t a n t C M ^ were a l m o s t l i n e a r and p a r a l l e l a t an a n g l e X t o t h e 0 a x i s . As a r e s u l t , a new c a r t e s i a n c o o r d i n a t e s y s t e m , 5 and c, , was e s t a b -l i s h e d w i t h 5 a x i s a l o n g t h e c o n t o u r l i n e C . . = 0 . T h e r e f o r e , t o a f i r s t a p p r o x i m a t i o n , C M q c a n be e x p r e s s e d as a f u n c t i o n o f E, w h i c h i s r e l a t e d t o 0 and 6 by t h e c o o r d i n a t e t r a n s f o r m -a t i o n (35) where s = sin A and c = cos A S i m i l a r . t o t h e p l u n g i n g c a s e , t h e a e r o d y n a m i c moment c o e f f i c i e n t i s e x p r e s s e d as a p o l y n o m i a l i n £ 179 C M = a , | + az%2 + a 3 ? 3 + . . .+ a N ? M (36) where S u b s t i t u t i n g (35) and (36) i n t o (33) and combining the two terms on the r i g h t hand s i d e , the nondimensional equation of motion reduces to the form © + © = /^M©,©) where J^Q - ^ g ^ i and fQ{®fQ) = U*{( VsJ2Uo)@ ~ C& + ^ ( § 5 - ©c)' (37) (38) + For a system i n a i r , y << 1. 6 Using the method of V a r i a t i o n of Parameters as i n the p l u n g i n g a n a l y s i s , the s o l u t i o n of (37) can be w r i t t e n as i® = - § i ( © ) d * 6 (39) ^ = - k (8) d t e where 2TT© ; .air <40> K ( © ) = -A f [© sinf, 0 cos^] sin 4" 4w> 0 2TT© J 9 0 180 Note, the v a r i a t i o n of phase with time which was not p r e s e n t i n the p l u n g i n g case. S u b s t i t u t i n g (38) i n t o (40) and c a r r y i n g out the i n t e g r a t i o n , 6. and K become + . . . + - + ( o ) K e ( § ) - £ l / [ c • | 3 { ( § f c t M + c ' t j ® where L - 3, S ,7, • - ; j = »,3, 5", • • ,1 The c o e f f i c i e n t s b. . and t. . ,, are given by i f ] 1,1+1 b- j = c . . d . . (41) + • • • + - 1 1 - 1 - - . - . • . - 7 1 i - I - j + | ( 4 2 ) (43) Here the c. . are the b i n o m i a l c o e f f i c i e n t s of the terms i n equation (38) , and d^ ^ + 1 and s^ + 1 are constants obtained from the i n t e g r a t i o n of equation (40). These c o e f f i c i e n t s can be determined from the f o l l o w i n g e x p r e s s i o n s : 181 - ( k - i j 9ij+2 "(i+O-O"-') 9iJ e. = J + 2 e. • (44) l,J + Z where I « I, 3, 5,-• j j = \, 3,5, • • •, I ; k = 1,2 ,3, • • t - I . The amplitudes of the l i m i t c y c l e s can be obtained from (39) u s i n g the c o n d i t i o n , dQ/dr = 0. The s t a b i l i t y of the sus-t a i n e d motions can be analyzed by examining the s i g n of 9 6 - / 9 0 e v a l u a t e d a t these l i m i t c y c l e s . o On determining the l i m i t c y c l e amplitude, ft can be e v a l -uated u s i n g equation (42) as <p = - K X + <f>0 (45) where 0 O i s the constant of i n t e g r a t i o n . Thus, the s t e a d y - s t a t e 182 amplitude of o s c i l l a t i o n i s given by © = ® s i n lp (46) where ^ = T + 0 = ( | - f < ) - C + 0o (47) From (47) i t appears t h a t the frequency of o s c i l l a t i o n i s reduced from the n a t u r a l frequency by the amount K Q. T h e r e f o r e , the second equation of (39) i n terms of the reduced frequency para-meter becomes I + # = I - K (48) 

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